Module: Mathlib.Algebra.DualQuaternion

Goal state before:
pp: case h2.e_re
R : Type u_1
inst✝ : CommRing R
x✝ y✝ : ℍ[DualNumber R]
⊢ TrivSqZeroExt.snd (x✝ * y✝).re =
    ({ re := TrivSqZeroExt.fst x✝.re, imI := TrivSqZeroExt.fst x✝.imI, imJ := TrivSqZeroExt.fst x✝.imJ,
            imK := TrivSqZeroExt.fst x✝.imK } *
          { re := TrivSqZeroExt.snd y✝.re, imI := TrivSqZeroExt.snd y✝.imI, imJ := TrivSqZeroExt.snd y✝.imJ,
            imK := TrivSqZeroExt.snd y✝.imK }).re +
      ({ re := TrivSqZeroExt.snd x✝.re, imI := TrivSqZeroExt.snd x✝.imI, imJ := TrivSqZeroExt.snd x✝.imJ,
            imK := TrivSqZeroExt.snd x✝.imK } *
          { re := TrivSqZeroExt.fst y✝.re, imI := TrivSqZeroExt.fst y✝.imI, imJ := TrivSqZeroExt.fst y✝.imJ,
            imK := TrivSqZeroExt.fst y✝.imK }).re

pp: case h2.e_imI
R : Type u_1
inst✝ : CommRing R
x✝ y✝ : ℍ[DualNumber R]
⊢ TrivSqZeroExt.snd (x✝ * y✝).imI =
    ({ re := TrivSqZeroExt.fst x✝.re, imI := TrivSqZeroExt.fst x✝.imI, imJ := TrivSqZeroExt.fst x✝.imJ,
            imK := TrivSqZeroExt.fst x✝.imK } *
          { re := TrivSqZeroExt.snd y✝.re, imI := TrivSqZeroExt.snd y✝.imI, imJ := TrivSqZeroExt.snd y✝.imJ,
            imK := TrivSqZeroExt.snd y✝.imK }).imI +
      ({ re := TrivSqZeroExt.snd x✝.re, imI := TrivSqZeroExt.snd x✝.imI, imJ := TrivSqZeroExt.snd x✝.imJ,
            imK := TrivSqZeroExt.snd x✝.imK } *
          { re := TrivSqZeroExt.fst y✝.re, imI := TrivSqZeroExt.fst y✝.imI, imJ := TrivSqZeroExt.fst y✝.imJ,
            imK := TrivSqZeroExt.fst y✝.imK }).imI

pp: case h2.e_imJ
R : Type u_1
inst✝ : CommRing R
x✝ y✝ : ℍ[DualNumber R]
⊢ TrivSqZeroExt.snd (x✝ * y✝).imJ =
    ({ re := TrivSqZeroExt.fst x✝.re, imI := TrivSqZeroExt.fst x✝.imI, imJ := TrivSqZeroExt.fst x✝.imJ,
            imK := TrivSqZeroExt.fst x✝.imK } *
          { re := TrivSqZeroExt.snd y✝.re, imI := TrivSqZeroExt.snd y✝.imI, imJ := TrivSqZeroExt.snd y✝.imJ,
            imK := TrivSqZeroExt.snd y✝.imK }).imJ +
      ({ re := TrivSqZeroExt.snd x✝.re, imI := TrivSqZeroExt.snd x✝.imI, imJ := TrivSqZeroExt.snd x✝.imJ,
            imK := TrivSqZeroExt.snd x✝.imK } *
          { re := TrivSqZeroExt.fst y✝.re, imI := TrivSqZeroExt.fst y✝.imI, imJ := TrivSqZeroExt.fst y✝.imJ,
            imK := TrivSqZeroExt.fst y✝.imK }).imJ

pp: case h2.e_imK
R : Type u_1
inst✝ : CommRing R
x✝ y✝ : ℍ[DualNumber R]
⊢ TrivSqZeroExt.snd (x✝ * y✝).imK =
    ({ re := TrivSqZeroExt.fst x✝.re, imI := TrivSqZeroExt.fst x✝.imI, imJ := TrivSqZeroExt.fst x✝.imJ,
            imK := TrivSqZeroExt.fst x✝.imK } *
          { re := TrivSqZeroExt.snd y✝.re, imI := TrivSqZeroExt.snd y✝.imI, imJ := TrivSqZeroExt.snd y✝.imJ,
            imK := TrivSqZeroExt.snd y✝.imK }).imK +
      ({ re := TrivSqZeroExt.snd x✝.re, imI := TrivSqZeroExt.snd x✝.imI, imJ := TrivSqZeroExt.snd x✝.imJ,
            imK := TrivSqZeroExt.snd x✝.imK } *
          { re := TrivSqZeroExt.fst y✝.re, imI := TrivSqZeroExt.fst y✝.imI, imJ := TrivSqZeroExt.fst y✝.imJ,
            imK := TrivSqZeroExt.fst y✝.imK }).imK

Tactic:
<;>

Goal state after:
case h2.e_re
R : Type u_1
inst✝ : CommRing R
x✝ y✝ : ℍ[DualNumber R]
⊢ TrivSqZeroExt.snd (x✝ * y✝).re =
    ({ re := TrivSqZeroExt.fst x✝.re, imI := TrivSqZeroExt.fst x✝.imI, imJ := TrivSqZeroExt.fst x✝.imJ,
            imK := TrivSqZeroExt.fst x✝.imK } *
          { re := TrivSqZeroExt.snd y✝.re, imI := TrivSqZeroExt.snd y✝.imI, imJ := TrivSqZeroExt.snd y✝.imJ,
            imK := TrivSqZeroExt.snd y✝.imK }).re +
      ({ re := TrivSqZeroExt.snd x✝.re, imI := TrivSqZeroExt.snd x✝.imI, imJ := TrivSqZeroExt.snd x✝.imJ,
            imK := TrivSqZeroExt.snd x✝.imK } *
          { re := TrivSqZeroExt.fst y✝.re, imI := TrivSqZeroExt.fst y✝.imI, imJ := TrivSqZeroExt.fst y✝.imJ,
            imK := TrivSqZeroExt.fst y✝.imK }).re

case h2.e_imI
R : Type u_1
inst✝ : CommRing R
x✝ y✝ : ℍ[DualNumber R]
⊢ TrivSqZeroExt.snd (x✝ * y✝).imI =
    ({ re := TrivSqZeroExt.fst x✝.re, imI := TrivSqZeroExt.fst x✝.imI, imJ := TrivSqZeroExt.fst x✝.imJ,
            imK := TrivSqZeroExt.fst x✝.imK } *
          { re := TrivSqZeroExt.snd y✝.re, imI := TrivSqZeroExt.snd y✝.imI, imJ := TrivSqZeroExt.snd y✝.imJ,
            imK := TrivSqZeroExt.snd y✝.imK }).imI +
      ({ re := TrivSqZeroExt.snd x✝.re, imI := TrivSqZeroExt.snd x✝.imI, imJ := TrivSqZeroExt.snd x✝.imJ,
            imK := TrivSqZeroExt.snd x✝.imK } *
          { re := TrivSqZeroExt.fst y✝.re, imI := TrivSqZeroExt.fst y✝.imI, imJ := TrivSqZeroExt.fst y✝.imJ,
            imK := TrivSqZeroExt.fst y✝.imK }).imI

case h2.e_imJ
R : Type u_1
inst✝ : CommRing R
x✝ y✝ : ℍ[DualNumber R]
⊢ TrivSqZeroExt.snd (x✝ * y✝).imJ =
    ({ re := TrivSqZeroExt.fst x✝.re, imI := TrivSqZeroExt.fst x✝.imI, imJ := TrivSqZeroExt.fst x✝.imJ,
            imK := TrivSqZeroExt.fst x✝.imK } *
          { re := TrivSqZeroExt.snd y✝.re, imI := TrivSqZeroExt.snd y✝.imI, imJ := TrivSqZeroExt.snd y✝.imJ,
            imK := TrivSqZeroExt.snd y✝.imK }).imJ +
      ({ re := TrivSqZeroExt.snd x✝.re, imI := TrivSqZeroExt.snd x✝.imI, imJ := TrivSqZeroExt.snd x✝.imJ,
            imK := TrivSqZeroExt.snd x✝.imK } *
          { re := TrivSqZeroExt.fst y✝.re, imI := TrivSqZeroExt.fst y✝.imI, imJ := TrivSqZeroExt.fst y✝.imJ,
            imK := TrivSqZeroExt.fst y✝.imK }).imJ

case h2.e_imK
R : Type u_1
inst✝ : CommRing R
x✝ y✝ : ℍ[DualNumber R]
⊢ TrivSqZeroExt.snd (x✝ * y✝).imK =
    ({ re := TrivSqZeroExt.fst x✝.re, imI := TrivSqZeroExt.fst x✝.imI, imJ := TrivSqZeroExt.fst x✝.imJ,
            imK := TrivSqZeroExt.fst x✝.imK } *
          { re := TrivSqZeroExt.snd y✝.re, imI := TrivSqZeroExt.snd y✝.imI, imJ := TrivSqZeroExt.snd y✝.imJ,
            imK := TrivSqZeroExt.snd y✝.imK }).imK +
      ({ re := TrivSqZeroExt.snd x✝.re, imI := TrivSqZeroExt.snd x✝.imI, imJ := TrivSqZeroExt.snd x✝.imJ,
            imK := TrivSqZeroExt.snd x✝.imK } *
          { re := TrivSqZeroExt.fst y✝.re, imI := TrivSqZeroExt.fst y✝.imI, imJ := TrivSqZeroExt.fst y✝.imJ,
            imK := TrivSqZeroExt.fst y✝.imK }).imK