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Theorem mat1rhmelval 20334
 Description: The value of the ring homomorphism 𝐹. (Contributed by AV, 22-Dec-2019.)
Hypotheses
Ref Expression
mat1rhmval.k 𝐾 = (Base‘𝑅)
mat1rhmval.a 𝐴 = ({𝐸} Mat 𝑅)
mat1rhmval.b 𝐵 = (Base‘𝐴)
mat1rhmval.o 𝑂 = ⟨𝐸, 𝐸
mat1rhmval.f 𝐹 = (𝑥𝐾 ↦ {⟨𝑂, 𝑥⟩})
Assertion
Ref Expression
mat1rhmelval ((𝑅 ∈ Ring ∧ 𝐸𝑉𝑋𝐾) → (𝐸(𝐹𝑋)𝐸) = 𝑋)
Distinct variable groups:   𝑥,𝐾   𝑥,𝑂   𝑥,𝐸   𝑥,𝑅   𝑥,𝑉   𝑥,𝑋
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐹(𝑥)

Proof of Theorem mat1rhmelval
StepHypRef Expression
1 df-ov 6693 . 2 (𝐸(𝐹𝑋)𝐸) = ((𝐹𝑋)‘⟨𝐸, 𝐸⟩)
2 mat1rhmval.k . . . . 5 𝐾 = (Base‘𝑅)
3 mat1rhmval.a . . . . 5 𝐴 = ({𝐸} Mat 𝑅)
4 mat1rhmval.b . . . . 5 𝐵 = (Base‘𝐴)
5 mat1rhmval.o . . . . 5 𝑂 = ⟨𝐸, 𝐸
6 mat1rhmval.f . . . . 5 𝐹 = (𝑥𝐾 ↦ {⟨𝑂, 𝑥⟩})
72, 3, 4, 5, 6mat1rhmval 20333 . . . 4 ((𝑅 ∈ Ring ∧ 𝐸𝑉𝑋𝐾) → (𝐹𝑋) = {⟨𝑂, 𝑋⟩})
87fveq1d 6231 . . 3 ((𝑅 ∈ Ring ∧ 𝐸𝑉𝑋𝐾) → ((𝐹𝑋)‘⟨𝐸, 𝐸⟩) = ({⟨𝑂, 𝑋⟩}‘⟨𝐸, 𝐸⟩))
95eqcomi 2660 . . . . 5 𝐸, 𝐸⟩ = 𝑂
109fveq2i 6232 . . . 4 ({⟨𝑂, 𝑋⟩}‘⟨𝐸, 𝐸⟩) = ({⟨𝑂, 𝑋⟩}‘𝑂)
11 opex 4962 . . . . . 6 𝐸, 𝐸⟩ ∈ V
125, 11eqeltri 2726 . . . . 5 𝑂 ∈ V
13 simp3 1083 . . . . 5 ((𝑅 ∈ Ring ∧ 𝐸𝑉𝑋𝐾) → 𝑋𝐾)
14 fvsng 6488 . . . . 5 ((𝑂 ∈ V ∧ 𝑋𝐾) → ({⟨𝑂, 𝑋⟩}‘𝑂) = 𝑋)
1512, 13, 14sylancr 696 . . . 4 ((𝑅 ∈ Ring ∧ 𝐸𝑉𝑋𝐾) → ({⟨𝑂, 𝑋⟩}‘𝑂) = 𝑋)
1610, 15syl5eq 2697 . . 3 ((𝑅 ∈ Ring ∧ 𝐸𝑉𝑋𝐾) → ({⟨𝑂, 𝑋⟩}‘⟨𝐸, 𝐸⟩) = 𝑋)
178, 16eqtrd 2685 . 2 ((𝑅 ∈ Ring ∧ 𝐸𝑉𝑋𝐾) → ((𝐹𝑋)‘⟨𝐸, 𝐸⟩) = 𝑋)
181, 17syl5eq 2697 1 ((𝑅 ∈ Ring ∧ 𝐸𝑉𝑋𝐾) → (𝐸(𝐹𝑋)𝐸) = 𝑋)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ w3a 1054   = wceq 1523   ∈ wcel 2030  Vcvv 3231  {csn 4210  ⟨cop 4216   ↦ cmpt 4762  ‘cfv 5926  (class class class)co 6690  Basecbs 15904  Ringcrg 18593   Mat cmat 20261 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-ov 6693 This theorem is referenced by:  mat1ghm  20337  mat1mhm  20338
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