Theorem mat1rhmelval 22423

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 = (𝑥 ∈ 𝐾 ↦ {⟨𝑂, 𝑥⟩})

Assertion

Ref	Expression
mat1rhmelval	⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾) → (𝐸(𝐹‘𝑋)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

Step	Hyp	Ref	Expression
1		df-ov 7413	. 2 ⊢ (𝐸(𝐹‘𝑋)𝐸) = ((𝐹‘𝑋)‘⟨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 = (𝑥 ∈ 𝐾 ↦ {⟨𝑂, 𝑥⟩})
7	2, 3, 4, 5, 6	mat1rhmval 22422	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾) → (𝐹‘𝑋) = {⟨𝑂, 𝑋⟩})
8	7	fveq1d 6883	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾) → ((𝐹‘𝑋)‘⟨𝐸, 𝐸⟩) = ({⟨𝑂, 𝑋⟩}‘⟨𝐸, 𝐸⟩))
9	5	eqcomi 2745	. . . . 5 ⊢ ⟨𝐸, 𝐸⟩ = 𝑂
10	9	fveq2i 6884	. . . 4 ⊢ ({⟨𝑂, 𝑋⟩}‘⟨𝐸, 𝐸⟩) = ({⟨𝑂, 𝑋⟩}‘𝑂)
11		opex 5444	. . . . . 6 ⊢ ⟨𝐸, 𝐸⟩ ∈ V
12	5, 11	eqeltri 2831	. . . . 5 ⊢ 𝑂 ∈ V
13		simp3 1138	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾) → 𝑋 ∈ 𝐾)
14		fvsng 7177	. . . . 5 ⊢ ((𝑂 ∈ V ∧ 𝑋 ∈ 𝐾) → ({⟨𝑂, 𝑋⟩}‘𝑂) = 𝑋)
15	12, 13, 14	sylancr 587	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾) → ({⟨𝑂, 𝑋⟩}‘𝑂) = 𝑋)
16	10, 15	eqtrid 2783	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾) → ({⟨𝑂, 𝑋⟩}‘⟨𝐸, 𝐸⟩) = 𝑋)
17	8, 16	eqtrd 2771	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾) → ((𝐹‘𝑋)‘⟨𝐸, 𝐸⟩) = 𝑋)
18	1, 17	eqtrid 2783	1 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾) → (𝐸(𝐹‘𝑋)𝐸) = 𝑋)

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  Vcvv 3464  {csn 4606  ⟨cop 4612   ↦ cmpt 5206  ‘cfv 6536  (class class class)co 7410  Basecbs 17233  Ringcrg 20198   Mat cmat 22350

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-iota 6489  df-fun 6538  df-fv 6544  df-ov 7413

This theorem is referenced by:  mat1ghm  22426  mat1mhm  22427