Theorem mat1rhmelval 22463

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 7359	. 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 22462	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾) → (𝐹‘𝑋) = {⟨𝑂, 𝑋⟩})
8	7	fveq1d 6829	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾) → ((𝐹‘𝑋)‘⟨𝐸, 𝐸⟩) = ({⟨𝑂, 𝑋⟩}‘⟨𝐸, 𝐸⟩))
9	5	eqcomi 2748	. . . . 5 ⊢ ⟨𝐸, 𝐸⟩ = 𝑂
10	9	fveq2i 6830	. . . 4 ⊢ ({⟨𝑂, 𝑋⟩}‘⟨𝐸, 𝐸⟩) = ({⟨𝑂, 𝑋⟩}‘𝑂)
11		opex 5403	. . . . . 6 ⊢ ⟨𝐸, 𝐸⟩ ∈ V
12	5, 11	eqeltri 2835	. . . . 5 ⊢ 𝑂 ∈ V
13		simp3 1144	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾) → 𝑋 ∈ 𝐾)
14		fvsng 7124	. . . . 5 ⊢ ((𝑂 ∈ V ∧ 𝑋 ∈ 𝐾) → ({⟨𝑂, 𝑋⟩}‘𝑂) = 𝑋)
15	12, 13, 14	sylancr 593	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾) → ({⟨𝑂, 𝑋⟩}‘𝑂) = 𝑋)
16	10, 15	eqtrid 2786	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾) → ({⟨𝑂, 𝑋⟩}‘⟨𝐸, 𝐸⟩) = 𝑋)
17	8, 16	eqtrd 2774	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾) → ((𝐹‘𝑋)‘⟨𝐸, 𝐸⟩) = 𝑋)
18	1, 17	eqtrid 2786	1 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾) → (𝐸(𝐹‘𝑋)𝐸) = 𝑋)

Syntax hints:   → wi 4   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  Vcvv 3431  {csn 4555  ⟨cop 4561   ↦ cmpt 5153  ‘cfv 6485  (class class class)co 7356  Basecbs 17170  Ringcrg 20205   Mat cmat 22390

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-iota 6441  df-fun 6487  df-fv 6493  df-ov 7359

This theorem is referenced by:  mat1ghm  22466  mat1mhm  22467