Theorem mat1rhmelval 21089

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 7159	. 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 21088	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾) → (𝐹‘𝑋) = {⟨𝑂, 𝑋⟩})
8	7	fveq1d 6672	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾) → ((𝐹‘𝑋)‘⟨𝐸, 𝐸⟩) = ({⟨𝑂, 𝑋⟩}‘⟨𝐸, 𝐸⟩))
9	5	eqcomi 2830	. . . . 5 ⊢ ⟨𝐸, 𝐸⟩ = 𝑂
10	9	fveq2i 6673	. . . 4 ⊢ ({⟨𝑂, 𝑋⟩}‘⟨𝐸, 𝐸⟩) = ({⟨𝑂, 𝑋⟩}‘𝑂)
11		opex 5356	. . . . . 6 ⊢ ⟨𝐸, 𝐸⟩ ∈ V
12	5, 11	eqeltri 2909	. . . . 5 ⊢ 𝑂 ∈ V
13		simp3 1134	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾) → 𝑋 ∈ 𝐾)
14		fvsng 6942	. . . . 5 ⊢ ((𝑂 ∈ V ∧ 𝑋 ∈ 𝐾) → ({⟨𝑂, 𝑋⟩}‘𝑂) = 𝑋)
15	12, 13, 14	sylancr 589	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾) → ({⟨𝑂, 𝑋⟩}‘𝑂) = 𝑋)
16	10, 15	syl5eq 2868	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾) → ({⟨𝑂, 𝑋⟩}‘⟨𝐸, 𝐸⟩) = 𝑋)
17	8, 16	eqtrd 2856	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾) → ((𝐹‘𝑋)‘⟨𝐸, 𝐸⟩) = 𝑋)
18	1, 17	syl5eq 2868	1 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾) → (𝐸(𝐹‘𝑋)𝐸) = 𝑋)

Syntax hints:   → wi 4   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114  Vcvv 3494  {csn 4567  ⟨cop 4573   ↦ cmpt 5146  ‘cfv 6355  (class class class)co 7156  Basecbs 16483  Ringcrg 19297   Mat cmat 21016

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-iota 6314  df-fun 6357  df-fv 6363  df-ov 7159

This theorem is referenced by:  mat1ghm  21092  mat1mhm  21093