Theorem mat1rhmelval 21629

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 7278	. 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 21628	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾) → (𝐹‘𝑋) = {⟨𝑂, 𝑋⟩})
8	7	fveq1d 6776	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾) → ((𝐹‘𝑋)‘⟨𝐸, 𝐸⟩) = ({⟨𝑂, 𝑋⟩}‘⟨𝐸, 𝐸⟩))
9	5	eqcomi 2747	. . . . 5 ⊢ ⟨𝐸, 𝐸⟩ = 𝑂
10	9	fveq2i 6777	. . . 4 ⊢ ({⟨𝑂, 𝑋⟩}‘⟨𝐸, 𝐸⟩) = ({⟨𝑂, 𝑋⟩}‘𝑂)
11		opex 5379	. . . . . 6 ⊢ ⟨𝐸, 𝐸⟩ ∈ V
12	5, 11	eqeltri 2835	. . . . 5 ⊢ 𝑂 ∈ V
13		simp3 1137	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾) → 𝑋 ∈ 𝐾)
14		fvsng 7052	. . . . 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 2790	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾) → ({⟨𝑂, 𝑋⟩}‘⟨𝐸, 𝐸⟩) = 𝑋)
17	8, 16	eqtrd 2778	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾) → ((𝐹‘𝑋)‘⟨𝐸, 𝐸⟩) = 𝑋)
18	1, 17	eqtrid 2790	1 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾) → (𝐸(𝐹‘𝑋)𝐸) = 𝑋)

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  Vcvv 3432  {csn 4561  ⟨cop 4567   ↦ cmpt 5157  ‘cfv 6433  (class class class)co 7275  Basecbs 16912  Ringcrg 19783   Mat cmat 21554

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-iota 6391  df-fun 6435  df-fv 6441  df-ov 7278

This theorem is referenced by:  mat1ghm  21632  mat1mhm  21633