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Theorem mat1rhmelval 21982
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 7412 . 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 21981 . . . 4 ((𝑅 ∈ Ring ∧ 𝐸𝑉𝑋𝐾) → (𝐹𝑋) = {⟨𝑂, 𝑋⟩})
87fveq1d 6894 . . 3 ((𝑅 ∈ Ring ∧ 𝐸𝑉𝑋𝐾) → ((𝐹𝑋)‘⟨𝐸, 𝐸⟩) = ({⟨𝑂, 𝑋⟩}‘⟨𝐸, 𝐸⟩))
95eqcomi 2742 . . . . 5 𝐸, 𝐸⟩ = 𝑂
109fveq2i 6895 . . . 4 ({⟨𝑂, 𝑋⟩}‘⟨𝐸, 𝐸⟩) = ({⟨𝑂, 𝑋⟩}‘𝑂)
11 opex 5465 . . . . . 6 𝐸, 𝐸⟩ ∈ V
125, 11eqeltri 2830 . . . . 5 𝑂 ∈ V
13 simp3 1139 . . . . 5 ((𝑅 ∈ Ring ∧ 𝐸𝑉𝑋𝐾) → 𝑋𝐾)
14 fvsng 7178 . . . . 5 ((𝑂 ∈ V ∧ 𝑋𝐾) → ({⟨𝑂, 𝑋⟩}‘𝑂) = 𝑋)
1512, 13, 14sylancr 588 . . . 4 ((𝑅 ∈ Ring ∧ 𝐸𝑉𝑋𝐾) → ({⟨𝑂, 𝑋⟩}‘𝑂) = 𝑋)
1610, 15eqtrid 2785 . . 3 ((𝑅 ∈ Ring ∧ 𝐸𝑉𝑋𝐾) → ({⟨𝑂, 𝑋⟩}‘⟨𝐸, 𝐸⟩) = 𝑋)
178, 16eqtrd 2773 . 2 ((𝑅 ∈ Ring ∧ 𝐸𝑉𝑋𝐾) → ((𝐹𝑋)‘⟨𝐸, 𝐸⟩) = 𝑋)
181, 17eqtrid 2785 1 ((𝑅 ∈ Ring ∧ 𝐸𝑉𝑋𝐾) → (𝐸(𝐹𝑋)𝐸) = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1088   = wceq 1542  wcel 2107  Vcvv 3475  {csn 4629  cop 4635  cmpt 5232  cfv 6544  (class class class)co 7409  Basecbs 17144  Ringcrg 20056   Mat cmat 21907
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-iota 6496  df-fun 6546  df-fv 6552  df-ov 7412
This theorem is referenced by:  mat1ghm  21985  mat1mhm  21986
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