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Theorem mat1rhmelval 21989
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 7414 . 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 21988 . . . 4 ((𝑅 ∈ Ring ∧ 𝐸𝑉𝑋𝐾) → (𝐹𝑋) = {⟨𝑂, 𝑋⟩})
87fveq1d 6893 . . 3 ((𝑅 ∈ Ring ∧ 𝐸𝑉𝑋𝐾) → ((𝐹𝑋)‘⟨𝐸, 𝐸⟩) = ({⟨𝑂, 𝑋⟩}‘⟨𝐸, 𝐸⟩))
95eqcomi 2741 . . . . 5 𝐸, 𝐸⟩ = 𝑂
109fveq2i 6894 . . . 4 ({⟨𝑂, 𝑋⟩}‘⟨𝐸, 𝐸⟩) = ({⟨𝑂, 𝑋⟩}‘𝑂)
11 opex 5464 . . . . . 6 𝐸, 𝐸⟩ ∈ V
125, 11eqeltri 2829 . . . . 5 𝑂 ∈ V
13 simp3 1138 . . . . 5 ((𝑅 ∈ Ring ∧ 𝐸𝑉𝑋𝐾) → 𝑋𝐾)
14 fvsng 7180 . . . . 5 ((𝑂 ∈ V ∧ 𝑋𝐾) → ({⟨𝑂, 𝑋⟩}‘𝑂) = 𝑋)
1512, 13, 14sylancr 587 . . . 4 ((𝑅 ∈ Ring ∧ 𝐸𝑉𝑋𝐾) → ({⟨𝑂, 𝑋⟩}‘𝑂) = 𝑋)
1610, 15eqtrid 2784 . . 3 ((𝑅 ∈ Ring ∧ 𝐸𝑉𝑋𝐾) → ({⟨𝑂, 𝑋⟩}‘⟨𝐸, 𝐸⟩) = 𝑋)
178, 16eqtrd 2772 . 2 ((𝑅 ∈ Ring ∧ 𝐸𝑉𝑋𝐾) → ((𝐹𝑋)‘⟨𝐸, 𝐸⟩) = 𝑋)
181, 17eqtrid 2784 1 ((𝑅 ∈ Ring ∧ 𝐸𝑉𝑋𝐾) → (𝐸(𝐹𝑋)𝐸) = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1087   = wceq 1541  wcel 2106  Vcvv 3474  {csn 4628  cop 4634  cmpt 5231  cfv 6543  (class class class)co 7411  Basecbs 17146  Ringcrg 20058   Mat cmat 21914
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7414
This theorem is referenced by:  mat1ghm  21992  mat1mhm  21993
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