MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mat1rhmelval Structured version   Visualization version   GIF version

Theorem mat1rhmelval 21244
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 7186 . 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 21243 . . . 4 ((𝑅 ∈ Ring ∧ 𝐸𝑉𝑋𝐾) → (𝐹𝑋) = {⟨𝑂, 𝑋⟩})
87fveq1d 6689 . . 3 ((𝑅 ∈ Ring ∧ 𝐸𝑉𝑋𝐾) → ((𝐹𝑋)‘⟨𝐸, 𝐸⟩) = ({⟨𝑂, 𝑋⟩}‘⟨𝐸, 𝐸⟩))
95eqcomi 2748 . . . . 5 𝐸, 𝐸⟩ = 𝑂
109fveq2i 6690 . . . 4 ({⟨𝑂, 𝑋⟩}‘⟨𝐸, 𝐸⟩) = ({⟨𝑂, 𝑋⟩}‘𝑂)
11 opex 5332 . . . . . 6 𝐸, 𝐸⟩ ∈ V
125, 11eqeltri 2830 . . . . 5 𝑂 ∈ V
13 simp3 1139 . . . . 5 ((𝑅 ∈ Ring ∧ 𝐸𝑉𝑋𝐾) → 𝑋𝐾)
14 fvsng 6965 . . . . 5 ((𝑂 ∈ V ∧ 𝑋𝐾) → ({⟨𝑂, 𝑋⟩}‘𝑂) = 𝑋)
1512, 13, 14sylancr 590 . . . 4 ((𝑅 ∈ Ring ∧ 𝐸𝑉𝑋𝐾) → ({⟨𝑂, 𝑋⟩}‘𝑂) = 𝑋)
1610, 15syl5eq 2786 . . 3 ((𝑅 ∈ Ring ∧ 𝐸𝑉𝑋𝐾) → ({⟨𝑂, 𝑋⟩}‘⟨𝐸, 𝐸⟩) = 𝑋)
178, 16eqtrd 2774 . 2 ((𝑅 ∈ Ring ∧ 𝐸𝑉𝑋𝐾) → ((𝐹𝑋)‘⟨𝐸, 𝐸⟩) = 𝑋)
181, 17syl5eq 2786 1 ((𝑅 ∈ Ring ∧ 𝐸𝑉𝑋𝐾) → (𝐸(𝐹𝑋)𝐸) = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1088   = wceq 1542  wcel 2114  Vcvv 3400  {csn 4526  cop 4532  cmpt 5120  cfv 6350  (class class class)co 7183  Basecbs 16599  Ringcrg 19429   Mat cmat 21171
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 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-sep 5177  ax-nul 5184  ax-pr 5306
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ral 3059  df-rex 3060  df-v 3402  df-sbc 3686  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4222  df-if 4425  df-sn 4527  df-pr 4529  df-op 4533  df-uni 4807  df-br 5041  df-opab 5103  df-mpt 5121  df-id 5439  df-xp 5541  df-rel 5542  df-cnv 5543  df-co 5544  df-dm 5545  df-iota 6308  df-fun 6352  df-fv 6358  df-ov 7186
This theorem is referenced by:  mat1ghm  21247  mat1mhm  21248
  Copyright terms: Public domain W3C validator