Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rhmsubclem2 Structured version   Visualization version   GIF version

Theorem rhmsubclem2 46085
Description: Lemma 2 for rhmsubc 46088. (Contributed by AV, 2-Mar-2020.)
Hypotheses
Ref Expression
rngcrescrhm.u (πœ‘ β†’ π‘ˆ ∈ 𝑉)
rngcrescrhm.c 𝐢 = (RngCatβ€˜π‘ˆ)
rngcrescrhm.r (πœ‘ β†’ 𝑅 = (Ring ∩ π‘ˆ))
rngcrescrhm.h 𝐻 = ( RingHom β†Ύ (𝑅 Γ— 𝑅))
Assertion
Ref Expression
rhmsubclem2 ((πœ‘ ∧ 𝑋 ∈ 𝑅 ∧ π‘Œ ∈ 𝑅) β†’ (π‘‹π»π‘Œ) = (𝑋 RingHom π‘Œ))

Proof of Theorem rhmsubclem2
StepHypRef Expression
1 opelxpi 5668 . . . 4 ((𝑋 ∈ 𝑅 ∧ π‘Œ ∈ 𝑅) β†’ βŸ¨π‘‹, π‘ŒβŸ© ∈ (𝑅 Γ— 𝑅))
213adant1 1131 . . 3 ((πœ‘ ∧ 𝑋 ∈ 𝑅 ∧ π‘Œ ∈ 𝑅) β†’ βŸ¨π‘‹, π‘ŒβŸ© ∈ (𝑅 Γ— 𝑅))
32fvresd 6858 . 2 ((πœ‘ ∧ 𝑋 ∈ 𝑅 ∧ π‘Œ ∈ 𝑅) β†’ (( RingHom β†Ύ (𝑅 Γ— 𝑅))β€˜βŸ¨π‘‹, π‘ŒβŸ©) = ( RingHom β€˜βŸ¨π‘‹, π‘ŒβŸ©))
4 df-ov 7353 . . 3 (π‘‹π»π‘Œ) = (π»β€˜βŸ¨π‘‹, π‘ŒβŸ©)
5 rngcrescrhm.h . . . 4 𝐻 = ( RingHom β†Ύ (𝑅 Γ— 𝑅))
65fveq1i 6839 . . 3 (π»β€˜βŸ¨π‘‹, π‘ŒβŸ©) = (( RingHom β†Ύ (𝑅 Γ— 𝑅))β€˜βŸ¨π‘‹, π‘ŒβŸ©)
74, 6eqtri 2766 . 2 (π‘‹π»π‘Œ) = (( RingHom β†Ύ (𝑅 Γ— 𝑅))β€˜βŸ¨π‘‹, π‘ŒβŸ©)
8 df-ov 7353 . 2 (𝑋 RingHom π‘Œ) = ( RingHom β€˜βŸ¨π‘‹, π‘ŒβŸ©)
93, 7, 83eqtr4g 2803 1 ((πœ‘ ∧ 𝑋 ∈ 𝑅 ∧ π‘Œ ∈ 𝑅) β†’ (π‘‹π»π‘Œ) = (𝑋 RingHom π‘Œ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   ∩ cin 3908  βŸ¨cop 4591   Γ— cxp 5629   β†Ύ cres 5633  β€˜cfv 6492  (class class class)co 7350  Ringcrg 19888   RingHom crh 20066  RngCatcrngc 45955
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-ext 2709  ax-sep 5255  ax-nul 5262  ax-pr 5383
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-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3064  df-rex 3073  df-rab 3407  df-v 3446  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-br 5105  df-opab 5167  df-xp 5637  df-res 5643  df-iota 6444  df-fv 6500  df-ov 7353
This theorem is referenced by:  rhmsubclem3  46086  rhmsubclem4  46087
  Copyright terms: Public domain W3C validator