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Theorem rhmsubclem2 46134
Description: Lemma 2 for rhmsubc 46137. (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 5667 . . . 4 ((𝑋 ∈ 𝑅 ∧ π‘Œ ∈ 𝑅) β†’ βŸ¨π‘‹, π‘ŒβŸ© ∈ (𝑅 Γ— 𝑅))
213adant1 1130 . . 3 ((πœ‘ ∧ 𝑋 ∈ 𝑅 ∧ π‘Œ ∈ 𝑅) β†’ βŸ¨π‘‹, π‘ŒβŸ© ∈ (𝑅 Γ— 𝑅))
32fvresd 6857 . 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 7352 . . 3 (π‘‹π»π‘Œ) = (π»β€˜βŸ¨π‘‹, π‘ŒβŸ©)
5 rngcrescrhm.h . . . 4 𝐻 = ( RingHom β†Ύ (𝑅 Γ— 𝑅))
65fveq1i 6838 . . 3 (π»β€˜βŸ¨π‘‹, π‘ŒβŸ©) = (( RingHom β†Ύ (𝑅 Γ— 𝑅))β€˜βŸ¨π‘‹, π‘ŒβŸ©)
74, 6eqtri 2765 . 2 (π‘‹π»π‘Œ) = (( RingHom β†Ύ (𝑅 Γ— 𝑅))β€˜βŸ¨π‘‹, π‘ŒβŸ©)
8 df-ov 7352 . 2 (𝑋 RingHom π‘Œ) = ( RingHom β€˜βŸ¨π‘‹, π‘ŒβŸ©)
93, 7, 83eqtr4g 2802 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 1087   = wceq 1541   ∈ wcel 2106   ∩ cin 3907  βŸ¨cop 4590   Γ— cxp 5628   β†Ύ cres 5632  β€˜cfv 6491  (class class class)co 7349  Ringcrg 19888   RingHom crh 20066  RngCatcrngc 46004
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-ext 2708  ax-sep 5254  ax-nul 5261  ax-pr 5382
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-sb 2068  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-br 5104  df-opab 5166  df-xp 5636  df-res 5642  df-iota 6443  df-fv 6499  df-ov 7352
This theorem is referenced by:  rhmsubclem3  46135  rhmsubclem4  46136
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