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Theorem rhmsubclem2 45624
Description: Lemma 2 for rhmsubc 45627. (Contributed by AV, 2-Mar-2020.)
Hypotheses
Ref Expression
rngcrescrhm.u (𝜑𝑈𝑉)
rngcrescrhm.c 𝐶 = (RngCat‘𝑈)
rngcrescrhm.r (𝜑𝑅 = (Ring ∩ 𝑈))
rngcrescrhm.h 𝐻 = ( RingHom ↾ (𝑅 × 𝑅))
Assertion
Ref Expression
rhmsubclem2 ((𝜑𝑋𝑅𝑌𝑅) → (𝑋𝐻𝑌) = (𝑋 RingHom 𝑌))

Proof of Theorem rhmsubclem2
StepHypRef Expression
1 opelxpi 5627 . . . 4 ((𝑋𝑅𝑌𝑅) → ⟨𝑋, 𝑌⟩ ∈ (𝑅 × 𝑅))
213adant1 1129 . . 3 ((𝜑𝑋𝑅𝑌𝑅) → ⟨𝑋, 𝑌⟩ ∈ (𝑅 × 𝑅))
32fvresd 6791 . 2 ((𝜑𝑋𝑅𝑌𝑅) → (( RingHom ↾ (𝑅 × 𝑅))‘⟨𝑋, 𝑌⟩) = ( RingHom ‘⟨𝑋, 𝑌⟩))
4 df-ov 7275 . . 3 (𝑋𝐻𝑌) = (𝐻‘⟨𝑋, 𝑌⟩)
5 rngcrescrhm.h . . . 4 𝐻 = ( RingHom ↾ (𝑅 × 𝑅))
65fveq1i 6772 . . 3 (𝐻‘⟨𝑋, 𝑌⟩) = (( RingHom ↾ (𝑅 × 𝑅))‘⟨𝑋, 𝑌⟩)
74, 6eqtri 2768 . 2 (𝑋𝐻𝑌) = (( RingHom ↾ (𝑅 × 𝑅))‘⟨𝑋, 𝑌⟩)
8 df-ov 7275 . 2 (𝑋 RingHom 𝑌) = ( RingHom ‘⟨𝑋, 𝑌⟩)
93, 7, 83eqtr4g 2805 1 ((𝜑𝑋𝑅𝑌𝑅) → (𝑋𝐻𝑌) = (𝑋 RingHom 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1086   = wceq 1542  wcel 2110  cin 3891  cop 4573   × cxp 5588  cres 5592  cfv 6432  (class class class)co 7272  Ringcrg 19794   RingHom crh 19967  RngCatcrngc 45494
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 2015  ax-8 2112  ax-9 2120  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pr 5356
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2072  df-clab 2718  df-cleq 2732  df-clel 2818  df-ral 3071  df-rex 3072  df-rab 3075  df-v 3433  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-br 5080  df-opab 5142  df-xp 5596  df-res 5602  df-iota 6390  df-fv 6440  df-ov 7275
This theorem is referenced by:  rhmsubclem3  45625  rhmsubclem4  45626
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