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Theorem rhmsubclem2 45645
Description: Lemma 2 for rhmsubc 45648. (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 5626 . . . 4 ((𝑋𝑅𝑌𝑅) → ⟨𝑋, 𝑌⟩ ∈ (𝑅 × 𝑅))
213adant1 1129 . . 3 ((𝜑𝑋𝑅𝑌𝑅) → ⟨𝑋, 𝑌⟩ ∈ (𝑅 × 𝑅))
32fvresd 6794 . 2 ((𝜑𝑋𝑅𝑌𝑅) → (( RingHom ↾ (𝑅 × 𝑅))‘⟨𝑋, 𝑌⟩) = ( RingHom ‘⟨𝑋, 𝑌⟩))
4 df-ov 7278 . . 3 (𝑋𝐻𝑌) = (𝐻‘⟨𝑋, 𝑌⟩)
5 rngcrescrhm.h . . . 4 𝐻 = ( RingHom ↾ (𝑅 × 𝑅))
65fveq1i 6775 . . 3 (𝐻‘⟨𝑋, 𝑌⟩) = (( RingHom ↾ (𝑅 × 𝑅))‘⟨𝑋, 𝑌⟩)
74, 6eqtri 2766 . 2 (𝑋𝐻𝑌) = (( RingHom ↾ (𝑅 × 𝑅))‘⟨𝑋, 𝑌⟩)
8 df-ov 7278 . 2 (𝑋 RingHom 𝑌) = ( RingHom ‘⟨𝑋, 𝑌⟩)
93, 7, 83eqtr4g 2803 1 ((𝜑𝑋𝑅𝑌𝑅) → (𝑋𝐻𝑌) = (𝑋 RingHom 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1086   = wceq 1539  wcel 2106  cin 3886  cop 4567   × cxp 5587  cres 5591  cfv 6433  (class class class)co 7275  Ringcrg 19783   RingHom crh 19956  RngCatcrngc 45515
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-xp 5595  df-res 5601  df-iota 6391  df-fv 6441  df-ov 7278
This theorem is referenced by:  rhmsubclem3  45646  rhmsubclem4  45647
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