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Theorem rhmsubcALTVlem2 48203
Description: Lemma 2 for rhmsubcALTV 48206. (Contributed by AV, 2-Mar-2020.) (New usage is discouraged.)
Hypotheses
Ref Expression
rngcrescrhmALTV.u (𝜑𝑈𝑉)
rngcrescrhmALTV.c 𝐶 = (RngCatALTV‘𝑈)
rngcrescrhmALTV.r (𝜑𝑅 = (Ring ∩ 𝑈))
rngcrescrhmALTV.h 𝐻 = ( RingHom ↾ (𝑅 × 𝑅))
Assertion
Ref Expression
rhmsubcALTVlem2 ((𝜑𝑋𝑅𝑌𝑅) → (𝑋𝐻𝑌) = (𝑋 RingHom 𝑌))

Proof of Theorem rhmsubcALTVlem2
StepHypRef Expression
1 opelxpi 5721 . . . 4 ((𝑋𝑅𝑌𝑅) → ⟨𝑋, 𝑌⟩ ∈ (𝑅 × 𝑅))
213adant1 1130 . . 3 ((𝜑𝑋𝑅𝑌𝑅) → ⟨𝑋, 𝑌⟩ ∈ (𝑅 × 𝑅))
32fvresd 6925 . 2 ((𝜑𝑋𝑅𝑌𝑅) → (( RingHom ↾ (𝑅 × 𝑅))‘⟨𝑋, 𝑌⟩) = ( RingHom ‘⟨𝑋, 𝑌⟩))
4 df-ov 7435 . . 3 (𝑋𝐻𝑌) = (𝐻‘⟨𝑋, 𝑌⟩)
5 rngcrescrhmALTV.h . . . 4 𝐻 = ( RingHom ↾ (𝑅 × 𝑅))
65fveq1i 6906 . . 3 (𝐻‘⟨𝑋, 𝑌⟩) = (( RingHom ↾ (𝑅 × 𝑅))‘⟨𝑋, 𝑌⟩)
74, 6eqtri 2764 . 2 (𝑋𝐻𝑌) = (( RingHom ↾ (𝑅 × 𝑅))‘⟨𝑋, 𝑌⟩)
8 df-ov 7435 . 2 (𝑋 RingHom 𝑌) = ( RingHom ‘⟨𝑋, 𝑌⟩)
93, 7, 83eqtr4g 2801 1 ((𝜑𝑋𝑅𝑌𝑅) → (𝑋𝐻𝑌) = (𝑋 RingHom 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1086   = wceq 1539  wcel 2107  cin 3949  cop 4631   × cxp 5682  cres 5686  cfv 6560  (class class class)co 7432  Ringcrg 20231   RingHom crh 20470  RngCatALTVcrngcALTV 48184
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-xp 5690  df-res 5696  df-iota 6513  df-fv 6568  df-ov 7435
This theorem is referenced by:  rhmsubcALTVlem3  48204  rhmsubcALTVlem4  48205
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