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 41879
 Description: Lemma 2 for rhmsubc 41882. (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 5072 . . . 4 ((𝑋𝑅𝑌𝑅) → ⟨𝑋, 𝑌⟩ ∈ (𝑅 × 𝑅))
213adant1 1072 . . 3 ((𝜑𝑋𝑅𝑌𝑅) → ⟨𝑋, 𝑌⟩ ∈ (𝑅 × 𝑅))
3 fvres 6117 . . 3 (⟨𝑋, 𝑌⟩ ∈ (𝑅 × 𝑅) → (( RingHom ↾ (𝑅 × 𝑅))‘⟨𝑋, 𝑌⟩) = ( RingHom ‘⟨𝑋, 𝑌⟩))
42, 3syl 17 . 2 ((𝜑𝑋𝑅𝑌𝑅) → (( RingHom ↾ (𝑅 × 𝑅))‘⟨𝑋, 𝑌⟩) = ( RingHom ‘⟨𝑋, 𝑌⟩))
5 df-ov 6552 . . 3 (𝑋𝐻𝑌) = (𝐻‘⟨𝑋, 𝑌⟩)
6 rngcrescrhm.h . . . 4 𝐻 = ( RingHom ↾ (𝑅 × 𝑅))
76fveq1i 6104 . . 3 (𝐻‘⟨𝑋, 𝑌⟩) = (( RingHom ↾ (𝑅 × 𝑅))‘⟨𝑋, 𝑌⟩)
85, 7eqtri 2632 . 2 (𝑋𝐻𝑌) = (( RingHom ↾ (𝑅 × 𝑅))‘⟨𝑋, 𝑌⟩)
9 df-ov 6552 . 2 (𝑋 RingHom 𝑌) = ( RingHom ‘⟨𝑋, 𝑌⟩)
104, 8, 93eqtr4g 2669 1 ((𝜑𝑋𝑅𝑌𝑅) → (𝑋𝐻𝑌) = (𝑋 RingHom 𝑌))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ∩ cin 3539  ⟨cop 4131   × cxp 5036   ↾ cres 5040  ‘cfv 5804  (class class class)co 6549  Ringcrg 18370   RingHom crh 18535  RngCatcrngc 41749 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-xp 5044  df-res 5050  df-iota 5768  df-fv 5812  df-ov 6552 This theorem is referenced by:  rhmsubclem3  41880  rhmsubclem4  41881
 Copyright terms: Public domain W3C validator