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Theorem crngocom 37987
Description: The multiplication operation of a commutative ring is commutative. (Contributed by Jeff Madsen, 8-Jun-2010.)
Hypotheses
Ref Expression
crngocom.1 𝐺 = (1st𝑅)
crngocom.2 𝐻 = (2nd𝑅)
crngocom.3 𝑋 = ran 𝐺
Assertion
Ref Expression
crngocom ((𝑅 ∈ CRingOps ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐻𝐵) = (𝐵𝐻𝐴))

Proof of Theorem crngocom
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 crngocom.1 . . . . 5 𝐺 = (1st𝑅)
2 crngocom.2 . . . . 5 𝐻 = (2nd𝑅)
3 crngocom.3 . . . . 5 𝑋 = ran 𝐺
41, 2, 3iscrngo2 37983 . . . 4 (𝑅 ∈ CRingOps ↔ (𝑅 ∈ RingOps ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝐻𝑦) = (𝑦𝐻𝑥)))
54simprbi 496 . . 3 (𝑅 ∈ CRingOps → ∀𝑥𝑋𝑦𝑋 (𝑥𝐻𝑦) = (𝑦𝐻𝑥))
6 oveq1 7437 . . . . 5 (𝑥 = 𝐴 → (𝑥𝐻𝑦) = (𝐴𝐻𝑦))
7 oveq2 7438 . . . . 5 (𝑥 = 𝐴 → (𝑦𝐻𝑥) = (𝑦𝐻𝐴))
86, 7eqeq12d 2750 . . . 4 (𝑥 = 𝐴 → ((𝑥𝐻𝑦) = (𝑦𝐻𝑥) ↔ (𝐴𝐻𝑦) = (𝑦𝐻𝐴)))
9 oveq2 7438 . . . . 5 (𝑦 = 𝐵 → (𝐴𝐻𝑦) = (𝐴𝐻𝐵))
10 oveq1 7437 . . . . 5 (𝑦 = 𝐵 → (𝑦𝐻𝐴) = (𝐵𝐻𝐴))
119, 10eqeq12d 2750 . . . 4 (𝑦 = 𝐵 → ((𝐴𝐻𝑦) = (𝑦𝐻𝐴) ↔ (𝐴𝐻𝐵) = (𝐵𝐻𝐴)))
128, 11rspc2v 3632 . . 3 ((𝐴𝑋𝐵𝑋) → (∀𝑥𝑋𝑦𝑋 (𝑥𝐻𝑦) = (𝑦𝐻𝑥) → (𝐴𝐻𝐵) = (𝐵𝐻𝐴)))
135, 12mpan9 506 . 2 ((𝑅 ∈ CRingOps ∧ (𝐴𝑋𝐵𝑋)) → (𝐴𝐻𝐵) = (𝐵𝐻𝐴))
14133impb 1114 1 ((𝑅 ∈ CRingOps ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐻𝐵) = (𝐵𝐻𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1536  wcel 2105  wral 3058  ran crn 5689  cfv 6562  (class class class)co 7430  1st c1st 8010  2nd c2nd 8011  RingOpscrngo 37880  CRingOpsccring 37979
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-iota 6515  df-fun 6564  df-fv 6570  df-ov 7433  df-1st 8012  df-2nd 8013  df-rngo 37881  df-com2 37976  df-crngo 37980
This theorem is referenced by:  crngm23  37988  crngohomfo  37992  isidlc  38001  dmncan2  38063
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