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Theorem crngocom 33930
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 33926 . . . 4 (𝑅 ∈ CRingOps ↔ (𝑅 ∈ RingOps ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝐻𝑦) = (𝑦𝐻𝑥)))
54simprbi 479 . . 3 (𝑅 ∈ CRingOps → ∀𝑥𝑋𝑦𝑋 (𝑥𝐻𝑦) = (𝑦𝐻𝑥))
6 oveq1 6697 . . . . 5 (𝑥 = 𝐴 → (𝑥𝐻𝑦) = (𝐴𝐻𝑦))
7 oveq2 6698 . . . . 5 (𝑥 = 𝐴 → (𝑦𝐻𝑥) = (𝑦𝐻𝐴))
86, 7eqeq12d 2666 . . . 4 (𝑥 = 𝐴 → ((𝑥𝐻𝑦) = (𝑦𝐻𝑥) ↔ (𝐴𝐻𝑦) = (𝑦𝐻𝐴)))
9 oveq2 6698 . . . . 5 (𝑦 = 𝐵 → (𝐴𝐻𝑦) = (𝐴𝐻𝐵))
10 oveq1 6697 . . . . 5 (𝑦 = 𝐵 → (𝑦𝐻𝐴) = (𝐵𝐻𝐴))
119, 10eqeq12d 2666 . . . 4 (𝑦 = 𝐵 → ((𝐴𝐻𝑦) = (𝑦𝐻𝐴) ↔ (𝐴𝐻𝐵) = (𝐵𝐻𝐴)))
128, 11rspc2v 3353 . . 3 ((𝐴𝑋𝐵𝑋) → (∀𝑥𝑋𝑦𝑋 (𝑥𝐻𝑦) = (𝑦𝐻𝑥) → (𝐴𝐻𝐵) = (𝐵𝐻𝐴)))
135, 12mpan9 485 . 2 ((𝑅 ∈ CRingOps ∧ (𝐴𝑋𝐵𝑋)) → (𝐴𝐻𝐵) = (𝐵𝐻𝐴))
14133impb 1279 1 ((𝑅 ∈ CRingOps ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐻𝐵) = (𝐵𝐻𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  ran crn 5144  cfv 5926  (class class class)co 6690  1st c1st 7208  2nd c2nd 7209  RingOpscrngo 33823  CRingOpsccring 33922
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-iota 5889  df-fun 5928  df-fv 5934  df-ov 6693  df-1st 7210  df-2nd 7211  df-rngo 33824  df-com2 33919  df-crngo 33923
This theorem is referenced by:  crngm23  33931  crngohomfo  33935  isidlc  33944  dmncan2  34006
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