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Theorem crngocom 34279
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 34275 . . . 4 (𝑅 ∈ CRingOps ↔ (𝑅 ∈ RingOps ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝐻𝑦) = (𝑦𝐻𝑥)))
54simprbi 491 . . 3 (𝑅 ∈ CRingOps → ∀𝑥𝑋𝑦𝑋 (𝑥𝐻𝑦) = (𝑦𝐻𝑥))
6 oveq1 6883 . . . . 5 (𝑥 = 𝐴 → (𝑥𝐻𝑦) = (𝐴𝐻𝑦))
7 oveq2 6884 . . . . 5 (𝑥 = 𝐴 → (𝑦𝐻𝑥) = (𝑦𝐻𝐴))
86, 7eqeq12d 2812 . . . 4 (𝑥 = 𝐴 → ((𝑥𝐻𝑦) = (𝑦𝐻𝑥) ↔ (𝐴𝐻𝑦) = (𝑦𝐻𝐴)))
9 oveq2 6884 . . . . 5 (𝑦 = 𝐵 → (𝐴𝐻𝑦) = (𝐴𝐻𝐵))
10 oveq1 6883 . . . . 5 (𝑦 = 𝐵 → (𝑦𝐻𝐴) = (𝐵𝐻𝐴))
119, 10eqeq12d 2812 . . . 4 (𝑦 = 𝐵 → ((𝐴𝐻𝑦) = (𝑦𝐻𝐴) ↔ (𝐴𝐻𝐵) = (𝐵𝐻𝐴)))
128, 11rspc2v 3508 . . 3 ((𝐴𝑋𝐵𝑋) → (∀𝑥𝑋𝑦𝑋 (𝑥𝐻𝑦) = (𝑦𝐻𝑥) → (𝐴𝐻𝐵) = (𝐵𝐻𝐴)))
135, 12mpan9 503 . 2 ((𝑅 ∈ CRingOps ∧ (𝐴𝑋𝐵𝑋)) → (𝐴𝐻𝐵) = (𝐵𝐻𝐴))
14133impb 1144 1 ((𝑅 ∈ CRingOps ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐻𝐵) = (𝐵𝐻𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385  w3a 1108   = wceq 1653  wcel 2157  wral 3087  ran crn 5311  cfv 6099  (class class class)co 6876  1st c1st 7397  2nd c2nd 7398  RingOpscrngo 34172  CRingOpsccring 34271
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-sep 4973  ax-nul 4981  ax-pow 5033  ax-pr 5095  ax-un 7181
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ral 3092  df-rex 3093  df-rab 3096  df-v 3385  df-sbc 3632  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-sn 4367  df-pr 4369  df-op 4373  df-uni 4627  df-br 4842  df-opab 4904  df-mpt 4921  df-id 5218  df-xp 5316  df-rel 5317  df-cnv 5318  df-co 5319  df-dm 5320  df-rn 5321  df-iota 6062  df-fun 6101  df-fv 6107  df-ov 6879  df-1st 7399  df-2nd 7400  df-rngo 34173  df-com2 34268  df-crngo 34272
This theorem is referenced by:  crngm23  34280  crngohomfo  34284  isidlc  34293  dmncan2  34355
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