Mathbox for Jeff Madsen < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  crngocom Structured version   Visualization version   GIF version

Theorem crngocom 35166
 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 35162 . . . 4 (𝑅 ∈ CRingOps ↔ (𝑅 ∈ RingOps ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝐻𝑦) = (𝑦𝐻𝑥)))
54simprbi 497 . . 3 (𝑅 ∈ CRingOps → ∀𝑥𝑋𝑦𝑋 (𝑥𝐻𝑦) = (𝑦𝐻𝑥))
6 oveq1 7157 . . . . 5 (𝑥 = 𝐴 → (𝑥𝐻𝑦) = (𝐴𝐻𝑦))
7 oveq2 7158 . . . . 5 (𝑥 = 𝐴 → (𝑦𝐻𝑥) = (𝑦𝐻𝐴))
86, 7eqeq12d 2842 . . . 4 (𝑥 = 𝐴 → ((𝑥𝐻𝑦) = (𝑦𝐻𝑥) ↔ (𝐴𝐻𝑦) = (𝑦𝐻𝐴)))
9 oveq2 7158 . . . . 5 (𝑦 = 𝐵 → (𝐴𝐻𝑦) = (𝐴𝐻𝐵))
10 oveq1 7157 . . . . 5 (𝑦 = 𝐵 → (𝑦𝐻𝐴) = (𝐵𝐻𝐴))
119, 10eqeq12d 2842 . . . 4 (𝑦 = 𝐵 → ((𝐴𝐻𝑦) = (𝑦𝐻𝐴) ↔ (𝐴𝐻𝐵) = (𝐵𝐻𝐴)))
128, 11rspc2v 3637 . . 3 ((𝐴𝑋𝐵𝑋) → (∀𝑥𝑋𝑦𝑋 (𝑥𝐻𝑦) = (𝑦𝐻𝑥) → (𝐴𝐻𝐵) = (𝐵𝐻𝐴)))
135, 12mpan9 507 . 2 ((𝑅 ∈ CRingOps ∧ (𝐴𝑋𝐵𝑋)) → (𝐴𝐻𝐵) = (𝐵𝐻𝐴))
14133impb 1109 1 ((𝑅 ∈ CRingOps ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐻𝐵) = (𝐵𝐻𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1081   = wceq 1530   ∈ wcel 2107  ∀wral 3143  ran crn 5555  ‘cfv 6354  (class class class)co 7150  1st c1st 7683  2nd c2nd 7684  RingOpscrngo 35059  CRingOpsccring 35158 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-iota 6313  df-fun 6356  df-fv 6362  df-ov 7153  df-1st 7685  df-2nd 7686  df-rngo 35060  df-com2 35155  df-crngo 35159 This theorem is referenced by:  crngm23  35167  crngohomfo  35171  isidlc  35180  dmncan2  35242
 Copyright terms: Public domain W3C validator