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Theorem iscrngo2 37321
Description: The predicate "is a commutative ring". (Contributed by Jeff Madsen, 8-Jun-2010.)
Hypotheses
Ref Expression
iscring2.1 𝐺 = (1st𝑅)
iscring2.2 𝐻 = (2nd𝑅)
iscring2.3 𝑋 = ran 𝐺
Assertion
Ref Expression
iscrngo2 (𝑅 ∈ CRingOps ↔ (𝑅 ∈ RingOps ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝐻𝑦) = (𝑦𝐻𝑥)))
Distinct variable groups:   𝑥,𝑅,𝑦   𝑥,𝑋,𝑦
Allowed substitution hints:   𝐺(𝑥,𝑦)   𝐻(𝑥,𝑦)

Proof of Theorem iscrngo2
StepHypRef Expression
1 iscrngo 37320 . 2 (𝑅 ∈ CRingOps ↔ (𝑅 ∈ RingOps ∧ 𝑅 ∈ Com2))
2 relrngo 37220 . . . . 5 Rel RingOps
3 1st2nd 8018 . . . . 5 ((Rel RingOps ∧ 𝑅 ∈ RingOps) → 𝑅 = ⟨(1st𝑅), (2nd𝑅)⟩)
42, 3mpan 687 . . . 4 (𝑅 ∈ RingOps → 𝑅 = ⟨(1st𝑅), (2nd𝑅)⟩)
5 eleq1 2813 . . . . 5 (𝑅 = ⟨(1st𝑅), (2nd𝑅)⟩ → (𝑅 ∈ Com2 ↔ ⟨(1st𝑅), (2nd𝑅)⟩ ∈ Com2))
6 iscring2.3 . . . . . . . 8 𝑋 = ran 𝐺
7 iscring2.1 . . . . . . . . 9 𝐺 = (1st𝑅)
87rneqi 5926 . . . . . . . 8 ran 𝐺 = ran (1st𝑅)
96, 8eqtri 2752 . . . . . . 7 𝑋 = ran (1st𝑅)
109raleqi 3315 . . . . . 6 (∀𝑥𝑋𝑦 ∈ ran (1st𝑅)(𝑥(2nd𝑅)𝑦) = (𝑦(2nd𝑅)𝑥) ↔ ∀𝑥 ∈ ran (1st𝑅)∀𝑦 ∈ ran (1st𝑅)(𝑥(2nd𝑅)𝑦) = (𝑦(2nd𝑅)𝑥))
11 iscring2.2 . . . . . . . . . 10 𝐻 = (2nd𝑅)
1211oveqi 7414 . . . . . . . . 9 (𝑥𝐻𝑦) = (𝑥(2nd𝑅)𝑦)
1311oveqi 7414 . . . . . . . . 9 (𝑦𝐻𝑥) = (𝑦(2nd𝑅)𝑥)
1412, 13eqeq12i 2742 . . . . . . . 8 ((𝑥𝐻𝑦) = (𝑦𝐻𝑥) ↔ (𝑥(2nd𝑅)𝑦) = (𝑦(2nd𝑅)𝑥))
159, 14raleqbii 3330 . . . . . . 7 (∀𝑦𝑋 (𝑥𝐻𝑦) = (𝑦𝐻𝑥) ↔ ∀𝑦 ∈ ran (1st𝑅)(𝑥(2nd𝑅)𝑦) = (𝑦(2nd𝑅)𝑥))
1615ralbii 3085 . . . . . 6 (∀𝑥𝑋𝑦𝑋 (𝑥𝐻𝑦) = (𝑦𝐻𝑥) ↔ ∀𝑥𝑋𝑦 ∈ ran (1st𝑅)(𝑥(2nd𝑅)𝑦) = (𝑦(2nd𝑅)𝑥))
17 fvex 6894 . . . . . . 7 (1st𝑅) ∈ V
18 fvex 6894 . . . . . . 7 (2nd𝑅) ∈ V
19 iscom2 37319 . . . . . . 7 (((1st𝑅) ∈ V ∧ (2nd𝑅) ∈ V) → (⟨(1st𝑅), (2nd𝑅)⟩ ∈ Com2 ↔ ∀𝑥 ∈ ran (1st𝑅)∀𝑦 ∈ ran (1st𝑅)(𝑥(2nd𝑅)𝑦) = (𝑦(2nd𝑅)𝑥)))
2017, 18, 19mp2an 689 . . . . . 6 (⟨(1st𝑅), (2nd𝑅)⟩ ∈ Com2 ↔ ∀𝑥 ∈ ran (1st𝑅)∀𝑦 ∈ ran (1st𝑅)(𝑥(2nd𝑅)𝑦) = (𝑦(2nd𝑅)𝑥))
2110, 16, 203bitr4ri 304 . . . . 5 (⟨(1st𝑅), (2nd𝑅)⟩ ∈ Com2 ↔ ∀𝑥𝑋𝑦𝑋 (𝑥𝐻𝑦) = (𝑦𝐻𝑥))
225, 21bitrdi 287 . . . 4 (𝑅 = ⟨(1st𝑅), (2nd𝑅)⟩ → (𝑅 ∈ Com2 ↔ ∀𝑥𝑋𝑦𝑋 (𝑥𝐻𝑦) = (𝑦𝐻𝑥)))
234, 22syl 17 . . 3 (𝑅 ∈ RingOps → (𝑅 ∈ Com2 ↔ ∀𝑥𝑋𝑦𝑋 (𝑥𝐻𝑦) = (𝑦𝐻𝑥)))
2423pm5.32i 574 . 2 ((𝑅 ∈ RingOps ∧ 𝑅 ∈ Com2) ↔ (𝑅 ∈ RingOps ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝐻𝑦) = (𝑦𝐻𝑥)))
251, 24bitri 275 1 (𝑅 ∈ CRingOps ↔ (𝑅 ∈ RingOps ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝐻𝑦) = (𝑦𝐻𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 395   = wceq 1533  wcel 2098  wral 3053  Vcvv 3466  cop 4626  ran crn 5667  Rel wrel 5671  cfv 6533  (class class class)co 7401  1st c1st 7966  2nd c2nd 7967  RingOpscrngo 37218  Com2ccm2 37313  CRingOpsccring 37317
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417  ax-un 7718
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-iota 6485  df-fun 6535  df-fv 6541  df-ov 7404  df-1st 7968  df-2nd 7969  df-rngo 37219  df-com2 37314  df-crngo 37318
This theorem is referenced by:  crngocom  37325  crngohomfo  37330
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