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Theorem iscrngo2 37984
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 37983 . 2 (𝑅 ∈ CRingOps ↔ (𝑅 ∈ RingOps ∧ 𝑅 ∈ Com2))
2 relrngo 37883 . . . . 5 Rel RingOps
3 1st2nd 8063 . . . . 5 ((Rel RingOps ∧ 𝑅 ∈ RingOps) → 𝑅 = ⟨(1st𝑅), (2nd𝑅)⟩)
42, 3mpan 690 . . . 4 (𝑅 ∈ RingOps → 𝑅 = ⟨(1st𝑅), (2nd𝑅)⟩)
5 eleq1 2827 . . . . 5 (𝑅 = ⟨(1st𝑅), (2nd𝑅)⟩ → (𝑅 ∈ Com2 ↔ ⟨(1st𝑅), (2nd𝑅)⟩ ∈ Com2))
6 iscring2.3 . . . . . . . 8 𝑋 = ran 𝐺
7 iscring2.1 . . . . . . . . 9 𝐺 = (1st𝑅)
87rneqi 5951 . . . . . . . 8 ran 𝐺 = ran (1st𝑅)
96, 8eqtri 2763 . . . . . . 7 𝑋 = ran (1st𝑅)
109raleqi 3322 . . . . . 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 7444 . . . . . . . . 9 (𝑥𝐻𝑦) = (𝑥(2nd𝑅)𝑦)
1311oveqi 7444 . . . . . . . . 9 (𝑦𝐻𝑥) = (𝑦(2nd𝑅)𝑥)
1412, 13eqeq12i 2753 . . . . . . . 8 ((𝑥𝐻𝑦) = (𝑦𝐻𝑥) ↔ (𝑥(2nd𝑅)𝑦) = (𝑦(2nd𝑅)𝑥))
159, 14raleqbii 3342 . . . . . . 7 (∀𝑦𝑋 (𝑥𝐻𝑦) = (𝑦𝐻𝑥) ↔ ∀𝑦 ∈ ran (1st𝑅)(𝑥(2nd𝑅)𝑦) = (𝑦(2nd𝑅)𝑥))
1615ralbii 3091 . . . . . 6 (∀𝑥𝑋𝑦𝑋 (𝑥𝐻𝑦) = (𝑦𝐻𝑥) ↔ ∀𝑥𝑋𝑦 ∈ ran (1st𝑅)(𝑥(2nd𝑅)𝑦) = (𝑦(2nd𝑅)𝑥))
17 fvex 6920 . . . . . . 7 (1st𝑅) ∈ V
18 fvex 6920 . . . . . . 7 (2nd𝑅) ∈ V
19 iscom2 37982 . . . . . . 7 (((1st𝑅) ∈ V ∧ (2nd𝑅) ∈ V) → (⟨(1st𝑅), (2nd𝑅)⟩ ∈ Com2 ↔ ∀𝑥 ∈ ran (1st𝑅)∀𝑦 ∈ ran (1st𝑅)(𝑥(2nd𝑅)𝑦) = (𝑦(2nd𝑅)𝑥)))
2017, 18, 19mp2an 692 . . . . . 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 206  wa 395   = wceq 1537  wcel 2106  wral 3059  Vcvv 3478  cop 4637  ran crn 5690  Rel wrel 5694  cfv 6563  (class class class)co 7431  1st c1st 8011  2nd c2nd 8012  RingOpscrngo 37881  Com2ccm2 37976  CRingOpsccring 37980
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-1st 8013  df-2nd 8014  df-rngo 37882  df-com2 37977  df-crngo 37981
This theorem is referenced by:  crngocom  37988  crngohomfo  37993
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