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Theorem iscrngo2 32966
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 32965 . 2 (𝑅 ∈ CRingOps ↔ (𝑅 ∈ RingOps ∧ 𝑅 ∈ Com2))
2 relrngo 32865 . . . . 5 Rel RingOps
3 1st2nd 7105 . . . . 5 ((Rel RingOps ∧ 𝑅 ∈ RingOps) → 𝑅 = ⟨(1st𝑅), (2nd𝑅)⟩)
42, 3mpan 702 . . . 4 (𝑅 ∈ RingOps → 𝑅 = ⟨(1st𝑅), (2nd𝑅)⟩)
5 eleq1 2676 . . . . 5 (𝑅 = ⟨(1st𝑅), (2nd𝑅)⟩ → (𝑅 ∈ Com2 ↔ ⟨(1st𝑅), (2nd𝑅)⟩ ∈ Com2))
6 iscring2.3 . . . . . . . 8 𝑋 = ran 𝐺
7 iscring2.1 . . . . . . . . 9 𝐺 = (1st𝑅)
87rneqi 5273 . . . . . . . 8 ran 𝐺 = ran (1st𝑅)
96, 8eqtri 2632 . . . . . . 7 𝑋 = ran (1st𝑅)
109raleqi 3119 . . . . . 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 6562 . . . . . . . . 9 (𝑥𝐻𝑦) = (𝑥(2nd𝑅)𝑦)
1311oveqi 6562 . . . . . . . . 9 (𝑦𝐻𝑥) = (𝑦(2nd𝑅)𝑥)
1412, 13eqeq12i 2624 . . . . . . . 8 ((𝑥𝐻𝑦) = (𝑦𝐻𝑥) ↔ (𝑥(2nd𝑅)𝑦) = (𝑦(2nd𝑅)𝑥))
159, 14raleqbii 2973 . . . . . . 7 (∀𝑦𝑋 (𝑥𝐻𝑦) = (𝑦𝐻𝑥) ↔ ∀𝑦 ∈ ran (1st𝑅)(𝑥(2nd𝑅)𝑦) = (𝑦(2nd𝑅)𝑥))
1615ralbii 2963 . . . . . 6 (∀𝑥𝑋𝑦𝑋 (𝑥𝐻𝑦) = (𝑦𝐻𝑥) ↔ ∀𝑥𝑋𝑦 ∈ ran (1st𝑅)(𝑥(2nd𝑅)𝑦) = (𝑦(2nd𝑅)𝑥))
17 fvex 6113 . . . . . . 7 (1st𝑅) ∈ V
18 fvex 6113 . . . . . . 7 (2nd𝑅) ∈ V
19 iscom2 32964 . . . . . . 7 (((1st𝑅) ∈ V ∧ (2nd𝑅) ∈ V) → (⟨(1st𝑅), (2nd𝑅)⟩ ∈ Com2 ↔ ∀𝑥 ∈ ran (1st𝑅)∀𝑦 ∈ ran (1st𝑅)(𝑥(2nd𝑅)𝑦) = (𝑦(2nd𝑅)𝑥)))
2017, 18, 19mp2an 704 . . . . . 6 (⟨(1st𝑅), (2nd𝑅)⟩ ∈ Com2 ↔ ∀𝑥 ∈ ran (1st𝑅)∀𝑦 ∈ ran (1st𝑅)(𝑥(2nd𝑅)𝑦) = (𝑦(2nd𝑅)𝑥))
2110, 16, 203bitr4ri 292 . . . . 5 (⟨(1st𝑅), (2nd𝑅)⟩ ∈ Com2 ↔ ∀𝑥𝑋𝑦𝑋 (𝑥𝐻𝑦) = (𝑦𝐻𝑥))
225, 21syl6bb 275 . . . 4 (𝑅 = ⟨(1st𝑅), (2nd𝑅)⟩ → (𝑅 ∈ Com2 ↔ ∀𝑥𝑋𝑦𝑋 (𝑥𝐻𝑦) = (𝑦𝐻𝑥)))
234, 22syl 17 . . 3 (𝑅 ∈ RingOps → (𝑅 ∈ Com2 ↔ ∀𝑥𝑋𝑦𝑋 (𝑥𝐻𝑦) = (𝑦𝐻𝑥)))
2423pm5.32i 667 . 2 ((𝑅 ∈ RingOps ∧ 𝑅 ∈ Com2) ↔ (𝑅 ∈ RingOps ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝐻𝑦) = (𝑦𝐻𝑥)))
251, 24bitri 263 1 (𝑅 ∈ CRingOps ↔ (𝑅 ∈ RingOps ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝐻𝑦) = (𝑦𝐻𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wb 195  wa 383   = wceq 1475  wcel 1977  wral 2896  Vcvv 3173  cop 4131  ran crn 5039  Rel wrel 5043  cfv 5804  (class class class)co 6549  1st c1st 7057  2nd c2nd 7058  RingOpscrngo 32863  Com2ccm2 32958  CRingOpsccring 32962
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-1st 7059  df-2nd 7060  df-rngo 32864  df-com2 32959  df-crngo 32963
This theorem is referenced by:  crngocom  32970  crngohomfo  32975
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