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Theorem iscrngo2 35156
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 35155 . 2 (𝑅 ∈ CRingOps ↔ (𝑅 ∈ RingOps ∧ 𝑅 ∈ Com2))
2 relrngo 35055 . . . . 5 Rel RingOps
3 1st2nd 7727 . . . . 5 ((Rel RingOps ∧ 𝑅 ∈ RingOps) → 𝑅 = ⟨(1st𝑅), (2nd𝑅)⟩)
42, 3mpan 686 . . . 4 (𝑅 ∈ RingOps → 𝑅 = ⟨(1st𝑅), (2nd𝑅)⟩)
5 eleq1 2897 . . . . 5 (𝑅 = ⟨(1st𝑅), (2nd𝑅)⟩ → (𝑅 ∈ Com2 ↔ ⟨(1st𝑅), (2nd𝑅)⟩ ∈ Com2))
6 iscring2.3 . . . . . . . 8 𝑋 = ran 𝐺
7 iscring2.1 . . . . . . . . 9 𝐺 = (1st𝑅)
87rneqi 5800 . . . . . . . 8 ran 𝐺 = ran (1st𝑅)
96, 8eqtri 2841 . . . . . . 7 𝑋 = ran (1st𝑅)
109raleqi 3411 . . . . . 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 7158 . . . . . . . . 9 (𝑥𝐻𝑦) = (𝑥(2nd𝑅)𝑦)
1311oveqi 7158 . . . . . . . . 9 (𝑦𝐻𝑥) = (𝑦(2nd𝑅)𝑥)
1412, 13eqeq12i 2833 . . . . . . . 8 ((𝑥𝐻𝑦) = (𝑦𝐻𝑥) ↔ (𝑥(2nd𝑅)𝑦) = (𝑦(2nd𝑅)𝑥))
159, 14raleqbii 3231 . . . . . . 7 (∀𝑦𝑋 (𝑥𝐻𝑦) = (𝑦𝐻𝑥) ↔ ∀𝑦 ∈ ran (1st𝑅)(𝑥(2nd𝑅)𝑦) = (𝑦(2nd𝑅)𝑥))
1615ralbii 3162 . . . . . 6 (∀𝑥𝑋𝑦𝑋 (𝑥𝐻𝑦) = (𝑦𝐻𝑥) ↔ ∀𝑥𝑋𝑦 ∈ ran (1st𝑅)(𝑥(2nd𝑅)𝑦) = (𝑦(2nd𝑅)𝑥))
17 fvex 6676 . . . . . . 7 (1st𝑅) ∈ V
18 fvex 6676 . . . . . . 7 (2nd𝑅) ∈ V
19 iscom2 35154 . . . . . . 7 (((1st𝑅) ∈ V ∧ (2nd𝑅) ∈ V) → (⟨(1st𝑅), (2nd𝑅)⟩ ∈ Com2 ↔ ∀𝑥 ∈ ran (1st𝑅)∀𝑦 ∈ ran (1st𝑅)(𝑥(2nd𝑅)𝑦) = (𝑦(2nd𝑅)𝑥)))
2017, 18, 19mp2an 688 . . . . . 6 (⟨(1st𝑅), (2nd𝑅)⟩ ∈ Com2 ↔ ∀𝑥 ∈ ran (1st𝑅)∀𝑦 ∈ ran (1st𝑅)(𝑥(2nd𝑅)𝑦) = (𝑦(2nd𝑅)𝑥))
2110, 16, 203bitr4ri 305 . . . . 5 (⟨(1st𝑅), (2nd𝑅)⟩ ∈ Com2 ↔ ∀𝑥𝑋𝑦𝑋 (𝑥𝐻𝑦) = (𝑦𝐻𝑥))
225, 21syl6bb 288 . . . 4 (𝑅 = ⟨(1st𝑅), (2nd𝑅)⟩ → (𝑅 ∈ Com2 ↔ ∀𝑥𝑋𝑦𝑋 (𝑥𝐻𝑦) = (𝑦𝐻𝑥)))
234, 22syl 17 . . 3 (𝑅 ∈ RingOps → (𝑅 ∈ Com2 ↔ ∀𝑥𝑋𝑦𝑋 (𝑥𝐻𝑦) = (𝑦𝐻𝑥)))
2423pm5.32i 575 . 2 ((𝑅 ∈ RingOps ∧ 𝑅 ∈ Com2) ↔ (𝑅 ∈ RingOps ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝐻𝑦) = (𝑦𝐻𝑥)))
251, 24bitri 276 1 (𝑅 ∈ CRingOps ↔ (𝑅 ∈ RingOps ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝐻𝑦) = (𝑦𝐻𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396   = wceq 1528  wcel 2105  wral 3135  Vcvv 3492  cop 4563  ran crn 5549  Rel wrel 5553  cfv 6348  (class class class)co 7145  1st c1st 7676  2nd c2nd 7677  RingOpscrngo 35053  Com2ccm2 35148  CRingOpsccring 35152
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-iota 6307  df-fun 6350  df-fv 6356  df-ov 7148  df-1st 7678  df-2nd 7679  df-rngo 35054  df-com2 35149  df-crngo 35153
This theorem is referenced by:  crngocom  35160  crngohomfo  35165
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