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Theorem iscrngo2 34105
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 34104 . 2 (𝑅 ∈ CRingOps ↔ (𝑅 ∈ RingOps ∧ 𝑅 ∈ Com2))
2 relrngo 34004 . . . . 5 Rel RingOps
3 1st2nd 7442 . . . . 5 ((Rel RingOps ∧ 𝑅 ∈ RingOps) → 𝑅 = ⟨(1st𝑅), (2nd𝑅)⟩)
42, 3mpan 673 . . . 4 (𝑅 ∈ RingOps → 𝑅 = ⟨(1st𝑅), (2nd𝑅)⟩)
5 eleq1 2873 . . . . 5 (𝑅 = ⟨(1st𝑅), (2nd𝑅)⟩ → (𝑅 ∈ Com2 ↔ ⟨(1st𝑅), (2nd𝑅)⟩ ∈ Com2))
6 iscring2.3 . . . . . . . 8 𝑋 = ran 𝐺
7 iscring2.1 . . . . . . . . 9 𝐺 = (1st𝑅)
87rneqi 5553 . . . . . . . 8 ran 𝐺 = ran (1st𝑅)
96, 8eqtri 2828 . . . . . . 7 𝑋 = ran (1st𝑅)
109raleqi 3331 . . . . . 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 6883 . . . . . . . . 9 (𝑥𝐻𝑦) = (𝑥(2nd𝑅)𝑦)
1311oveqi 6883 . . . . . . . . 9 (𝑦𝐻𝑥) = (𝑦(2nd𝑅)𝑥)
1412, 13eqeq12i 2820 . . . . . . . 8 ((𝑥𝐻𝑦) = (𝑦𝐻𝑥) ↔ (𝑥(2nd𝑅)𝑦) = (𝑦(2nd𝑅)𝑥))
159, 14raleqbii 3178 . . . . . . 7 (∀𝑦𝑋 (𝑥𝐻𝑦) = (𝑦𝐻𝑥) ↔ ∀𝑦 ∈ ran (1st𝑅)(𝑥(2nd𝑅)𝑦) = (𝑦(2nd𝑅)𝑥))
1615ralbii 3168 . . . . . 6 (∀𝑥𝑋𝑦𝑋 (𝑥𝐻𝑦) = (𝑦𝐻𝑥) ↔ ∀𝑥𝑋𝑦 ∈ ran (1st𝑅)(𝑥(2nd𝑅)𝑦) = (𝑦(2nd𝑅)𝑥))
17 fvex 6417 . . . . . . 7 (1st𝑅) ∈ V
18 fvex 6417 . . . . . . 7 (2nd𝑅) ∈ V
19 iscom2 34103 . . . . . . 7 (((1st𝑅) ∈ V ∧ (2nd𝑅) ∈ V) → (⟨(1st𝑅), (2nd𝑅)⟩ ∈ Com2 ↔ ∀𝑥 ∈ ran (1st𝑅)∀𝑦 ∈ ran (1st𝑅)(𝑥(2nd𝑅)𝑦) = (𝑦(2nd𝑅)𝑥)))
2017, 18, 19mp2an 675 . . . . . 6 (⟨(1st𝑅), (2nd𝑅)⟩ ∈ Com2 ↔ ∀𝑥 ∈ ran (1st𝑅)∀𝑦 ∈ ran (1st𝑅)(𝑥(2nd𝑅)𝑦) = (𝑦(2nd𝑅)𝑥))
2110, 16, 203bitr4ri 295 . . . . 5 (⟨(1st𝑅), (2nd𝑅)⟩ ∈ Com2 ↔ ∀𝑥𝑋𝑦𝑋 (𝑥𝐻𝑦) = (𝑦𝐻𝑥))
225, 21syl6bb 278 . . . 4 (𝑅 = ⟨(1st𝑅), (2nd𝑅)⟩ → (𝑅 ∈ Com2 ↔ ∀𝑥𝑋𝑦𝑋 (𝑥𝐻𝑦) = (𝑦𝐻𝑥)))
234, 22syl 17 . . 3 (𝑅 ∈ RingOps → (𝑅 ∈ Com2 ↔ ∀𝑥𝑋𝑦𝑋 (𝑥𝐻𝑦) = (𝑦𝐻𝑥)))
2423pm5.32i 566 . 2 ((𝑅 ∈ RingOps ∧ 𝑅 ∈ Com2) ↔ (𝑅 ∈ RingOps ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝐻𝑦) = (𝑦𝐻𝑥)))
251, 24bitri 266 1 (𝑅 ∈ CRingOps ↔ (𝑅 ∈ RingOps ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝐻𝑦) = (𝑦𝐻𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wb 197  wa 384   = wceq 1637  wcel 2156  wral 3096  Vcvv 3391  cop 4376  ran crn 5312  Rel wrel 5316  cfv 6097  (class class class)co 6870  1st c1st 7392  2nd c2nd 7393  RingOpscrngo 34002  Com2ccm2 34097  CRingOpsccring 34101
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7175
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ral 3101  df-rex 3102  df-rab 3105  df-v 3393  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-sn 4371  df-pr 4373  df-op 4377  df-uni 4631  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-iota 6060  df-fun 6099  df-fv 6105  df-ov 6873  df-1st 7394  df-2nd 7395  df-rngo 34003  df-com2 34098  df-crngo 34102
This theorem is referenced by:  crngocom  34109  crngohomfo  34114
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