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Theorem isdrngo3 36827
Description: A division ring is a ring in which 1 ≠ 0 and every nonzero element is invertible. (Contributed by Jeff Madsen, 10-Jun-2010.)
Hypotheses
Ref Expression
isdivrng1.1 𝐺 = (1st𝑅)
isdivrng1.2 𝐻 = (2nd𝑅)
isdivrng1.3 𝑍 = (GId‘𝐺)
isdivrng1.4 𝑋 = ran 𝐺
isdivrng2.5 𝑈 = (GId‘𝐻)
Assertion
Ref Expression
isdrngo3 (𝑅 ∈ DivRingOps ↔ (𝑅 ∈ RingOps ∧ (𝑈𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦𝑋 (𝑦𝐻𝑥) = 𝑈)))
Distinct variable groups:   𝑥,𝐻,𝑦   𝑥,𝑋,𝑦   𝑥,𝑍,𝑦   𝑥,𝑅,𝑦   𝑥,𝑈,𝑦
Allowed substitution hints:   𝐺(𝑥,𝑦)

Proof of Theorem isdrngo3
StepHypRef Expression
1 isdivrng1.1 . . 3 𝐺 = (1st𝑅)
2 isdivrng1.2 . . 3 𝐻 = (2nd𝑅)
3 isdivrng1.3 . . 3 𝑍 = (GId‘𝐺)
4 isdivrng1.4 . . 3 𝑋 = ran 𝐺
5 isdivrng2.5 . . 3 𝑈 = (GId‘𝐻)
61, 2, 3, 4, 5isdrngo2 36826 . 2 (𝑅 ∈ DivRingOps ↔ (𝑅 ∈ RingOps ∧ (𝑈𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈)))
7 eldifi 4127 . . . . . 6 (𝑥 ∈ (𝑋 ∖ {𝑍}) → 𝑥𝑋)
8 difss 4132 . . . . . . . 8 (𝑋 ∖ {𝑍}) ⊆ 𝑋
9 ssrexv 4052 . . . . . . . 8 ((𝑋 ∖ {𝑍}) ⊆ 𝑋 → (∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈 → ∃𝑦𝑋 (𝑦𝐻𝑥) = 𝑈))
108, 9ax-mp 5 . . . . . . 7 (∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈 → ∃𝑦𝑋 (𝑦𝐻𝑥) = 𝑈)
11 neeq1 3004 . . . . . . . . . . . . . . . 16 ((𝑦𝐻𝑥) = 𝑈 → ((𝑦𝐻𝑥) ≠ 𝑍𝑈𝑍))
1211biimparc 481 . . . . . . . . . . . . . . 15 ((𝑈𝑍 ∧ (𝑦𝐻𝑥) = 𝑈) → (𝑦𝐻𝑥) ≠ 𝑍)
133, 4, 1, 2rngolz 36790 . . . . . . . . . . . . . . . . . 18 ((𝑅 ∈ RingOps ∧ 𝑥𝑋) → (𝑍𝐻𝑥) = 𝑍)
14 oveq1 7416 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑍 → (𝑦𝐻𝑥) = (𝑍𝐻𝑥))
1514eqeq1d 2735 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑍 → ((𝑦𝐻𝑥) = 𝑍 ↔ (𝑍𝐻𝑥) = 𝑍))
1613, 15syl5ibrcom 246 . . . . . . . . . . . . . . . . 17 ((𝑅 ∈ RingOps ∧ 𝑥𝑋) → (𝑦 = 𝑍 → (𝑦𝐻𝑥) = 𝑍))
1716necon3d 2962 . . . . . . . . . . . . . . . 16 ((𝑅 ∈ RingOps ∧ 𝑥𝑋) → ((𝑦𝐻𝑥) ≠ 𝑍𝑦𝑍))
1817imp 408 . . . . . . . . . . . . . . 15 (((𝑅 ∈ RingOps ∧ 𝑥𝑋) ∧ (𝑦𝐻𝑥) ≠ 𝑍) → 𝑦𝑍)
1912, 18sylan2 594 . . . . . . . . . . . . . 14 (((𝑅 ∈ RingOps ∧ 𝑥𝑋) ∧ (𝑈𝑍 ∧ (𝑦𝐻𝑥) = 𝑈)) → 𝑦𝑍)
2019an4s 659 . . . . . . . . . . . . 13 (((𝑅 ∈ RingOps ∧ 𝑈𝑍) ∧ (𝑥𝑋 ∧ (𝑦𝐻𝑥) = 𝑈)) → 𝑦𝑍)
2120anassrs 469 . . . . . . . . . . . 12 ((((𝑅 ∈ RingOps ∧ 𝑈𝑍) ∧ 𝑥𝑋) ∧ (𝑦𝐻𝑥) = 𝑈) → 𝑦𝑍)
22 pm3.2 471 . . . . . . . . . . . 12 (𝑦𝑋 → (𝑦𝑍 → (𝑦𝑋𝑦𝑍)))
2321, 22syl5com 31 . . . . . . . . . . 11 ((((𝑅 ∈ RingOps ∧ 𝑈𝑍) ∧ 𝑥𝑋) ∧ (𝑦𝐻𝑥) = 𝑈) → (𝑦𝑋 → (𝑦𝑋𝑦𝑍)))
24 eldifsn 4791 . . . . . . . . . . 11 (𝑦 ∈ (𝑋 ∖ {𝑍}) ↔ (𝑦𝑋𝑦𝑍))
2523, 24syl6ibr 252 . . . . . . . . . 10 ((((𝑅 ∈ RingOps ∧ 𝑈𝑍) ∧ 𝑥𝑋) ∧ (𝑦𝐻𝑥) = 𝑈) → (𝑦𝑋𝑦 ∈ (𝑋 ∖ {𝑍})))
2625imdistanda 573 . . . . . . . . 9 (((𝑅 ∈ RingOps ∧ 𝑈𝑍) ∧ 𝑥𝑋) → (((𝑦𝐻𝑥) = 𝑈𝑦𝑋) → ((𝑦𝐻𝑥) = 𝑈𝑦 ∈ (𝑋 ∖ {𝑍}))))
27 ancom 462 . . . . . . . . 9 ((𝑦𝑋 ∧ (𝑦𝐻𝑥) = 𝑈) ↔ ((𝑦𝐻𝑥) = 𝑈𝑦𝑋))
28 ancom 462 . . . . . . . . 9 ((𝑦 ∈ (𝑋 ∖ {𝑍}) ∧ (𝑦𝐻𝑥) = 𝑈) ↔ ((𝑦𝐻𝑥) = 𝑈𝑦 ∈ (𝑋 ∖ {𝑍})))
2926, 27, 283imtr4g 296 . . . . . . . 8 (((𝑅 ∈ RingOps ∧ 𝑈𝑍) ∧ 𝑥𝑋) → ((𝑦𝑋 ∧ (𝑦𝐻𝑥) = 𝑈) → (𝑦 ∈ (𝑋 ∖ {𝑍}) ∧ (𝑦𝐻𝑥) = 𝑈)))
3029reximdv2 3165 . . . . . . 7 (((𝑅 ∈ RingOps ∧ 𝑈𝑍) ∧ 𝑥𝑋) → (∃𝑦𝑋 (𝑦𝐻𝑥) = 𝑈 → ∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈))
3110, 30impbid2 225 . . . . . 6 (((𝑅 ∈ RingOps ∧ 𝑈𝑍) ∧ 𝑥𝑋) → (∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈 ↔ ∃𝑦𝑋 (𝑦𝐻𝑥) = 𝑈))
327, 31sylan2 594 . . . . 5 (((𝑅 ∈ RingOps ∧ 𝑈𝑍) ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → (∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈 ↔ ∃𝑦𝑋 (𝑦𝐻𝑥) = 𝑈))
3332ralbidva 3176 . . . 4 ((𝑅 ∈ RingOps ∧ 𝑈𝑍) → (∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈 ↔ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦𝑋 (𝑦𝐻𝑥) = 𝑈))
3433pm5.32da 580 . . 3 (𝑅 ∈ RingOps → ((𝑈𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈) ↔ (𝑈𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦𝑋 (𝑦𝐻𝑥) = 𝑈)))
3534pm5.32i 576 . 2 ((𝑅 ∈ RingOps ∧ (𝑈𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈)) ↔ (𝑅 ∈ RingOps ∧ (𝑈𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦𝑋 (𝑦𝐻𝑥) = 𝑈)))
366, 35bitri 275 1 (𝑅 ∈ DivRingOps ↔ (𝑅 ∈ RingOps ∧ (𝑈𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦𝑋 (𝑦𝐻𝑥) = 𝑈)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  wne 2941  wral 3062  wrex 3071  cdif 3946  wss 3949  {csn 4629  ran crn 5678  cfv 6544  (class class class)co 7409  1st c1st 7973  2nd c2nd 7974  GIdcgi 29743  RingOpscrngo 36762  DivRingOpscdrng 36816
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-1st 7975  df-2nd 7976  df-1o 8466  df-en 8940  df-grpo 29746  df-gid 29747  df-ginv 29748  df-ablo 29798  df-ass 36711  df-exid 36713  df-mgmOLD 36717  df-sgrOLD 36729  df-mndo 36735  df-rngo 36763  df-drngo 36817
This theorem is referenced by:  isfldidl  36936
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