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Theorem isdrngo3 37130
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 37129 . 2 (𝑅 ∈ DivRingOps ↔ (𝑅 ∈ RingOps ∧ (𝑈𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈)))
7 eldifi 4125 . . . . . 6 (𝑥 ∈ (𝑋 ∖ {𝑍}) → 𝑥𝑋)
8 difss 4130 . . . . . . . 8 (𝑋 ∖ {𝑍}) ⊆ 𝑋
9 ssrexv 4050 . . . . . . . 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 3001 . . . . . . . . . . . . . . . 16 ((𝑦𝐻𝑥) = 𝑈 → ((𝑦𝐻𝑥) ≠ 𝑍𝑈𝑍))
1211biimparc 478 . . . . . . . . . . . . . . 15 ((𝑈𝑍 ∧ (𝑦𝐻𝑥) = 𝑈) → (𝑦𝐻𝑥) ≠ 𝑍)
133, 4, 1, 2rngolz 37093 . . . . . . . . . . . . . . . . . 18 ((𝑅 ∈ RingOps ∧ 𝑥𝑋) → (𝑍𝐻𝑥) = 𝑍)
14 oveq1 7418 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑍 → (𝑦𝐻𝑥) = (𝑍𝐻𝑥))
1514eqeq1d 2732 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑍 → ((𝑦𝐻𝑥) = 𝑍 ↔ (𝑍𝐻𝑥) = 𝑍))
1613, 15syl5ibrcom 246 . . . . . . . . . . . . . . . . 17 ((𝑅 ∈ RingOps ∧ 𝑥𝑋) → (𝑦 = 𝑍 → (𝑦𝐻𝑥) = 𝑍))
1716necon3d 2959 . . . . . . . . . . . . . . . 16 ((𝑅 ∈ RingOps ∧ 𝑥𝑋) → ((𝑦𝐻𝑥) ≠ 𝑍𝑦𝑍))
1817imp 405 . . . . . . . . . . . . . . 15 (((𝑅 ∈ RingOps ∧ 𝑥𝑋) ∧ (𝑦𝐻𝑥) ≠ 𝑍) → 𝑦𝑍)
1912, 18sylan2 591 . . . . . . . . . . . . . 14 (((𝑅 ∈ RingOps ∧ 𝑥𝑋) ∧ (𝑈𝑍 ∧ (𝑦𝐻𝑥) = 𝑈)) → 𝑦𝑍)
2019an4s 656 . . . . . . . . . . . . 13 (((𝑅 ∈ RingOps ∧ 𝑈𝑍) ∧ (𝑥𝑋 ∧ (𝑦𝐻𝑥) = 𝑈)) → 𝑦𝑍)
2120anassrs 466 . . . . . . . . . . . 12 ((((𝑅 ∈ RingOps ∧ 𝑈𝑍) ∧ 𝑥𝑋) ∧ (𝑦𝐻𝑥) = 𝑈) → 𝑦𝑍)
22 pm3.2 468 . . . . . . . . . . . 12 (𝑦𝑋 → (𝑦𝑍 → (𝑦𝑋𝑦𝑍)))
2321, 22syl5com 31 . . . . . . . . . . 11 ((((𝑅 ∈ RingOps ∧ 𝑈𝑍) ∧ 𝑥𝑋) ∧ (𝑦𝐻𝑥) = 𝑈) → (𝑦𝑋 → (𝑦𝑋𝑦𝑍)))
24 eldifsn 4789 . . . . . . . . . . 11 (𝑦 ∈ (𝑋 ∖ {𝑍}) ↔ (𝑦𝑋𝑦𝑍))
2523, 24imbitrrdi 251 . . . . . . . . . 10 ((((𝑅 ∈ RingOps ∧ 𝑈𝑍) ∧ 𝑥𝑋) ∧ (𝑦𝐻𝑥) = 𝑈) → (𝑦𝑋𝑦 ∈ (𝑋 ∖ {𝑍})))
2625imdistanda 570 . . . . . . . . 9 (((𝑅 ∈ RingOps ∧ 𝑈𝑍) ∧ 𝑥𝑋) → (((𝑦𝐻𝑥) = 𝑈𝑦𝑋) → ((𝑦𝐻𝑥) = 𝑈𝑦 ∈ (𝑋 ∖ {𝑍}))))
27 ancom 459 . . . . . . . . 9 ((𝑦𝑋 ∧ (𝑦𝐻𝑥) = 𝑈) ↔ ((𝑦𝐻𝑥) = 𝑈𝑦𝑋))
28 ancom 459 . . . . . . . . 9 ((𝑦 ∈ (𝑋 ∖ {𝑍}) ∧ (𝑦𝐻𝑥) = 𝑈) ↔ ((𝑦𝐻𝑥) = 𝑈𝑦 ∈ (𝑋 ∖ {𝑍})))
2926, 27, 283imtr4g 295 . . . . . . . 8 (((𝑅 ∈ RingOps ∧ 𝑈𝑍) ∧ 𝑥𝑋) → ((𝑦𝑋 ∧ (𝑦𝐻𝑥) = 𝑈) → (𝑦 ∈ (𝑋 ∖ {𝑍}) ∧ (𝑦𝐻𝑥) = 𝑈)))
3029reximdv2 3162 . . . . . . 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 591 . . . . 5 (((𝑅 ∈ RingOps ∧ 𝑈𝑍) ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → (∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈 ↔ ∃𝑦𝑋 (𝑦𝐻𝑥) = 𝑈))
3332ralbidva 3173 . . . 4 ((𝑅 ∈ RingOps ∧ 𝑈𝑍) → (∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈 ↔ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦𝑋 (𝑦𝐻𝑥) = 𝑈))
3433pm5.32da 577 . . 3 (𝑅 ∈ RingOps → ((𝑈𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈) ↔ (𝑈𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦𝑋 (𝑦𝐻𝑥) = 𝑈)))
3534pm5.32i 573 . 2 ((𝑅 ∈ RingOps ∧ (𝑈𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈)) ↔ (𝑅 ∈ RingOps ∧ (𝑈𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦𝑋 (𝑦𝐻𝑥) = 𝑈)))
366, 35bitri 274 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 394   = wceq 1539  wcel 2104  wne 2938  wral 3059  wrex 3068  cdif 3944  wss 3947  {csn 4627  ran crn 5676  cfv 6542  (class class class)co 7411  1st c1st 7975  2nd c2nd 7976  GIdcgi 30010  RingOpscrngo 37065  DivRingOpscdrng 37119
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7367  df-ov 7414  df-1st 7977  df-2nd 7978  df-1o 8468  df-en 8942  df-grpo 30013  df-gid 30014  df-ginv 30015  df-ablo 30065  df-ass 37014  df-exid 37016  df-mgmOLD 37020  df-sgrOLD 37032  df-mndo 37038  df-rngo 37066  df-drngo 37120
This theorem is referenced by:  isfldidl  37239
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