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Theorem isdrngo3 33425
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 33424 . 2 (𝑅 ∈ DivRingOps ↔ (𝑅 ∈ RingOps ∧ (𝑈𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈)))
7 eldifi 3715 . . . . . 6 (𝑥 ∈ (𝑋 ∖ {𝑍}) → 𝑥𝑋)
8 difss 3720 . . . . . . . 8 (𝑋 ∖ {𝑍}) ⊆ 𝑋
9 ssrexv 3651 . . . . . . . 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 2852 . . . . . . . . . . . . . . . 16 ((𝑦𝐻𝑥) = 𝑈 → ((𝑦𝐻𝑥) ≠ 𝑍𝑈𝑍))
1211biimparc 504 . . . . . . . . . . . . . . 15 ((𝑈𝑍 ∧ (𝑦𝐻𝑥) = 𝑈) → (𝑦𝐻𝑥) ≠ 𝑍)
133, 4, 1, 2rngolz 33388 . . . . . . . . . . . . . . . . . 18 ((𝑅 ∈ RingOps ∧ 𝑥𝑋) → (𝑍𝐻𝑥) = 𝑍)
14 oveq1 6617 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑍 → (𝑦𝐻𝑥) = (𝑍𝐻𝑥))
1514eqeq1d 2623 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑍 → ((𝑦𝐻𝑥) = 𝑍 ↔ (𝑍𝐻𝑥) = 𝑍))
1613, 15syl5ibrcom 237 . . . . . . . . . . . . . . . . 17 ((𝑅 ∈ RingOps ∧ 𝑥𝑋) → (𝑦 = 𝑍 → (𝑦𝐻𝑥) = 𝑍))
1716necon3d 2811 . . . . . . . . . . . . . . . 16 ((𝑅 ∈ RingOps ∧ 𝑥𝑋) → ((𝑦𝐻𝑥) ≠ 𝑍𝑦𝑍))
1817imp 445 . . . . . . . . . . . . . . 15 (((𝑅 ∈ RingOps ∧ 𝑥𝑋) ∧ (𝑦𝐻𝑥) ≠ 𝑍) → 𝑦𝑍)
1912, 18sylan2 491 . . . . . . . . . . . . . 14 (((𝑅 ∈ RingOps ∧ 𝑥𝑋) ∧ (𝑈𝑍 ∧ (𝑦𝐻𝑥) = 𝑈)) → 𝑦𝑍)
2019an4s 868 . . . . . . . . . . . . 13 (((𝑅 ∈ RingOps ∧ 𝑈𝑍) ∧ (𝑥𝑋 ∧ (𝑦𝐻𝑥) = 𝑈)) → 𝑦𝑍)
2120anassrs 679 . . . . . . . . . . . 12 ((((𝑅 ∈ RingOps ∧ 𝑈𝑍) ∧ 𝑥𝑋) ∧ (𝑦𝐻𝑥) = 𝑈) → 𝑦𝑍)
22 pm3.2 463 . . . . . . . . . . . 12 (𝑦𝑋 → (𝑦𝑍 → (𝑦𝑋𝑦𝑍)))
2321, 22syl5com 31 . . . . . . . . . . 11 ((((𝑅 ∈ RingOps ∧ 𝑈𝑍) ∧ 𝑥𝑋) ∧ (𝑦𝐻𝑥) = 𝑈) → (𝑦𝑋 → (𝑦𝑋𝑦𝑍)))
24 eldifsn 4292 . . . . . . . . . . 11 (𝑦 ∈ (𝑋 ∖ {𝑍}) ↔ (𝑦𝑋𝑦𝑍))
2523, 24syl6ibr 242 . . . . . . . . . 10 ((((𝑅 ∈ RingOps ∧ 𝑈𝑍) ∧ 𝑥𝑋) ∧ (𝑦𝐻𝑥) = 𝑈) → (𝑦𝑋𝑦 ∈ (𝑋 ∖ {𝑍})))
2625imdistanda 728 . . . . . . . . 9 (((𝑅 ∈ RingOps ∧ 𝑈𝑍) ∧ 𝑥𝑋) → (((𝑦𝐻𝑥) = 𝑈𝑦𝑋) → ((𝑦𝐻𝑥) = 𝑈𝑦 ∈ (𝑋 ∖ {𝑍}))))
27 ancom 466 . . . . . . . . 9 ((𝑦𝑋 ∧ (𝑦𝐻𝑥) = 𝑈) ↔ ((𝑦𝐻𝑥) = 𝑈𝑦𝑋))
28 ancom 466 . . . . . . . . 9 ((𝑦 ∈ (𝑋 ∖ {𝑍}) ∧ (𝑦𝐻𝑥) = 𝑈) ↔ ((𝑦𝐻𝑥) = 𝑈𝑦 ∈ (𝑋 ∖ {𝑍})))
2926, 27, 283imtr4g 285 . . . . . . . 8 (((𝑅 ∈ RingOps ∧ 𝑈𝑍) ∧ 𝑥𝑋) → ((𝑦𝑋 ∧ (𝑦𝐻𝑥) = 𝑈) → (𝑦 ∈ (𝑋 ∖ {𝑍}) ∧ (𝑦𝐻𝑥) = 𝑈)))
3029reximdv2 3009 . . . . . . 7 (((𝑅 ∈ RingOps ∧ 𝑈𝑍) ∧ 𝑥𝑋) → (∃𝑦𝑋 (𝑦𝐻𝑥) = 𝑈 → ∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈))
3110, 30impbid2 216 . . . . . 6 (((𝑅 ∈ RingOps ∧ 𝑈𝑍) ∧ 𝑥𝑋) → (∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈 ↔ ∃𝑦𝑋 (𝑦𝐻𝑥) = 𝑈))
327, 31sylan2 491 . . . . 5 (((𝑅 ∈ RingOps ∧ 𝑈𝑍) ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → (∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈 ↔ ∃𝑦𝑋 (𝑦𝐻𝑥) = 𝑈))
3332ralbidva 2980 . . . 4 ((𝑅 ∈ RingOps ∧ 𝑈𝑍) → (∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈 ↔ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦𝑋 (𝑦𝐻𝑥) = 𝑈))
3433pm5.32da 672 . . 3 (𝑅 ∈ RingOps → ((𝑈𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈) ↔ (𝑈𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦𝑋 (𝑦𝐻𝑥) = 𝑈)))
3534pm5.32i 668 . 2 ((𝑅 ∈ RingOps ∧ (𝑈𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈)) ↔ (𝑅 ∈ RingOps ∧ (𝑈𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦𝑋 (𝑦𝐻𝑥) = 𝑈)))
366, 35bitri 264 1 (𝑅 ∈ DivRingOps ↔ (𝑅 ∈ RingOps ∧ (𝑈𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦𝑋 (𝑦𝐻𝑥) = 𝑈)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wne 2790  wral 2907  wrex 2908  cdif 3556  wss 3559  {csn 4153  ran crn 5080  cfv 5852  (class class class)co 6610  1st c1st 7118  2nd c2nd 7119  GIdcgi 27214  RingOpscrngo 33360  DivRingOpscdrng 33414
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-om 7020  df-1st 7120  df-2nd 7121  df-1o 7512  df-er 7694  df-en 7908  df-dom 7909  df-sdom 7910  df-fin 7911  df-grpo 27217  df-gid 27218  df-ginv 27219  df-ablo 27269  df-ass 33309  df-exid 33311  df-mgmOLD 33315  df-sgrOLD 33327  df-mndo 33333  df-rngo 33361  df-drngo 33415
This theorem is referenced by:  isfldidl  33534
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