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Theorem isdrngo3 33888
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 33887 . 2 (𝑅 ∈ DivRingOps ↔ (𝑅 ∈ RingOps ∧ (𝑈𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈)))
7 eldifi 3765 . . . . . 6 (𝑥 ∈ (𝑋 ∖ {𝑍}) → 𝑥𝑋)
8 difss 3770 . . . . . . . 8 (𝑋 ∖ {𝑍}) ⊆ 𝑋
9 ssrexv 3700 . . . . . . . 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 2885 . . . . . . . . . . . . . . . 16 ((𝑦𝐻𝑥) = 𝑈 → ((𝑦𝐻𝑥) ≠ 𝑍𝑈𝑍))
1211biimparc 503 . . . . . . . . . . . . . . 15 ((𝑈𝑍 ∧ (𝑦𝐻𝑥) = 𝑈) → (𝑦𝐻𝑥) ≠ 𝑍)
133, 4, 1, 2rngolz 33851 . . . . . . . . . . . . . . . . . 18 ((𝑅 ∈ RingOps ∧ 𝑥𝑋) → (𝑍𝐻𝑥) = 𝑍)
14 oveq1 6697 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑍 → (𝑦𝐻𝑥) = (𝑍𝐻𝑥))
1514eqeq1d 2653 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑍 → ((𝑦𝐻𝑥) = 𝑍 ↔ (𝑍𝐻𝑥) = 𝑍))
1613, 15syl5ibrcom 237 . . . . . . . . . . . . . . . . 17 ((𝑅 ∈ RingOps ∧ 𝑥𝑋) → (𝑦 = 𝑍 → (𝑦𝐻𝑥) = 𝑍))
1716necon3d 2844 . . . . . . . . . . . . . . . 16 ((𝑅 ∈ RingOps ∧ 𝑥𝑋) → ((𝑦𝐻𝑥) ≠ 𝑍𝑦𝑍))
1817imp 444 . . . . . . . . . . . . . . 15 (((𝑅 ∈ RingOps ∧ 𝑥𝑋) ∧ (𝑦𝐻𝑥) ≠ 𝑍) → 𝑦𝑍)
1912, 18sylan2 490 . . . . . . . . . . . . . 14 (((𝑅 ∈ RingOps ∧ 𝑥𝑋) ∧ (𝑈𝑍 ∧ (𝑦𝐻𝑥) = 𝑈)) → 𝑦𝑍)
2019an4s 886 . . . . . . . . . . . . 13 (((𝑅 ∈ RingOps ∧ 𝑈𝑍) ∧ (𝑥𝑋 ∧ (𝑦𝐻𝑥) = 𝑈)) → 𝑦𝑍)
2120anassrs 681 . . . . . . . . . . . 12 ((((𝑅 ∈ RingOps ∧ 𝑈𝑍) ∧ 𝑥𝑋) ∧ (𝑦𝐻𝑥) = 𝑈) → 𝑦𝑍)
22 pm3.2 462 . . . . . . . . . . . 12 (𝑦𝑋 → (𝑦𝑍 → (𝑦𝑋𝑦𝑍)))
2321, 22syl5com 31 . . . . . . . . . . 11 ((((𝑅 ∈ RingOps ∧ 𝑈𝑍) ∧ 𝑥𝑋) ∧ (𝑦𝐻𝑥) = 𝑈) → (𝑦𝑋 → (𝑦𝑋𝑦𝑍)))
24 eldifsn 4350 . . . . . . . . . . 11 (𝑦 ∈ (𝑋 ∖ {𝑍}) ↔ (𝑦𝑋𝑦𝑍))
2523, 24syl6ibr 242 . . . . . . . . . 10 ((((𝑅 ∈ RingOps ∧ 𝑈𝑍) ∧ 𝑥𝑋) ∧ (𝑦𝐻𝑥) = 𝑈) → (𝑦𝑋𝑦 ∈ (𝑋 ∖ {𝑍})))
2625imdistanda 729 . . . . . . . . 9 (((𝑅 ∈ RingOps ∧ 𝑈𝑍) ∧ 𝑥𝑋) → (((𝑦𝐻𝑥) = 𝑈𝑦𝑋) → ((𝑦𝐻𝑥) = 𝑈𝑦 ∈ (𝑋 ∖ {𝑍}))))
27 ancom 465 . . . . . . . . 9 ((𝑦𝑋 ∧ (𝑦𝐻𝑥) = 𝑈) ↔ ((𝑦𝐻𝑥) = 𝑈𝑦𝑋))
28 ancom 465 . . . . . . . . 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 3043 . . . . . . 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 490 . . . . 5 (((𝑅 ∈ RingOps ∧ 𝑈𝑍) ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → (∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈 ↔ ∃𝑦𝑋 (𝑦𝐻𝑥) = 𝑈))
3332ralbidva 3014 . . . 4 ((𝑅 ∈ RingOps ∧ 𝑈𝑍) → (∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈 ↔ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦𝑋 (𝑦𝐻𝑥) = 𝑈))
3433pm5.32da 674 . . 3 (𝑅 ∈ RingOps → ((𝑈𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈) ↔ (𝑈𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦𝑋 (𝑦𝐻𝑥) = 𝑈)))
3534pm5.32i 670 . 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 383   = wceq 1523  wcel 2030  wne 2823  wral 2941  wrex 2942  cdif 3604  wss 3607  {csn 4210  ran crn 5144  cfv 5926  (class class class)co 6690  1st c1st 7208  2nd c2nd 7209  GIdcgi 27472  RingOpscrngo 33823  DivRingOpscdrng 33877
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-om 7108  df-1st 7210  df-2nd 7211  df-1o 7605  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-grpo 27475  df-gid 27476  df-ginv 27477  df-ablo 27527  df-ass 33772  df-exid 33774  df-mgmOLD 33778  df-sgrOLD 33790  df-mndo 33796  df-rngo 33824  df-drngo 33878
This theorem is referenced by:  isfldidl  33997
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