Theorem dvrunz 36039

Description: In a division ring the unit is different from the zero. (Contributed by FL, 14-Feb-2010.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
dvrunz.1	⊢ 𝐺 = (1st ‘𝑅)
dvrunz.2	⊢ 𝐻 = (2nd ‘𝑅)
dvrunz.3	⊢ 𝑋 = ran 𝐺
dvrunz.4	⊢ 𝑍 = (GId‘𝐺)
dvrunz.5	⊢ 𝑈 = (GId‘𝐻)

Assertion

Ref	Expression
dvrunz	⊢ (𝑅 ∈ DivRingOps → 𝑈 ≠ 𝑍)

Proof of Theorem dvrunz

Step	Hyp	Ref	Expression
1		dvrunz.4	. . . 4 ⊢ 𝑍 = (GId‘𝐺)
2	1	fvexi 6770	. . 3 ⊢ 𝑍 ∈ V
3	2	zrdivrng 36038	. 2 ⊢ ¬ ⟨{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}, {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩ ∈ DivRingOps
4		dvrunz.1	. . . . . . 7 ⊢ 𝐺 = (1st ‘𝑅)
5		dvrunz.2	. . . . . . 7 ⊢ 𝐻 = (2nd ‘𝑅)
6		dvrunz.3	. . . . . . 7 ⊢ 𝑋 = ran 𝐺
7	4, 5, 6, 1	drngoi 36036	. . . . . 6 ⊢ (𝑅 ∈ DivRingOps → (𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp))
8	7	simpld 494	. . . . 5 ⊢ (𝑅 ∈ DivRingOps → 𝑅 ∈ RingOps)
9		dvrunz.5	. . . . . 6 ⊢ 𝑈 = (GId‘𝐻)
10	4, 5, 1, 9, 6	rngoueqz 36025	. . . . 5 ⊢ (𝑅 ∈ RingOps → (𝑋 ≈ 1o ↔ 𝑈 = 𝑍))
11	8, 10	syl 17	. . . 4 ⊢ (𝑅 ∈ DivRingOps → (𝑋 ≈ 1o ↔ 𝑈 = 𝑍))
12	4, 6, 1	rngosn6 36011	. . . . . . 7 ⊢ (𝑅 ∈ RingOps → (𝑋 ≈ 1o ↔ 𝑅 = ⟨{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}, {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩))
13	8, 12	syl 17	. . . . . 6 ⊢ (𝑅 ∈ DivRingOps → (𝑋 ≈ 1o ↔ 𝑅 = ⟨{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}, {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩))
14		eleq1 2826	. . . . . . 7 ⊢ (𝑅 = ⟨{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}, {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩ → (𝑅 ∈ DivRingOps ↔ ⟨{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}, {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩ ∈ DivRingOps))
15	14	biimpd 228	. . . . . 6 ⊢ (𝑅 = ⟨{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}, {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩ → (𝑅 ∈ DivRingOps → ⟨{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}, {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩ ∈ DivRingOps))
16	13, 15	syl6bi 252	. . . . 5 ⊢ (𝑅 ∈ DivRingOps → (𝑋 ≈ 1o → (𝑅 ∈ DivRingOps → ⟨{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}, {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩ ∈ DivRingOps)))
17	16	pm2.43a 54	. . . 4 ⊢ (𝑅 ∈ DivRingOps → (𝑋 ≈ 1o → ⟨{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}, {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩ ∈ DivRingOps))
18	11, 17	sylbird 259	. . 3 ⊢ (𝑅 ∈ DivRingOps → (𝑈 = 𝑍 → ⟨{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}, {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩ ∈ DivRingOps))
19	18	necon3bd 2956	. 2 ⊢ (𝑅 ∈ DivRingOps → (¬ ⟨{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}, {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩ ∈ DivRingOps → 𝑈 ≠ 𝑍))
20	3, 19	mpi 20	1 ⊢ (𝑅 ∈ DivRingOps → 𝑈 ≠ 𝑍)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   = wceq 1539   ∈ wcel 2108   ≠ wne 2942   ∖ cdif 3880  {csn 4558  ⟨cop 4564   class class class wbr 5070   × cxp 5578  ran crn 5581   ↾ cres 5582  ‘cfv 6418  1^st c1st 7802  2^nd c2nd 7803  1_oc1o 8260   ≈ cen 8688  GrpOpcgr 28752  GIdcgi 28753  RingOpscrngo 35979  DivRingOpscdrng 36033

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-om 7688  df-1st 7804  df-2nd 7805  df-1o 8267  df-er 8456  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-grpo 28756  df-gid 28757  df-ablo 28808  df-ass 35928  df-exid 35930  df-mgmOLD 35934  df-sgrOLD 35946  df-mndo 35952  df-rngo 35980  df-drngo 36034

This theorem is referenced by:  isdrngo2  36043  divrngpr  36138  isfldidl  36153