Theorem dvrunz 36112

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 6788	. . 3 ⊢ 𝑍 ∈ V
3	2	zrdivrng 36111	. 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 36109	. . . . . 6 ⊢ (𝑅 ∈ DivRingOps → (𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp))
8	7	simpld 495	. . . . 5 ⊢ (𝑅 ∈ DivRingOps → 𝑅 ∈ RingOps)
9		dvrunz.5	. . . . . 6 ⊢ 𝑈 = (GId‘𝐻)
10	4, 5, 1, 9, 6	rngoueqz 36098	. . . . 5 ⊢ (𝑅 ∈ RingOps → (𝑋 ≈ 1o ↔ 𝑈 = 𝑍))
11	8, 10	syl 17	. . . 4 ⊢ (𝑅 ∈ DivRingOps → (𝑋 ≈ 1o ↔ 𝑈 = 𝑍))
12	4, 6, 1	rngosn6 36084	. . . . . . 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 2957	. 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 2106   ≠ wne 2943   ∖ cdif 3884  {csn 4561  ⟨cop 4567   class class class wbr 5074   × cxp 5587  ran crn 5590   ↾ cres 5591  ‘cfv 6433  1^st c1st 7829  2^nd c2nd 7830  1_oc1o 8290   ≈ cen 8730  GrpOpcgr 28851  GIdcgi 28852  RingOpscrngo 36052  DivRingOpscdrng 36106

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-om 7713  df-1st 7831  df-2nd 7832  df-1o 8297  df-er 8498  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-grpo 28855  df-gid 28856  df-ablo 28907  df-ass 36001  df-exid 36003  df-mgmOLD 36007  df-sgrOLD 36019  df-mndo 36025  df-rngo 36053  df-drngo 36107

This theorem is referenced by:  isdrngo2  36116  divrngpr  36211  isfldidl  36226