Theorem dvrunz 38586

Description: In a division ring the ring 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 6897	. . 3 ⊢ 𝑍 ∈ V
3	2	zrdivrng 38585	. 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 38583	. . . . . 6 ⊢ (𝑅 ∈ DivRingOps → (𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp))
8	7	simpld 499	. . . . 5 ⊢ (𝑅 ∈ DivRingOps → 𝑅 ∈ RingOps)
9		dvrunz.5	. . . . . 6 ⊢ 𝑈 = (GId‘𝐻)
10	4, 5, 1, 9, 6	rngoueqz 38572	. . . . 5 ⊢ (𝑅 ∈ RingOps → (𝑋 ≈ 1o ↔ 𝑈 = 𝑍))
11	8, 10	syl 18	. . . 4 ⊢ (𝑅 ∈ DivRingOps → (𝑋 ≈ 1o ↔ 𝑈 = 𝑍))
12	4, 6, 1	rngosn6 38558	. . . . . . 7 ⊢ (𝑅 ∈ RingOps → (𝑋 ≈ 1o ↔ 𝑅 = ⟨{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}, {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩))
13	8, 12	syl 18	. . . . . 6 ⊢ (𝑅 ∈ DivRingOps → (𝑋 ≈ 1o ↔ 𝑅 = ⟨{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}, {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩))
14		eleq1 2851	. . . . . . 7 ⊢ (𝑅 = ⟨{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}, {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩ → (𝑅 ∈ DivRingOps ↔ ⟨{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}, {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩ ∈ DivRingOps))
15	14	biimpd 232	. . . . . 6 ⊢ (𝑅 = ⟨{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}, {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩ → (𝑅 ∈ DivRingOps → ⟨{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}, {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩ ∈ DivRingOps))
16	13, 15	biimtrdi 256	. . . . 5 ⊢ (𝑅 ∈ DivRingOps → (𝑋 ≈ 1o → (𝑅 ∈ DivRingOps → ⟨{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}, {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩ ∈ DivRingOps)))
17	16	pm2.43a 55	. . . 4 ⊢ (𝑅 ∈ DivRingOps → (𝑋 ≈ 1o → ⟨{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}, {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩ ∈ DivRingOps))
18	11, 17	sylbird 263	. . 3 ⊢ (𝑅 ∈ DivRingOps → (𝑈 = 𝑍 → ⟨{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}, {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩ ∈ DivRingOps))
19	18	necon3bd 2972	. 2 ⊢ (𝑅 ∈ DivRingOps → (¬ ⟨{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}, {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩ ∈ DivRingOps → 𝑈 ≠ 𝑍))
20	3, 19	mpi 21	1 ⊢ (𝑅 ∈ DivRingOps → 𝑈 ≠ 𝑍)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2143   ≠ wne 2958   ∖ cdif 3903  {csn 4590  ⟨cop 4596   class class class wbr 5110   × cxp 5661  ran crn 5664   ↾ cres 5665  ‘cfv 6538  1^st c1st 7985  2^nd c2nd 7986  1_oc1o 8447   ≈ cen 8941  GrpOpcgr 30822  GIdcgi 30823  RingOpscrngo 38526  DivRingOpscdrng 38580

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7369  df-ov 7415  df-1st 7987  df-2nd 7988  df-1o 8454  df-en 8945  df-grpo 30826  df-gid 30827  df-ablo 30878  df-ass 38475  df-exid 38477  df-mgmOLD 38481  df-sgrOLD 38493  df-mndo 38499  df-rngo 38527  df-drngo 38581

This theorem is referenced by:  isdrngo2  38590  divrngpr  38685  isfldidl  38700