Theorem dvrunz 38067

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 6845	. . 3 ⊢ 𝑍 ∈ V
3	2	zrdivrng 38066	. 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 38064	. . . . . 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 38053	. . . . 5 ⊢ (𝑅 ∈ RingOps → (𝑋 ≈ 1o ↔ 𝑈 = 𝑍))
11	8, 10	syl 17	. . . 4 ⊢ (𝑅 ∈ DivRingOps → (𝑋 ≈ 1o ↔ 𝑈 = 𝑍))
12	4, 6, 1	rngosn6 38039	. . . . . . 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 2821	. . . . . . 7 ⊢ (𝑅 = ⟨{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}, {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩ → (𝑅 ∈ DivRingOps ↔ ⟨{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}, {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩ ∈ DivRingOps))
15	14	biimpd 229	. . . . . 6 ⊢ (𝑅 = ⟨{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}, {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩ → (𝑅 ∈ DivRingOps → ⟨{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}, {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩ ∈ DivRingOps))
16	13, 15	biimtrdi 253	. . . . 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 260	. . 3 ⊢ (𝑅 ∈ DivRingOps → (𝑈 = 𝑍 → ⟨{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}, {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩ ∈ DivRingOps))
19	18	necon3bd 2943	. 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 206   = wceq 1541   ∈ wcel 2113   ≠ wne 2929   ∖ cdif 3895  {csn 4577  ⟨cop 4583   class class class wbr 5095   × cxp 5619  ran crn 5622   ↾ cres 5623  ‘cfv 6489  1^st c1st 7928  2^nd c2nd 7929  1_oc1o 8387   ≈ cen 8876  GrpOpcgr 30490  GIdcgi 30491  RingOpscrngo 38007  DivRingOpscdrng 38061

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7677

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rmo 3347  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-riota 7312  df-ov 7358  df-1st 7930  df-2nd 7931  df-1o 8394  df-en 8880  df-grpo 30494  df-gid 30495  df-ablo 30546  df-ass 37956  df-exid 37958  df-mgmOLD 37962  df-sgrOLD 37974  df-mndo 37980  df-rngo 38008  df-drngo 38062

This theorem is referenced by:  isdrngo2  38071  divrngpr  38166  isfldidl  38181