Theorem dvrunz 38414

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 6876	. . 3 ⊢ 𝑍 ∈ V
3	2	zrdivrng 38413	. 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 38411	. . . . . 6 ⊢ (𝑅 ∈ DivRingOps → (𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp))
8	7	simpld 498	. . . . 5 ⊢ (𝑅 ∈ DivRingOps → 𝑅 ∈ RingOps)
9		dvrunz.5	. . . . . 6 ⊢ 𝑈 = (GId‘𝐻)
10	4, 5, 1, 9, 6	rngoueqz 38400	. . . . 5 ⊢ (𝑅 ∈ RingOps → (𝑋 ≈ 1o ↔ 𝑈 = 𝑍))
11	8, 10	syl 17	. . . 4 ⊢ (𝑅 ∈ DivRingOps → (𝑋 ≈ 1o ↔ 𝑈 = 𝑍))
12	4, 6, 1	rngosn6 38386	. . . . . . 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 2849	. . . . . . 7 ⊢ (𝑅 = ⟨{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}, {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩ → (𝑅 ∈ DivRingOps ↔ ⟨{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}, {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩ ∈ DivRingOps))
15	14	biimpd 231	. . . . . 6 ⊢ (𝑅 = ⟨{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}, {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩ → (𝑅 ∈ DivRingOps → ⟨{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}, {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩ ∈ DivRingOps))
16	13, 15	biimtrdi 255	. . . . 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 262	. . 3 ⊢ (𝑅 ∈ DivRingOps → (𝑈 = 𝑍 → ⟨{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}, {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩ ∈ DivRingOps))
19	18	necon3bd 2970	. 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 208   = wceq 1559   ∈ wcel 2141   ≠ wne 2956   ∖ cdif 3899  {csn 4579  ⟨cop 4585   class class class wbr 5097   × cxp 5641  ran crn 5644   ↾ cres 5645  ‘cfv 6516  1^st c1st 7963  2^nd c2nd 7964  1_oc1o 8424   ≈ cen 8918  GrpOpcgr 30649  GIdcgi 30650  RingOpscrngo 38354  DivRingOpscdrng 38408

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5224  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7713

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-suc 6347  df-iota 6472  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-riota 7348  df-ov 7394  df-1st 7965  df-2nd 7966  df-1o 8431  df-en 8922  df-grpo 30653  df-gid 30654  df-ablo 30705  df-ass 38303  df-exid 38305  df-mgmOLD 38309  df-sgrOLD 38321  df-mndo 38327  df-rngo 38355  df-drngo 38409

This theorem is referenced by:  isdrngo2  38418  divrngpr  38513  isfldidl  38528