Theorem zrdivrng 37933

Description: The zero ring is not a division ring. (Contributed by FL, 24-Jan-2010.) (Proof shortened by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)

Hypothesis

Ref	Expression
zrdivrng.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
zrdivrng	⊢ ¬ ⟨{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}, {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}⟩ ∈ DivRingOps

Proof of Theorem zrdivrng

Step	Hyp	Ref	Expression
1		0ngrp 30455	. 2 ⊢ ¬ ∅ ∈ GrpOp
2		opex 5407	. . . . . . . . . 10 ⊢ ⟨𝐴, 𝐴⟩ ∈ V
3	2	rnsnop 6173	. . . . . . . . 9 ⊢ ran {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} = {𝐴}
4		zrdivrng.1	. . . . . . . . . . 11 ⊢ 𝐴 ∈ V
5	4	gidsn 37932	. . . . . . . . . 10 ⊢ (GId‘{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}) = 𝐴
6	5	sneqi 4588	. . . . . . . . 9 ⊢ {(GId‘{⟨⟨𝐴, 𝐴⟩, 𝐴⟩})} = {𝐴}
7	3, 6	difeq12i 4075	. . . . . . . 8 ⊢ (ran {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ∖ {(GId‘{⟨⟨𝐴, 𝐴⟩, 𝐴⟩})}) = ({𝐴} ∖ {𝐴})
8		difid 4327	. . . . . . . 8 ⊢ ({𝐴} ∖ {𝐴}) = ∅
9	7, 8	eqtri 2752	. . . . . . 7 ⊢ (ran {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ∖ {(GId‘{⟨⟨𝐴, 𝐴⟩, 𝐴⟩})}) = ∅
10	9	xpeq2i 5646	. . . . . 6 ⊢ ((ran {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ∖ {(GId‘{⟨⟨𝐴, 𝐴⟩, 𝐴⟩})}) × (ran {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ∖ {(GId‘{⟨⟨𝐴, 𝐴⟩, 𝐴⟩})})) = ((ran {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ∖ {(GId‘{⟨⟨𝐴, 𝐴⟩, 𝐴⟩})}) × ∅)
11		xp0 6107	. . . . . 6 ⊢ ((ran {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ∖ {(GId‘{⟨⟨𝐴, 𝐴⟩, 𝐴⟩})}) × ∅) = ∅
12	10, 11	eqtri 2752	. . . . 5 ⊢ ((ran {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ∖ {(GId‘{⟨⟨𝐴, 𝐴⟩, 𝐴⟩})}) × (ran {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ∖ {(GId‘{⟨⟨𝐴, 𝐴⟩, 𝐴⟩})})) = ∅
13	12	reseq2i 5927	. . . 4 ⊢ ({⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ↾ ((ran {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ∖ {(GId‘{⟨⟨𝐴, 𝐴⟩, 𝐴⟩})}) × (ran {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ∖ {(GId‘{⟨⟨𝐴, 𝐴⟩, 𝐴⟩})}))) = ({⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ↾ ∅)
14		res0 5934	. . . 4 ⊢ ({⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ↾ ∅) = ∅
15	13, 14	eqtri 2752	. . 3 ⊢ ({⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ↾ ((ran {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ∖ {(GId‘{⟨⟨𝐴, 𝐴⟩, 𝐴⟩})}) × (ran {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ∖ {(GId‘{⟨⟨𝐴, 𝐴⟩, 𝐴⟩})}))) = ∅
16		snex 5375	. . . . 5 ⊢ {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ∈ V
17		isdivrngo 37930	. . . . 5 ⊢ ({⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ∈ V → (⟨{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}, {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}⟩ ∈ DivRingOps ↔ (⟨{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}, {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}⟩ ∈ RingOps ∧ ({⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ↾ ((ran {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ∖ {(GId‘{⟨⟨𝐴, 𝐴⟩, 𝐴⟩})}) × (ran {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ∖ {(GId‘{⟨⟨𝐴, 𝐴⟩, 𝐴⟩})}))) ∈ GrpOp)))
18	16, 17	ax-mp 5	. . . 4 ⊢ (⟨{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}, {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}⟩ ∈ DivRingOps ↔ (⟨{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}, {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}⟩ ∈ RingOps ∧ ({⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ↾ ((ran {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ∖ {(GId‘{⟨⟨𝐴, 𝐴⟩, 𝐴⟩})}) × (ran {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ∖ {(GId‘{⟨⟨𝐴, 𝐴⟩, 𝐴⟩})}))) ∈ GrpOp))
19	18	simprbi 496	. . 3 ⊢ (⟨{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}, {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}⟩ ∈ DivRingOps → ({⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ↾ ((ran {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ∖ {(GId‘{⟨⟨𝐴, 𝐴⟩, 𝐴⟩})}) × (ran {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ∖ {(GId‘{⟨⟨𝐴, 𝐴⟩, 𝐴⟩})}))) ∈ GrpOp)
20	15, 19	eqeltrrid 2833	. 2 ⊢ (⟨{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}, {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}⟩ ∈ DivRingOps → ∅ ∈ GrpOp)
21	1, 20	mto 197	1 ⊢ ¬ ⟨{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}, {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}⟩ ∈ DivRingOps

Syntax hints:  ¬ wn 3   ↔ wb 206   ∧ wa 395   ∈ wcel 2109  Vcvv 3436   ∖ cdif 3900  ∅c0 4284  {csn 4577  ⟨cop 4583   × cxp 5617  ran crn 5620   ↾ cres 5621  ‘cfv 6482  GrpOpcgr 30433  GIdcgi 30434  RingOpscrngo 37874  DivRingOpscdrng 37928

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-riota 7306  df-ov 7352  df-1st 7924  df-2nd 7925  df-grpo 30437  df-gid 30438  df-rngo 37875  df-drngo 37929

This theorem is referenced by:  dvrunz  37934