Theorem zrdivrng 38491

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 30803	. 2 ⊢ ¬ ∅ ∈ GrpOp
2		opex 5446	. . . . . . . . . 10 ⊢ ⟨𝐴, 𝐴⟩ ∈ V
3	2	rnsnop 6226	. . . . . . . . 9 ⊢ ran {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} = {𝐴}
4		zrdivrng.1	. . . . . . . . . . 11 ⊢ 𝐴 ∈ V
5	4	gidsn 38490	. . . . . . . . . 10 ⊢ (GId‘{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}) = 𝐴
6	5	sneqi 4605	. . . . . . . . 9 ⊢ {(GId‘{⟨⟨𝐴, 𝐴⟩, 𝐴⟩})} = {𝐴}
7	3, 6	difeq12i 4087	. . . . . . . 8 ⊢ (ran {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ∖ {(GId‘{⟨⟨𝐴, 𝐴⟩, 𝐴⟩})}) = ({𝐴} ∖ {𝐴})
8		difid 4339	. . . . . . . 8 ⊢ ({𝐴} ∖ {𝐴}) = ∅
9	7, 8	eqtri 2792	. . . . . . 7 ⊢ (ran {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ∖ {(GId‘{⟨⟨𝐴, 𝐴⟩, 𝐴⟩})}) = ∅
10	9	xpeq2i 5689	. . . . . 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 5762	. . . . . 6 ⊢ ((ran {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ∖ {(GId‘{⟨⟨𝐴, 𝐴⟩, 𝐴⟩})}) × ∅) = ∅
12	10, 11	eqtri 2792	. . . . 5 ⊢ ((ran {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ∖ {(GId‘{⟨⟨𝐴, 𝐴⟩, 𝐴⟩})}) × (ran {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ∖ {(GId‘{⟨⟨𝐴, 𝐴⟩, 𝐴⟩})})) = ∅
13	12	reseq2i 5976	. . . 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 5983	. . . 4 ⊢ ({⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ↾ ∅) = ∅
15	13, 14	eqtri 2792	. . 3 ⊢ ({⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ↾ ((ran {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ∖ {(GId‘{⟨⟨𝐴, 𝐴⟩, 𝐴⟩})}) × (ran {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ∖ {(GId‘{⟨⟨𝐴, 𝐴⟩, 𝐴⟩})}))) = ∅
16		snex 5411	. . . . 5 ⊢ {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ∈ V
17		isdivrngo 38488	. . . . 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 502	. . 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 2874	. 2 ⊢ (⟨{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}, {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}⟩ ∈ DivRingOps → ∅ ∈ GrpOp)
21	1, 20	mto 200	1 ⊢ ¬ ⟨{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}, {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}⟩ ∈ DivRingOps

Syntax hints:  ¬ wn 3   ↔ wb 209   ∧ wa 400   ∈ wcel 2149  Vcvv 3463   ∖ cdif 3910  ∅c0 4294  {csn 4594  ⟨cop 4600   × cxp 5660  ran crn 5663   ↾ cres 5664  ‘cfv 6537  GrpOpcgr 30781  GIdcgi 30782  RingOpscrngo 38432  DivRingOpscdrng 38486

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7368  df-ov 7414  df-1st 7985  df-2nd 7986  df-grpo 30785  df-gid 30786  df-rngo 38433  df-drngo 38487

This theorem is referenced by:  dvrunz  38492