Theorem zrdivrng 38320

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 30600	. 2 ⊢ ¬ ∅ ∈ GrpOp
2		opex 5403	. . . . . . . . . 10 ⊢ ⟨𝐴, 𝐴⟩ ∈ V
3	2	rnsnop 6175	. . . . . . . . 9 ⊢ ran {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} = {𝐴}
4		zrdivrng.1	. . . . . . . . . . 11 ⊢ 𝐴 ∈ V
5	4	gidsn 38319	. . . . . . . . . 10 ⊢ (GId‘{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}) = 𝐴
6	5	sneqi 4566	. . . . . . . . 9 ⊢ {(GId‘{⟨⟨𝐴, 𝐴⟩, 𝐴⟩})} = {𝐴}
7	3, 6	difeq12i 4055	. . . . . . . 8 ⊢ (ran {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ∖ {(GId‘{⟨⟨𝐴, 𝐴⟩, 𝐴⟩})}) = ({𝐴} ∖ {𝐴})
8		difid 4304	. . . . . . . 8 ⊢ ({𝐴} ∖ {𝐴}) = ∅
9	7, 8	eqtri 2762	. . . . . . 7 ⊢ (ran {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ∖ {(GId‘{⟨⟨𝐴, 𝐴⟩, 𝐴⟩})}) = ∅
10	9	xpeq2i 5645	. . . . . 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 5718	. . . . . 6 ⊢ ((ran {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ∖ {(GId‘{⟨⟨𝐴, 𝐴⟩, 𝐴⟩})}) × ∅) = ∅
12	10, 11	eqtri 2762	. . . . 5 ⊢ ((ran {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ∖ {(GId‘{⟨⟨𝐴, 𝐴⟩, 𝐴⟩})}) × (ran {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ∖ {(GId‘{⟨⟨𝐴, 𝐴⟩, 𝐴⟩})})) = ∅
13	12	reseq2i 5928	. . . 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 5935	. . . 4 ⊢ ({⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ↾ ∅) = ∅
15	13, 14	eqtri 2762	. . 3 ⊢ ({⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ↾ ((ran {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ∖ {(GId‘{⟨⟨𝐴, 𝐴⟩, 𝐴⟩})}) × (ran {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ∖ {(GId‘{⟨⟨𝐴, 𝐴⟩, 𝐴⟩})}))) = ∅
16		snex 5368	. . . . 5 ⊢ {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ∈ V
17		isdivrngo 38317	. . . . 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 498	. . 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 2844	. 2 ⊢ (⟨{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}, {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}⟩ ∈ DivRingOps → ∅ ∈ GrpOp)
21	1, 20	mto 198	1 ⊢ ¬ ⟨{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}, {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}⟩ ∈ DivRingOps

Syntax hints:  ¬ wn 3   ↔ wb 207   ∧ wa 396   ∈ wcel 2119  Vcvv 3431   ∖ cdif 3880  ∅c0 4261  {csn 4555  ⟨cop 4561   × cxp 5616  ran crn 5619   ↾ cres 5620  ‘cfv 6485  GrpOpcgr 30578  GIdcgi 30579  RingOpscrngo 38261  DivRingOpscdrng 38315

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-riota 7313  df-ov 7359  df-1st 7931  df-2nd 7932  df-grpo 30582  df-gid 30583  df-rngo 38262  df-drngo 38316

This theorem is referenced by:  dvrunz  38321