Theorem zrdivrng 36821

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 29764	. 2 ⊢ ¬ ∅ ∈ GrpOp
2		opex 5465	. . . . . . . . . 10 ⊢ ⟨𝐴, 𝐴⟩ ∈ V
3	2	rnsnop 6224	. . . . . . . . 9 ⊢ ran {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} = {𝐴}
4		zrdivrng.1	. . . . . . . . . . 11 ⊢ 𝐴 ∈ V
5	4	gidsn 36820	. . . . . . . . . 10 ⊢ (GId‘{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}) = 𝐴
6	5	sneqi 4640	. . . . . . . . 9 ⊢ {(GId‘{⟨⟨𝐴, 𝐴⟩, 𝐴⟩})} = {𝐴}
7	3, 6	difeq12i 4121	. . . . . . . 8 ⊢ (ran {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ∖ {(GId‘{⟨⟨𝐴, 𝐴⟩, 𝐴⟩})}) = ({𝐴} ∖ {𝐴})
8		difid 4371	. . . . . . . 8 ⊢ ({𝐴} ∖ {𝐴}) = ∅
9	7, 8	eqtri 2761	. . . . . . 7 ⊢ (ran {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ∖ {(GId‘{⟨⟨𝐴, 𝐴⟩, 𝐴⟩})}) = ∅
10	9	xpeq2i 5704	. . . . . 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 6158	. . . . . 6 ⊢ ((ran {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ∖ {(GId‘{⟨⟨𝐴, 𝐴⟩, 𝐴⟩})}) × ∅) = ∅
12	10, 11	eqtri 2761	. . . . 5 ⊢ ((ran {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ∖ {(GId‘{⟨⟨𝐴, 𝐴⟩, 𝐴⟩})}) × (ran {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ∖ {(GId‘{⟨⟨𝐴, 𝐴⟩, 𝐴⟩})})) = ∅
13	12	reseq2i 5979	. . . 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 5986	. . . 4 ⊢ ({⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ↾ ∅) = ∅
15	13, 14	eqtri 2761	. . 3 ⊢ ({⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ↾ ((ran {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ∖ {(GId‘{⟨⟨𝐴, 𝐴⟩, 𝐴⟩})}) × (ran {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ∖ {(GId‘{⟨⟨𝐴, 𝐴⟩, 𝐴⟩})}))) = ∅
16		snex 5432	. . . . 5 ⊢ {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ∈ V
17		isdivrngo 36818	. . . . 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 2839	. 2 ⊢ (⟨{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}, {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}⟩ ∈ DivRingOps → ∅ ∈ GrpOp)
21	1, 20	mto 196	1 ⊢ ¬ ⟨{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}, {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}⟩ ∈ DivRingOps

Syntax hints:  ¬ wn 3   ↔ wb 205   ∧ wa 397   ∈ wcel 2107  Vcvv 3475   ∖ cdif 3946  ∅c0 4323  {csn 4629  ⟨cop 4635   × cxp 5675  ran crn 5678   ↾ cres 5679  ‘cfv 6544  GrpOpcgr 29742  GIdcgi 29743  RingOpscrngo 36762  DivRingOpscdrng 36816

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-1st 7975  df-2nd 7976  df-grpo 29746  df-gid 29747  df-rngo 36763  df-drngo 36817

This theorem is referenced by:  dvrunz  36822