Theorem zrdivrng 38887

Description: Obsolete theorem, use zrdrng 21026 instead. The zero ring is not a division ring. (Contributed by FL, 24-Jan-2010.) (Proof shortened by Mario Carneiro, 15-Dec-2013.) (Proof modification is discouraged.) (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 31113	. 2 ⊢ ¬ ∅ ∈ GrpOp
2		opex 5432	. . . . . . . . . 10 ⊢ ⟨𝐴, 𝐴⟩ ∈ V
3	2	rnsnop 6225	. . . . . . . . 9 ⊢ ran {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} = {𝐴}
4		zrdivrng.1	. . . . . . . . . . 11 ⊢ 𝐴 ∈ V
5	4	gidsn 38886	. . . . . . . . . 10 ⊢ (GId‘{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}) = 𝐴
6	5	sneqi 4595	. . . . . . . . 9 ⊢ {(GId‘{⟨⟨𝐴, 𝐴⟩, 𝐴⟩})} = {𝐴}
7	3, 6	difeq12i 4072	. . . . . . . 8 ⊢ (ran {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ∖ {(GId‘{⟨⟨𝐴, 𝐴⟩, 𝐴⟩})}) = ({𝐴} ∖ {𝐴})
8		difid 4325	. . . . . . . 8 ⊢ ({𝐴} ∖ {𝐴}) = ∅
9	7, 8	eqtri 2784	. . . . . . 7 ⊢ (ran {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ∖ {(GId‘{⟨⟨𝐴, 𝐴⟩, 𝐴⟩})}) = ∅
10	9	xpeq2i 5678	. . . . . 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 5751	. . . . . 6 ⊢ ((ran {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ∖ {(GId‘{⟨⟨𝐴, 𝐴⟩, 𝐴⟩})}) × ∅) = ∅
12	10, 11	eqtri 2784	. . . . 5 ⊢ ((ran {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ∖ {(GId‘{⟨⟨𝐴, 𝐴⟩, 𝐴⟩})}) × (ran {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ∖ {(GId‘{⟨⟨𝐴, 𝐴⟩, 𝐴⟩})})) = ∅
13	12	reseq2i 5967	. . . 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 5974	. . . 4 ⊢ ({⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ↾ ∅) = ∅
15	13, 14	eqtri 2784	. . 3 ⊢ ({⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ↾ ((ran {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ∖ {(GId‘{⟨⟨𝐴, 𝐴⟩, 𝐴⟩})}) × (ran {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ∖ {(GId‘{⟨⟨𝐴, 𝐴⟩, 𝐴⟩})}))) = ∅
16		snex 5397	. . . . 5 ⊢ {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ∈ V
17		isdivrngo 38884	. . . . 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 503	. . 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 2866	. 2 ⊢ (⟨{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}, {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}⟩ ∈ DivRingOps → ∅ ∈ GrpOp)
21	1, 20	mto 200	1 ⊢ ¬ ⟨{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}, {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}⟩ ∈ DivRingOps

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   ∧ wa 401   ∈ wcel 2145  Vcvv 3451   ∖ cdif 3896  ∅c0 4279  {csn 4584  ⟨cop 4590   × cxp 5649  ran crn 5652   ↾ cres 5653  ‘cfv 6538  GrpOpcgr 31091  GIdcgi 31092  RingOpscrngo 38828  DivRingOpscdrng 38882

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-1st 8001  df-2nd 8002  df-grpo 31095  df-gid 31096  df-rngo 38829  df-drngo 38883

This theorem is used by:  dvrunz  38888