Theorem divrngcl 38278

Description: The product of two nonzero elements of a division ring is nonzero. (Contributed by Jeff Madsen, 9-Jun-2010.)

Hypotheses

Ref	Expression
isdivrng1.1	⊢ 𝐺 = (1st ‘𝑅)
isdivrng1.2	⊢ 𝐻 = (2nd ‘𝑅)
isdivrng1.3	⊢ 𝑍 = (GId‘𝐺)
isdivrng1.4	⊢ 𝑋 = ran 𝐺

Assertion

Ref	Expression
divrngcl	⊢ ((𝑅 ∈ DivRingOps ∧ 𝐴 ∈ (𝑋 ∖ {𝑍}) ∧ 𝐵 ∈ (𝑋 ∖ {𝑍})) → (𝐴𝐻𝐵) ∈ (𝑋 ∖ {𝑍}))

Proof of Theorem divrngcl

Step	Hyp	Ref	Expression
1		isdivrng1.1	. . 3 ⊢ 𝐺 = (1st ‘𝑅)
2		isdivrng1.2	. . 3 ⊢ 𝐻 = (2nd ‘𝑅)
3		isdivrng1.3	. . 3 ⊢ 𝑍 = (GId‘𝐺)
4		isdivrng1.4	. . 3 ⊢ 𝑋 = ran 𝐺
5	1, 2, 3, 4	isdrngo1 38277	. 2 ⊢ (𝑅 ∈ DivRingOps ↔ (𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp))
6		ovres 7533	. . . . 5 ⊢ ((𝐴 ∈ (𝑋 ∖ {𝑍}) ∧ 𝐵 ∈ (𝑋 ∖ {𝑍})) → (𝐴(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝐵) = (𝐴𝐻𝐵))
7	6	adantl 481	. . . 4 ⊢ (((𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) ∧ (𝐴 ∈ (𝑋 ∖ {𝑍}) ∧ 𝐵 ∈ (𝑋 ∖ {𝑍}))) → (𝐴(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝐵) = (𝐴𝐻𝐵))
8		eqid 2736	. . . . . . . . 9 ⊢ ran (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) = ran (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))
9	8	grpocl 30571	. . . . . . . 8 ⊢ (((𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp ∧ 𝐴 ∈ ran (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∧ 𝐵 ∈ ran (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))) → (𝐴(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝐵) ∈ ran (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))))
10	9	3expib 1123	. . . . . . 7 ⊢ ((𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp → ((𝐴 ∈ ran (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∧ 𝐵 ∈ ran (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))) → (𝐴(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝐵) ∈ ran (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))))
11	10	adantl 481	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) → ((𝐴 ∈ ran (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∧ 𝐵 ∈ ran (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))) → (𝐴(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝐵) ∈ ran (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))))
12		grporndm 30581	. . . . . . . . . 10 ⊢ ((𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp → ran (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) = dom dom (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))))
13	12	adantl 481	. . . . . . . . 9 ⊢ ((𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) → ran (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) = dom dom (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))))
14		difss 4076	. . . . . . . . . . . . . . 15 ⊢ (𝑋 ∖ {𝑍}) ⊆ 𝑋
15		xpss12 5646	. . . . . . . . . . . . . . 15 ⊢ (((𝑋 ∖ {𝑍}) ⊆ 𝑋 ∧ (𝑋 ∖ {𝑍}) ⊆ 𝑋) → ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})) ⊆ (𝑋 × 𝑋))
16	14, 14, 15	mp2an 693	. . . . . . . . . . . . . 14 ⊢ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})) ⊆ (𝑋 × 𝑋)
17	1, 2, 4	rngosm 38221	. . . . . . . . . . . . . . 15 ⊢ (𝑅 ∈ RingOps → 𝐻:(𝑋 × 𝑋)⟶𝑋)
18	17	fdmd 6678	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ RingOps → dom 𝐻 = (𝑋 × 𝑋))
19	16, 18	sseqtrrid 3965	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ RingOps → ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})) ⊆ dom 𝐻)
20		ssdmres 5978	. . . . . . . . . . . . 13 ⊢ (((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})) ⊆ dom 𝐻 ↔ dom (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) = ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))
21	19, 20	sylib 218	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ RingOps → dom (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) = ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))
22	21	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) → dom (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) = ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))
23	22	dmeqd 5860	. . . . . . . . . 10 ⊢ ((𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) → dom dom (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) = dom ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))
24		dmxpid 5885	. . . . . . . . . 10 ⊢ dom ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})) = (𝑋 ∖ {𝑍})
25	23, 24	eqtrdi 2787	. . . . . . . . 9 ⊢ ((𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) → dom dom (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) = (𝑋 ∖ {𝑍}))
26	13, 25	eqtrd 2771	. . . . . . . 8 ⊢ ((𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) → ran (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) = (𝑋 ∖ {𝑍}))
27	26	eleq2d 2822	. . . . . . 7 ⊢ ((𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) → (𝐴 ∈ ran (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ↔ 𝐴 ∈ (𝑋 ∖ {𝑍})))
28	26	eleq2d 2822	. . . . . . 7 ⊢ ((𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) → (𝐵 ∈ ran (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ↔ 𝐵 ∈ (𝑋 ∖ {𝑍})))
29	27, 28	anbi12d 633	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) → ((𝐴 ∈ ran (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∧ 𝐵 ∈ ran (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))) ↔ (𝐴 ∈ (𝑋 ∖ {𝑍}) ∧ 𝐵 ∈ (𝑋 ∖ {𝑍}))))
30	26	eleq2d 2822	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) → ((𝐴(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝐵) ∈ ran (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ↔ (𝐴(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝐵) ∈ (𝑋 ∖ {𝑍})))
31	11, 29, 30	3imtr3d 293	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) → ((𝐴 ∈ (𝑋 ∖ {𝑍}) ∧ 𝐵 ∈ (𝑋 ∖ {𝑍})) → (𝐴(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝐵) ∈ (𝑋 ∖ {𝑍})))
32	31	imp 406	. . . 4 ⊢ (((𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) ∧ (𝐴 ∈ (𝑋 ∖ {𝑍}) ∧ 𝐵 ∈ (𝑋 ∖ {𝑍}))) → (𝐴(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝐵) ∈ (𝑋 ∖ {𝑍}))
33	7, 32	eqeltrrd 2837	. . 3 ⊢ (((𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) ∧ (𝐴 ∈ (𝑋 ∖ {𝑍}) ∧ 𝐵 ∈ (𝑋 ∖ {𝑍}))) → (𝐴𝐻𝐵) ∈ (𝑋 ∖ {𝑍}))
34	33	3impb 1115	. 2 ⊢ (((𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) ∧ 𝐴 ∈ (𝑋 ∖ {𝑍}) ∧ 𝐵 ∈ (𝑋 ∖ {𝑍})) → (𝐴𝐻𝐵) ∈ (𝑋 ∖ {𝑍}))
35	5, 34	syl3an1b 1406	1 ⊢ ((𝑅 ∈ DivRingOps ∧ 𝐴 ∈ (𝑋 ∖ {𝑍}) ∧ 𝐵 ∈ (𝑋 ∖ {𝑍})) → (𝐴𝐻𝐵) ∈ (𝑋 ∖ {𝑍}))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ∖ cdif 3886   ⊆ wss 3889  {csn 4567   × cxp 5629  dom cdm 5631  ran crn 5632   ↾ cres 5633  ‘cfv 6498  (class class class)co 7367  1^st c1st 7940  2^nd c2nd 7941  GrpOpcgr 30560  GIdcgi 30561  RingOpscrngo 38215  DivRingOpscdrng 38269

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-fo 6504  df-fv 6506  df-ov 7370  df-1st 7942  df-2nd 7943  df-grpo 30564  df-rngo 38216  df-drngo 38270

This theorem is referenced by: (None)