Theorem divrngcl 37936

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 37935	. 2 ⊢ (𝑅 ∈ DivRingOps ↔ (𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp))
6		ovres 7519	. . . . 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 2729	. . . . . . . . 9 ⊢ ran (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) = ran (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))
9	8	grpocl 30462	. . . . . . . 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 1122	. . . . . . 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 30472	. . . . . . . . . 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 4089	. . . . . . . . . . . . . . 15 ⊢ (𝑋 ∖ {𝑍}) ⊆ 𝑋
15		xpss12 5638	. . . . . . . . . . . . . . 15 ⊢ (((𝑋 ∖ {𝑍}) ⊆ 𝑋 ∧ (𝑋 ∖ {𝑍}) ⊆ 𝑋) → ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})) ⊆ (𝑋 × 𝑋))
16	14, 14, 15	mp2an 692	. . . . . . . . . . . . . 14 ⊢ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})) ⊆ (𝑋 × 𝑋)
17	1, 2, 4	rngosm 37879	. . . . . . . . . . . . . . 15 ⊢ (𝑅 ∈ RingOps → 𝐻:(𝑋 × 𝑋)⟶𝑋)
18	17	fdmd 6666	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ RingOps → dom 𝐻 = (𝑋 × 𝑋))
19	16, 18	sseqtrrid 3981	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ RingOps → ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})) ⊆ dom 𝐻)
20		ssdmres 5968	. . . . . . . . . . . . 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 5852	. . . . . . . . . 10 ⊢ ((𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) → dom dom (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) = dom ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))
24		dmxpid 5876	. . . . . . . . . 10 ⊢ dom ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})) = (𝑋 ∖ {𝑍})
25	23, 24	eqtrdi 2780	. . . . . . . . 9 ⊢ ((𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) → dom dom (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) = (𝑋 ∖ {𝑍}))
26	13, 25	eqtrd 2764	. . . . . . . 8 ⊢ ((𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) → ran (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) = (𝑋 ∖ {𝑍}))
27	26	eleq2d 2814	. . . . . . 7 ⊢ ((𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) → (𝐴 ∈ ran (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ↔ 𝐴 ∈ (𝑋 ∖ {𝑍})))
28	26	eleq2d 2814	. . . . . . 7 ⊢ ((𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) → (𝐵 ∈ ran (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ↔ 𝐵 ∈ (𝑋 ∖ {𝑍})))
29	27, 28	anbi12d 632	. . . . . 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 2814	. . . . . 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 2829	. . 3 ⊢ (((𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) ∧ (𝐴 ∈ (𝑋 ∖ {𝑍}) ∧ 𝐵 ∈ (𝑋 ∖ {𝑍}))) → (𝐴𝐻𝐵) ∈ (𝑋 ∖ {𝑍}))
34	33	3impb 1114	. 2 ⊢ (((𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) ∧ 𝐴 ∈ (𝑋 ∖ {𝑍}) ∧ 𝐵 ∈ (𝑋 ∖ {𝑍})) → (𝐴𝐻𝐵) ∈ (𝑋 ∖ {𝑍}))
35	5, 34	syl3an1b 1405	1 ⊢ ((𝑅 ∈ DivRingOps ∧ 𝐴 ∈ (𝑋 ∖ {𝑍}) ∧ 𝐵 ∈ (𝑋 ∖ {𝑍})) → (𝐴𝐻𝐵) ∈ (𝑋 ∖ {𝑍}))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ∖ cdif 3902   ⊆ wss 3905  {csn 4579   × cxp 5621  dom cdm 5623  ran crn 5624   ↾ cres 5625  ‘cfv 6486  (class class class)co 7353  1^st c1st 7929  2^nd c2nd 7930  GrpOpcgr 30451  GIdcgi 30452  RingOpscrngo 37873  DivRingOpscdrng 37927

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374  ax-un 7675

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-fo 6492  df-fv 6494  df-ov 7356  df-1st 7931  df-2nd 7932  df-grpo 30455  df-rngo 37874  df-drngo 37928

This theorem is referenced by: (None)