Theorem lsmdisj2a 17923

Description: Association of the disjointness constraint in a subgroup sum. (Contributed by Mario Carneiro, 21-Apr-2016.)

Hypotheses

Ref	Expression
lsmcntz.p	⊢ ⊕ = (LSSum‘𝐺)
lsmcntz.s	⊢ (𝜑 → 𝑆 ∈ (SubGrp‘𝐺))
lsmcntz.t	⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺))
lsmcntz.u	⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺))
lsmdisj.o	⊢ 0 = (0g‘𝐺)

Assertion

Ref	Expression
lsmdisj2a	⊢ (𝜑 → ((((𝑆 ⊕ 𝑇) ∩ 𝑈) = { 0 } ∧ (𝑆 ∩ 𝑇) = { 0 }) ↔ ((𝑇 ∩ (𝑆 ⊕ 𝑈)) = { 0 } ∧ (𝑆 ∩ 𝑈) = { 0 })))

Proof of Theorem lsmdisj2a

Step	Hyp	Ref	Expression
1		lsmcntz.p	. . . 4 ⊢ ⊕ = (LSSum‘𝐺)
2		lsmcntz.s	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ (SubGrp‘𝐺))
3	2	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (((𝑆 ⊕ 𝑇) ∩ 𝑈) = { 0 } ∧ (𝑆 ∩ 𝑇) = { 0 })) → 𝑆 ∈ (SubGrp‘𝐺))
4		lsmcntz.t	. . . . 5 ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺))
5	4	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (((𝑆 ⊕ 𝑇) ∩ 𝑈) = { 0 } ∧ (𝑆 ∩ 𝑇) = { 0 })) → 𝑇 ∈ (SubGrp‘𝐺))
6		lsmcntz.u	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺))
7	6	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (((𝑆 ⊕ 𝑇) ∩ 𝑈) = { 0 } ∧ (𝑆 ∩ 𝑇) = { 0 })) → 𝑈 ∈ (SubGrp‘𝐺))
8		lsmdisj.o	. . . 4 ⊢ 0 = (0g‘𝐺)
9		simprl 790	. . . 4 ⊢ ((𝜑 ∧ (((𝑆 ⊕ 𝑇) ∩ 𝑈) = { 0 } ∧ (𝑆 ∩ 𝑇) = { 0 })) → ((𝑆 ⊕ 𝑇) ∩ 𝑈) = { 0 })
10		simprr 792	. . . 4 ⊢ ((𝜑 ∧ (((𝑆 ⊕ 𝑇) ∩ 𝑈) = { 0 } ∧ (𝑆 ∩ 𝑇) = { 0 })) → (𝑆 ∩ 𝑇) = { 0 })
11	1, 3, 5, 7, 8, 9, 10	lsmdisj2 17918	. . 3 ⊢ ((𝜑 ∧ (((𝑆 ⊕ 𝑇) ∩ 𝑈) = { 0 } ∧ (𝑆 ∩ 𝑇) = { 0 })) → (𝑇 ∩ (𝑆 ⊕ 𝑈)) = { 0 })
12	1, 3, 5, 7, 8, 9	lsmdisj 17917	. . . 4 ⊢ ((𝜑 ∧ (((𝑆 ⊕ 𝑇) ∩ 𝑈) = { 0 } ∧ (𝑆 ∩ 𝑇) = { 0 })) → ((𝑆 ∩ 𝑈) = { 0 } ∧ (𝑇 ∩ 𝑈) = { 0 }))
13	12	simpld 474	. . 3 ⊢ ((𝜑 ∧ (((𝑆 ⊕ 𝑇) ∩ 𝑈) = { 0 } ∧ (𝑆 ∩ 𝑇) = { 0 })) → (𝑆 ∩ 𝑈) = { 0 })
14	11, 13	jca 553	. 2 ⊢ ((𝜑 ∧ (((𝑆 ⊕ 𝑇) ∩ 𝑈) = { 0 } ∧ (𝑆 ∩ 𝑇) = { 0 })) → ((𝑇 ∩ (𝑆 ⊕ 𝑈)) = { 0 } ∧ (𝑆 ∩ 𝑈) = { 0 }))
15		incom 3767	. . . 4 ⊢ ((𝑆 ⊕ 𝑇) ∩ 𝑈) = (𝑈 ∩ (𝑆 ⊕ 𝑇))
16	2	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ((𝑇 ∩ (𝑆 ⊕ 𝑈)) = { 0 } ∧ (𝑆 ∩ 𝑈) = { 0 })) → 𝑆 ∈ (SubGrp‘𝐺))
17	6	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ((𝑇 ∩ (𝑆 ⊕ 𝑈)) = { 0 } ∧ (𝑆 ∩ 𝑈) = { 0 })) → 𝑈 ∈ (SubGrp‘𝐺))
18	4	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ((𝑇 ∩ (𝑆 ⊕ 𝑈)) = { 0 } ∧ (𝑆 ∩ 𝑈) = { 0 })) → 𝑇 ∈ (SubGrp‘𝐺))
19		incom 3767	. . . . . 6 ⊢ ((𝑆 ⊕ 𝑈) ∩ 𝑇) = (𝑇 ∩ (𝑆 ⊕ 𝑈))
20		simprl 790	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑇 ∩ (𝑆 ⊕ 𝑈)) = { 0 } ∧ (𝑆 ∩ 𝑈) = { 0 })) → (𝑇 ∩ (𝑆 ⊕ 𝑈)) = { 0 })
21	19, 20	syl5eq 2656	. . . . 5 ⊢ ((𝜑 ∧ ((𝑇 ∩ (𝑆 ⊕ 𝑈)) = { 0 } ∧ (𝑆 ∩ 𝑈) = { 0 })) → ((𝑆 ⊕ 𝑈) ∩ 𝑇) = { 0 })
22		simprr 792	. . . . 5 ⊢ ((𝜑 ∧ ((𝑇 ∩ (𝑆 ⊕ 𝑈)) = { 0 } ∧ (𝑆 ∩ 𝑈) = { 0 })) → (𝑆 ∩ 𝑈) = { 0 })
23	1, 16, 17, 18, 8, 21, 22	lsmdisj2 17918	. . . 4 ⊢ ((𝜑 ∧ ((𝑇 ∩ (𝑆 ⊕ 𝑈)) = { 0 } ∧ (𝑆 ∩ 𝑈) = { 0 })) → (𝑈 ∩ (𝑆 ⊕ 𝑇)) = { 0 })
24	15, 23	syl5eq 2656	. . 3 ⊢ ((𝜑 ∧ ((𝑇 ∩ (𝑆 ⊕ 𝑈)) = { 0 } ∧ (𝑆 ∩ 𝑈) = { 0 })) → ((𝑆 ⊕ 𝑇) ∩ 𝑈) = { 0 })
25		incom 3767	. . . 4 ⊢ (𝑆 ∩ 𝑇) = (𝑇 ∩ 𝑆)
26	1, 18, 16, 17, 8, 20	lsmdisjr 17920	. . . . 5 ⊢ ((𝜑 ∧ ((𝑇 ∩ (𝑆 ⊕ 𝑈)) = { 0 } ∧ (𝑆 ∩ 𝑈) = { 0 })) → ((𝑇 ∩ 𝑆) = { 0 } ∧ (𝑇 ∩ 𝑈) = { 0 }))
27	26	simpld 474	. . . 4 ⊢ ((𝜑 ∧ ((𝑇 ∩ (𝑆 ⊕ 𝑈)) = { 0 } ∧ (𝑆 ∩ 𝑈) = { 0 })) → (𝑇 ∩ 𝑆) = { 0 })
28	25, 27	syl5eq 2656	. . 3 ⊢ ((𝜑 ∧ ((𝑇 ∩ (𝑆 ⊕ 𝑈)) = { 0 } ∧ (𝑆 ∩ 𝑈) = { 0 })) → (𝑆 ∩ 𝑇) = { 0 })
29	24, 28	jca 553	. 2 ⊢ ((𝜑 ∧ ((𝑇 ∩ (𝑆 ⊕ 𝑈)) = { 0 } ∧ (𝑆 ∩ 𝑈) = { 0 })) → (((𝑆 ⊕ 𝑇) ∩ 𝑈) = { 0 } ∧ (𝑆 ∩ 𝑇) = { 0 }))
30	14, 29	impbida 873	1 ⊢ (𝜑 → ((((𝑆 ⊕ 𝑇) ∩ 𝑈) = { 0 } ∧ (𝑆 ∩ 𝑇) = { 0 }) ↔ ((𝑇 ∩ (𝑆 ⊕ 𝑈)) = { 0 } ∧ (𝑆 ∩ 𝑈) = { 0 })))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ∩ cin 3539  {csn 4125  ‘cfv 5804  (class class class)co 6549  0_gc0g 15923  SubGrpcsubg 17411  LSSumclsm 17872

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-0g 15925  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-submnd 17159  df-grp 17248  df-minusg 17249  df-subg 17414  df-lsm 17874

This theorem is referenced by:  lsmdisj3a  17925