Theorem lsmdisj2a 18100

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 481	. . . 4 ⊢ ((𝜑 ∧ (((𝑆 ⊕ 𝑇) ∩ 𝑈) = { 0 } ∧ (𝑆 ∩ 𝑇) = { 0 })) → 𝑆 ∈ (SubGrp‘𝐺))
4		lsmcntz.t	. . . . 5 ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺))
5	4	adantr 481	. . . 4 ⊢ ((𝜑 ∧ (((𝑆 ⊕ 𝑇) ∩ 𝑈) = { 0 } ∧ (𝑆 ∩ 𝑇) = { 0 })) → 𝑇 ∈ (SubGrp‘𝐺))
6		lsmcntz.u	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺))
7	6	adantr 481	. . . 4 ⊢ ((𝜑 ∧ (((𝑆 ⊕ 𝑇) ∩ 𝑈) = { 0 } ∧ (𝑆 ∩ 𝑇) = { 0 })) → 𝑈 ∈ (SubGrp‘𝐺))
8		lsmdisj.o	. . . 4 ⊢ 0 = (0g‘𝐺)
9		simprl 794	. . . 4 ⊢ ((𝜑 ∧ (((𝑆 ⊕ 𝑇) ∩ 𝑈) = { 0 } ∧ (𝑆 ∩ 𝑇) = { 0 })) → ((𝑆 ⊕ 𝑇) ∩ 𝑈) = { 0 })
10		simprr 796	. . . 4 ⊢ ((𝜑 ∧ (((𝑆 ⊕ 𝑇) ∩ 𝑈) = { 0 } ∧ (𝑆 ∩ 𝑇) = { 0 })) → (𝑆 ∩ 𝑇) = { 0 })
11	1, 3, 5, 7, 8, 9, 10	lsmdisj2 18095	. . 3 ⊢ ((𝜑 ∧ (((𝑆 ⊕ 𝑇) ∩ 𝑈) = { 0 } ∧ (𝑆 ∩ 𝑇) = { 0 })) → (𝑇 ∩ (𝑆 ⊕ 𝑈)) = { 0 })
12	1, 3, 5, 7, 8, 9	lsmdisj 18094	. . . 4 ⊢ ((𝜑 ∧ (((𝑆 ⊕ 𝑇) ∩ 𝑈) = { 0 } ∧ (𝑆 ∩ 𝑇) = { 0 })) → ((𝑆 ∩ 𝑈) = { 0 } ∧ (𝑇 ∩ 𝑈) = { 0 }))
13	12	simpld 475	. . 3 ⊢ ((𝜑 ∧ (((𝑆 ⊕ 𝑇) ∩ 𝑈) = { 0 } ∧ (𝑆 ∩ 𝑇) = { 0 })) → (𝑆 ∩ 𝑈) = { 0 })
14	11, 13	jca 554	. 2 ⊢ ((𝜑 ∧ (((𝑆 ⊕ 𝑇) ∩ 𝑈) = { 0 } ∧ (𝑆 ∩ 𝑇) = { 0 })) → ((𝑇 ∩ (𝑆 ⊕ 𝑈)) = { 0 } ∧ (𝑆 ∩ 𝑈) = { 0 }))
15		incom 3805	. . . 4 ⊢ ((𝑆 ⊕ 𝑇) ∩ 𝑈) = (𝑈 ∩ (𝑆 ⊕ 𝑇))
16	2	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ ((𝑇 ∩ (𝑆 ⊕ 𝑈)) = { 0 } ∧ (𝑆 ∩ 𝑈) = { 0 })) → 𝑆 ∈ (SubGrp‘𝐺))
17	6	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ ((𝑇 ∩ (𝑆 ⊕ 𝑈)) = { 0 } ∧ (𝑆 ∩ 𝑈) = { 0 })) → 𝑈 ∈ (SubGrp‘𝐺))
18	4	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ ((𝑇 ∩ (𝑆 ⊕ 𝑈)) = { 0 } ∧ (𝑆 ∩ 𝑈) = { 0 })) → 𝑇 ∈ (SubGrp‘𝐺))
19		incom 3805	. . . . . 6 ⊢ ((𝑆 ⊕ 𝑈) ∩ 𝑇) = (𝑇 ∩ (𝑆 ⊕ 𝑈))
20		simprl 794	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑇 ∩ (𝑆 ⊕ 𝑈)) = { 0 } ∧ (𝑆 ∩ 𝑈) = { 0 })) → (𝑇 ∩ (𝑆 ⊕ 𝑈)) = { 0 })
21	19, 20	syl5eq 2668	. . . . 5 ⊢ ((𝜑 ∧ ((𝑇 ∩ (𝑆 ⊕ 𝑈)) = { 0 } ∧ (𝑆 ∩ 𝑈) = { 0 })) → ((𝑆 ⊕ 𝑈) ∩ 𝑇) = { 0 })
22		simprr 796	. . . . 5 ⊢ ((𝜑 ∧ ((𝑇 ∩ (𝑆 ⊕ 𝑈)) = { 0 } ∧ (𝑆 ∩ 𝑈) = { 0 })) → (𝑆 ∩ 𝑈) = { 0 })
23	1, 16, 17, 18, 8, 21, 22	lsmdisj2 18095	. . . 4 ⊢ ((𝜑 ∧ ((𝑇 ∩ (𝑆 ⊕ 𝑈)) = { 0 } ∧ (𝑆 ∩ 𝑈) = { 0 })) → (𝑈 ∩ (𝑆 ⊕ 𝑇)) = { 0 })
24	15, 23	syl5eq 2668	. . 3 ⊢ ((𝜑 ∧ ((𝑇 ∩ (𝑆 ⊕ 𝑈)) = { 0 } ∧ (𝑆 ∩ 𝑈) = { 0 })) → ((𝑆 ⊕ 𝑇) ∩ 𝑈) = { 0 })
25		incom 3805	. . . 4 ⊢ (𝑆 ∩ 𝑇) = (𝑇 ∩ 𝑆)
26	1, 18, 16, 17, 8, 20	lsmdisjr 18097	. . . . 5 ⊢ ((𝜑 ∧ ((𝑇 ∩ (𝑆 ⊕ 𝑈)) = { 0 } ∧ (𝑆 ∩ 𝑈) = { 0 })) → ((𝑇 ∩ 𝑆) = { 0 } ∧ (𝑇 ∩ 𝑈) = { 0 }))
27	26	simpld 475	. . . 4 ⊢ ((𝜑 ∧ ((𝑇 ∩ (𝑆 ⊕ 𝑈)) = { 0 } ∧ (𝑆 ∩ 𝑈) = { 0 })) → (𝑇 ∩ 𝑆) = { 0 })
28	25, 27	syl5eq 2668	. . 3 ⊢ ((𝜑 ∧ ((𝑇 ∩ (𝑆 ⊕ 𝑈)) = { 0 } ∧ (𝑆 ∩ 𝑈) = { 0 })) → (𝑆 ∩ 𝑇) = { 0 })
29	24, 28	jca 554	. 2 ⊢ ((𝜑 ∧ ((𝑇 ∩ (𝑆 ⊕ 𝑈)) = { 0 } ∧ (𝑆 ∩ 𝑈) = { 0 })) → (((𝑆 ⊕ 𝑇) ∩ 𝑈) = { 0 } ∧ (𝑆 ∩ 𝑇) = { 0 }))
30	14, 29	impbida 877	1 ⊢ (𝜑 → ((((𝑆 ⊕ 𝑇) ∩ 𝑈) = { 0 } ∧ (𝑆 ∩ 𝑇) = { 0 }) ↔ ((𝑇 ∩ (𝑆 ⊕ 𝑈)) = { 0 } ∧ (𝑆 ∩ 𝑈) = { 0 })))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990   ∩ cin 3573  {csn 4177  ‘cfv 5888  (class class class)co 6650  0_gc0g 16100  SubGrpcsubg 17588  LSSumclsm 18049

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-grp 17425  df-minusg 17426  df-subg 17591  df-lsm 18051

This theorem is referenced by:  lsmdisj3a  18102