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Theorem lsmdisj 19587

Description: Disjointness from 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 = (0_g‘𝐺)
lsmdisj.i	⊢ (𝜑 → ((𝑆 ⊕ 𝑇) ∩ 𝑈) = { 0 })

**Assertion**
Ref	Expression
lsmdisj	⊢ (𝜑 → ((𝑆 ∩ 𝑈) = { 0 } ∧ (𝑇 ∩ 𝑈) = { 0 }))

**Proof of Theorem lsmdisj**
Step	Hyp	Ref	Expression
1		lsmcntz.s	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ (SubGrp‘𝐺))
2		lsmcntz.t	. . . . . 6 ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺))
3		lsmcntz.p	. . . . . . 7 ⊢ ⊕ = (LSSum‘𝐺)
4	3	lsmub1 19563	. . . . . 6 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ (SubGrp‘𝐺)) → 𝑆 ⊆ (𝑆 ⊕ 𝑇))
5	1, 2, 4	syl2anc 584	. . . . 5 ⊢ (𝜑 → 𝑆 ⊆ (𝑆 ⊕ 𝑇))
6	5	ssrind 4203	. . . 4 ⊢ (𝜑 → (𝑆 ∩ 𝑈) ⊆ ((𝑆 ⊕ 𝑇) ∩ 𝑈))
7		lsmdisj.i	. . . 4 ⊢ (𝜑 → ((𝑆 ⊕ 𝑇) ∩ 𝑈) = { 0 })
8	6, 7	sseqtrd 3980	. . 3 ⊢ (𝜑 → (𝑆 ∩ 𝑈) ⊆ { 0 })
9		lsmdisj.o	. . . . . . 7 ⊢ 0 = (0_g‘𝐺)
10	9	subg0cl 19042	. . . . . 6 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 0 ∈ 𝑆)
11	1, 10	syl 17	. . . . 5 ⊢ (𝜑 → 0 ∈ 𝑆)
12		lsmcntz.u	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺))
13	9	subg0cl 19042	. . . . . 6 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → 0 ∈ 𝑈)
14	12, 13	syl 17	. . . . 5 ⊢ (𝜑 → 0 ∈ 𝑈)
15	11, 14	elind 4159	. . . 4 ⊢ (𝜑 → 0 ∈ (𝑆 ∩ 𝑈))
16	15	snssd 4769	. . 3 ⊢ (𝜑 → { 0 } ⊆ (𝑆 ∩ 𝑈))
17	8, 16	eqssd 3961	. 2 ⊢ (𝜑 → (𝑆 ∩ 𝑈) = { 0 })
18	3	lsmub2 19564	. . . . . 6 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ (SubGrp‘𝐺)) → 𝑇 ⊆ (𝑆 ⊕ 𝑇))
19	1, 2, 18	syl2anc 584	. . . . 5 ⊢ (𝜑 → 𝑇 ⊆ (𝑆 ⊕ 𝑇))
20	19	ssrind 4203	. . . 4 ⊢ (𝜑 → (𝑇 ∩ 𝑈) ⊆ ((𝑆 ⊕ 𝑇) ∩ 𝑈))
21	20, 7	sseqtrd 3980	. . 3 ⊢ (𝜑 → (𝑇 ∩ 𝑈) ⊆ { 0 })
22	9	subg0cl 19042	. . . . . 6 ⊢ (𝑇 ∈ (SubGrp‘𝐺) → 0 ∈ 𝑇)
23	2, 22	syl 17	. . . . 5 ⊢ (𝜑 → 0 ∈ 𝑇)
24	23, 14	elind 4159	. . . 4 ⊢ (𝜑 → 0 ∈ (𝑇 ∩ 𝑈))
25	24	snssd 4769	. . 3 ⊢ (𝜑 → { 0 } ⊆ (𝑇 ∩ 𝑈))
26	21, 25	eqssd 3961	. 2 ⊢ (𝜑 → (𝑇 ∩ 𝑈) = { 0 })
27	17, 26	jca 511	1 ⊢ (𝜑 → ((𝑆 ∩ 𝑈) = { 0 } ∧ (𝑇 ∩ 𝑈) = { 0 }))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ∩ cin 3910 ⊆ wss 3911 {csn 4585 ‘cfv 6499 (class class class)co 7369 0_gc0g 17378 SubGrpcsubg 19028 LSSumclsm 19540

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-rep 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 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-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4868 df-iun 4953 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-om 7823 df-1st 7947 df-2nd 7948 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-er 8648 df-en 8896 df-dom 8897 df-sdom 8898 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-nn 12163 df-2 12225 df-sets 17110 df-slot 17128 df-ndx 17140 df-base 17156 df-ress 17177 df-plusg 17209 df-0g 17380 df-mgm 18543 df-sgrp 18622 df-mnd 18638 df-submnd 18687 df-grp 18844 df-minusg 18845 df-subg 19031 df-lsm 19542

This theorem is referenced by: lsmdisjr 19590 lsmdisj2a 19593 lsmdisj2b 19594

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