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Theorem lsmdisj 19723

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 19699	. . . . . 6 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ (SubGrp‘𝐺)) → 𝑆 ⊆ (𝑆 ⊕ 𝑇))
5	1, 2, 4	syl2anc 584	. . . . 5 ⊢ (𝜑 → 𝑆 ⊆ (𝑆 ⊕ 𝑇))
6	5	ssrind 4255	. . . 4 ⊢ (𝜑 → (𝑆 ∩ 𝑈) ⊆ ((𝑆 ⊕ 𝑇) ∩ 𝑈))
7		lsmdisj.i	. . . 4 ⊢ (𝜑 → ((𝑆 ⊕ 𝑇) ∩ 𝑈) = { 0 })
8	6, 7	sseqtrd 4039	. . 3 ⊢ (𝜑 → (𝑆 ∩ 𝑈) ⊆ { 0 })
9		lsmdisj.o	. . . . . . 7 ⊢ 0 = (0_g‘𝐺)
10	9	subg0cl 19174	. . . . . 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 19174	. . . . . 6 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → 0 ∈ 𝑈)
14	12, 13	syl 17	. . . . 5 ⊢ (𝜑 → 0 ∈ 𝑈)
15	11, 14	elind 4213	. . . 4 ⊢ (𝜑 → 0 ∈ (𝑆 ∩ 𝑈))
16	15	snssd 4817	. . 3 ⊢ (𝜑 → { 0 } ⊆ (𝑆 ∩ 𝑈))
17	8, 16	eqssd 4016	. 2 ⊢ (𝜑 → (𝑆 ∩ 𝑈) = { 0 })
18	3	lsmub2 19700	. . . . . 6 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ (SubGrp‘𝐺)) → 𝑇 ⊆ (𝑆 ⊕ 𝑇))
19	1, 2, 18	syl2anc 584	. . . . 5 ⊢ (𝜑 → 𝑇 ⊆ (𝑆 ⊕ 𝑇))
20	19	ssrind 4255	. . . 4 ⊢ (𝜑 → (𝑇 ∩ 𝑈) ⊆ ((𝑆 ⊕ 𝑇) ∩ 𝑈))
21	20, 7	sseqtrd 4039	. . 3 ⊢ (𝜑 → (𝑇 ∩ 𝑈) ⊆ { 0 })
22	9	subg0cl 19174	. . . . . 6 ⊢ (𝑇 ∈ (SubGrp‘𝐺) → 0 ∈ 𝑇)
23	2, 22	syl 17	. . . . 5 ⊢ (𝜑 → 0 ∈ 𝑇)
24	23, 14	elind 4213	. . . 4 ⊢ (𝜑 → 0 ∈ (𝑇 ∩ 𝑈))
25	24	snssd 4817	. . 3 ⊢ (𝜑 → { 0 } ⊆ (𝑇 ∩ 𝑈))
26	21, 25	eqssd 4016	. 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 1539 ∈ wcel 2108 ∩ cin 3965 ⊆ wss 3966 {csn 4634 ‘cfv 6569 (class class class)co 7438 0_gc0g 17495 SubGrpcsubg 19160 LSSumclsm 19676

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5288 ax-sep 5305 ax-nul 5315 ax-pow 5374 ax-pr 5441 ax-un 7761 ax-cnex 11218 ax-resscn 11219 ax-1cn 11220 ax-icn 11221 ax-addcl 11222 ax-addrcl 11223 ax-mulcl 11224 ax-mulrcl 11225 ax-mulcom 11226 ax-addass 11227 ax-mulass 11228 ax-distr 11229 ax-i2m1 11230 ax-1ne0 11231 ax-1rid 11232 ax-rnegex 11233 ax-rrecex 11234 ax-cnre 11235 ax-pre-lttri 11236 ax-pre-lttrn 11237 ax-pre-ltadd 11238 ax-pre-mulgt0 11239

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3483 df-sbc 3795 df-csb 3912 df-dif 3969 df-un 3971 df-in 3973 df-ss 3983 df-pss 3986 df-nul 4343 df-if 4535 df-pw 4610 df-sn 4635 df-pr 4637 df-op 4641 df-uni 4916 df-iun 5001 df-br 5152 df-opab 5214 df-mpt 5235 df-tr 5269 df-id 5587 df-eprel 5593 df-po 5601 df-so 5602 df-fr 5645 df-we 5647 df-xp 5699 df-rel 5700 df-cnv 5701 df-co 5702 df-dm 5703 df-rn 5704 df-res 5705 df-ima 5706 df-pred 6329 df-ord 6395 df-on 6396 df-lim 6397 df-suc 6398 df-iota 6522 df-fun 6571 df-fn 6572 df-f 6573 df-f1 6574 df-fo 6575 df-f1o 6576 df-fv 6577 df-riota 7395 df-ov 7441 df-oprab 7442 df-mpo 7443 df-om 7895 df-1st 8022 df-2nd 8023 df-frecs 8314 df-wrecs 8345 df-recs 8419 df-rdg 8458 df-er 8753 df-en 8994 df-dom 8995 df-sdom 8996 df-pnf 11304 df-mnf 11305 df-xr 11306 df-ltxr 11307 df-le 11308 df-sub 11501 df-neg 11502 df-nn 12274 df-2 12336 df-sets 17207 df-slot 17225 df-ndx 17237 df-base 17255 df-ress 17284 df-plusg 17320 df-0g 17497 df-mgm 18675 df-sgrp 18754 df-mnd 18770 df-submnd 18819 df-grp 18976 df-minusg 18977 df-subg 19163 df-lsm 19678

This theorem is referenced by: lsmdisjr 19726 lsmdisj2a 19729 lsmdisj2b 19730

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