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Mirrors > Home > MPE Home > Th. List > lsmdisj | Structured version Visualization version GIF version |
Description: Disjointness from a subgroup sum. (Contributed by Mario Carneiro, 21-Apr-2016.) |
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) |
lsmdisj.i | ⊢ (𝜑 → ((𝑆 ⊕ 𝑇) ∩ 𝑈) = { 0 }) |
Ref | Expression |
---|---|
lsmdisj | ⊢ (𝜑 → ((𝑆 ∩ 𝑈) = { 0 } ∧ (𝑇 ∩ 𝑈) = { 0 })) |
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 18785 | . . . . . 6 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ (SubGrp‘𝐺)) → 𝑆 ⊆ (𝑆 ⊕ 𝑇)) |
5 | 1, 2, 4 | syl2anc 586 | . . . . 5 ⊢ (𝜑 → 𝑆 ⊆ (𝑆 ⊕ 𝑇)) |
6 | 5 | ssrind 4215 | . . . 4 ⊢ (𝜑 → (𝑆 ∩ 𝑈) ⊆ ((𝑆 ⊕ 𝑇) ∩ 𝑈)) |
7 | lsmdisj.i | . . . 4 ⊢ (𝜑 → ((𝑆 ⊕ 𝑇) ∩ 𝑈) = { 0 }) | |
8 | 6, 7 | sseqtrd 4010 | . . 3 ⊢ (𝜑 → (𝑆 ∩ 𝑈) ⊆ { 0 }) |
9 | lsmdisj.o | . . . . . . 7 ⊢ 0 = (0g‘𝐺) | |
10 | 9 | subg0cl 18290 | . . . . . 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 18290 | . . . . . 6 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → 0 ∈ 𝑈) |
14 | 12, 13 | syl 17 | . . . . 5 ⊢ (𝜑 → 0 ∈ 𝑈) |
15 | 11, 14 | elind 4174 | . . . 4 ⊢ (𝜑 → 0 ∈ (𝑆 ∩ 𝑈)) |
16 | 15 | snssd 4745 | . . 3 ⊢ (𝜑 → { 0 } ⊆ (𝑆 ∩ 𝑈)) |
17 | 8, 16 | eqssd 3987 | . 2 ⊢ (𝜑 → (𝑆 ∩ 𝑈) = { 0 }) |
18 | 3 | lsmub2 18786 | . . . . . 6 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ (SubGrp‘𝐺)) → 𝑇 ⊆ (𝑆 ⊕ 𝑇)) |
19 | 1, 2, 18 | syl2anc 586 | . . . . 5 ⊢ (𝜑 → 𝑇 ⊆ (𝑆 ⊕ 𝑇)) |
20 | 19 | ssrind 4215 | . . . 4 ⊢ (𝜑 → (𝑇 ∩ 𝑈) ⊆ ((𝑆 ⊕ 𝑇) ∩ 𝑈)) |
21 | 20, 7 | sseqtrd 4010 | . . 3 ⊢ (𝜑 → (𝑇 ∩ 𝑈) ⊆ { 0 }) |
22 | 9 | subg0cl 18290 | . . . . . 6 ⊢ (𝑇 ∈ (SubGrp‘𝐺) → 0 ∈ 𝑇) |
23 | 2, 22 | syl 17 | . . . . 5 ⊢ (𝜑 → 0 ∈ 𝑇) |
24 | 23, 14 | elind 4174 | . . . 4 ⊢ (𝜑 → 0 ∈ (𝑇 ∩ 𝑈)) |
25 | 24 | snssd 4745 | . . 3 ⊢ (𝜑 → { 0 } ⊆ (𝑇 ∩ 𝑈)) |
26 | 21, 25 | eqssd 3987 | . 2 ⊢ (𝜑 → (𝑇 ∩ 𝑈) = { 0 }) |
27 | 17, 26 | jca 514 | 1 ⊢ (𝜑 → ((𝑆 ∩ 𝑈) = { 0 } ∧ (𝑇 ∩ 𝑈) = { 0 })) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ∩ cin 3938 ⊆ wss 3939 {csn 4570 ‘cfv 6358 (class class class)co 7159 0gc0g 16716 SubGrpcsubg 18276 LSSumclsm 18762 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 df-2 11703 df-ndx 16489 df-slot 16490 df-base 16492 df-sets 16493 df-ress 16494 df-plusg 16581 df-0g 16718 df-mgm 17855 df-sgrp 17904 df-mnd 17915 df-submnd 17960 df-grp 18109 df-minusg 18110 df-subg 18279 df-lsm 18764 |
This theorem is referenced by: lsmdisjr 18813 lsmdisj2a 18816 lsmdisj2b 18817 |
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