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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 19000 | . . . . . 6 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ (SubGrp‘𝐺)) → 𝑆 ⊆ (𝑆 ⊕ 𝑇)) |
5 | 1, 2, 4 | syl2anc 587 | . . . . 5 ⊢ (𝜑 → 𝑆 ⊆ (𝑆 ⊕ 𝑇)) |
6 | 5 | ssrind 4136 | . . . 4 ⊢ (𝜑 → (𝑆 ∩ 𝑈) ⊆ ((𝑆 ⊕ 𝑇) ∩ 𝑈)) |
7 | lsmdisj.i | . . . 4 ⊢ (𝜑 → ((𝑆 ⊕ 𝑇) ∩ 𝑈) = { 0 }) | |
8 | 6, 7 | sseqtrd 3927 | . . 3 ⊢ (𝜑 → (𝑆 ∩ 𝑈) ⊆ { 0 }) |
9 | lsmdisj.o | . . . . . . 7 ⊢ 0 = (0g‘𝐺) | |
10 | 9 | subg0cl 18505 | . . . . . 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 18505 | . . . . . 6 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → 0 ∈ 𝑈) |
14 | 12, 13 | syl 17 | . . . . 5 ⊢ (𝜑 → 0 ∈ 𝑈) |
15 | 11, 14 | elind 4094 | . . . 4 ⊢ (𝜑 → 0 ∈ (𝑆 ∩ 𝑈)) |
16 | 15 | snssd 4708 | . . 3 ⊢ (𝜑 → { 0 } ⊆ (𝑆 ∩ 𝑈)) |
17 | 8, 16 | eqssd 3904 | . 2 ⊢ (𝜑 → (𝑆 ∩ 𝑈) = { 0 }) |
18 | 3 | lsmub2 19001 | . . . . . 6 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ (SubGrp‘𝐺)) → 𝑇 ⊆ (𝑆 ⊕ 𝑇)) |
19 | 1, 2, 18 | syl2anc 587 | . . . . 5 ⊢ (𝜑 → 𝑇 ⊆ (𝑆 ⊕ 𝑇)) |
20 | 19 | ssrind 4136 | . . . 4 ⊢ (𝜑 → (𝑇 ∩ 𝑈) ⊆ ((𝑆 ⊕ 𝑇) ∩ 𝑈)) |
21 | 20, 7 | sseqtrd 3927 | . . 3 ⊢ (𝜑 → (𝑇 ∩ 𝑈) ⊆ { 0 }) |
22 | 9 | subg0cl 18505 | . . . . . 6 ⊢ (𝑇 ∈ (SubGrp‘𝐺) → 0 ∈ 𝑇) |
23 | 2, 22 | syl 17 | . . . . 5 ⊢ (𝜑 → 0 ∈ 𝑇) |
24 | 23, 14 | elind 4094 | . . . 4 ⊢ (𝜑 → 0 ∈ (𝑇 ∩ 𝑈)) |
25 | 24 | snssd 4708 | . . 3 ⊢ (𝜑 → { 0 } ⊆ (𝑇 ∩ 𝑈)) |
26 | 21, 25 | eqssd 3904 | . 2 ⊢ (𝜑 → (𝑇 ∩ 𝑈) = { 0 }) |
27 | 17, 26 | jca 515 | 1 ⊢ (𝜑 → ((𝑆 ∩ 𝑈) = { 0 } ∧ (𝑇 ∩ 𝑈) = { 0 })) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ∩ cin 3852 ⊆ wss 3853 {csn 4527 ‘cfv 6358 (class class class)co 7191 0gc0g 16898 SubGrpcsubg 18491 LSSumclsm 18977 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-nn 11796 df-2 11858 df-ndx 16669 df-slot 16670 df-base 16672 df-sets 16673 df-ress 16674 df-plusg 16762 df-0g 16900 df-mgm 18068 df-sgrp 18117 df-mnd 18128 df-submnd 18173 df-grp 18322 df-minusg 18323 df-subg 18494 df-lsm 18979 |
This theorem is referenced by: lsmdisjr 19028 lsmdisj2a 19031 lsmdisj2b 19032 |
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