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Mirrors > Home > MPE Home > Th. List > lsmdisj3 | Structured version Visualization version GIF version |
Description: Association of the disjointness constraint in 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 }) |
lsmdisj2.i | ⊢ (𝜑 → (𝑆 ∩ 𝑇) = { 0 }) |
lsmdisj3.z | ⊢ 𝑍 = (Cntz‘𝐺) |
lsmdisj3.s | ⊢ (𝜑 → 𝑆 ⊆ (𝑍‘𝑇)) |
Ref | Expression |
---|---|
lsmdisj3 | ⊢ (𝜑 → (𝑆 ∩ (𝑇 ⊕ 𝑈)) = { 0 }) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lsmcntz.p | . 2 ⊢ ⊕ = (LSSum‘𝐺) | |
2 | lsmcntz.t | . 2 ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺)) | |
3 | lsmcntz.s | . 2 ⊢ (𝜑 → 𝑆 ∈ (SubGrp‘𝐺)) | |
4 | lsmcntz.u | . 2 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺)) | |
5 | lsmdisj.o | . 2 ⊢ 0 = (0g‘𝐺) | |
6 | lsmdisj3.s | . . . . 5 ⊢ (𝜑 → 𝑆 ⊆ (𝑍‘𝑇)) | |
7 | lsmdisj3.z | . . . . . 6 ⊢ 𝑍 = (Cntz‘𝐺) | |
8 | 1, 7 | lsmcom2 18510 | . . . . 5 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑆 ⊆ (𝑍‘𝑇)) → (𝑆 ⊕ 𝑇) = (𝑇 ⊕ 𝑆)) |
9 | 3, 2, 6, 8 | syl3anc 1364 | . . . 4 ⊢ (𝜑 → (𝑆 ⊕ 𝑇) = (𝑇 ⊕ 𝑆)) |
10 | 9 | ineq1d 4108 | . . 3 ⊢ (𝜑 → ((𝑆 ⊕ 𝑇) ∩ 𝑈) = ((𝑇 ⊕ 𝑆) ∩ 𝑈)) |
11 | lsmdisj.i | . . 3 ⊢ (𝜑 → ((𝑆 ⊕ 𝑇) ∩ 𝑈) = { 0 }) | |
12 | 10, 11 | eqtr3d 2833 | . 2 ⊢ (𝜑 → ((𝑇 ⊕ 𝑆) ∩ 𝑈) = { 0 }) |
13 | incom 4099 | . . 3 ⊢ (𝑇 ∩ 𝑆) = (𝑆 ∩ 𝑇) | |
14 | lsmdisj2.i | . . 3 ⊢ (𝜑 → (𝑆 ∩ 𝑇) = { 0 }) | |
15 | 13, 14 | syl5eq 2843 | . 2 ⊢ (𝜑 → (𝑇 ∩ 𝑆) = { 0 }) |
16 | 1, 2, 3, 4, 5, 12, 15 | lsmdisj2 18535 | 1 ⊢ (𝜑 → (𝑆 ∩ (𝑇 ⊕ 𝑈)) = { 0 }) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1522 ∈ wcel 2081 ∩ cin 3858 ⊆ wss 3859 {csn 4472 ‘cfv 6225 (class class class)co 7016 0gc0g 16542 SubGrpcsubg 18027 Cntzccntz 18186 LSSumclsm 18489 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5081 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 ax-cnex 10439 ax-resscn 10440 ax-1cn 10441 ax-icn 10442 ax-addcl 10443 ax-addrcl 10444 ax-mulcl 10445 ax-mulrcl 10446 ax-mulcom 10447 ax-addass 10448 ax-mulass 10449 ax-distr 10450 ax-i2m1 10451 ax-1ne0 10452 ax-1rid 10453 ax-rnegex 10454 ax-rrecex 10455 ax-cnre 10456 ax-pre-lttri 10457 ax-pre-lttrn 10458 ax-pre-ltadd 10459 ax-pre-mulgt0 10460 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-pss 3876 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-tp 4477 df-op 4479 df-uni 4746 df-iun 4827 df-br 4963 df-opab 5025 df-mpt 5042 df-tr 5064 df-id 5348 df-eprel 5353 df-po 5362 df-so 5363 df-fr 5402 df-we 5404 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-pred 6023 df-ord 6069 df-on 6070 df-lim 6071 df-suc 6072 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-riota 6977 df-ov 7019 df-oprab 7020 df-mpo 7021 df-om 7437 df-1st 7545 df-2nd 7546 df-wrecs 7798 df-recs 7860 df-rdg 7898 df-er 8139 df-en 8358 df-dom 8359 df-sdom 8360 df-pnf 10523 df-mnf 10524 df-xr 10525 df-ltxr 10526 df-le 10527 df-sub 10719 df-neg 10720 df-nn 11487 df-2 11548 df-ndx 16315 df-slot 16316 df-base 16318 df-sets 16319 df-ress 16320 df-plusg 16407 df-0g 16544 df-mgm 17681 df-sgrp 17723 df-mnd 17734 df-submnd 17775 df-grp 17864 df-minusg 17865 df-subg 18030 df-cntz 18188 df-lsm 18491 |
This theorem is referenced by: dmdprdsplit2lem 18884 |
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