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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 18780 | . . . . 5 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑆 ⊆ (𝑍‘𝑇)) → (𝑆 ⊕ 𝑇) = (𝑇 ⊕ 𝑆)) |
9 | 3, 2, 6, 8 | syl3anc 1367 | . . . 4 ⊢ (𝜑 → (𝑆 ⊕ 𝑇) = (𝑇 ⊕ 𝑆)) |
10 | 9 | ineq1d 4188 | . . 3 ⊢ (𝜑 → ((𝑆 ⊕ 𝑇) ∩ 𝑈) = ((𝑇 ⊕ 𝑆) ∩ 𝑈)) |
11 | lsmdisj.i | . . 3 ⊢ (𝜑 → ((𝑆 ⊕ 𝑇) ∩ 𝑈) = { 0 }) | |
12 | 10, 11 | eqtr3d 2858 | . 2 ⊢ (𝜑 → ((𝑇 ⊕ 𝑆) ∩ 𝑈) = { 0 }) |
13 | incom 4178 | . . 3 ⊢ (𝑇 ∩ 𝑆) = (𝑆 ∩ 𝑇) | |
14 | lsmdisj2.i | . . 3 ⊢ (𝜑 → (𝑆 ∩ 𝑇) = { 0 }) | |
15 | 13, 14 | syl5eq 2868 | . 2 ⊢ (𝜑 → (𝑇 ∩ 𝑆) = { 0 }) |
16 | 1, 2, 3, 4, 5, 12, 15 | lsmdisj2 18808 | 1 ⊢ (𝜑 → (𝑆 ∩ (𝑇 ⊕ 𝑈)) = { 0 }) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ∩ cin 3935 ⊆ wss 3936 {csn 4567 ‘cfv 6355 (class class class)co 7156 0gc0g 16713 SubGrpcsubg 18273 Cntzccntz 18445 LSSumclsm 18759 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-0g 16715 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-submnd 17957 df-grp 18106 df-minusg 18107 df-subg 18276 df-cntz 18447 df-lsm 18761 |
This theorem is referenced by: dmdprdsplit2lem 19167 |
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