Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > lsmdisj3r | Structured version Visualization version GIF version |
Description: Association of the disjointness constraint in a subgroup sum. (Contributed by Mario Carneiro, 22-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) |
lsmdisjr.i | ⊢ (𝜑 → (𝑆 ∩ (𝑇 ⊕ 𝑈)) = { 0 }) |
lsmdisj2r.i | ⊢ (𝜑 → (𝑇 ∩ 𝑈) = { 0 }) |
lsmdisj3r.z | ⊢ 𝑍 = (Cntz‘𝐺) |
lsmdisj3r.s | ⊢ (𝜑 → 𝑇 ⊆ (𝑍‘𝑈)) |
Ref | Expression |
---|---|
lsmdisj3r | ⊢ (𝜑 → ((𝑆 ⊕ 𝑇) ∩ 𝑈) = { 0 }) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lsmcntz.p | . 2 ⊢ ⊕ = (LSSum‘𝐺) | |
2 | lsmcntz.s | . 2 ⊢ (𝜑 → 𝑆 ∈ (SubGrp‘𝐺)) | |
3 | lsmcntz.u | . 2 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺)) | |
4 | lsmcntz.t | . 2 ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺)) | |
5 | lsmdisj.o | . 2 ⊢ 0 = (0g‘𝐺) | |
6 | lsmdisj3r.s | . . . . 5 ⊢ (𝜑 → 𝑇 ⊆ (𝑍‘𝑈)) | |
7 | lsmdisj3r.z | . . . . . 6 ⊢ 𝑍 = (Cntz‘𝐺) | |
8 | 1, 7 | lsmcom2 18782 | . . . . 5 ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) → (𝑇 ⊕ 𝑈) = (𝑈 ⊕ 𝑇)) |
9 | 4, 3, 6, 8 | syl3anc 1367 | . . . 4 ⊢ (𝜑 → (𝑇 ⊕ 𝑈) = (𝑈 ⊕ 𝑇)) |
10 | 9 | ineq2d 4191 | . . 3 ⊢ (𝜑 → (𝑆 ∩ (𝑇 ⊕ 𝑈)) = (𝑆 ∩ (𝑈 ⊕ 𝑇))) |
11 | lsmdisjr.i | . . 3 ⊢ (𝜑 → (𝑆 ∩ (𝑇 ⊕ 𝑈)) = { 0 }) | |
12 | 10, 11 | eqtr3d 2860 | . 2 ⊢ (𝜑 → (𝑆 ∩ (𝑈 ⊕ 𝑇)) = { 0 }) |
13 | incom 4180 | . . 3 ⊢ (𝑈 ∩ 𝑇) = (𝑇 ∩ 𝑈) | |
14 | lsmdisj2r.i | . . 3 ⊢ (𝜑 → (𝑇 ∩ 𝑈) = { 0 }) | |
15 | 13, 14 | syl5eq 2870 | . 2 ⊢ (𝜑 → (𝑈 ∩ 𝑇) = { 0 }) |
16 | 1, 2, 3, 4, 5, 12, 15 | lsmdisj2r 18813 | 1 ⊢ (𝜑 → ((𝑆 ⊕ 𝑇) ∩ 𝑈) = { 0 }) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ∩ cin 3937 ⊆ wss 3938 {csn 4569 ‘cfv 6357 (class class class)co 7158 0gc0g 16715 SubGrpcsubg 18275 Cntzccntz 18447 LSSumclsm 18761 |
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 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-tpos 7894 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-0g 16717 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-submnd 17959 df-grp 18108 df-minusg 18109 df-subg 18278 df-cntz 18449 df-oppg 18476 df-lsm 18763 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |