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Mirrors > Home > MPE Home > Th. List > lsmdisj2r | 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 }) |
Ref | Expression |
---|---|
lsmdisj2r | ⊢ (𝜑 → ((𝑆 ⊕ 𝑈) ∩ 𝑇) = { 0 }) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2798 | . . . . 5 ⊢ (oppg‘𝐺) = (oppg‘𝐺) | |
2 | lsmcntz.p | . . . . 5 ⊢ ⊕ = (LSSum‘𝐺) | |
3 | 1, 2 | oppglsm 18759 | . . . 4 ⊢ (𝑈(LSSum‘(oppg‘𝐺))𝑆) = (𝑆 ⊕ 𝑈) |
4 | 3 | ineq2i 4136 | . . 3 ⊢ (𝑇 ∩ (𝑈(LSSum‘(oppg‘𝐺))𝑆)) = (𝑇 ∩ (𝑆 ⊕ 𝑈)) |
5 | incom 4128 | . . 3 ⊢ (𝑇 ∩ (𝑆 ⊕ 𝑈)) = ((𝑆 ⊕ 𝑈) ∩ 𝑇) | |
6 | 4, 5 | eqtri 2821 | . 2 ⊢ (𝑇 ∩ (𝑈(LSSum‘(oppg‘𝐺))𝑆)) = ((𝑆 ⊕ 𝑈) ∩ 𝑇) |
7 | eqid 2798 | . . 3 ⊢ (LSSum‘(oppg‘𝐺)) = (LSSum‘(oppg‘𝐺)) | |
8 | lsmcntz.u | . . . 4 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺)) | |
9 | 1 | oppgsubg 18483 | . . . 4 ⊢ (SubGrp‘𝐺) = (SubGrp‘(oppg‘𝐺)) |
10 | 8, 9 | eleqtrdi 2900 | . . 3 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘(oppg‘𝐺))) |
11 | lsmcntz.t | . . . 4 ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺)) | |
12 | 11, 9 | eleqtrdi 2900 | . . 3 ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘(oppg‘𝐺))) |
13 | lsmcntz.s | . . . 4 ⊢ (𝜑 → 𝑆 ∈ (SubGrp‘𝐺)) | |
14 | 13, 9 | eleqtrdi 2900 | . . 3 ⊢ (𝜑 → 𝑆 ∈ (SubGrp‘(oppg‘𝐺))) |
15 | lsmdisj.o | . . . 4 ⊢ 0 = (0g‘𝐺) | |
16 | 1, 15 | oppgid 18476 | . . 3 ⊢ 0 = (0g‘(oppg‘𝐺)) |
17 | 1, 2 | oppglsm 18759 | . . . . . 6 ⊢ (𝑈(LSSum‘(oppg‘𝐺))𝑇) = (𝑇 ⊕ 𝑈) |
18 | 17 | ineq1i 4135 | . . . . 5 ⊢ ((𝑈(LSSum‘(oppg‘𝐺))𝑇) ∩ 𝑆) = ((𝑇 ⊕ 𝑈) ∩ 𝑆) |
19 | incom 4128 | . . . . 5 ⊢ ((𝑇 ⊕ 𝑈) ∩ 𝑆) = (𝑆 ∩ (𝑇 ⊕ 𝑈)) | |
20 | 18, 19 | eqtri 2821 | . . . 4 ⊢ ((𝑈(LSSum‘(oppg‘𝐺))𝑇) ∩ 𝑆) = (𝑆 ∩ (𝑇 ⊕ 𝑈)) |
21 | lsmdisjr.i | . . . 4 ⊢ (𝜑 → (𝑆 ∩ (𝑇 ⊕ 𝑈)) = { 0 }) | |
22 | 20, 21 | syl5eq 2845 | . . 3 ⊢ (𝜑 → ((𝑈(LSSum‘(oppg‘𝐺))𝑇) ∩ 𝑆) = { 0 }) |
23 | incom 4128 | . . . 4 ⊢ (𝑇 ∩ 𝑈) = (𝑈 ∩ 𝑇) | |
24 | lsmdisj2r.i | . . . 4 ⊢ (𝜑 → (𝑇 ∩ 𝑈) = { 0 }) | |
25 | 23, 24 | syl5eqr 2847 | . . 3 ⊢ (𝜑 → (𝑈 ∩ 𝑇) = { 0 }) |
26 | 7, 10, 12, 14, 16, 22, 25 | lsmdisj2 18800 | . 2 ⊢ (𝜑 → (𝑇 ∩ (𝑈(LSSum‘(oppg‘𝐺))𝑆)) = { 0 }) |
27 | 6, 26 | syl5eqr 2847 | 1 ⊢ (𝜑 → ((𝑆 ⊕ 𝑈) ∩ 𝑇) = { 0 }) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 ∩ cin 3880 {csn 4525 ‘cfv 6324 (class class class)co 7135 0gc0g 16705 SubGrpcsubg 18265 oppgcoppg 18465 LSSumclsm 18751 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-tpos 7875 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-0g 16707 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-grp 18098 df-minusg 18099 df-subg 18268 df-oppg 18466 df-lsm 18753 |
This theorem is referenced by: lsmdisj3r 18804 lsmdisj2b 18806 |
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