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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 2818 | . . . . 5 ⊢ (oppg‘𝐺) = (oppg‘𝐺) | |
2 | lsmcntz.p | . . . . 5 ⊢ ⊕ = (LSSum‘𝐺) | |
3 | 1, 2 | oppglsm 18696 | . . . 4 ⊢ (𝑈(LSSum‘(oppg‘𝐺))𝑆) = (𝑆 ⊕ 𝑈) |
4 | 3 | ineq2i 4183 | . . 3 ⊢ (𝑇 ∩ (𝑈(LSSum‘(oppg‘𝐺))𝑆)) = (𝑇 ∩ (𝑆 ⊕ 𝑈)) |
5 | incom 4175 | . . 3 ⊢ (𝑇 ∩ (𝑆 ⊕ 𝑈)) = ((𝑆 ⊕ 𝑈) ∩ 𝑇) | |
6 | 4, 5 | eqtri 2841 | . 2 ⊢ (𝑇 ∩ (𝑈(LSSum‘(oppg‘𝐺))𝑆)) = ((𝑆 ⊕ 𝑈) ∩ 𝑇) |
7 | eqid 2818 | . . 3 ⊢ (LSSum‘(oppg‘𝐺)) = (LSSum‘(oppg‘𝐺)) | |
8 | lsmcntz.u | . . . 4 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺)) | |
9 | 1 | oppgsubg 18429 | . . . 4 ⊢ (SubGrp‘𝐺) = (SubGrp‘(oppg‘𝐺)) |
10 | 8, 9 | eleqtrdi 2920 | . . 3 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘(oppg‘𝐺))) |
11 | lsmcntz.t | . . . 4 ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺)) | |
12 | 11, 9 | eleqtrdi 2920 | . . 3 ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘(oppg‘𝐺))) |
13 | lsmcntz.s | . . . 4 ⊢ (𝜑 → 𝑆 ∈ (SubGrp‘𝐺)) | |
14 | 13, 9 | eleqtrdi 2920 | . . 3 ⊢ (𝜑 → 𝑆 ∈ (SubGrp‘(oppg‘𝐺))) |
15 | lsmdisj.o | . . . 4 ⊢ 0 = (0g‘𝐺) | |
16 | 1, 15 | oppgid 18422 | . . 3 ⊢ 0 = (0g‘(oppg‘𝐺)) |
17 | 1, 2 | oppglsm 18696 | . . . . . 6 ⊢ (𝑈(LSSum‘(oppg‘𝐺))𝑇) = (𝑇 ⊕ 𝑈) |
18 | 17 | ineq1i 4182 | . . . . 5 ⊢ ((𝑈(LSSum‘(oppg‘𝐺))𝑇) ∩ 𝑆) = ((𝑇 ⊕ 𝑈) ∩ 𝑆) |
19 | incom 4175 | . . . . 5 ⊢ ((𝑇 ⊕ 𝑈) ∩ 𝑆) = (𝑆 ∩ (𝑇 ⊕ 𝑈)) | |
20 | 18, 19 | eqtri 2841 | . . . 4 ⊢ ((𝑈(LSSum‘(oppg‘𝐺))𝑇) ∩ 𝑆) = (𝑆 ∩ (𝑇 ⊕ 𝑈)) |
21 | lsmdisjr.i | . . . 4 ⊢ (𝜑 → (𝑆 ∩ (𝑇 ⊕ 𝑈)) = { 0 }) | |
22 | 20, 21 | syl5eq 2865 | . . 3 ⊢ (𝜑 → ((𝑈(LSSum‘(oppg‘𝐺))𝑇) ∩ 𝑆) = { 0 }) |
23 | incom 4175 | . . . 4 ⊢ (𝑇 ∩ 𝑈) = (𝑈 ∩ 𝑇) | |
24 | lsmdisj2r.i | . . . 4 ⊢ (𝜑 → (𝑇 ∩ 𝑈) = { 0 }) | |
25 | 23, 24 | syl5eqr 2867 | . . 3 ⊢ (𝜑 → (𝑈 ∩ 𝑇) = { 0 }) |
26 | 7, 10, 12, 14, 16, 22, 25 | lsmdisj2 18737 | . 2 ⊢ (𝜑 → (𝑇 ∩ (𝑈(LSSum‘(oppg‘𝐺))𝑆)) = { 0 }) |
27 | 6, 26 | syl5eqr 2867 | 1 ⊢ (𝜑 → ((𝑆 ⊕ 𝑈) ∩ 𝑇) = { 0 }) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 ∩ cin 3932 {csn 4557 ‘cfv 6348 (class class class)co 7145 0gc0g 16701 SubGrpcsubg 18211 oppgcoppg 18411 LSSumclsm 18688 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-tpos 7881 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-0g 16703 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-submnd 17945 df-grp 18044 df-minusg 18045 df-subg 18214 df-oppg 18412 df-lsm 18690 |
This theorem is referenced by: lsmdisj3r 18741 lsmdisj2b 18743 |
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