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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 2738 | . . . . 5 ⊢ (oppg‘𝐺) = (oppg‘𝐺) | |
2 | lsmcntz.p | . . . . 5 ⊢ ⊕ = (LSSum‘𝐺) | |
3 | 1, 2 | oppglsm 18878 | . . . 4 ⊢ (𝑈(LSSum‘(oppg‘𝐺))𝑆) = (𝑆 ⊕ 𝑈) |
4 | 3 | ineq2i 4098 | . . 3 ⊢ (𝑇 ∩ (𝑈(LSSum‘(oppg‘𝐺))𝑆)) = (𝑇 ∩ (𝑆 ⊕ 𝑈)) |
5 | incom 4089 | . . 3 ⊢ (𝑇 ∩ (𝑆 ⊕ 𝑈)) = ((𝑆 ⊕ 𝑈) ∩ 𝑇) | |
6 | 4, 5 | eqtri 2761 | . 2 ⊢ (𝑇 ∩ (𝑈(LSSum‘(oppg‘𝐺))𝑆)) = ((𝑆 ⊕ 𝑈) ∩ 𝑇) |
7 | eqid 2738 | . . 3 ⊢ (LSSum‘(oppg‘𝐺)) = (LSSum‘(oppg‘𝐺)) | |
8 | lsmcntz.u | . . . 4 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺)) | |
9 | 1 | oppgsubg 18602 | . . . 4 ⊢ (SubGrp‘𝐺) = (SubGrp‘(oppg‘𝐺)) |
10 | 8, 9 | eleqtrdi 2843 | . . 3 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘(oppg‘𝐺))) |
11 | lsmcntz.t | . . . 4 ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺)) | |
12 | 11, 9 | eleqtrdi 2843 | . . 3 ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘(oppg‘𝐺))) |
13 | lsmcntz.s | . . . 4 ⊢ (𝜑 → 𝑆 ∈ (SubGrp‘𝐺)) | |
14 | 13, 9 | eleqtrdi 2843 | . . 3 ⊢ (𝜑 → 𝑆 ∈ (SubGrp‘(oppg‘𝐺))) |
15 | lsmdisj.o | . . . 4 ⊢ 0 = (0g‘𝐺) | |
16 | 1, 15 | oppgid 18595 | . . 3 ⊢ 0 = (0g‘(oppg‘𝐺)) |
17 | 1, 2 | oppglsm 18878 | . . . . . 6 ⊢ (𝑈(LSSum‘(oppg‘𝐺))𝑇) = (𝑇 ⊕ 𝑈) |
18 | 17 | ineq1i 4097 | . . . . 5 ⊢ ((𝑈(LSSum‘(oppg‘𝐺))𝑇) ∩ 𝑆) = ((𝑇 ⊕ 𝑈) ∩ 𝑆) |
19 | incom 4089 | . . . . 5 ⊢ ((𝑇 ⊕ 𝑈) ∩ 𝑆) = (𝑆 ∩ (𝑇 ⊕ 𝑈)) | |
20 | 18, 19 | eqtri 2761 | . . . 4 ⊢ ((𝑈(LSSum‘(oppg‘𝐺))𝑇) ∩ 𝑆) = (𝑆 ∩ (𝑇 ⊕ 𝑈)) |
21 | lsmdisjr.i | . . . 4 ⊢ (𝜑 → (𝑆 ∩ (𝑇 ⊕ 𝑈)) = { 0 }) | |
22 | 20, 21 | syl5eq 2785 | . . 3 ⊢ (𝜑 → ((𝑈(LSSum‘(oppg‘𝐺))𝑇) ∩ 𝑆) = { 0 }) |
23 | incom 4089 | . . . 4 ⊢ (𝑇 ∩ 𝑈) = (𝑈 ∩ 𝑇) | |
24 | lsmdisj2r.i | . . . 4 ⊢ (𝜑 → (𝑇 ∩ 𝑈) = { 0 }) | |
25 | 23, 24 | eqtr3id 2787 | . . 3 ⊢ (𝜑 → (𝑈 ∩ 𝑇) = { 0 }) |
26 | 7, 10, 12, 14, 16, 22, 25 | lsmdisj2 18919 | . 2 ⊢ (𝜑 → (𝑇 ∩ (𝑈(LSSum‘(oppg‘𝐺))𝑆)) = { 0 }) |
27 | 6, 26 | eqtr3id 2787 | 1 ⊢ (𝜑 → ((𝑆 ⊕ 𝑈) ∩ 𝑇) = { 0 }) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2113 ∩ cin 3840 {csn 4513 ‘cfv 6333 (class class class)co 7164 0gc0g 16809 SubGrpcsubg 18384 oppgcoppg 18584 LSSumclsm 18870 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-rep 5151 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 ax-cnex 10664 ax-resscn 10665 ax-1cn 10666 ax-icn 10667 ax-addcl 10668 ax-addrcl 10669 ax-mulcl 10670 ax-mulrcl 10671 ax-mulcom 10672 ax-addass 10673 ax-mulass 10674 ax-distr 10675 ax-i2m1 10676 ax-1ne0 10677 ax-1rid 10678 ax-rnegex 10679 ax-rrecex 10680 ax-cnre 10681 ax-pre-lttri 10682 ax-pre-lttrn 10683 ax-pre-ltadd 10684 ax-pre-mulgt0 10685 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-pss 3860 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-tp 4518 df-op 4520 df-uni 4794 df-iun 4880 df-br 5028 df-opab 5090 df-mpt 5108 df-tr 5134 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 6123 df-ord 6169 df-on 6170 df-lim 6171 df-suc 6172 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-riota 7121 df-ov 7167 df-oprab 7168 df-mpo 7169 df-om 7594 df-1st 7707 df-2nd 7708 df-tpos 7914 df-wrecs 7969 df-recs 8030 df-rdg 8068 df-er 8313 df-en 8549 df-dom 8550 df-sdom 8551 df-pnf 10748 df-mnf 10749 df-xr 10750 df-ltxr 10751 df-le 10752 df-sub 10943 df-neg 10944 df-nn 11710 df-2 11772 df-ndx 16582 df-slot 16583 df-base 16585 df-sets 16586 df-ress 16587 df-plusg 16674 df-0g 16811 df-mgm 17961 df-sgrp 18010 df-mnd 18021 df-submnd 18066 df-grp 18215 df-minusg 18216 df-subg 18387 df-oppg 18585 df-lsm 18872 |
This theorem is referenced by: lsmdisj3r 18923 lsmdisj2b 18925 |
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