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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 2726 | . . . . 5 ⊢ (oppg‘𝐺) = (oppg‘𝐺) | |
2 | lsmcntz.p | . . . . 5 ⊢ ⊕ = (LSSum‘𝐺) | |
3 | 1, 2 | oppglsm 19562 | . . . 4 ⊢ (𝑈(LSSum‘(oppg‘𝐺))𝑆) = (𝑆 ⊕ 𝑈) |
4 | 3 | ineq2i 4204 | . . 3 ⊢ (𝑇 ∩ (𝑈(LSSum‘(oppg‘𝐺))𝑆)) = (𝑇 ∩ (𝑆 ⊕ 𝑈)) |
5 | incom 4196 | . . 3 ⊢ (𝑇 ∩ (𝑆 ⊕ 𝑈)) = ((𝑆 ⊕ 𝑈) ∩ 𝑇) | |
6 | 4, 5 | eqtri 2754 | . 2 ⊢ (𝑇 ∩ (𝑈(LSSum‘(oppg‘𝐺))𝑆)) = ((𝑆 ⊕ 𝑈) ∩ 𝑇) |
7 | eqid 2726 | . . 3 ⊢ (LSSum‘(oppg‘𝐺)) = (LSSum‘(oppg‘𝐺)) | |
8 | lsmcntz.u | . . . 4 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺)) | |
9 | 1 | oppgsubg 19282 | . . . 4 ⊢ (SubGrp‘𝐺) = (SubGrp‘(oppg‘𝐺)) |
10 | 8, 9 | eleqtrdi 2837 | . . 3 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘(oppg‘𝐺))) |
11 | lsmcntz.t | . . . 4 ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺)) | |
12 | 11, 9 | eleqtrdi 2837 | . . 3 ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘(oppg‘𝐺))) |
13 | lsmcntz.s | . . . 4 ⊢ (𝜑 → 𝑆 ∈ (SubGrp‘𝐺)) | |
14 | 13, 9 | eleqtrdi 2837 | . . 3 ⊢ (𝜑 → 𝑆 ∈ (SubGrp‘(oppg‘𝐺))) |
15 | lsmdisj.o | . . . 4 ⊢ 0 = (0g‘𝐺) | |
16 | 1, 15 | oppgid 19275 | . . 3 ⊢ 0 = (0g‘(oppg‘𝐺)) |
17 | 1, 2 | oppglsm 19562 | . . . . . 6 ⊢ (𝑈(LSSum‘(oppg‘𝐺))𝑇) = (𝑇 ⊕ 𝑈) |
18 | 17 | ineq1i 4203 | . . . . 5 ⊢ ((𝑈(LSSum‘(oppg‘𝐺))𝑇) ∩ 𝑆) = ((𝑇 ⊕ 𝑈) ∩ 𝑆) |
19 | incom 4196 | . . . . 5 ⊢ ((𝑇 ⊕ 𝑈) ∩ 𝑆) = (𝑆 ∩ (𝑇 ⊕ 𝑈)) | |
20 | 18, 19 | eqtri 2754 | . . . 4 ⊢ ((𝑈(LSSum‘(oppg‘𝐺))𝑇) ∩ 𝑆) = (𝑆 ∩ (𝑇 ⊕ 𝑈)) |
21 | lsmdisjr.i | . . . 4 ⊢ (𝜑 → (𝑆 ∩ (𝑇 ⊕ 𝑈)) = { 0 }) | |
22 | 20, 21 | eqtrid 2778 | . . 3 ⊢ (𝜑 → ((𝑈(LSSum‘(oppg‘𝐺))𝑇) ∩ 𝑆) = { 0 }) |
23 | incom 4196 | . . . 4 ⊢ (𝑇 ∩ 𝑈) = (𝑈 ∩ 𝑇) | |
24 | lsmdisj2r.i | . . . 4 ⊢ (𝜑 → (𝑇 ∩ 𝑈) = { 0 }) | |
25 | 23, 24 | eqtr3id 2780 | . . 3 ⊢ (𝜑 → (𝑈 ∩ 𝑇) = { 0 }) |
26 | 7, 10, 12, 14, 16, 22, 25 | lsmdisj2 19602 | . 2 ⊢ (𝜑 → (𝑇 ∩ (𝑈(LSSum‘(oppg‘𝐺))𝑆)) = { 0 }) |
27 | 6, 26 | eqtr3id 2780 | 1 ⊢ (𝜑 → ((𝑆 ⊕ 𝑈) ∩ 𝑇) = { 0 }) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ∩ cin 3942 {csn 4623 ‘cfv 6537 (class class class)co 7405 0gc0g 17394 SubGrpcsubg 19047 oppgcoppg 19261 LSSumclsm 19554 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-tpos 8212 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-0g 17396 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-submnd 18714 df-grp 18866 df-minusg 18867 df-subg 19050 df-oppg 19262 df-lsm 19556 |
This theorem is referenced by: lsmdisj3r 19606 lsmdisj2b 19608 |
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