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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 2727 | . . . . 5 ⊢ (oppg‘𝐺) = (oppg‘𝐺) | |
2 | lsmcntz.p | . . . . 5 ⊢ ⊕ = (LSSum‘𝐺) | |
3 | 1, 2 | oppglsm 19581 | . . . 4 ⊢ (𝑈(LSSum‘(oppg‘𝐺))𝑆) = (𝑆 ⊕ 𝑈) |
4 | 3 | ineq2i 4205 | . . 3 ⊢ (𝑇 ∩ (𝑈(LSSum‘(oppg‘𝐺))𝑆)) = (𝑇 ∩ (𝑆 ⊕ 𝑈)) |
5 | incom 4197 | . . 3 ⊢ (𝑇 ∩ (𝑆 ⊕ 𝑈)) = ((𝑆 ⊕ 𝑈) ∩ 𝑇) | |
6 | 4, 5 | eqtri 2755 | . 2 ⊢ (𝑇 ∩ (𝑈(LSSum‘(oppg‘𝐺))𝑆)) = ((𝑆 ⊕ 𝑈) ∩ 𝑇) |
7 | eqid 2727 | . . 3 ⊢ (LSSum‘(oppg‘𝐺)) = (LSSum‘(oppg‘𝐺)) | |
8 | lsmcntz.u | . . . 4 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺)) | |
9 | 1 | oppgsubg 19301 | . . . 4 ⊢ (SubGrp‘𝐺) = (SubGrp‘(oppg‘𝐺)) |
10 | 8, 9 | eleqtrdi 2838 | . . 3 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘(oppg‘𝐺))) |
11 | lsmcntz.t | . . . 4 ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺)) | |
12 | 11, 9 | eleqtrdi 2838 | . . 3 ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘(oppg‘𝐺))) |
13 | lsmcntz.s | . . . 4 ⊢ (𝜑 → 𝑆 ∈ (SubGrp‘𝐺)) | |
14 | 13, 9 | eleqtrdi 2838 | . . 3 ⊢ (𝜑 → 𝑆 ∈ (SubGrp‘(oppg‘𝐺))) |
15 | lsmdisj.o | . . . 4 ⊢ 0 = (0g‘𝐺) | |
16 | 1, 15 | oppgid 19294 | . . 3 ⊢ 0 = (0g‘(oppg‘𝐺)) |
17 | 1, 2 | oppglsm 19581 | . . . . . 6 ⊢ (𝑈(LSSum‘(oppg‘𝐺))𝑇) = (𝑇 ⊕ 𝑈) |
18 | 17 | ineq1i 4204 | . . . . 5 ⊢ ((𝑈(LSSum‘(oppg‘𝐺))𝑇) ∩ 𝑆) = ((𝑇 ⊕ 𝑈) ∩ 𝑆) |
19 | incom 4197 | . . . . 5 ⊢ ((𝑇 ⊕ 𝑈) ∩ 𝑆) = (𝑆 ∩ (𝑇 ⊕ 𝑈)) | |
20 | 18, 19 | eqtri 2755 | . . . 4 ⊢ ((𝑈(LSSum‘(oppg‘𝐺))𝑇) ∩ 𝑆) = (𝑆 ∩ (𝑇 ⊕ 𝑈)) |
21 | lsmdisjr.i | . . . 4 ⊢ (𝜑 → (𝑆 ∩ (𝑇 ⊕ 𝑈)) = { 0 }) | |
22 | 20, 21 | eqtrid 2779 | . . 3 ⊢ (𝜑 → ((𝑈(LSSum‘(oppg‘𝐺))𝑇) ∩ 𝑆) = { 0 }) |
23 | incom 4197 | . . . 4 ⊢ (𝑇 ∩ 𝑈) = (𝑈 ∩ 𝑇) | |
24 | lsmdisj2r.i | . . . 4 ⊢ (𝜑 → (𝑇 ∩ 𝑈) = { 0 }) | |
25 | 23, 24 | eqtr3id 2781 | . . 3 ⊢ (𝜑 → (𝑈 ∩ 𝑇) = { 0 }) |
26 | 7, 10, 12, 14, 16, 22, 25 | lsmdisj2 19621 | . 2 ⊢ (𝜑 → (𝑇 ∩ (𝑈(LSSum‘(oppg‘𝐺))𝑆)) = { 0 }) |
27 | 6, 26 | eqtr3id 2781 | 1 ⊢ (𝜑 → ((𝑆 ⊕ 𝑈) ∩ 𝑇) = { 0 }) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ∩ cin 3943 {csn 4624 ‘cfv 6542 (class class class)co 7414 0gc0g 17406 SubGrpcsubg 19059 oppgcoppg 19280 LSSumclsm 19573 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7863 df-1st 7985 df-2nd 7986 df-tpos 8223 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-er 8716 df-en 8954 df-dom 8955 df-sdom 8956 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 df-sub 11462 df-neg 11463 df-nn 12229 df-2 12291 df-sets 17118 df-slot 17136 df-ndx 17148 df-base 17166 df-ress 17195 df-plusg 17231 df-0g 17408 df-mgm 18585 df-sgrp 18664 df-mnd 18680 df-submnd 18726 df-grp 18878 df-minusg 18879 df-subg 19062 df-oppg 19281 df-lsm 19575 |
This theorem is referenced by: lsmdisj3r 19625 lsmdisj2b 19627 |
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