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Mirrors > Home > MPE Home > Th. List > lsmcom | Structured version Visualization version GIF version |
Description: Subgroup sum commutes. (Contributed by NM, 6-Feb-2014.) (Revised by Mario Carneiro, 21-Jun-2014.) |
Ref | Expression |
---|---|
lsmcom.s | ⊢ ⊕ = (LSSum‘𝐺) |
Ref | Expression |
---|---|
lsmcom | ⊢ ((𝐺 ∈ Abel ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝑇 ⊕ 𝑈) = (𝑈 ⊕ 𝑇)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . 2 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Abel) | |
2 | eqid 2825 | . . 3 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
3 | 2 | subgss 17953 | . 2 ⊢ (𝑇 ∈ (SubGrp‘𝐺) → 𝑇 ⊆ (Base‘𝐺)) |
4 | 2 | subgss 17953 | . 2 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 ⊆ (Base‘𝐺)) |
5 | lsmcom.s | . . 3 ⊢ ⊕ = (LSSum‘𝐺) | |
6 | 2, 5 | lsmcomx 18619 | . 2 ⊢ ((𝐺 ∈ Abel ∧ 𝑇 ⊆ (Base‘𝐺) ∧ 𝑈 ⊆ (Base‘𝐺)) → (𝑇 ⊕ 𝑈) = (𝑈 ⊕ 𝑇)) |
7 | 1, 3, 4, 6 | syl3an 1203 | 1 ⊢ ((𝐺 ∈ Abel ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝑇 ⊕ 𝑈) = (𝑈 ⊕ 𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1111 = wceq 1656 ∈ wcel 2164 ⊆ wss 3798 ‘cfv 6127 (class class class)co 6910 Basecbs 16229 SubGrpcsubg 17946 LSSumclsm 18407 Abelcabl 18554 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-id 5252 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-1st 7433 df-2nd 7434 df-subg 17949 df-lsm 18409 df-cmn 18555 df-abl 18556 |
This theorem is referenced by: lsm4 18623 pgpfac1lem4 18838 pgpfaclem1 18841 lspprabs 19461 ocvpj 20431 lcvexchlem3 35106 lcvexchlem4 35107 lcvexchlem5 35108 lsatcvatlem 35119 lsatcvat 35120 lsatcvat3 35122 l1cvat 35125 dia2dimlem5 37138 dihjatc3 37383 dihmeetlem9N 37385 dihjat 37493 lclkrlem2b 37578 |
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