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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 2821 | . . 3 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
3 | 2 | subgss 18274 | . 2 ⊢ (𝑇 ∈ (SubGrp‘𝐺) → 𝑇 ⊆ (Base‘𝐺)) |
4 | 2 | subgss 18274 | . 2 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 ⊆ (Base‘𝐺)) |
5 | lsmcom.s | . . 3 ⊢ ⊕ = (LSSum‘𝐺) | |
6 | 2, 5 | lsmcomx 18970 | . 2 ⊢ ((𝐺 ∈ Abel ∧ 𝑇 ⊆ (Base‘𝐺) ∧ 𝑈 ⊆ (Base‘𝐺)) → (𝑇 ⊕ 𝑈) = (𝑈 ⊕ 𝑇)) |
7 | 1, 3, 4, 6 | syl3an 1156 | 1 ⊢ ((𝐺 ∈ Abel ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝑇 ⊕ 𝑈) = (𝑈 ⊕ 𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ⊆ wss 3936 ‘cfv 6350 (class class class)co 7150 Basecbs 16477 SubGrpcsubg 18267 LSSumclsm 18753 Abelcabl 18901 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-ov 7153 df-oprab 7154 df-mpo 7155 df-1st 7683 df-2nd 7684 df-subg 18270 df-lsm 18755 df-cmn 18902 df-abl 18903 |
This theorem is referenced by: lsm4 18974 pgpfac1lem4 19194 pgpfaclem1 19197 lspprabs 19861 ocvpj 20855 lcvexchlem3 36166 lcvexchlem4 36167 lcvexchlem5 36168 lsatcvatlem 36179 lsatcvat 36180 lsatcvat3 36182 l1cvat 36185 dia2dimlem5 38198 dihjatc3 38443 dihmeetlem9N 38445 dihjat 38553 lclkrlem2b 38638 |
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