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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 2824 | . . 3 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
3 | 2 | subgss 18283 | . 2 ⊢ (𝑇 ∈ (SubGrp‘𝐺) → 𝑇 ⊆ (Base‘𝐺)) |
4 | 2 | subgss 18283 | . 2 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 ⊆ (Base‘𝐺)) |
5 | lsmcom.s | . . 3 ⊢ ⊕ = (LSSum‘𝐺) | |
6 | 2, 5 | lsmcomx 18979 | . 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 1536 ∈ wcel 2113 ⊆ wss 3939 ‘cfv 6358 (class class class)co 7159 Basecbs 16486 SubGrpcsubg 18276 LSSumclsm 18762 Abelcabl 18910 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7162 df-oprab 7163 df-mpo 7164 df-1st 7692 df-2nd 7693 df-subg 18279 df-lsm 18764 df-cmn 18911 df-abl 18912 |
This theorem is referenced by: lsm4 18983 pgpfac1lem4 19203 pgpfaclem1 19206 lspprabs 19870 ocvpj 20864 lcvexchlem3 36176 lcvexchlem4 36177 lcvexchlem5 36178 lsatcvatlem 36189 lsatcvat 36190 lsatcvat3 36192 l1cvat 36195 dia2dimlem5 38208 dihjatc3 38453 dihmeetlem9N 38455 dihjat 38563 lclkrlem2b 38648 |
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