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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 2728 | . . 3 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
3 | 2 | subgss 19075 | . 2 ⊢ (𝑇 ∈ (SubGrp‘𝐺) → 𝑇 ⊆ (Base‘𝐺)) |
4 | 2 | subgss 19075 | . 2 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 ⊆ (Base‘𝐺)) |
5 | lsmcom.s | . . 3 ⊢ ⊕ = (LSSum‘𝐺) | |
6 | 2, 5 | lsmcomx 19804 | . 2 ⊢ ((𝐺 ∈ Abel ∧ 𝑇 ⊆ (Base‘𝐺) ∧ 𝑈 ⊆ (Base‘𝐺)) → (𝑇 ⊕ 𝑈) = (𝑈 ⊕ 𝑇)) |
7 | 1, 3, 4, 6 | syl3an 1158 | 1 ⊢ ((𝐺 ∈ Abel ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝑇 ⊕ 𝑈) = (𝑈 ⊕ 𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ⊆ wss 3945 ‘cfv 6542 (class class class)co 7414 Basecbs 17173 SubGrpcsubg 19068 LSSumclsm 19582 Abelcabl 19729 |
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 2167 ax-ext 2699 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 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-id 5570 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-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7417 df-oprab 7418 df-mpo 7419 df-1st 7987 df-2nd 7988 df-subg 19071 df-lsm 19584 df-cmn 19730 df-abl 19731 |
This theorem is referenced by: lsm4 19808 pgpfac1lem4 20028 pgpfaclem1 20031 lspprabs 20973 ocvpj 21644 idlsrgcmnd 33220 lcvexchlem3 38502 lcvexchlem4 38503 lcvexchlem5 38504 lsatcvatlem 38515 lsatcvat 38516 lsatcvat3 38518 l1cvat 38521 dia2dimlem5 40535 dihjatc3 40780 dihmeetlem9N 40782 dihjat 40890 lclkrlem2b 40975 |
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