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Mirrors > Home > MPE Home > Th. List > lsmelvali | Structured version Visualization version GIF version |
Description: Subgroup sum membership (for a left module or left vector space). (Contributed by NM, 4-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.) |
Ref | Expression |
---|---|
lsmelval.a | ⊢ + = (+g‘𝐺) |
lsmelval.p | ⊢ ⊕ = (LSSum‘𝐺) |
Ref | Expression |
---|---|
lsmelvali | ⊢ (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ (𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝑈)) → (𝑋 + 𝑌) ∈ (𝑇 ⊕ 𝑈)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | subgrcl 18278 | . . . 4 ⊢ (𝑇 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp) | |
2 | 1 | adantr 483 | . . 3 ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → 𝐺 ∈ Grp) |
3 | eqid 2821 | . . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
4 | 3 | subgss 18274 | . . . 4 ⊢ (𝑇 ∈ (SubGrp‘𝐺) → 𝑇 ⊆ (Base‘𝐺)) |
5 | 4 | adantr 483 | . . 3 ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → 𝑇 ⊆ (Base‘𝐺)) |
6 | 3 | subgss 18274 | . . . 4 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 ⊆ (Base‘𝐺)) |
7 | 6 | adantl 484 | . . 3 ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → 𝑈 ⊆ (Base‘𝐺)) |
8 | 2, 5, 7 | 3jca 1124 | . 2 ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝐺 ∈ Grp ∧ 𝑇 ⊆ (Base‘𝐺) ∧ 𝑈 ⊆ (Base‘𝐺))) |
9 | lsmelval.a | . . 3 ⊢ + = (+g‘𝐺) | |
10 | lsmelval.p | . . 3 ⊢ ⊕ = (LSSum‘𝐺) | |
11 | 3, 9, 10 | lsmelvalix 18760 | . 2 ⊢ (((𝐺 ∈ Grp ∧ 𝑇 ⊆ (Base‘𝐺) ∧ 𝑈 ⊆ (Base‘𝐺)) ∧ (𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝑈)) → (𝑋 + 𝑌) ∈ (𝑇 ⊕ 𝑈)) |
12 | 8, 11 | sylan 582 | 1 ⊢ (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ (𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝑈)) → (𝑋 + 𝑌) ∈ (𝑇 ⊕ 𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ⊆ wss 3935 ‘cfv 6349 (class class class)co 7150 Basecbs 16477 +gcplusg 16559 Grpcgrp 18097 SubGrpcsubg 18267 LSSumclsm 18753 |
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 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 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 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-1st 7683 df-2nd 7684 df-subg 18270 df-lsm 18755 |
This theorem is referenced by: lsmsubg 18773 lsmmod 18795 lsmdisj2 18802 lsmhash 18825 ablfacrp 19182 lsmcl 19849 lsmelval2 19851 lsppreli 19856 lspprabs 19861 lspabs3 19887 pjthlem2 24035 lkrlsp 36232 dia2dimlem5 38198 mapdindp0 38849 hdmaprnlem3eN 38988 |
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