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Mirrors > Home > MPE Home > Th. List > lsmless12 | Structured version Visualization version GIF version |
Description: Subset implies subgroup sum subset. (Contributed by NM, 14-Jan-2015.) (Revised by Mario Carneiro, 19-Apr-2016.) |
Ref | Expression |
---|---|
lsmub1.p | ⊢ ⊕ = (LSSum‘𝐺) |
Ref | Expression |
---|---|
lsmless12 | ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ (𝑅 ⊆ 𝑆 ∧ 𝑇 ⊆ 𝑈)) → (𝑅 ⊕ 𝑇) ⊆ (𝑆 ⊕ 𝑈)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | subgrcl 18222 | . . . 4 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp) | |
2 | 1 | ad2antrr 722 | . . 3 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ (𝑅 ⊆ 𝑆 ∧ 𝑇 ⊆ 𝑈)) → 𝐺 ∈ Grp) |
3 | eqid 2818 | . . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
4 | 3 | subgss 18218 | . . . 4 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ (Base‘𝐺)) |
5 | 4 | ad2antrr 722 | . . 3 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ (𝑅 ⊆ 𝑆 ∧ 𝑇 ⊆ 𝑈)) → 𝑆 ⊆ (Base‘𝐺)) |
6 | simprr 769 | . . . 4 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ (𝑅 ⊆ 𝑆 ∧ 𝑇 ⊆ 𝑈)) → 𝑇 ⊆ 𝑈) | |
7 | 3 | subgss 18218 | . . . . 5 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 ⊆ (Base‘𝐺)) |
8 | 7 | ad2antlr 723 | . . . 4 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ (𝑅 ⊆ 𝑆 ∧ 𝑇 ⊆ 𝑈)) → 𝑈 ⊆ (Base‘𝐺)) |
9 | 6, 8 | sstrd 3974 | . . 3 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ (𝑅 ⊆ 𝑆 ∧ 𝑇 ⊆ 𝑈)) → 𝑇 ⊆ (Base‘𝐺)) |
10 | simprl 767 | . . 3 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ (𝑅 ⊆ 𝑆 ∧ 𝑇 ⊆ 𝑈)) → 𝑅 ⊆ 𝑆) | |
11 | lsmub1.p | . . . 4 ⊢ ⊕ = (LSSum‘𝐺) | |
12 | 3, 11 | lsmless1x 18698 | . . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝑆 ⊆ (Base‘𝐺) ∧ 𝑇 ⊆ (Base‘𝐺)) ∧ 𝑅 ⊆ 𝑆) → (𝑅 ⊕ 𝑇) ⊆ (𝑆 ⊕ 𝑇)) |
13 | 2, 5, 9, 10, 12 | syl31anc 1365 | . 2 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ (𝑅 ⊆ 𝑆 ∧ 𝑇 ⊆ 𝑈)) → (𝑅 ⊕ 𝑇) ⊆ (𝑆 ⊕ 𝑇)) |
14 | simpll 763 | . . 3 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ (𝑅 ⊆ 𝑆 ∧ 𝑇 ⊆ 𝑈)) → 𝑆 ∈ (SubGrp‘𝐺)) | |
15 | simplr 765 | . . 3 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ (𝑅 ⊆ 𝑆 ∧ 𝑇 ⊆ 𝑈)) → 𝑈 ∈ (SubGrp‘𝐺)) | |
16 | 11 | lsmless2 18715 | . . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ 𝑈) → (𝑆 ⊕ 𝑇) ⊆ (𝑆 ⊕ 𝑈)) |
17 | 14, 15, 6, 16 | syl3anc 1363 | . 2 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ (𝑅 ⊆ 𝑆 ∧ 𝑇 ⊆ 𝑈)) → (𝑆 ⊕ 𝑇) ⊆ (𝑆 ⊕ 𝑈)) |
18 | 13, 17 | sstrd 3974 | 1 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ (𝑅 ⊆ 𝑆 ∧ 𝑇 ⊆ 𝑈)) → (𝑅 ⊕ 𝑇) ⊆ (𝑆 ⊕ 𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ⊆ wss 3933 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 Grpcgrp 18041 SubGrpcsubg 18211 LSSumclsm 18688 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-1st 7678 df-2nd 7679 df-subg 18214 df-lsm 18690 |
This theorem is referenced by: lsmlub 18719 dochexmidlem2 38477 |
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