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Mirrors > Home > MPE Home > Th. List > lsmless2 | Structured version Visualization version GIF version |
Description: Subset implies subgroup sum subset. (Contributed by NM, 25-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.) |
Ref | Expression |
---|---|
lsmub1.p | ⊢ ⊕ = (LSSum‘𝐺) |
Ref | Expression |
---|---|
lsmless2 | ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ 𝑈) → (𝑆 ⊕ 𝑇) ⊆ (𝑆 ⊕ 𝑈)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | subgrcl 17599 | . . 3 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp) | |
2 | 1 | 3ad2ant1 1082 | . 2 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ 𝑈) → 𝐺 ∈ Grp) |
3 | eqid 2622 | . . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
4 | 3 | subgss 17595 | . . 3 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ (Base‘𝐺)) |
5 | 4 | 3ad2ant1 1082 | . 2 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ 𝑈) → 𝑆 ⊆ (Base‘𝐺)) |
6 | 3 | subgss 17595 | . . 3 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 ⊆ (Base‘𝐺)) |
7 | 6 | 3ad2ant2 1083 | . 2 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ 𝑈) → 𝑈 ⊆ (Base‘𝐺)) |
8 | simp3 1063 | . 2 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ 𝑈) → 𝑇 ⊆ 𝑈) | |
9 | lsmub1.p | . . 3 ⊢ ⊕ = (LSSum‘𝐺) | |
10 | 3, 9 | lsmless2x 18060 | . 2 ⊢ (((𝐺 ∈ Grp ∧ 𝑆 ⊆ (Base‘𝐺) ∧ 𝑈 ⊆ (Base‘𝐺)) ∧ 𝑇 ⊆ 𝑈) → (𝑆 ⊕ 𝑇) ⊆ (𝑆 ⊕ 𝑈)) |
11 | 2, 5, 7, 8, 10 | syl31anc 1329 | 1 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ 𝑈) → (𝑆 ⊕ 𝑇) ⊆ (𝑆 ⊕ 𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ⊆ wss 3574 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 Grpcgrp 17422 SubGrpcsubg 17588 LSSumclsm 18049 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-subg 17591 df-lsm 18051 |
This theorem is referenced by: lsmless12 18076 lsmmod 18088 lsmelval2 19085 lsmsat 34295 lsatcvat3 34339 cdlemn5pre 36489 dvh3dim3N 36738 |
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