Theorem lsmless2 18785

Description: Subset implies subgroup sum subset. (Contributed by NM, 25-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)

Hypothesis

Ref	Expression
lsmub1.p	⊢ ⊕ = (LSSum‘𝐺)

Assertion

Ref	Expression
lsmless2	⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ 𝑈) → (𝑆 ⊕ 𝑇) ⊆ (𝑆 ⊕ 𝑈))

Proof of Theorem lsmless2

Step	Hyp	Ref	Expression
1		subgrcl 18283	. . 3 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
2	1	3ad2ant1 1129	. 2 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ 𝑈) → 𝐺 ∈ Grp)
3		eqid 2821	. . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺)
4	3	subgss 18279	. . 3 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ (Base‘𝐺))
5	4	3ad2ant1 1129	. 2 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ 𝑈) → 𝑆 ⊆ (Base‘𝐺))
6	3	subgss 18279	. . 3 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 ⊆ (Base‘𝐺))
7	6	3ad2ant2 1130	. 2 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ 𝑈) → 𝑈 ⊆ (Base‘𝐺))
8		simp3 1134	. 2 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ 𝑈) → 𝑇 ⊆ 𝑈)
9		lsmub1.p	. . 3 ⊢ ⊕ = (LSSum‘𝐺)
10	3, 9	lsmless2x 18769	. 2 ⊢ (((𝐺 ∈ Grp ∧ 𝑆 ⊆ (Base‘𝐺) ∧ 𝑈 ⊆ (Base‘𝐺)) ∧ 𝑇 ⊆ 𝑈) → (𝑆 ⊕ 𝑇) ⊆ (𝑆 ⊕ 𝑈))
11	2, 5, 7, 8, 10	syl31anc 1369	1 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ 𝑈) → (𝑆 ⊕ 𝑇) ⊆ (𝑆 ⊕ 𝑈))

Syntax hints:   → wi 4   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110   ⊆ wss 3935  ‘cfv 6354  (class class class)co 7155  Basecbs 16482  Grpcgrp 18102  SubGrpcsubg 18272  LSSumclsm 18758

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 5189  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460

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 4567  df-pr 4569  df-op 4573  df-uni 4838  df-iun 4920  df-br 5066  df-opab 5128  df-mpt 5146  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-ov 7158  df-oprab 7159  df-mpo 7160  df-1st 7688  df-2nd 7689  df-subg 18275  df-lsm 18760

This theorem is referenced by:  lsmless12  18786  lsmmod  18800  lsmelval2  19856  lsmsat  36143  lsatcvat3  36187  cdlemn5pre  38335  dvh3dim3N  38584