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Theorem lsmless2 18781
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) ∧ 𝑇𝑈) → (𝑆 𝑇) ⊆ (𝑆 𝑈))

Proof of Theorem lsmless2
StepHypRef Expression
1 subgrcl 18279 . . 3 (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
213ad2ant1 1130 . 2 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇𝑈) → 𝐺 ∈ Grp)
3 eqid 2801 . . . 4 (Base‘𝐺) = (Base‘𝐺)
43subgss 18275 . . 3 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ (Base‘𝐺))
543ad2ant1 1130 . 2 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇𝑈) → 𝑆 ⊆ (Base‘𝐺))
63subgss 18275 . . 3 (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 ⊆ (Base‘𝐺))
763ad2ant2 1131 . 2 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇𝑈) → 𝑈 ⊆ (Base‘𝐺))
8 simp3 1135 . 2 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇𝑈) → 𝑇𝑈)
9 lsmub1.p . . 3 = (LSSum‘𝐺)
103, 9lsmless2x 18765 . 2 (((𝐺 ∈ Grp ∧ 𝑆 ⊆ (Base‘𝐺) ∧ 𝑈 ⊆ (Base‘𝐺)) ∧ 𝑇𝑈) → (𝑆 𝑇) ⊆ (𝑆 𝑈))
112, 5, 7, 8, 10syl31anc 1370 1 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇𝑈) → (𝑆 𝑇) ⊆ (𝑆 𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1084   = wceq 1538  wcel 2112  wss 3884  cfv 6328  (class class class)co 7139  Basecbs 16478  Grpcgrp 18098  SubGrpcsubg 18268  LSSumclsm 18754
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-1st 7675  df-2nd 7676  df-subg 18271  df-lsm 18756
This theorem is referenced by:  lsmless12  18782  lsmmod  18796  lsmelval2  19853  lsmsat  36297  lsatcvat3  36341  cdlemn5pre  38489  dvh3dim3N  38738
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