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Theorem lsmless1 18788
Description: Subset implies subgroup sum subset. (Contributed by NM, 6-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
Hypothesis
Ref Expression
lsmub1.p = (LSSum‘𝐺)
Assertion
Ref Expression
lsmless1 ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑆𝑇) → (𝑆 𝑈) ⊆ (𝑇 𝑈))

Proof of Theorem lsmless1
StepHypRef Expression
1 subgrcl 18287 . . 3 (𝑇 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
213ad2ant1 1130 . 2 ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑆𝑇) → 𝐺 ∈ Grp)
3 eqid 2824 . . . 4 (Base‘𝐺) = (Base‘𝐺)
43subgss 18283 . . 3 (𝑇 ∈ (SubGrp‘𝐺) → 𝑇 ⊆ (Base‘𝐺))
543ad2ant1 1130 . 2 ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑆𝑇) → 𝑇 ⊆ (Base‘𝐺))
63subgss 18283 . . 3 (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 ⊆ (Base‘𝐺))
763ad2ant2 1131 . 2 ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑆𝑇) → 𝑈 ⊆ (Base‘𝐺))
8 simp3 1135 . 2 ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑆𝑇) → 𝑆𝑇)
9 lsmub1.p . . 3 = (LSSum‘𝐺)
103, 9lsmless1x 18772 . 2 (((𝐺 ∈ Grp ∧ 𝑇 ⊆ (Base‘𝐺) ∧ 𝑈 ⊆ (Base‘𝐺)) ∧ 𝑆𝑇) → (𝑆 𝑈) ⊆ (𝑇 𝑈))
112, 5, 7, 8, 10syl31anc 1370 1 ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑆𝑇) → (𝑆 𝑈) ⊆ (𝑇 𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1084   = wceq 1538  wcel 2115  wss 3920  cfv 6344  (class class class)co 7150  Basecbs 16486  Grpcgrp 18106  SubGrpcsubg 18276  LSSumclsm 18762
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5177  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7456
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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3483  df-sbc 3760  df-csb 3868  df-dif 3923  df-un 3925  df-in 3927  df-ss 3937  df-nul 4278  df-if 4452  df-pw 4525  df-sn 4552  df-pr 4554  df-op 4558  df-uni 4826  df-iun 4908  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-iota 6303  df-fun 6346  df-fn 6347  df-f 6348  df-f1 6349  df-fo 6350  df-f1o 6351  df-fv 6352  df-ov 7153  df-oprab 7154  df-mpo 7155  df-1st 7685  df-2nd 7686  df-subg 18279  df-lsm 18764
This theorem is referenced by:  lsmelval2  19860  lcvexchlem4  36279
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