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Theorem lsmless12 18716
Description: Subset implies subgroup sum subset. (Contributed by NM, 14-Jan-2015.) (Revised by Mario Carneiro, 19-Apr-2016.)
Hypothesis
Ref Expression
lsmub1.p = (LSSum‘𝐺)
Assertion
Ref Expression
lsmless12 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ (𝑅𝑆𝑇𝑈)) → (𝑅 𝑇) ⊆ (𝑆 𝑈))

Proof of Theorem lsmless12
StepHypRef Expression
1 subgrcl 18222 . . . 4 (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
21ad2antrr 722 . . 3 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ (𝑅𝑆𝑇𝑈)) → 𝐺 ∈ Grp)
3 eqid 2818 . . . . 5 (Base‘𝐺) = (Base‘𝐺)
43subgss 18218 . . . 4 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ (Base‘𝐺))
54ad2antrr 722 . . 3 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ (𝑅𝑆𝑇𝑈)) → 𝑆 ⊆ (Base‘𝐺))
6 simprr 769 . . . 4 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ (𝑅𝑆𝑇𝑈)) → 𝑇𝑈)
73subgss 18218 . . . . 5 (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 ⊆ (Base‘𝐺))
87ad2antlr 723 . . . 4 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ (𝑅𝑆𝑇𝑈)) → 𝑈 ⊆ (Base‘𝐺))
96, 8sstrd 3974 . . 3 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ (𝑅𝑆𝑇𝑈)) → 𝑇 ⊆ (Base‘𝐺))
10 simprl 767 . . 3 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ (𝑅𝑆𝑇𝑈)) → 𝑅𝑆)
11 lsmub1.p . . . 4 = (LSSum‘𝐺)
123, 11lsmless1x 18698 . . 3 (((𝐺 ∈ Grp ∧ 𝑆 ⊆ (Base‘𝐺) ∧ 𝑇 ⊆ (Base‘𝐺)) ∧ 𝑅𝑆) → (𝑅 𝑇) ⊆ (𝑆 𝑇))
132, 5, 9, 10, 12syl31anc 1365 . 2 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ (𝑅𝑆𝑇𝑈)) → (𝑅 𝑇) ⊆ (𝑆 𝑇))
14 simpll 763 . . 3 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ (𝑅𝑆𝑇𝑈)) → 𝑆 ∈ (SubGrp‘𝐺))
15 simplr 765 . . 3 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ (𝑅𝑆𝑇𝑈)) → 𝑈 ∈ (SubGrp‘𝐺))
1611lsmless2 18715 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇𝑈) → (𝑆 𝑇) ⊆ (𝑆 𝑈))
1714, 15, 6, 16syl3anc 1363 . 2 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ (𝑅𝑆𝑇𝑈)) → (𝑆 𝑇) ⊆ (𝑆 𝑈))
1813, 17sstrd 3974 1 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ (𝑅𝑆𝑇𝑈)) → (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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