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Theorem lsmmod2 18869
 Description: Modular law dual for subgroup sum. Similar to part of Theorem 16.9 of [MaedaMaeda] p. 70. (Contributed by NM, 8-Jan-2015.) (Revised by Mario Carneiro, 21-Apr-2016.)
Hypothesis
Ref Expression
lsmmod.p = (LSSum‘𝐺)
Assertion
Ref Expression
lsmmod2 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ 𝑈𝑆) → (𝑆 ∩ (𝑇 𝑈)) = ((𝑆𝑇) 𝑈))

Proof of Theorem lsmmod2
StepHypRef Expression
1 simpl3 1190 . . . . . 6 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ 𝑈𝑆) → 𝑈 ∈ (SubGrp‘𝐺))
2 eqid 2758 . . . . . . 7 (oppg𝐺) = (oppg𝐺)
32oppgsubg 18558 . . . . . 6 (SubGrp‘𝐺) = (SubGrp‘(oppg𝐺))
41, 3eleqtrdi 2862 . . . . 5 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ 𝑈𝑆) → 𝑈 ∈ (SubGrp‘(oppg𝐺)))
5 simpl2 1189 . . . . . 6 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ 𝑈𝑆) → 𝑇 ∈ (SubGrp‘𝐺))
65, 3eleqtrdi 2862 . . . . 5 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ 𝑈𝑆) → 𝑇 ∈ (SubGrp‘(oppg𝐺)))
7 simpl1 1188 . . . . . 6 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ 𝑈𝑆) → 𝑆 ∈ (SubGrp‘𝐺))
87, 3eleqtrdi 2862 . . . . 5 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ 𝑈𝑆) → 𝑆 ∈ (SubGrp‘(oppg𝐺)))
9 simpr 488 . . . . 5 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ 𝑈𝑆) → 𝑈𝑆)
10 eqid 2758 . . . . . 6 (LSSum‘(oppg𝐺)) = (LSSum‘(oppg𝐺))
1110lsmmod 18868 . . . . 5 (((𝑈 ∈ (SubGrp‘(oppg𝐺)) ∧ 𝑇 ∈ (SubGrp‘(oppg𝐺)) ∧ 𝑆 ∈ (SubGrp‘(oppg𝐺))) ∧ 𝑈𝑆) → (𝑈(LSSum‘(oppg𝐺))(𝑇𝑆)) = ((𝑈(LSSum‘(oppg𝐺))𝑇) ∩ 𝑆))
124, 6, 8, 9, 11syl31anc 1370 . . . 4 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ 𝑈𝑆) → (𝑈(LSSum‘(oppg𝐺))(𝑇𝑆)) = ((𝑈(LSSum‘(oppg𝐺))𝑇) ∩ 𝑆))
1312eqcomd 2764 . . 3 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ 𝑈𝑆) → ((𝑈(LSSum‘(oppg𝐺))𝑇) ∩ 𝑆) = (𝑈(LSSum‘(oppg𝐺))(𝑇𝑆)))
14 incom 4106 . . 3 ((𝑈(LSSum‘(oppg𝐺))𝑇) ∩ 𝑆) = (𝑆 ∩ (𝑈(LSSum‘(oppg𝐺))𝑇))
15 incom 4106 . . . 4 (𝑇𝑆) = (𝑆𝑇)
1615oveq2i 7161 . . 3 (𝑈(LSSum‘(oppg𝐺))(𝑇𝑆)) = (𝑈(LSSum‘(oppg𝐺))(𝑆𝑇))
1713, 14, 163eqtr3g 2816 . 2 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ 𝑈𝑆) → (𝑆 ∩ (𝑈(LSSum‘(oppg𝐺))𝑇)) = (𝑈(LSSum‘(oppg𝐺))(𝑆𝑇)))
18 lsmmod.p . . . 4 = (LSSum‘𝐺)
192, 18oppglsm 18834 . . 3 (𝑈(LSSum‘(oppg𝐺))𝑇) = (𝑇 𝑈)
2019ineq2i 4114 . 2 (𝑆 ∩ (𝑈(LSSum‘(oppg𝐺))𝑇)) = (𝑆 ∩ (𝑇 𝑈))
212, 18oppglsm 18834 . 2 (𝑈(LSSum‘(oppg𝐺))(𝑆𝑇)) = ((𝑆𝑇) 𝑈)
2217, 20, 213eqtr3g 2816 1 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ 𝑈𝑆) → (𝑆 ∩ (𝑇 𝑈)) = ((𝑆𝑇) 𝑈))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ∩ cin 3857   ⊆ wss 3858  ‘cfv 6335  (class class class)co 7150  SubGrpcsubg 18340  oppgcoppg 18540  LSSumclsm 18826 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 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5156  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459  ax-cnex 10631  ax-resscn 10632  ax-1cn 10633  ax-icn 10634  ax-addcl 10635  ax-addrcl 10636  ax-mulcl 10637  ax-mulrcl 10638  ax-mulcom 10639  ax-addass 10640  ax-mulass 10641  ax-distr 10642  ax-i2m1 10643  ax-1ne0 10644  ax-1rid 10645  ax-rnegex 10646  ax-rrecex 10647  ax-cnre 10648  ax-pre-lttri 10649  ax-pre-lttrn 10650  ax-pre-ltadd 10651  ax-pre-mulgt0 10652 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-pss 3877  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-tp 4527  df-op 4529  df-uni 4799  df-int 4839  df-iun 4885  df-iin 4886  df-br 5033  df-opab 5095  df-mpt 5113  df-tr 5139  df-id 5430  df-eprel 5435  df-po 5443  df-so 5444  df-fr 5483  df-we 5485  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7580  df-1st 7693  df-2nd 7694  df-tpos 7902  df-wrecs 7957  df-recs 8018  df-rdg 8056  df-1o 8112  df-er 8299  df-en 8528  df-dom 8529  df-sdom 8530  df-fin 8531  df-pnf 10715  df-mnf 10716  df-xr 10717  df-ltxr 10718  df-le 10719  df-sub 10910  df-neg 10911  df-nn 11675  df-2 11737  df-ndx 16544  df-slot 16545  df-base 16547  df-sets 16548  df-ress 16549  df-plusg 16636  df-0g 16773  df-mre 16915  df-mrc 16916  df-acs 16918  df-mgm 17918  df-sgrp 17967  df-mnd 17978  df-submnd 18023  df-grp 18172  df-minusg 18173  df-subg 18343  df-oppg 18541  df-lsm 18828 This theorem is referenced by:  lcvexchlem3  36612  lcfrlem23  39141
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