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Theorem lsmmod2 19638
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 2728 . . . . . . 7 (oppg𝐺) = (oppg𝐺)
32oppgsubg 19324 . . . . . 6 (SubGrp‘𝐺) = (SubGrp‘(oppg𝐺))
41, 3eleqtrdi 2839 . . . . 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 2839 . . . . 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 2839 . . . . 5 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ 𝑈𝑆) → 𝑆 ∈ (SubGrp‘(oppg𝐺)))
9 simpr 483 . . . . 5 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ 𝑈𝑆) → 𝑈𝑆)
10 eqid 2728 . . . . . 6 (LSSum‘(oppg𝐺)) = (LSSum‘(oppg𝐺))
1110lsmmod 19637 . . . . 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 2734 . . 3 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ 𝑈𝑆) → ((𝑈(LSSum‘(oppg𝐺))𝑇) ∩ 𝑆) = (𝑈(LSSum‘(oppg𝐺))(𝑇𝑆)))
14 incom 4203 . . 3 ((𝑈(LSSum‘(oppg𝐺))𝑇) ∩ 𝑆) = (𝑆 ∩ (𝑈(LSSum‘(oppg𝐺))𝑇))
15 incom 4203 . . . 4 (𝑇𝑆) = (𝑆𝑇)
1615oveq2i 7437 . . 3 (𝑈(LSSum‘(oppg𝐺))(𝑇𝑆)) = (𝑈(LSSum‘(oppg𝐺))(𝑆𝑇))
1713, 14, 163eqtr3g 2791 . 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 19604 . . 3 (𝑈(LSSum‘(oppg𝐺))𝑇) = (𝑇 𝑈)
2019ineq2i 4211 . 2 (𝑆 ∩ (𝑈(LSSum‘(oppg𝐺))𝑇)) = (𝑆 ∩ (𝑇 𝑈))
212, 18oppglsm 19604 . 2 (𝑈(LSSum‘(oppg𝐺))(𝑆𝑇)) = ((𝑆𝑇) 𝑈)
2217, 20, 213eqtr3g 2791 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 394  w3a 1084   = wceq 1533  wcel 2098  cin 3948  wss 3949  cfv 6553  (class class class)co 7426  SubGrpcsubg 19082  oppgcoppg 19303  LSSumclsm 19596
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11202  ax-resscn 11203  ax-1cn 11204  ax-icn 11205  ax-addcl 11206  ax-addrcl 11207  ax-mulcl 11208  ax-mulrcl 11209  ax-mulcom 11210  ax-addass 11211  ax-mulass 11212  ax-distr 11213  ax-i2m1 11214  ax-1ne0 11215  ax-1rid 11216  ax-rnegex 11217  ax-rrecex 11218  ax-cnre 11219  ax-pre-lttri 11220  ax-pre-lttrn 11221  ax-pre-ltadd 11222  ax-pre-mulgt0 11223
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-int 4954  df-iun 5002  df-iin 5003  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7877  df-1st 7999  df-2nd 8000  df-tpos 8238  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-rdg 8437  df-1o 8493  df-er 8731  df-en 8971  df-dom 8972  df-sdom 8973  df-fin 8974  df-pnf 11288  df-mnf 11289  df-xr 11290  df-ltxr 11291  df-le 11292  df-sub 11484  df-neg 11485  df-nn 12251  df-2 12313  df-sets 17140  df-slot 17158  df-ndx 17170  df-base 17188  df-ress 17217  df-plusg 17253  df-0g 17430  df-mre 17573  df-mrc 17574  df-acs 17576  df-mgm 18607  df-sgrp 18686  df-mnd 18702  df-submnd 18748  df-grp 18900  df-minusg 18901  df-subg 19085  df-oppg 19304  df-lsm 19598
This theorem is referenced by:  lcvexchlem3  38540  lcfrlem23  41070
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