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Theorem lsmmod2 17912
 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 1059 . . . . . 6 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ 𝑈𝑆) → 𝑈 ∈ (SubGrp‘𝐺))
2 eqid 2610 . . . . . . 7 (oppg𝐺) = (oppg𝐺)
32oppgsubg 17616 . . . . . 6 (SubGrp‘𝐺) = (SubGrp‘(oppg𝐺))
41, 3syl6eleq 2698 . . . . 5 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ 𝑈𝑆) → 𝑈 ∈ (SubGrp‘(oppg𝐺)))
5 simpl2 1058 . . . . . 6 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ 𝑈𝑆) → 𝑇 ∈ (SubGrp‘𝐺))
65, 3syl6eleq 2698 . . . . 5 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ 𝑈𝑆) → 𝑇 ∈ (SubGrp‘(oppg𝐺)))
7 simpl1 1057 . . . . . 6 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ 𝑈𝑆) → 𝑆 ∈ (SubGrp‘𝐺))
87, 3syl6eleq 2698 . . . . 5 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ 𝑈𝑆) → 𝑆 ∈ (SubGrp‘(oppg𝐺)))
9 simpr 476 . . . . 5 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ 𝑈𝑆) → 𝑈𝑆)
10 eqid 2610 . . . . . 6 (LSSum‘(oppg𝐺)) = (LSSum‘(oppg𝐺))
1110lsmmod 17911 . . . . 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 1321 . . . 4 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ 𝑈𝑆) → (𝑈(LSSum‘(oppg𝐺))(𝑇𝑆)) = ((𝑈(LSSum‘(oppg𝐺))𝑇) ∩ 𝑆))
1312eqcomd 2616 . . 3 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ 𝑈𝑆) → ((𝑈(LSSum‘(oppg𝐺))𝑇) ∩ 𝑆) = (𝑈(LSSum‘(oppg𝐺))(𝑇𝑆)))
14 incom 3767 . . 3 ((𝑈(LSSum‘(oppg𝐺))𝑇) ∩ 𝑆) = (𝑆 ∩ (𝑈(LSSum‘(oppg𝐺))𝑇))
15 incom 3767 . . . 4 (𝑇𝑆) = (𝑆𝑇)
1615oveq2i 6560 . . 3 (𝑈(LSSum‘(oppg𝐺))(𝑇𝑆)) = (𝑈(LSSum‘(oppg𝐺))(𝑆𝑇))
1713, 14, 163eqtr3g 2667 . 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 17880 . . 3 (𝑈(LSSum‘(oppg𝐺))𝑇) = (𝑇 𝑈)
2019ineq2i 3773 . 2 (𝑆 ∩ (𝑈(LSSum‘(oppg𝐺))𝑇)) = (𝑆 ∩ (𝑇 𝑈))
212, 18oppglsm 17880 . 2 (𝑈(LSSum‘(oppg𝐺))(𝑆𝑇)) = ((𝑆𝑇) 𝑈)
2217, 20, 213eqtr3g 2667 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 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ∩ cin 3539   ⊆ wss 3540  ‘cfv 5804  (class class class)co 6549  SubGrpcsubg 17411  oppgcoppg 17598  LSSumclsm 17872 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-tpos 7239  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-0g 15925  df-mre 16069  df-mrc 16070  df-acs 16072  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-submnd 17159  df-grp 17248  df-minusg 17249  df-subg 17414  df-oppg 17599  df-lsm 17874 This theorem is referenced by:  lcvexchlem3  33341  lcfrlem23  35872
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