Theorem lsmmod2 19543

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)) ∧ 𝑈 ⊆ 𝑆) → (𝑆 ∩ (𝑇 ⊕ 𝑈)) = ((𝑆 ∩ 𝑇) ⊕ 𝑈))

Proof of Theorem lsmmod2

Step	Hyp	Ref	Expression
1		simpl3 1193	. . . . . 6 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ 𝑈 ⊆ 𝑆) → 𝑈 ∈ (SubGrp‘𝐺))
2		eqid 2732	. . . . . . 7 ⊢ (oppg‘𝐺) = (oppg‘𝐺)
3	2	oppgsubg 19229	. . . . . 6 ⊢ (SubGrp‘𝐺) = (SubGrp‘(oppg‘𝐺))
4	1, 3	eleqtrdi 2843	. . . . 5 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ 𝑈 ⊆ 𝑆) → 𝑈 ∈ (SubGrp‘(oppg‘𝐺)))
5		simpl2 1192	. . . . . 6 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ 𝑈 ⊆ 𝑆) → 𝑇 ∈ (SubGrp‘𝐺))
6	5, 3	eleqtrdi 2843	. . . . 5 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ 𝑈 ⊆ 𝑆) → 𝑇 ∈ (SubGrp‘(oppg‘𝐺)))
7		simpl1 1191	. . . . . 6 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ 𝑈 ⊆ 𝑆) → 𝑆 ∈ (SubGrp‘𝐺))
8	7, 3	eleqtrdi 2843	. . . . 5 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ 𝑈 ⊆ 𝑆) → 𝑆 ∈ (SubGrp‘(oppg‘𝐺)))
9		simpr 485	. . . . 5 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ 𝑈 ⊆ 𝑆) → 𝑈 ⊆ 𝑆)
10		eqid 2732	. . . . . 6 ⊢ (LSSum‘(oppg‘𝐺)) = (LSSum‘(oppg‘𝐺))
11	10	lsmmod 19542	. . . . 5 ⊢ (((𝑈 ∈ (SubGrp‘(oppg‘𝐺)) ∧ 𝑇 ∈ (SubGrp‘(oppg‘𝐺)) ∧ 𝑆 ∈ (SubGrp‘(oppg‘𝐺))) ∧ 𝑈 ⊆ 𝑆) → (𝑈(LSSum‘(oppg‘𝐺))(𝑇 ∩ 𝑆)) = ((𝑈(LSSum‘(oppg‘𝐺))𝑇) ∩ 𝑆))
12	4, 6, 8, 9, 11	syl31anc 1373	. . . 4 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ 𝑈 ⊆ 𝑆) → (𝑈(LSSum‘(oppg‘𝐺))(𝑇 ∩ 𝑆)) = ((𝑈(LSSum‘(oppg‘𝐺))𝑇) ∩ 𝑆))
13	12	eqcomd 2738	. . 3 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ 𝑈 ⊆ 𝑆) → ((𝑈(LSSum‘(oppg‘𝐺))𝑇) ∩ 𝑆) = (𝑈(LSSum‘(oppg‘𝐺))(𝑇 ∩ 𝑆)))
14		incom 4201	. . 3 ⊢ ((𝑈(LSSum‘(oppg‘𝐺))𝑇) ∩ 𝑆) = (𝑆 ∩ (𝑈(LSSum‘(oppg‘𝐺))𝑇))
15		incom 4201	. . . 4 ⊢ (𝑇 ∩ 𝑆) = (𝑆 ∩ 𝑇)
16	15	oveq2i 7419	. . 3 ⊢ (𝑈(LSSum‘(oppg‘𝐺))(𝑇 ∩ 𝑆)) = (𝑈(LSSum‘(oppg‘𝐺))(𝑆 ∩ 𝑇))
17	13, 14, 16	3eqtr3g 2795	. 2 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ 𝑈 ⊆ 𝑆) → (𝑆 ∩ (𝑈(LSSum‘(oppg‘𝐺))𝑇)) = (𝑈(LSSum‘(oppg‘𝐺))(𝑆 ∩ 𝑇)))
18		lsmmod.p	. . . 4 ⊢ ⊕ = (LSSum‘𝐺)
19	2, 18	oppglsm 19509	. . 3 ⊢ (𝑈(LSSum‘(oppg‘𝐺))𝑇) = (𝑇 ⊕ 𝑈)
20	19	ineq2i 4209	. 2 ⊢ (𝑆 ∩ (𝑈(LSSum‘(oppg‘𝐺))𝑇)) = (𝑆 ∩ (𝑇 ⊕ 𝑈))
21	2, 18	oppglsm 19509	. 2 ⊢ (𝑈(LSSum‘(oppg‘𝐺))(𝑆 ∩ 𝑇)) = ((𝑆 ∩ 𝑇) ⊕ 𝑈)
22	17, 20, 21	3eqtr3g 2795	1 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ 𝑈 ⊆ 𝑆) → (𝑆 ∩ (𝑇 ⊕ 𝑈)) = ((𝑆 ∩ 𝑇) ⊕ 𝑈))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   ∩ cin 3947   ⊆ wss 3948  ‘cfv 6543  (class class class)co 7408  SubGrpcsubg 18999  opp_gcoppg 19208  LSSumclsm 19501

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724  ax-cnex 11165  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185  ax-pre-mulgt0 11186

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-iin 5000  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7855  df-1st 7974  df-2nd 7975  df-tpos 8210  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-1o 8465  df-er 8702  df-en 8939  df-dom 8940  df-sdom 8941  df-fin 8942  df-pnf 11249  df-mnf 11250  df-xr 11251  df-ltxr 11252  df-le 11253  df-sub 11445  df-neg 11446  df-nn 12212  df-2 12274  df-sets 17096  df-slot 17114  df-ndx 17126  df-base 17144  df-ress 17173  df-plusg 17209  df-0g 17386  df-mre 17529  df-mrc 17530  df-acs 17532  df-mgm 18560  df-sgrp 18609  df-mnd 18625  df-submnd 18671  df-grp 18821  df-minusg 18822  df-subg 19002  df-oppg 19209  df-lsm 19503

This theorem is referenced by:  lcvexchlem3  37901  lcfrlem23  40431