Theorem lsmmod2 19792

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 1212	. . . . . 6 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ 𝑈 ⊆ 𝑆) → 𝑈 ∈ (SubGrp‘𝐺))
2		eqid 2765	. . . . . . 7 ⊢ (oppg‘𝐺) = (oppg‘𝐺)
3	2	oppgsubg 19479	. . . . . 6 ⊢ (SubGrp‘𝐺) = (SubGrp‘(oppg‘𝐺))
4	1, 3	eleqtrdi 2875	. . . . 5 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ 𝑈 ⊆ 𝑆) → 𝑈 ∈ (SubGrp‘(oppg‘𝐺)))
5		simpl2 1211	. . . . . 6 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ 𝑈 ⊆ 𝑆) → 𝑇 ∈ (SubGrp‘𝐺))
6	5, 3	eleqtrdi 2875	. . . . 5 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ 𝑈 ⊆ 𝑆) → 𝑇 ∈ (SubGrp‘(oppg‘𝐺)))
7		simpl1 1210	. . . . . 6 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ 𝑈 ⊆ 𝑆) → 𝑆 ∈ (SubGrp‘𝐺))
8	7, 3	eleqtrdi 2875	. . . . 5 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ 𝑈 ⊆ 𝑆) → 𝑆 ∈ (SubGrp‘(oppg‘𝐺)))
9		simpr 490	. . . . 5 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ 𝑈 ⊆ 𝑆) → 𝑈 ⊆ 𝑆)
10		eqid 2765	. . . . . 6 ⊢ (LSSum‘(oppg‘𝐺)) = (LSSum‘(oppg‘𝐺))
11	10	lsmmod 19791	. . . . 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 1400	. . . 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 2771	. . 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 4162	. . 3 ⊢ ((𝑈(LSSum‘(oppg‘𝐺))𝑇) ∩ 𝑆) = (𝑆 ∩ (𝑈(LSSum‘(oppg‘𝐺))𝑇))
15		incom 4162	. . . 4 ⊢ (𝑇 ∩ 𝑆) = (𝑆 ∩ 𝑇)
16	15	oveq2i 7430	. . 3 ⊢ (𝑈(LSSum‘(oppg‘𝐺))(𝑇 ∩ 𝑆)) = (𝑈(LSSum‘(oppg‘𝐺))(𝑆 ∩ 𝑇))
17	13, 14, 16	3eqtr3g 2823	. 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 19758	. . 3 ⊢ (𝑈(LSSum‘(oppg‘𝐺))𝑇) = (𝑇 ⊕ 𝑈)
20	19	ineq2i 4170	. 2 ⊢ (𝑆 ∩ (𝑈(LSSum‘(oppg‘𝐺))𝑇)) = (𝑆 ∩ (𝑇 ⊕ 𝑈))
21	2, 18	oppglsm 19758	. 2 ⊢ (𝑈(LSSum‘(oppg‘𝐺))(𝑆 ∩ 𝑇)) = ((𝑆 ∩ 𝑇) ⊕ 𝑈)
22	17, 20, 21	3eqtr3g 2823	1 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ 𝑈 ⊆ 𝑆) → (𝑆 ∩ (𝑇 ⊕ 𝑈)) = ((𝑆 ∩ 𝑇) ⊕ 𝑈))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146   ∩ cin 3905   ⊆ wss 3906  ‘cfv 6540  (class class class)co 7419  SubGrpcsubg 19232  opp_gcoppg 19461  LSSumclsm 19750

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742  ax-cnex 11173  ax-resscn 11174  ax-1cn 11175  ax-icn 11176  ax-addcl 11177  ax-addrcl 11178  ax-mulcl 11179  ax-mulrcl 11180  ax-mulcom 11181  ax-addass 11182  ax-mulass 11183  ax-distr 11184  ax-i2m1 11185  ax-1ne0 11186  ax-1rid 11187  ax-rnegex 11188  ax-rrecex 11189  ax-cnre 11190  ax-pre-lttri 11191  ax-pre-lttrn 11192  ax-pre-ltadd 11193  ax-pre-mulgt0 11194

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-int 4915  df-iun 4960  df-iin 4961  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7376  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7869  df-1st 7992  df-2nd 7993  df-tpos 8228  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-1o 8459  df-2o 8460  df-er 8700  df-en 8950  df-dom 8951  df-sdom 8952  df-fin 8953  df-pnf 11262  df-mnf 11263  df-xr 11264  df-ltxr 11265  df-le 11266  df-sub 11460  df-neg 11461  df-nn 12251  df-2 12320  df-sets 17248  df-slot 17266  df-ndx 17278  df-base 17294  df-ress 17315  df-plusg 17347  df-0g 17518  df-mre 17662  df-mrc 17663  df-acs 17665  df-mgm 18722  df-sgrp 18811  df-mnd 18827  df-submnd 18881  df-grp 19049  df-minusg 19050  df-subg 19235  df-oppg 19462  df-lsm 19752

This theorem is used by:  lcvexchlem3  39870  lcfrlem23  42399