Theorem lsmmod2 18533

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 1186	. . . . . 6 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ 𝑈 ⊆ 𝑆) → 𝑈 ∈ (SubGrp‘𝐺))
2		eqid 2797	. . . . . . 7 ⊢ (oppg‘𝐺) = (oppg‘𝐺)
3	2	oppgsubg 18236	. . . . . 6 ⊢ (SubGrp‘𝐺) = (SubGrp‘(oppg‘𝐺))
4	1, 3	syl6eleq 2895	. . . . 5 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ 𝑈 ⊆ 𝑆) → 𝑈 ∈ (SubGrp‘(oppg‘𝐺)))
5		simpl2 1185	. . . . . 6 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ 𝑈 ⊆ 𝑆) → 𝑇 ∈ (SubGrp‘𝐺))
6	5, 3	syl6eleq 2895	. . . . 5 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ 𝑈 ⊆ 𝑆) → 𝑇 ∈ (SubGrp‘(oppg‘𝐺)))
7		simpl1 1184	. . . . . 6 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ 𝑈 ⊆ 𝑆) → 𝑆 ∈ (SubGrp‘𝐺))
8	7, 3	syl6eleq 2895	. . . . 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 2797	. . . . . 6 ⊢ (LSSum‘(oppg‘𝐺)) = (LSSum‘(oppg‘𝐺))
11	10	lsmmod 18532	. . . . 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 1366	. . . 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 2803	. . 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 4105	. . 3 ⊢ ((𝑈(LSSum‘(oppg‘𝐺))𝑇) ∩ 𝑆) = (𝑆 ∩ (𝑈(LSSum‘(oppg‘𝐺))𝑇))
15		incom 4105	. . . 4 ⊢ (𝑇 ∩ 𝑆) = (𝑆 ∩ 𝑇)
16	15	oveq2i 7034	. . 3 ⊢ (𝑈(LSSum‘(oppg‘𝐺))(𝑇 ∩ 𝑆)) = (𝑈(LSSum‘(oppg‘𝐺))(𝑆 ∩ 𝑇))
17	13, 14, 16	3eqtr3g 2856	. 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 18501	. . 3 ⊢ (𝑈(LSSum‘(oppg‘𝐺))𝑇) = (𝑇 ⊕ 𝑈)
20	19	ineq2i 4112	. 2 ⊢ (𝑆 ∩ (𝑈(LSSum‘(oppg‘𝐺))𝑇)) = (𝑆 ∩ (𝑇 ⊕ 𝑈))
21	2, 18	oppglsm 18501	. 2 ⊢ (𝑈(LSSum‘(oppg‘𝐺))(𝑆 ∩ 𝑇)) = ((𝑆 ∩ 𝑇) ⊕ 𝑈)
22	17, 20, 21	3eqtr3g 2856	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 1080   = wceq 1525   ∈ wcel 2083   ∩ cin 3864   ⊆ wss 3865  ‘cfv 6232  (class class class)co 7023  SubGrpcsubg 18031  opp_gcoppg 18218  LSSumclsm 18493

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-rep 5088  ax-sep 5101  ax-nul 5108  ax-pow 5164  ax-pr 5228  ax-un 7326  ax-cnex 10446  ax-resscn 10447  ax-1cn 10448  ax-icn 10449  ax-addcl 10450  ax-addrcl 10451  ax-mulcl 10452  ax-mulrcl 10453  ax-mulcom 10454  ax-addass 10455  ax-mulass 10456  ax-distr 10457  ax-i2m1 10458  ax-1ne0 10459  ax-1rid 10460  ax-rnegex 10461  ax-rrecex 10462  ax-cnre 10463  ax-pre-lttri 10464  ax-pre-lttrn 10465  ax-pre-ltadd 10466  ax-pre-mulgt0 10467

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-nel 3093  df-ral 3112  df-rex 3113  df-reu 3114  df-rmo 3115  df-rab 3116  df-v 3442  df-sbc 3712  df-csb 3818  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-pss 3882  df-nul 4218  df-if 4388  df-pw 4461  df-sn 4479  df-pr 4481  df-tp 4483  df-op 4485  df-uni 4752  df-int 4789  df-iun 4833  df-iin 4834  df-br 4969  df-opab 5031  df-mpt 5048  df-tr 5071  df-id 5355  df-eprel 5360  df-po 5369  df-so 5370  df-fr 5409  df-we 5411  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-pred 6030  df-ord 6076  df-on 6077  df-lim 6078  df-suc 6079  df-iota 6196  df-fun 6234  df-fn 6235  df-f 6236  df-f1 6237  df-fo 6238  df-f1o 6239  df-fv 6240  df-riota 6984  df-ov 7026  df-oprab 7027  df-mpo 7028  df-om 7444  df-1st 7552  df-2nd 7553  df-tpos 7750  df-wrecs 7805  df-recs 7867  df-rdg 7905  df-1o 7960  df-oadd 7964  df-er 8146  df-en 8365  df-dom 8366  df-sdom 8367  df-fin 8368  df-pnf 10530  df-mnf 10531  df-xr 10532  df-ltxr 10533  df-le 10534  df-sub 10725  df-neg 10726  df-nn 11493  df-2 11554  df-ndx 16319  df-slot 16320  df-base 16322  df-sets 16323  df-ress 16324  df-plusg 16411  df-0g 16548  df-mre 16690  df-mrc 16691  df-acs 16693  df-mgm 17685  df-sgrp 17727  df-mnd 17738  df-submnd 17779  df-grp 17868  df-minusg 17869  df-subg 18034  df-oppg 18219  df-lsm 18495

This theorem is referenced by:  lcvexchlem3  35724  lcfrlem23  38253