Theorem lsmmod2 19716

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 1207	. . . . . 6 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ 𝑈 ⊆ 𝑆) → 𝑈 ∈ (SubGrp‘𝐺))
2		eqid 2762	. . . . . . 7 ⊢ (oppg‘𝐺) = (oppg‘𝐺)
3	2	oppgsubg 19403	. . . . . 6 ⊢ (SubGrp‘𝐺) = (SubGrp‘(oppg‘𝐺))
4	1, 3	eleqtrdi 2872	. . . . 5 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ 𝑈 ⊆ 𝑆) → 𝑈 ∈ (SubGrp‘(oppg‘𝐺)))
5		simpl2 1206	. . . . . 6 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ 𝑈 ⊆ 𝑆) → 𝑇 ∈ (SubGrp‘𝐺))
6	5, 3	eleqtrdi 2872	. . . . 5 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ 𝑈 ⊆ 𝑆) → 𝑇 ∈ (SubGrp‘(oppg‘𝐺)))
7		simpl1 1205	. . . . . 6 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ 𝑈 ⊆ 𝑆) → 𝑆 ∈ (SubGrp‘𝐺))
8	7, 3	eleqtrdi 2872	. . . . 5 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ 𝑈 ⊆ 𝑆) → 𝑆 ∈ (SubGrp‘(oppg‘𝐺)))
9		simpr 488	. . . . 5 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ 𝑈 ⊆ 𝑆) → 𝑈 ⊆ 𝑆)
10		eqid 2762	. . . . . 6 ⊢ (LSSum‘(oppg‘𝐺)) = (LSSum‘(oppg‘𝐺))
11	10	lsmmod 19715	. . . . 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 1392	. . . 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 2768	. . 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 4161	. . 3 ⊢ ((𝑈(LSSum‘(oppg‘𝐺))𝑇) ∩ 𝑆) = (𝑆 ∩ (𝑈(LSSum‘(oppg‘𝐺))𝑇))
15		incom 4161	. . . 4 ⊢ (𝑇 ∩ 𝑆) = (𝑆 ∩ 𝑇)
16	15	oveq2i 7407	. . 3 ⊢ (𝑈(LSSum‘(oppg‘𝐺))(𝑇 ∩ 𝑆)) = (𝑈(LSSum‘(oppg‘𝐺))(𝑆 ∩ 𝑇))
17	13, 14, 16	3eqtr3g 2820	. 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 19682	. . 3 ⊢ (𝑈(LSSum‘(oppg‘𝐺))𝑇) = (𝑇 ⊕ 𝑈)
20	19	ineq2i 4169	. 2 ⊢ (𝑆 ∩ (𝑈(LSSum‘(oppg‘𝐺))𝑇)) = (𝑆 ∩ (𝑇 ⊕ 𝑈))
21	2, 18	oppglsm 19682	. 2 ⊢ (𝑈(LSSum‘(oppg‘𝐺))(𝑆 ∩ 𝑇)) = ((𝑆 ∩ 𝑇) ⊕ 𝑈)
22	17, 20, 21	3eqtr3g 2820	1 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ 𝑈 ⊆ 𝑆) → (𝑆 ∩ (𝑇 ⊕ 𝑈)) = ((𝑆 ∩ 𝑇) ⊕ 𝑈))

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1098   = wceq 1560   ∈ wcel 2142   ∩ cin 3903   ⊆ wss 3904  ‘cfv 6521  (class class class)co 7396  SubGrpcsubg 19162  opp_gcoppg 19385  LSSumclsm 19674

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718  ax-cnex 11129  ax-resscn 11130  ax-1cn 11131  ax-icn 11132  ax-addcl 11133  ax-addrcl 11134  ax-mulcl 11135  ax-mulrcl 11136  ax-mulcom 11137  ax-addass 11138  ax-mulass 11139  ax-distr 11140  ax-i2m1 11141  ax-1ne0 11142  ax-1rid 11143  ax-rnegex 11144  ax-rrecex 11145  ax-cnre 11146  ax-pre-lttri 11147  ax-pre-lttrn 11148  ax-pre-ltadd 11149  ax-pre-mulgt0 11150

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4906  df-iun 4951  df-iin 4952  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6288  df-ord 6349  df-on 6350  df-lim 6351  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-riota 7353  df-ov 7399  df-oprab 7400  df-mpo 7401  df-om 7847  df-1st 7970  df-2nd 7971  df-tpos 8206  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8381  df-1o 8437  df-2o 8438  df-er 8678  df-en 8928  df-dom 8929  df-sdom 8930  df-fin 8931  df-pnf 11218  df-mnf 11219  df-xr 11220  df-ltxr 11221  df-le 11222  df-sub 11416  df-neg 11417  df-nn 12211  df-2 12280  df-sets 17200  df-slot 17218  df-ndx 17230  df-base 17246  df-ress 17267  df-plusg 17299  df-0g 17470  df-mre 17614  df-mrc 17615  df-acs 17617  df-mgm 18674  df-sgrp 18753  df-mnd 18769  df-submnd 18818  df-grp 18978  df-minusg 18979  df-subg 19165  df-oppg 19386  df-lsm 19676

This theorem is referenced by:  lcvexchlem3  39660  lcfrlem23  42189