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Theorem lsmmod2 16567
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  e.  (SubGrp `  G )  /\  T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G ) )  /\  U  C_  S )  -> 
( S  i^i  ( T  .(+)  U ) )  =  ( ( S  i^i  T )  .(+)  U ) )

Proof of Theorem lsmmod2
StepHypRef Expression
1 simpl3 1001 . . . . . 6  |-  ( ( ( S  e.  (SubGrp `  G )  /\  T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G ) )  /\  U  C_  S )  ->  U  e.  (SubGrp `  G
) )
2 eqid 2467 . . . . . . 7  |-  (oppg `  G
)  =  (oppg `  G
)
32oppgsubg 16270 . . . . . 6  |-  (SubGrp `  G )  =  (SubGrp `  (oppg
`  G ) )
41, 3syl6eleq 2565 . . . . 5  |-  ( ( ( S  e.  (SubGrp `  G )  /\  T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G ) )  /\  U  C_  S )  ->  U  e.  (SubGrp `  (oppg `  G
) ) )
5 simpl2 1000 . . . . . 6  |-  ( ( ( S  e.  (SubGrp `  G )  /\  T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G ) )  /\  U  C_  S )  ->  T  e.  (SubGrp `  G
) )
65, 3syl6eleq 2565 . . . . 5  |-  ( ( ( S  e.  (SubGrp `  G )  /\  T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G ) )  /\  U  C_  S )  ->  T  e.  (SubGrp `  (oppg `  G
) ) )
7 simpl1 999 . . . . . 6  |-  ( ( ( S  e.  (SubGrp `  G )  /\  T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G ) )  /\  U  C_  S )  ->  S  e.  (SubGrp `  G
) )
87, 3syl6eleq 2565 . . . . 5  |-  ( ( ( S  e.  (SubGrp `  G )  /\  T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G ) )  /\  U  C_  S )  ->  S  e.  (SubGrp `  (oppg `  G
) ) )
9 simpr 461 . . . . 5  |-  ( ( ( S  e.  (SubGrp `  G )  /\  T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G ) )  /\  U  C_  S )  ->  U  C_  S )
10 eqid 2467 . . . . . 6  |-  ( LSSum `  (oppg
`  G ) )  =  ( LSSum `  (oppg `  G
) )
1110lsmmod 16566 . . . . 5  |-  ( ( ( U  e.  (SubGrp `  (oppg
`  G ) )  /\  T  e.  (SubGrp `  (oppg
`  G ) )  /\  S  e.  (SubGrp `  (oppg
`  G ) ) )  /\  U  C_  S )  ->  ( U ( LSSum `  (oppg `  G
) ) ( T  i^i  S ) )  =  ( ( U ( LSSum `  (oppg
`  G ) ) T )  i^i  S
) )
124, 6, 8, 9, 11syl31anc 1231 . . . 4  |-  ( ( ( S  e.  (SubGrp `  G )  /\  T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G ) )  /\  U  C_  S )  -> 
( U ( LSSum `  (oppg
`  G ) ) ( T  i^i  S
) )  =  ( ( U ( LSSum `  (oppg
`  G ) ) T )  i^i  S
) )
1312eqcomd 2475 . . 3  |-  ( ( ( S  e.  (SubGrp `  G )  /\  T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G ) )  /\  U  C_  S )  -> 
( ( U (
LSSum `  (oppg
`  G ) ) T )  i^i  S
)  =  ( U ( LSSum `  (oppg
`  G ) ) ( T  i^i  S
) ) )
14 incom 3696 . . 3  |-  ( ( U ( LSSum `  (oppg `  G
) ) T )  i^i  S )  =  ( S  i^i  ( U ( LSSum `  (oppg `  G
) ) T ) )
15 incom 3696 . . . 4  |-  ( T  i^i  S )  =  ( S  i^i  T
)
1615oveq2i 6306 . . 3  |-  ( U ( LSSum `  (oppg
`  G ) ) ( T  i^i  S
) )  =  ( U ( LSSum `  (oppg `  G
) ) ( S  i^i  T ) )
1713, 14, 163eqtr3g 2531 . 2  |-  ( ( ( S  e.  (SubGrp `  G )  /\  T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G ) )  /\  U  C_  S )  -> 
( S  i^i  ( U ( LSSum `  (oppg `  G
) ) T ) )  =  ( U ( LSSum `  (oppg
`  G ) ) ( S  i^i  T
) ) )
18 lsmmod.p . . . 4  |-  .(+)  =  (
LSSum `  G )
192, 18oppglsm 16535 . . 3  |-  ( U ( LSSum `  (oppg
`  G ) ) T )  =  ( T  .(+)  U )
2019ineq2i 3702 . 2  |-  ( S  i^i  ( U (
LSSum `  (oppg
`  G ) ) T ) )  =  ( S  i^i  ( T  .(+)  U ) )
212, 18oppglsm 16535 . 2  |-  ( U ( LSSum `  (oppg
`  G ) ) ( S  i^i  T
) )  =  ( ( S  i^i  T
)  .(+)  U )
2217, 20, 213eqtr3g 2531 1  |-  ( ( ( S  e.  (SubGrp `  G )  /\  T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G ) )  /\  U  C_  S )  -> 
( S  i^i  ( T  .(+)  U ) )  =  ( ( S  i^i  T )  .(+)  U ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    i^i cin 3480    C_ wss 3481   ` cfv 5594  (class class class)co 6295  SubGrpcsubg 16067  oppgcoppg 16252   LSSumclsm 16527
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-iin 4334  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-tpos 6967  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-ress 14514  df-plusg 14585  df-0g 14714  df-mre 14858  df-mrc 14859  df-acs 14861  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-submnd 15840  df-grp 15929  df-minusg 15930  df-subg 16070  df-oppg 16253  df-lsm 16529
This theorem is referenced by:  lcvexchlem3  34234  lcfrlem23  36763
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