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Theorem lsmmod2 16166
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 988 . . . . . 6  |-  ( ( ( S  e.  (SubGrp `  G )  /\  T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G ) )  /\  U  C_  S )  ->  U  e.  (SubGrp `  G
) )
2 eqid 2441 . . . . . . 7  |-  (oppg `  G
)  =  (oppg `  G
)
32oppgsubg 15871 . . . . . 6  |-  (SubGrp `  G )  =  (SubGrp `  (oppg
`  G ) )
41, 3syl6eleq 2531 . . . . 5  |-  ( ( ( S  e.  (SubGrp `  G )  /\  T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G ) )  /\  U  C_  S )  ->  U  e.  (SubGrp `  (oppg `  G
) ) )
5 simpl2 987 . . . . . 6  |-  ( ( ( S  e.  (SubGrp `  G )  /\  T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G ) )  /\  U  C_  S )  ->  T  e.  (SubGrp `  G
) )
65, 3syl6eleq 2531 . . . . 5  |-  ( ( ( S  e.  (SubGrp `  G )  /\  T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G ) )  /\  U  C_  S )  ->  T  e.  (SubGrp `  (oppg `  G
) ) )
7 simpl1 986 . . . . . 6  |-  ( ( ( S  e.  (SubGrp `  G )  /\  T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G ) )  /\  U  C_  S )  ->  S  e.  (SubGrp `  G
) )
87, 3syl6eleq 2531 . . . . 5  |-  ( ( ( S  e.  (SubGrp `  G )  /\  T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G ) )  /\  U  C_  S )  ->  S  e.  (SubGrp `  (oppg `  G
) ) )
9 simpr 458 . . . . 5  |-  ( ( ( S  e.  (SubGrp `  G )  /\  T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G ) )  /\  U  C_  S )  ->  U  C_  S )
10 eqid 2441 . . . . . 6  |-  ( LSSum `  (oppg
`  G ) )  =  ( LSSum `  (oppg `  G
) )
1110lsmmod 16165 . . . . 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 1216 . . . 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 2446 . . 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 3540 . . 3  |-  ( ( U ( LSSum `  (oppg `  G
) ) T )  i^i  S )  =  ( S  i^i  ( U ( LSSum `  (oppg `  G
) ) T ) )
15 incom 3540 . . . 4  |-  ( T  i^i  S )  =  ( S  i^i  T
)
1615oveq2i 6101 . . 3  |-  ( U ( LSSum `  (oppg
`  G ) ) ( T  i^i  S
) )  =  ( U ( LSSum `  (oppg `  G
) ) ( S  i^i  T ) )
1713, 14, 163eqtr3g 2496 . 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 16134 . . 3  |-  ( U ( LSSum `  (oppg
`  G ) ) T )  =  ( T  .(+)  U )
2019ineq2i 3546 . 2  |-  ( S  i^i  ( U (
LSSum `  (oppg
`  G ) ) T ) )  =  ( S  i^i  ( T  .(+)  U ) )
212, 18oppglsm 16134 . 2  |-  ( U ( LSSum `  (oppg
`  G ) ) ( S  i^i  T
) )  =  ( ( S  i^i  T
)  .(+)  U )
2217, 20, 213eqtr3g 2496 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 960    = wceq 1364    e. wcel 1761    i^i cin 3324    C_ wss 3325   ` cfv 5415  (class class class)co 6090  SubGrpcsubg 15668  oppgcoppg 15853   LSSumclsm 16126
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-iin 4171  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-tpos 6744  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-2 10376  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-plusg 14247  df-0g 14376  df-mre 14520  df-mrc 14521  df-acs 14523  df-mnd 15411  df-submnd 15461  df-grp 15538  df-minusg 15539  df-subg 15671  df-oppg 15854  df-lsm 16128
This theorem is referenced by:  lcvexchlem3  32403  lcfrlem23  34932
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