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Theorem lsmmod2 14979
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 962 . . . . . 6  |-  ( ( ( S  e.  (SubGrp `  G )  /\  T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G ) )  /\  U  C_  S )  ->  U  e.  (SubGrp `  G
) )
2 eqid 2284 . . . . . . 7  |-  (oppg `  G
)  =  (oppg `  G
)
32oppgsubg 14830 . . . . . 6  |-  (SubGrp `  G )  =  (SubGrp `  (oppg
`  G ) )
41, 3syl6eleq 2374 . . . . 5  |-  ( ( ( S  e.  (SubGrp `  G )  /\  T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G ) )  /\  U  C_  S )  ->  U  e.  (SubGrp `  (oppg `  G
) ) )
5 simpl2 961 . . . . . 6  |-  ( ( ( S  e.  (SubGrp `  G )  /\  T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G ) )  /\  U  C_  S )  ->  T  e.  (SubGrp `  G
) )
65, 3syl6eleq 2374 . . . . 5  |-  ( ( ( S  e.  (SubGrp `  G )  /\  T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G ) )  /\  U  C_  S )  ->  T  e.  (SubGrp `  (oppg `  G
) ) )
7 simpl1 960 . . . . . 6  |-  ( ( ( S  e.  (SubGrp `  G )  /\  T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G ) )  /\  U  C_  S )  ->  S  e.  (SubGrp `  G
) )
87, 3syl6eleq 2374 . . . . 5  |-  ( ( ( S  e.  (SubGrp `  G )  /\  T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G ) )  /\  U  C_  S )  ->  S  e.  (SubGrp `  (oppg `  G
) ) )
9 simpr 449 . . . . 5  |-  ( ( ( S  e.  (SubGrp `  G )  /\  T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G ) )  /\  U  C_  S )  ->  U  C_  S )
10 eqid 2284 . . . . . 6  |-  ( LSSum `  (oppg
`  G ) )  =  ( LSSum `  (oppg `  G
) )
1110lsmmod 14978 . . . . 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 1187 . . . 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 2289 . . 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 3362 . . 3  |-  ( ( U ( LSSum `  (oppg `  G
) ) T )  i^i  S )  =  ( S  i^i  ( U ( LSSum `  (oppg `  G
) ) T ) )
15 incom 3362 . . . 4  |-  ( T  i^i  S )  =  ( S  i^i  T
)
1615oveq2i 5830 . . 3  |-  ( U ( LSSum `  (oppg
`  G ) ) ( T  i^i  S
) )  =  ( U ( LSSum `  (oppg `  G
) ) ( S  i^i  T ) )
1713, 14, 163eqtr3g 2339 . 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 14947 . . 3  |-  ( U ( LSSum `  (oppg
`  G ) ) T )  =  ( T  .(+)  U )
2019ineq2i 3368 . 2  |-  ( S  i^i  ( U (
LSSum `  (oppg
`  G ) ) T ) )  =  ( S  i^i  ( T  .(+)  U ) )
212, 18oppglsm 14947 . 2  |-  ( U ( LSSum `  (oppg
`  G ) ) ( S  i^i  T
) )  =  ( ( S  i^i  T
)  .(+)  U )
2217, 20, 213eqtr3g 2339 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 set class
Syntax hints:    -> wi 6    /\ wa 360    /\ w3a 936    = wceq 1624    e. wcel 1685    i^i cin 3152    C_ wss 3153   ` cfv 5221  (class class class)co 5819  SubGrpcsubg 14609  oppgcoppg 14812   LSSumclsm 14939
This theorem is referenced by:  lcvexchlem3  28493  lcfrlem23  31022
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-cnex 8788  ax-resscn 8789  ax-1cn 8790  ax-icn 8791  ax-addcl 8792  ax-addrcl 8793  ax-mulcl 8794  ax-mulrcl 8795  ax-mulcom 8796  ax-addass 8797  ax-mulass 8798  ax-distr 8799  ax-i2m1 8800  ax-1ne0 8801  ax-1rid 8802  ax-rnegex 8803  ax-rrecex 8804  ax-cnre 8805  ax-pre-lttri 8806  ax-pre-lttrn 8807  ax-pre-ltadd 8808  ax-pre-mulgt0 8809
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-int 3864  df-iun 3908  df-iin 3909  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-1st 6083  df-2nd 6084  df-tpos 6195  df-iota 6252  df-riota 6299  df-recs 6383  df-rdg 6418  df-1o 6474  df-oadd 6478  df-er 6655  df-en 6859  df-dom 6860  df-sdom 6861  df-fin 6862  df-pnf 8864  df-mnf 8865  df-xr 8866  df-ltxr 8867  df-le 8868  df-sub 9034  df-neg 9035  df-nn 9742  df-2 9799  df-ndx 13145  df-slot 13146  df-base 13147  df-sets 13148  df-ress 13149  df-plusg 13215  df-0g 13398  df-mre 13482  df-mrc 13483  df-acs 13485  df-mnd 14361  df-submnd 14410  df-grp 14483  df-minusg 14484  df-subg 14612  df-oppg 14813  df-lsm 14941
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