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Theorem lsmmod2 14985
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 960 . . . . . 6  |-  ( ( ( S  e.  (SubGrp `  G )  /\  T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G ) )  /\  U  C_  S )  ->  U  e.  (SubGrp `  G
) )
2 eqid 2283 . . . . . . 7  |-  (oppg `  G
)  =  (oppg `  G
)
32oppgsubg 14836 . . . . . 6  |-  (SubGrp `  G )  =  (SubGrp `  (oppg
`  G ) )
41, 3syl6eleq 2373 . . . . 5  |-  ( ( ( S  e.  (SubGrp `  G )  /\  T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G ) )  /\  U  C_  S )  ->  U  e.  (SubGrp `  (oppg `  G
) ) )
5 simpl2 959 . . . . . 6  |-  ( ( ( S  e.  (SubGrp `  G )  /\  T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G ) )  /\  U  C_  S )  ->  T  e.  (SubGrp `  G
) )
65, 3syl6eleq 2373 . . . . 5  |-  ( ( ( S  e.  (SubGrp `  G )  /\  T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G ) )  /\  U  C_  S )  ->  T  e.  (SubGrp `  (oppg `  G
) ) )
7 simpl1 958 . . . . . 6  |-  ( ( ( S  e.  (SubGrp `  G )  /\  T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G ) )  /\  U  C_  S )  ->  S  e.  (SubGrp `  G
) )
87, 3syl6eleq 2373 . . . . 5  |-  ( ( ( S  e.  (SubGrp `  G )  /\  T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G ) )  /\  U  C_  S )  ->  S  e.  (SubGrp `  (oppg `  G
) ) )
9 simpr 447 . . . . 5  |-  ( ( ( S  e.  (SubGrp `  G )  /\  T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G ) )  /\  U  C_  S )  ->  U  C_  S )
10 eqid 2283 . . . . . 6  |-  ( LSSum `  (oppg
`  G ) )  =  ( LSSum `  (oppg `  G
) )
1110lsmmod 14984 . . . . 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 1185 . . . 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 2288 . . 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 3361 . . 3  |-  ( ( U ( LSSum `  (oppg `  G
) ) T )  i^i  S )  =  ( S  i^i  ( U ( LSSum `  (oppg `  G
) ) T ) )
15 incom 3361 . . . 4  |-  ( T  i^i  S )  =  ( S  i^i  T
)
1615oveq2i 5869 . . 3  |-  ( U ( LSSum `  (oppg
`  G ) ) ( T  i^i  S
) )  =  ( U ( LSSum `  (oppg `  G
) ) ( S  i^i  T ) )
1713, 14, 163eqtr3g 2338 . 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 14953 . . 3  |-  ( U ( LSSum `  (oppg
`  G ) ) T )  =  ( T  .(+)  U )
2019ineq2i 3367 . 2  |-  ( S  i^i  ( U (
LSSum `  (oppg
`  G ) ) T ) )  =  ( S  i^i  ( T  .(+)  U ) )
212, 18oppglsm 14953 . 2  |-  ( U ( LSSum `  (oppg
`  G ) ) ( S  i^i  T
) )  =  ( ( S  i^i  T
)  .(+)  U )
2217, 20, 213eqtr3g 2338 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 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    i^i cin 3151    C_ wss 3152   ` cfv 5255  (class class class)co 5858  SubGrpcsubg 14615  oppgcoppg 14818   LSSumclsm 14945
This theorem is referenced by:  lcvexchlem3  28599  lcfrlem23  31128
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-tpos 6234  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-0g 13404  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-grp 14489  df-minusg 14490  df-subg 14618  df-oppg 14819  df-lsm 14947
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