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Theorem lsmmod2 16178
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 993 . . . . . 6  |-  ( ( ( S  e.  (SubGrp `  G )  /\  T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G ) )  /\  U  C_  S )  ->  U  e.  (SubGrp `  G
) )
2 eqid 2443 . . . . . . 7  |-  (oppg `  G
)  =  (oppg `  G
)
32oppgsubg 15883 . . . . . 6  |-  (SubGrp `  G )  =  (SubGrp `  (oppg
`  G ) )
41, 3syl6eleq 2533 . . . . 5  |-  ( ( ( S  e.  (SubGrp `  G )  /\  T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G ) )  /\  U  C_  S )  ->  U  e.  (SubGrp `  (oppg `  G
) ) )
5 simpl2 992 . . . . . 6  |-  ( ( ( S  e.  (SubGrp `  G )  /\  T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G ) )  /\  U  C_  S )  ->  T  e.  (SubGrp `  G
) )
65, 3syl6eleq 2533 . . . . 5  |-  ( ( ( S  e.  (SubGrp `  G )  /\  T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G ) )  /\  U  C_  S )  ->  T  e.  (SubGrp `  (oppg `  G
) ) )
7 simpl1 991 . . . . . 6  |-  ( ( ( S  e.  (SubGrp `  G )  /\  T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G ) )  /\  U  C_  S )  ->  S  e.  (SubGrp `  G
) )
87, 3syl6eleq 2533 . . . . 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 2443 . . . . . 6  |-  ( LSSum `  (oppg
`  G ) )  =  ( LSSum `  (oppg `  G
) )
1110lsmmod 16177 . . . . 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 1221 . . . 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 2448 . . 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 3548 . . 3  |-  ( ( U ( LSSum `  (oppg `  G
) ) T )  i^i  S )  =  ( S  i^i  ( U ( LSSum `  (oppg `  G
) ) T ) )
15 incom 3548 . . . 4  |-  ( T  i^i  S )  =  ( S  i^i  T
)
1615oveq2i 6107 . . 3  |-  ( U ( LSSum `  (oppg
`  G ) ) ( T  i^i  S
) )  =  ( U ( LSSum `  (oppg `  G
) ) ( S  i^i  T ) )
1713, 14, 163eqtr3g 2498 . 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 16146 . . 3  |-  ( U ( LSSum `  (oppg
`  G ) ) T )  =  ( T  .(+)  U )
2019ineq2i 3554 . 2  |-  ( S  i^i  ( U (
LSSum `  (oppg
`  G ) ) T ) )  =  ( S  i^i  ( T  .(+)  U ) )
212, 18oppglsm 16146 . 2  |-  ( U ( LSSum `  (oppg
`  G ) ) ( S  i^i  T
) )  =  ( ( S  i^i  T
)  .(+)  U )
2217, 20, 213eqtr3g 2498 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 965    = wceq 1369    e. wcel 1756    i^i cin 3332    C_ wss 3333   ` cfv 5423  (class class class)co 6096  SubGrpcsubg 15680  oppgcoppg 15865   LSSumclsm 16138
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-iin 4179  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-tpos 6750  df-recs 6837  df-rdg 6871  df-1o 6925  df-oadd 6929  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-nn 10328  df-2 10385  df-ndx 14182  df-slot 14183  df-base 14184  df-sets 14185  df-ress 14186  df-plusg 14256  df-0g 14385  df-mre 14529  df-mrc 14530  df-acs 14532  df-mnd 15420  df-submnd 15470  df-grp 15550  df-minusg 15551  df-subg 15683  df-oppg 15866  df-lsm 16140
This theorem is referenced by:  lcvexchlem3  32686  lcfrlem23  35215
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