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Theorem subglsm 16482
Description: The subgroup sum evaluated within a subgroup. (Contributed by Mario Carneiro, 27-Apr-2016.)
Hypotheses
Ref Expression
subglsm.h  |-  H  =  ( Gs  S )
subglsm.s  |-  .(+)  =  (
LSSum `  G )
subglsm.a  |-  A  =  ( LSSum `  H )
Assertion
Ref Expression
subglsm  |-  ( ( S  e.  (SubGrp `  G )  /\  T  C_  S  /\  U  C_  S )  ->  ( T  .(+)  U )  =  ( T A U ) )

Proof of Theorem subglsm
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp11 1021 . . . . . 6  |-  ( ( ( S  e.  (SubGrp `  G )  /\  T  C_  S  /\  U  C_  S )  /\  x  e.  T  /\  y  e.  U )  ->  S  e.  (SubGrp `  G )
)
2 subglsm.h . . . . . . 7  |-  H  =  ( Gs  S )
3 eqid 2462 . . . . . . 7  |-  ( +g  `  G )  =  ( +g  `  G )
42, 3ressplusg 14588 . . . . . 6  |-  ( S  e.  (SubGrp `  G
)  ->  ( +g  `  G )  =  ( +g  `  H ) )
51, 4syl 16 . . . . 5  |-  ( ( ( S  e.  (SubGrp `  G )  /\  T  C_  S  /\  U  C_  S )  /\  x  e.  T  /\  y  e.  U )  ->  ( +g  `  G )  =  ( +g  `  H
) )
65oveqd 6294 . . . 4  |-  ( ( ( S  e.  (SubGrp `  G )  /\  T  C_  S  /\  U  C_  S )  /\  x  e.  T  /\  y  e.  U )  ->  (
x ( +g  `  G
) y )  =  ( x ( +g  `  H ) y ) )
76mpt2eq3dva 6338 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  T  C_  S  /\  U  C_  S )  ->  (
x  e.  T , 
y  e.  U  |->  ( x ( +g  `  G
) y ) )  =  ( x  e.  T ,  y  e.  U  |->  ( x ( +g  `  H ) y ) ) )
87rneqd 5223 . 2  |-  ( ( S  e.  (SubGrp `  G )  /\  T  C_  S  /\  U  C_  S )  ->  ran  ( x  e.  T ,  y  e.  U  |->  ( x ( +g  `  G ) y ) )  =  ran  (
x  e.  T , 
y  e.  U  |->  ( x ( +g  `  H
) y ) ) )
9 subgrcl 15996 . . . 4  |-  ( S  e.  (SubGrp `  G
)  ->  G  e.  Grp )
1093ad2ant1 1012 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  T  C_  S  /\  U  C_  S )  ->  G  e.  Grp )
11 simp2 992 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  T  C_  S  /\  U  C_  S )  ->  T  C_  S )
12 eqid 2462 . . . . . 6  |-  ( Base `  G )  =  (
Base `  G )
1312subgss 15992 . . . . 5  |-  ( S  e.  (SubGrp `  G
)  ->  S  C_  ( Base `  G ) )
14133ad2ant1 1012 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  T  C_  S  /\  U  C_  S )  ->  S  C_  ( Base `  G
) )
1511, 14sstrd 3509 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  T  C_  S  /\  U  C_  S )  ->  T  C_  ( Base `  G
) )
16 simp3 993 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  T  C_  S  /\  U  C_  S )  ->  U  C_  S )
1716, 14sstrd 3509 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  T  C_  S  /\  U  C_  S )  ->  U  C_  ( Base `  G
) )
18 subglsm.s . . . 4  |-  .(+)  =  (
LSSum `  G )
1912, 3, 18lsmvalx 16450 . . 3  |-  ( ( G  e.  Grp  /\  T  C_  ( Base `  G
)  /\  U  C_  ( Base `  G ) )  ->  ( T  .(+)  U )  =  ran  (
x  e.  T , 
y  e.  U  |->  ( x ( +g  `  G
) y ) ) )
2010, 15, 17, 19syl3anc 1223 . 2  |-  ( ( S  e.  (SubGrp `  G )  /\  T  C_  S  /\  U  C_  S )  ->  ( T  .(+)  U )  =  ran  ( x  e.  T ,  y  e.  U  |->  ( x ( +g  `  G ) y ) ) )
212subggrp 15994 . . . 4  |-  ( S  e.  (SubGrp `  G
)  ->  H  e.  Grp )
22213ad2ant1 1012 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  T  C_  S  /\  U  C_  S )  ->  H  e.  Grp )
232subgbas 15995 . . . . 5  |-  ( S  e.  (SubGrp `  G
)  ->  S  =  ( Base `  H )
)
24233ad2ant1 1012 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  T  C_  S  /\  U  C_  S )  ->  S  =  ( Base `  H
) )
2511, 24sseqtrd 3535 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  T  C_  S  /\  U  C_  S )  ->  T  C_  ( Base `  H
) )
2616, 24sseqtrd 3535 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  T  C_  S  /\  U  C_  S )  ->  U  C_  ( Base `  H
) )
27 eqid 2462 . . . 4  |-  ( Base `  H )  =  (
Base `  H )
28 eqid 2462 . . . 4  |-  ( +g  `  H )  =  ( +g  `  H )
29 subglsm.a . . . 4  |-  A  =  ( LSSum `  H )
3027, 28, 29lsmvalx 16450 . . 3  |-  ( ( H  e.  Grp  /\  T  C_  ( Base `  H
)  /\  U  C_  ( Base `  H ) )  ->  ( T A U )  =  ran  ( x  e.  T ,  y  e.  U  |->  ( x ( +g  `  H ) y ) ) )
3122, 25, 26, 30syl3anc 1223 . 2  |-  ( ( S  e.  (SubGrp `  G )  /\  T  C_  S  /\  U  C_  S )  ->  ( T A U )  =  ran  ( x  e.  T ,  y  e.  U  |->  ( x ( +g  `  H ) y ) ) )
328, 20, 313eqtr4d 2513 1  |-  ( ( S  e.  (SubGrp `  G )  /\  T  C_  S  /\  U  C_  S )  ->  ( T  .(+)  U )  =  ( T A U ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 968    = wceq 1374    e. wcel 1762    C_ wss 3471   ran crn 4995   ` cfv 5581  (class class class)co 6277    |-> cmpt2 6279   Basecbs 14481   ↾s cress 14482   +g cplusg 14546   Grpcgrp 15718  SubGrpcsubg 15985   LSSumclsm 16445
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 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6674  df-1st 6776  df-2nd 6777  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7509  df-dom 7510  df-sdom 7511  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-nn 10528  df-2 10585  df-ndx 14484  df-slot 14485  df-base 14486  df-sets 14487  df-ress 14488  df-plusg 14559  df-subg 15988  df-lsm 16447
This theorem is referenced by:  pgpfaclem1  16917
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