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Theorem subglsm 14998
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 985 . . . . . 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 2296 . . . . . . 7  |-  ( +g  `  G )  =  ( +g  `  G )
42, 3ressplusg 13266 . . . . . 6  |-  ( S  e.  (SubGrp `  G
)  ->  ( +g  `  G )  =  ( +g  `  H ) )
51, 4syl 15 . . . . 5  |-  ( ( ( S  e.  (SubGrp `  G )  /\  T  C_  S  /\  U  C_  S )  /\  x  e.  T  /\  y  e.  U )  ->  ( +g  `  G )  =  ( +g  `  H
) )
65oveqd 5891 . . . 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 5928 . . 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 4922 . 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 14642 . . . 4  |-  ( S  e.  (SubGrp `  G
)  ->  G  e.  Grp )
1093ad2ant1 976 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  T  C_  S  /\  U  C_  S )  ->  G  e.  Grp )
11 simp2 956 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  T  C_  S  /\  U  C_  S )  ->  T  C_  S )
12 eqid 2296 . . . . . 6  |-  ( Base `  G )  =  (
Base `  G )
1312subgss 14638 . . . . 5  |-  ( S  e.  (SubGrp `  G
)  ->  S  C_  ( Base `  G ) )
14133ad2ant1 976 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  T  C_  S  /\  U  C_  S )  ->  S  C_  ( Base `  G
) )
1511, 14sstrd 3202 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  T  C_  S  /\  U  C_  S )  ->  T  C_  ( Base `  G
) )
16 simp3 957 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  T  C_  S  /\  U  C_  S )  ->  U  C_  S )
1716, 14sstrd 3202 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  T  C_  S  /\  U  C_  S )  ->  U  C_  ( Base `  G
) )
18 subglsm.s . . . 4  |-  .(+)  =  (
LSSum `  G )
1912, 3, 18lsmvalx 14966 . . 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 1182 . 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 14640 . . . 4  |-  ( S  e.  (SubGrp `  G
)  ->  H  e.  Grp )
22213ad2ant1 976 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  T  C_  S  /\  U  C_  S )  ->  H  e.  Grp )
232subgbas 14641 . . . . 5  |-  ( S  e.  (SubGrp `  G
)  ->  S  =  ( Base `  H )
)
24233ad2ant1 976 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  T  C_  S  /\  U  C_  S )  ->  S  =  ( Base `  H
) )
2511, 24sseqtrd 3227 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  T  C_  S  /\  U  C_  S )  ->  T  C_  ( Base `  H
) )
2616, 24sseqtrd 3227 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  T  C_  S  /\  U  C_  S )  ->  U  C_  ( Base `  H
) )
27 eqid 2296 . . . 4  |-  ( Base `  H )  =  (
Base `  H )
28 eqid 2296 . . . 4  |-  ( +g  `  H )  =  ( +g  `  H )
29 subglsm.a . . . 4  |-  A  =  ( LSSum `  H )
3027, 28, 29lsmvalx 14966 . . 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 1182 . 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 2338 1  |-  ( ( S  e.  (SubGrp `  G )  /\  T  C_  S  /\  U  C_  S )  ->  ( T  .(+)  U )  =  ( T A U ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1632    e. wcel 1696    C_ wss 3165   ran crn 4706   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   Basecbs 13164   ↾s cress 13165   +g cplusg 13224   Grpcgrp 14378  SubGrpcsubg 14631   LSSumclsm 14961
This theorem is referenced by:  pgpfaclem1  15332
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-subg 14634  df-lsm 14963
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