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Theorem subglsm 14982
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 2283 . . . . . . 7  |-  ( +g  `  G )  =  ( +g  `  G )
42, 3ressplusg 13250 . . . . . 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 5875 . . . 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 5912 . . 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 4906 . 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 14626 . . . 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 2283 . . . . . 6  |-  ( Base `  G )  =  (
Base `  G )
1312subgss 14622 . . . . 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 3189 . . 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 3189 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  T  C_  S  /\  U  C_  S )  ->  U  C_  ( Base `  G
) )
18 subglsm.s . . . 4  |-  .(+)  =  (
LSSum `  G )
1912, 3, 18lsmvalx 14950 . . 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 14624 . . . 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 14625 . . . . 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 3214 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  T  C_  S  /\  U  C_  S )  ->  T  C_  ( Base `  H
) )
2616, 24sseqtrd 3214 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  T  C_  S  /\  U  C_  S )  ->  U  C_  ( Base `  H
) )
27 eqid 2283 . . . 4  |-  ( Base `  H )  =  (
Base `  H )
28 eqid 2283 . . . 4  |-  ( +g  `  H )  =  ( +g  `  H )
29 subglsm.a . . . 4  |-  A  =  ( LSSum `  H )
3027, 28, 29lsmvalx 14950 . . 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 2325 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 1623    e. wcel 1684    C_ wss 3152   ran crn 4690   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   Basecbs 13148   ↾s cress 13149   +g cplusg 13208   Grpcgrp 14362  SubGrpcsubg 14615   LSSumclsm 14945
This theorem is referenced by:  pgpfaclem1  15316
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-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-iun 3907  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-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  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-subg 14618  df-lsm 14947
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