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Theorem subglsm 17013
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 1027 . . . . . 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 2402 . . . . . . 7  |-  ( +g  `  G )  =  ( +g  `  G )
42, 3ressplusg 14953 . . . . . 6  |-  ( S  e.  (SubGrp `  G
)  ->  ( +g  `  G )  =  ( +g  `  H ) )
51, 4syl 17 . . . . 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 6341 . . 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 5050 . 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 16528 . . . 4  |-  ( S  e.  (SubGrp `  G
)  ->  G  e.  Grp )
1093ad2ant1 1018 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  T  C_  S  /\  U  C_  S )  ->  G  e.  Grp )
11 simp2 998 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  T  C_  S  /\  U  C_  S )  ->  T  C_  S )
12 eqid 2402 . . . . . 6  |-  ( Base `  G )  =  (
Base `  G )
1312subgss 16524 . . . . 5  |-  ( S  e.  (SubGrp `  G
)  ->  S  C_  ( Base `  G ) )
14133ad2ant1 1018 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  T  C_  S  /\  U  C_  S )  ->  S  C_  ( Base `  G
) )
1511, 14sstrd 3451 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  T  C_  S  /\  U  C_  S )  ->  T  C_  ( Base `  G
) )
16 simp3 999 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  T  C_  S  /\  U  C_  S )  ->  U  C_  S )
1716, 14sstrd 3451 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  T  C_  S  /\  U  C_  S )  ->  U  C_  ( Base `  G
) )
18 subglsm.s . . . 4  |-  .(+)  =  (
LSSum `  G )
1912, 3, 18lsmvalx 16981 . . 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 1230 . 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 16526 . . . 4  |-  ( S  e.  (SubGrp `  G
)  ->  H  e.  Grp )
22213ad2ant1 1018 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  T  C_  S  /\  U  C_  S )  ->  H  e.  Grp )
232subgbas 16527 . . . . 5  |-  ( S  e.  (SubGrp `  G
)  ->  S  =  ( Base `  H )
)
24233ad2ant1 1018 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  T  C_  S  /\  U  C_  S )  ->  S  =  ( Base `  H
) )
2511, 24sseqtrd 3477 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  T  C_  S  /\  U  C_  S )  ->  T  C_  ( Base `  H
) )
2616, 24sseqtrd 3477 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  T  C_  S  /\  U  C_  S )  ->  U  C_  ( Base `  H
) )
27 eqid 2402 . . . 4  |-  ( Base `  H )  =  (
Base `  H )
28 eqid 2402 . . . 4  |-  ( +g  `  H )  =  ( +g  `  H )
29 subglsm.a . . . 4  |-  A  =  ( LSSum `  H )
3027, 28, 29lsmvalx 16981 . . 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 1230 . 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 2453 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 974    = wceq 1405    e. wcel 1842    C_ wss 3413   ran crn 4823   ` cfv 5568  (class class class)co 6277    |-> cmpt2 6279   Basecbs 14839   ↾s cress 14840   +g cplusg 14907   Grpcgrp 16375  SubGrpcsubg 16517   LSSumclsm 16976
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597  ax-pre-mulgt0 9598
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6683  df-1st 6783  df-2nd 6784  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-er 7347  df-en 7554  df-dom 7555  df-sdom 7556  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-le 9663  df-sub 9842  df-neg 9843  df-nn 10576  df-2 10634  df-ndx 14842  df-slot 14843  df-base 14844  df-sets 14845  df-ress 14846  df-plusg 14920  df-subg 16520  df-lsm 16978
This theorem is referenced by:  pgpfaclem1  17450
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