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Theorem subglsm 15305
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 987 . . . . . 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 2436 . . . . . . 7  |-  ( +g  `  G )  =  ( +g  `  G )
42, 3ressplusg 13571 . . . . . 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 6098 . . . 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 6138 . . 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 5097 . 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 14949 . . . 4  |-  ( S  e.  (SubGrp `  G
)  ->  G  e.  Grp )
1093ad2ant1 978 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  T  C_  S  /\  U  C_  S )  ->  G  e.  Grp )
11 simp2 958 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  T  C_  S  /\  U  C_  S )  ->  T  C_  S )
12 eqid 2436 . . . . . 6  |-  ( Base `  G )  =  (
Base `  G )
1312subgss 14945 . . . . 5  |-  ( S  e.  (SubGrp `  G
)  ->  S  C_  ( Base `  G ) )
14133ad2ant1 978 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  T  C_  S  /\  U  C_  S )  ->  S  C_  ( Base `  G
) )
1511, 14sstrd 3358 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  T  C_  S  /\  U  C_  S )  ->  T  C_  ( Base `  G
) )
16 simp3 959 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  T  C_  S  /\  U  C_  S )  ->  U  C_  S )
1716, 14sstrd 3358 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  T  C_  S  /\  U  C_  S )  ->  U  C_  ( Base `  G
) )
18 subglsm.s . . . 4  |-  .(+)  =  (
LSSum `  G )
1912, 3, 18lsmvalx 15273 . . 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 1184 . 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 14947 . . . 4  |-  ( S  e.  (SubGrp `  G
)  ->  H  e.  Grp )
22213ad2ant1 978 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  T  C_  S  /\  U  C_  S )  ->  H  e.  Grp )
232subgbas 14948 . . . . 5  |-  ( S  e.  (SubGrp `  G
)  ->  S  =  ( Base `  H )
)
24233ad2ant1 978 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  T  C_  S  /\  U  C_  S )  ->  S  =  ( Base `  H
) )
2511, 24sseqtrd 3384 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  T  C_  S  /\  U  C_  S )  ->  T  C_  ( Base `  H
) )
2616, 24sseqtrd 3384 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  T  C_  S  /\  U  C_  S )  ->  U  C_  ( Base `  H
) )
27 eqid 2436 . . . 4  |-  ( Base `  H )  =  (
Base `  H )
28 eqid 2436 . . . 4  |-  ( +g  `  H )  =  ( +g  `  H )
29 subglsm.a . . . 4  |-  A  =  ( LSSum `  H )
3027, 28, 29lsmvalx 15273 . . 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 1184 . 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 2478 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 936    = wceq 1652    e. wcel 1725    C_ wss 3320   ran crn 4879   ` cfv 5454  (class class class)co 6081    e. cmpt2 6083   Basecbs 13469   ↾s cress 13470   +g cplusg 13529   Grpcgrp 14685  SubGrpcsubg 14938   LSSumclsm 15268
This theorem is referenced by:  pgpfaclem1  15639
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-plusg 13542  df-subg 14941  df-lsm 15270
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