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Theorem lsmelvalm 16798
Description: Subgroup sum membership analog of lsmelval 16796 using vector subtraction. TODO: any way to shorten proof? (Contributed by NM, 16-Mar-2015.) (Revised by Mario Carneiro, 19-Apr-2016.)
Hypotheses
Ref Expression
lsmelvalm.m  |-  .-  =  ( -g `  G )
lsmelvalm.p  |-  .(+)  =  (
LSSum `  G )
lsmelvalm.t  |-  ( ph  ->  T  e.  (SubGrp `  G ) )
lsmelvalm.u  |-  ( ph  ->  U  e.  (SubGrp `  G ) )
Assertion
Ref Expression
lsmelvalm  |-  ( ph  ->  ( X  e.  ( T  .(+)  U )  <->  E. y  e.  T  E. z  e.  U  X  =  ( y  .-  z ) ) )
Distinct variable groups:    y, z,  .-    y, G, z    ph, y,
z    y, T, z    y, U, z    y, X, z
Allowed substitution hints:    .(+) ( y, z)

Proof of Theorem lsmelvalm
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 lsmelvalm.t . . 3  |-  ( ph  ->  T  e.  (SubGrp `  G ) )
2 lsmelvalm.u . . 3  |-  ( ph  ->  U  e.  (SubGrp `  G ) )
3 eqid 2457 . . . 4  |-  ( +g  `  G )  =  ( +g  `  G )
4 lsmelvalm.p . . . 4  |-  .(+)  =  (
LSSum `  G )
53, 4lsmelval 16796 . . 3  |-  ( ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )
)  ->  ( X  e.  ( T  .(+)  U )  <->  E. y  e.  T  E. x  e.  U  X  =  ( y
( +g  `  G ) x ) ) )
61, 2, 5syl2anc 661 . 2  |-  ( ph  ->  ( X  e.  ( T  .(+)  U )  <->  E. y  e.  T  E. x  e.  U  X  =  ( y ( +g  `  G ) x ) ) )
72adantr 465 . . . . . . . 8  |-  ( (
ph  /\  y  e.  T )  ->  U  e.  (SubGrp `  G )
)
8 eqid 2457 . . . . . . . . 9  |-  ( invg `  G )  =  ( invg `  G )
98subginvcl 16337 . . . . . . . 8  |-  ( ( U  e.  (SubGrp `  G )  /\  x  e.  U )  ->  (
( invg `  G ) `  x
)  e.  U )
107, 9sylan 471 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  T )  /\  x  e.  U )  ->  (
( invg `  G ) `  x
)  e.  U )
11 eqid 2457 . . . . . . . . 9  |-  ( Base `  G )  =  (
Base `  G )
12 lsmelvalm.m . . . . . . . . 9  |-  .-  =  ( -g `  G )
13 subgrcl 16333 . . . . . . . . . . 11  |-  ( T  e.  (SubGrp `  G
)  ->  G  e.  Grp )
141, 13syl 16 . . . . . . . . . 10  |-  ( ph  ->  G  e.  Grp )
1514ad2antrr 725 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  T )  /\  x  e.  U )  ->  G  e.  Grp )
1611subgss 16329 . . . . . . . . . . . 12  |-  ( T  e.  (SubGrp `  G
)  ->  T  C_  ( Base `  G ) )
171, 16syl 16 . . . . . . . . . . 11  |-  ( ph  ->  T  C_  ( Base `  G ) )
1817sselda 3499 . . . . . . . . . 10  |-  ( (
ph  /\  y  e.  T )  ->  y  e.  ( Base `  G
) )
1918adantr 465 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  T )  /\  x  e.  U )  ->  y  e.  ( Base `  G
) )
2011subgss 16329 . . . . . . . . . . 11  |-  ( U  e.  (SubGrp `  G
)  ->  U  C_  ( Base `  G ) )
217, 20syl 16 . . . . . . . . . 10  |-  ( (
ph  /\  y  e.  T )  ->  U  C_  ( Base `  G
) )
2221sselda 3499 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  T )  /\  x  e.  U )  ->  x  e.  ( Base `  G
) )
2311, 3, 12, 8, 15, 19, 22grpsubinv 16238 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  T )  /\  x  e.  U )  ->  (
y  .-  ( ( invg `  G ) `
 x ) )  =  ( y ( +g  `  G ) x ) )
2423eqcomd 2465 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  T )  /\  x  e.  U )  ->  (
y ( +g  `  G
) x )  =  ( y  .-  (
( invg `  G ) `  x
) ) )
25 oveq2 6304 . . . . . . . . 9  |-  ( z  =  ( ( invg `  G ) `
 x )  -> 
( y  .-  z
)  =  ( y 
.-  ( ( invg `  G ) `
 x ) ) )
2625eqeq2d 2471 . . . . . . . 8  |-  ( z  =  ( ( invg `  G ) `
 x )  -> 
( ( y ( +g  `  G ) x )  =  ( y  .-  z )  <-> 
( y ( +g  `  G ) x )  =  ( y  .-  ( ( invg `  G ) `  x
) ) ) )
2726rspcev 3210 . . . . . . 7  |-  ( ( ( ( invg `  G ) `  x
)  e.  U  /\  ( y ( +g  `  G ) x )  =  ( y  .-  ( ( invg `  G ) `  x
) ) )  ->  E. z  e.  U  ( y ( +g  `  G ) x )  =  ( y  .-  z ) )
2810, 24, 27syl2anc 661 . . . . . 6  |-  ( ( ( ph  /\  y  e.  T )  /\  x  e.  U )  ->  E. z  e.  U  ( y
( +g  `  G ) x )  =  ( y  .-  z ) )
29 eqeq1 2461 . . . . . . 7  |-  ( X  =  ( y ( +g  `  G ) x )  ->  ( X  =  ( y  .-  z )  <->  ( y
( +g  `  G ) x )  =  ( y  .-  z ) ) )
3029rexbidv 2968 . . . . . 6  |-  ( X  =  ( y ( +g  `  G ) x )  ->  ( E. z  e.  U  X  =  ( y  .-  z )  <->  E. z  e.  U  ( y
( +g  `  G ) x )  =  ( y  .-  z ) ) )
3128, 30syl5ibrcom 222 . . . . 5  |-  ( ( ( ph  /\  y  e.  T )  /\  x  e.  U )  ->  ( X  =  ( y
( +g  `  G ) x )  ->  E. z  e.  U  X  =  ( y  .-  z
) ) )
3231rexlimdva 2949 . . . 4  |-  ( (
ph  /\  y  e.  T )  ->  ( E. x  e.  U  X  =  ( y
( +g  `  G ) x )  ->  E. z  e.  U  X  =  ( y  .-  z
) ) )
338subginvcl 16337 . . . . . . . 8  |-  ( ( U  e.  (SubGrp `  G )  /\  z  e.  U )  ->  (
( invg `  G ) `  z
)  e.  U )
347, 33sylan 471 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  T )  /\  z  e.  U )  ->  (
( invg `  G ) `  z
)  e.  U )
3518adantr 465 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  T )  /\  z  e.  U )  ->  y  e.  ( Base `  G
) )
3621sselda 3499 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  T )  /\  z  e.  U )  ->  z  e.  ( Base `  G
) )
3711, 3, 8, 12grpsubval 16220 . . . . . . . 8  |-  ( ( y  e.  ( Base `  G )  /\  z  e.  ( Base `  G
) )  ->  (
y  .-  z )  =  ( y ( +g  `  G ) ( ( invg `  G ) `  z
) ) )
3835, 36, 37syl2anc 661 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  T )  /\  z  e.  U )  ->  (
y  .-  z )  =  ( y ( +g  `  G ) ( ( invg `  G ) `  z
) ) )
39 oveq2 6304 . . . . . . . . 9  |-  ( x  =  ( ( invg `  G ) `
 z )  -> 
( y ( +g  `  G ) x )  =  ( y ( +g  `  G ) ( ( invg `  G ) `  z
) ) )
4039eqeq2d 2471 . . . . . . . 8  |-  ( x  =  ( ( invg `  G ) `
 z )  -> 
( ( y  .-  z )  =  ( y ( +g  `  G
) x )  <->  ( y  .-  z )  =  ( y ( +g  `  G
) ( ( invg `  G ) `
 z ) ) ) )
4140rspcev 3210 . . . . . . 7  |-  ( ( ( ( invg `  G ) `  z
)  e.  U  /\  ( y  .-  z
)  =  ( y ( +g  `  G
) ( ( invg `  G ) `
 z ) ) )  ->  E. x  e.  U  ( y  .-  z )  =  ( y ( +g  `  G
) x ) )
4234, 38, 41syl2anc 661 . . . . . 6  |-  ( ( ( ph  /\  y  e.  T )  /\  z  e.  U )  ->  E. x  e.  U  ( y  .-  z )  =  ( y ( +g  `  G
) x ) )
43 eqeq1 2461 . . . . . . 7  |-  ( X  =  ( y  .-  z )  ->  ( X  =  ( y
( +g  `  G ) x )  <->  ( y  .-  z )  =  ( y ( +g  `  G
) x ) ) )
4443rexbidv 2968 . . . . . 6  |-  ( X  =  ( y  .-  z )  ->  ( E. x  e.  U  X  =  ( y
( +g  `  G ) x )  <->  E. x  e.  U  ( y  .-  z )  =  ( y ( +g  `  G
) x ) ) )
4542, 44syl5ibrcom 222 . . . . 5  |-  ( ( ( ph  /\  y  e.  T )  /\  z  e.  U )  ->  ( X  =  ( y  .-  z )  ->  E. x  e.  U  X  =  ( y ( +g  `  G ) x ) ) )
4645rexlimdva 2949 . . . 4  |-  ( (
ph  /\  y  e.  T )  ->  ( E. z  e.  U  X  =  ( y  .-  z )  ->  E. x  e.  U  X  =  ( y ( +g  `  G ) x ) ) )
4732, 46impbid 191 . . 3  |-  ( (
ph  /\  y  e.  T )  ->  ( E. x  e.  U  X  =  ( y
( +g  `  G ) x )  <->  E. z  e.  U  X  =  ( y  .-  z
) ) )
4847rexbidva 2965 . 2  |-  ( ph  ->  ( E. y  e.  T  E. x  e.  U  X  =  ( y ( +g  `  G
) x )  <->  E. y  e.  T  E. z  e.  U  X  =  ( y  .-  z
) ) )
496, 48bitrd 253 1  |-  ( ph  ->  ( X  e.  ( T  .(+)  U )  <->  E. y  e.  T  E. z  e.  U  X  =  ( y  .-  z ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819   E.wrex 2808    C_ wss 3471   ` cfv 5594  (class class class)co 6296   Basecbs 14644   +g cplusg 14712   Grpcgrp 16180   invgcminusg 16181   -gcsg 16182  SubGrpcsubg 16322   LSSumclsm 16781
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-ndx 14647  df-slot 14648  df-base 14649  df-sets 14650  df-ress 14651  df-plusg 14725  df-0g 14859  df-mgm 15999  df-sgrp 16038  df-mnd 16048  df-grp 16184  df-minusg 16185  df-sbg 16186  df-subg 16325  df-lsm 16783
This theorem is referenced by:  lsmelvalmi  16799  pgpfac1lem2  17253  pgpfac1lem3  17255  pgpfac1lem4  17256  mapdpglem3  37545
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