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Theorem lsmelvalm 16150
Description: Subgroup sum membership analog of lsmelval 16148 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 2443 . . . 4  |-  ( +g  `  G )  =  ( +g  `  G )
4 lsmelvalm.p . . . 4  |-  .(+)  =  (
LSSum `  G )
53, 4lsmelval 16148 . . 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 2443 . . . . . . . . 9  |-  ( invg `  G )  =  ( invg `  G )
98subginvcl 15690 . . . . . . . 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 2443 . . . . . . . . 9  |-  ( Base `  G )  =  (
Base `  G )
12 lsmelvalm.m . . . . . . . . 9  |-  .-  =  ( -g `  G )
13 subgrcl 15686 . . . . . . . . . . 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 15682 . . . . . . . . . . . 12  |-  ( T  e.  (SubGrp `  G
)  ->  T  C_  ( Base `  G ) )
171, 16syl 16 . . . . . . . . . . 11  |-  ( ph  ->  T  C_  ( Base `  G ) )
1817sselda 3356 . . . . . . . . . 10  |-  ( (
ph  /\  y  e.  T )  ->  y  e.  ( Base `  G
) )
1918adantr 465 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  T )  /\  x  e.  U )  ->  y  e.  ( Base `  G
) )
2011subgss 15682 . . . . . . . . . . 11  |-  ( U  e.  (SubGrp `  G
)  ->  U  C_  ( Base `  G ) )
217, 20syl 16 . . . . . . . . . 10  |-  ( (
ph  /\  y  e.  T )  ->  U  C_  ( Base `  G
) )
2221sselda 3356 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  T )  /\  x  e.  U )  ->  x  e.  ( Base `  G
) )
2311, 3, 12, 8, 15, 19, 22grpsubinv 15599 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  T )  /\  x  e.  U )  ->  (
y  .-  ( ( invg `  G ) `
 x ) )  =  ( y ( +g  `  G ) x ) )
2423eqcomd 2448 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  T )  /\  x  e.  U )  ->  (
y ( +g  `  G
) x )  =  ( y  .-  (
( invg `  G ) `  x
) ) )
25 oveq2 6099 . . . . . . . . 9  |-  ( z  =  ( ( invg `  G ) `
 x )  -> 
( y  .-  z
)  =  ( y 
.-  ( ( invg `  G ) `
 x ) ) )
2625eqeq2d 2454 . . . . . . . 8  |-  ( z  =  ( ( invg `  G ) `
 x )  -> 
( ( y ( +g  `  G ) x )  =  ( y  .-  z )  <-> 
( y ( +g  `  G ) x )  =  ( y  .-  ( ( invg `  G ) `  x
) ) ) )
2726rspcev 3073 . . . . . . 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 2449 . . . . . . 7  |-  ( X  =  ( y ( +g  `  G ) x )  ->  ( X  =  ( y  .-  z )  <->  ( y
( +g  `  G ) x )  =  ( y  .-  z ) ) )
3029rexbidv 2736 . . . . . 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 2841 . . . 4  |-  ( (
ph  /\  y  e.  T )  ->  ( E. x  e.  U  X  =  ( y
( +g  `  G ) x )  ->  E. z  e.  U  X  =  ( y  .-  z
) ) )
338subginvcl 15690 . . . . . . . 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 3356 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  T )  /\  z  e.  U )  ->  z  e.  ( Base `  G
) )
3711, 3, 8, 12grpsubval 15581 . . . . . . . 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 6099 . . . . . . . . 9  |-  ( x  =  ( ( invg `  G ) `
 z )  -> 
( y ( +g  `  G ) x )  =  ( y ( +g  `  G ) ( ( invg `  G ) `  z
) ) )
4039eqeq2d 2454 . . . . . . . 8  |-  ( x  =  ( ( invg `  G ) `
 z )  -> 
( ( y  .-  z )  =  ( y ( +g  `  G
) x )  <->  ( y  .-  z )  =  ( y ( +g  `  G
) ( ( invg `  G ) `
 z ) ) ) )
4140rspcev 3073 . . . . . . 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 2449 . . . . . . 7  |-  ( X  =  ( y  .-  z )  ->  ( X  =  ( y
( +g  `  G ) x )  <->  ( y  .-  z )  =  ( y ( +g  `  G
) x ) ) )
4443rexbidv 2736 . . . . . 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 2841 . . . 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 2732 . 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 1369    e. wcel 1756   E.wrex 2716    C_ wss 3328   ` cfv 5418  (class class class)co 6091   Basecbs 14174   +g cplusg 14238   Grpcgrp 15410   invgcminusg 15411   -gcsg 15413  SubGrpcsubg 15675   LSSumclsm 16133
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-nn 10323  df-2 10380  df-ndx 14177  df-slot 14178  df-base 14179  df-sets 14180  df-ress 14181  df-plusg 14251  df-0g 14380  df-mnd 15415  df-grp 15545  df-minusg 15546  df-sbg 15547  df-subg 15678  df-lsm 16135
This theorem is referenced by:  lsmelvalmi  16151  pgpfac1lem2  16576  pgpfac1lem3  16578  pgpfac1lem4  16579  mapdpglem3  35320
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