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Theorem lshpkrlem3 29302
Description: Lemma for lshpkrex 29308. Defining property of  G `  X. (Contributed by NM, 15-Jul-2014.)
Hypotheses
Ref Expression
lshpkrlem.v  |-  V  =  ( Base `  W
)
lshpkrlem.a  |-  .+  =  ( +g  `  W )
lshpkrlem.n  |-  N  =  ( LSpan `  W )
lshpkrlem.p  |-  .(+)  =  (
LSSum `  W )
lshpkrlem.h  |-  H  =  (LSHyp `  W )
lshpkrlem.w  |-  ( ph  ->  W  e.  LVec )
lshpkrlem.u  |-  ( ph  ->  U  e.  H )
lshpkrlem.z  |-  ( ph  ->  Z  e.  V )
lshpkrlem.x  |-  ( ph  ->  X  e.  V )
lshpkrlem.e  |-  ( ph  ->  ( U  .(+)  ( N `
 { Z }
) )  =  V )
lshpkrlem.d  |-  D  =  (Scalar `  W )
lshpkrlem.k  |-  K  =  ( Base `  D
)
lshpkrlem.t  |-  .x.  =  ( .s `  W )
lshpkrlem.o  |-  .0.  =  ( 0g `  D )
lshpkrlem.g  |-  G  =  ( x  e.  V  |->  ( iota_ k  e.  K E. y  e.  U  x  =  ( y  .+  ( k  .x.  Z
) ) ) )
Assertion
Ref Expression
lshpkrlem3  |-  ( ph  ->  E. z  e.  U  X  =  ( z  .+  ( ( G `  X )  .x.  Z
) ) )
Distinct variable groups:    x, k,
y,  .+    k, K, x    .0. , k    .x. , k, x, y    U, k, x, y    x, V    k, X, x, y   
k, Z, x, y   
z,  .+    z, G    z, U    z, X    z, Z, k, x, y    z,  .x.
Allowed substitution hints:    ph( x, y, z, k)    D( x, y, z, k)    .(+) ( x, y, z, k)    G( x, y, k)    H( x, y, z, k)    K( y, z)    N( x, y, z, k)    V( y, z, k)    W( x, y, z, k)    .0. ( x, y, z)

Proof of Theorem lshpkrlem3
Dummy variable  l is distinct from all other variables.
StepHypRef Expression
1 lshpkrlem.v . . . . 5  |-  V  =  ( Base `  W
)
2 lshpkrlem.a . . . . 5  |-  .+  =  ( +g  `  W )
3 lshpkrlem.n . . . . 5  |-  N  =  ( LSpan `  W )
4 lshpkrlem.p . . . . 5  |-  .(+)  =  (
LSSum `  W )
5 lshpkrlem.h . . . . 5  |-  H  =  (LSHyp `  W )
6 lshpkrlem.w . . . . 5  |-  ( ph  ->  W  e.  LVec )
7 lshpkrlem.u . . . . 5  |-  ( ph  ->  U  e.  H )
8 lshpkrlem.z . . . . 5  |-  ( ph  ->  Z  e.  V )
9 lshpkrlem.x . . . . 5  |-  ( ph  ->  X  e.  V )
10 lshpkrlem.e . . . . 5  |-  ( ph  ->  ( U  .(+)  ( N `
 { Z }
) )  =  V )
11 lshpkrlem.d . . . . 5  |-  D  =  (Scalar `  W )
12 lshpkrlem.k . . . . 5  |-  K  =  ( Base `  D
)
13 lshpkrlem.t . . . . 5  |-  .x.  =  ( .s `  W )
141, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13lshpsmreu 29299 . . . 4  |-  ( ph  ->  E! l  e.  K  E. z  e.  U  X  =  ( z  .+  ( l  .x.  Z
) ) )
15 riotasbc 6320 . . . 4  |-  ( E! l  e.  K  E. z  e.  U  X  =  ( z  .+  ( l  .x.  Z
) )  ->  [. ( iota_ l  e.  K E. z  e.  U  X  =  ( z  .+  ( l  .x.  Z
) ) )  / 
l ]. E. z  e.  U  X  =  ( z  .+  ( l 
.x.  Z ) ) )
1614, 15syl 15 . . 3  |-  ( ph  ->  [. ( iota_ l  e.  K E. z  e.  U  X  =  ( z  .+  ( l 
.x.  Z ) ) )  /  l ]. E. z  e.  U  X  =  ( z  .+  ( l  .x.  Z
) ) )
17 eqeq1 2289 . . . . . . 7  |-  ( x  =  X  ->  (
x  =  ( z 
.+  ( l  .x.  Z ) )  <->  X  =  ( z  .+  (
l  .x.  Z )
) ) )
1817rexbidv 2564 . . . . . 6  |-  ( x  =  X  ->  ( E. z  e.  U  x  =  ( z  .+  ( l  .x.  Z
) )  <->  E. z  e.  U  X  =  ( z  .+  (
l  .x.  Z )
) ) )
1918riotabidv 6306 . . . . 5  |-  ( x  =  X  ->  ( iota_ l  e.  K E. z  e.  U  x  =  ( z  .+  ( l  .x.  Z
) ) )  =  ( iota_ l  e.  K E. z  e.  U  X  =  ( z  .+  ( l  .x.  Z
) ) ) )
20 lshpkrlem.g . . . . . 6  |-  G  =  ( x  e.  V  |->  ( iota_ k  e.  K E. y  e.  U  x  =  ( y  .+  ( k  .x.  Z
) ) ) )
21 oveq1 5865 . . . . . . . . . . . 12  |-  ( k  =  l  ->  (
k  .x.  Z )  =  ( l  .x.  Z ) )
2221oveq2d 5874 . . . . . . . . . . 11  |-  ( k  =  l  ->  (
y  .+  ( k  .x.  Z ) )  =  ( y  .+  (
l  .x.  Z )
) )
2322eqeq2d 2294 . . . . . . . . . 10  |-  ( k  =  l  ->  (
x  =  ( y 
.+  ( k  .x.  Z ) )  <->  x  =  ( y  .+  (
l  .x.  Z )
) ) )
2423rexbidv 2564 . . . . . . . . 9  |-  ( k  =  l  ->  ( E. y  e.  U  x  =  ( y  .+  ( k  .x.  Z
) )  <->  E. y  e.  U  x  =  ( y  .+  (
l  .x.  Z )
) ) )
25 oveq1 5865 . . . . . . . . . . 11  |-  ( y  =  z  ->  (
y  .+  ( l  .x.  Z ) )  =  ( z  .+  (
l  .x.  Z )
) )
2625eqeq2d 2294 . . . . . . . . . 10  |-  ( y  =  z  ->  (
x  =  ( y 
.+  ( l  .x.  Z ) )  <->  x  =  ( z  .+  (
l  .x.  Z )
) ) )
2726cbvrexv 2765 . . . . . . . . 9  |-  ( E. y  e.  U  x  =  ( y  .+  ( l  .x.  Z
) )  <->  E. z  e.  U  x  =  ( z  .+  (
l  .x.  Z )
) )
2824, 27syl6bb 252 . . . . . . . 8  |-  ( k  =  l  ->  ( E. y  e.  U  x  =  ( y  .+  ( k  .x.  Z
) )  <->  E. z  e.  U  x  =  ( z  .+  (
l  .x.  Z )
) ) )
2928cbvriotav 6316 . . . . . . 7  |-  ( iota_ k  e.  K E. y  e.  U  x  =  ( y  .+  (
k  .x.  Z )
) )  =  (
iota_ l  e.  K E. z  e.  U  x  =  ( z  .+  ( l  .x.  Z
) ) )
3029mpteq2i 4103 . . . . . 6  |-  ( x  e.  V  |->  ( iota_ k  e.  K E. y  e.  U  x  =  ( y  .+  (
k  .x.  Z )
) ) )  =  ( x  e.  V  |->  ( iota_ l  e.  K E. z  e.  U  x  =  ( z  .+  ( l  .x.  Z
) ) ) )
3120, 30eqtri 2303 . . . . 5  |-  G  =  ( x  e.  V  |->  ( iota_ l  e.  K E. z  e.  U  x  =  ( z  .+  ( l  .x.  Z
) ) ) )
32 riotaex 6308 . . . . 5  |-  ( iota_ l  e.  K E. z  e.  U  X  =  ( z  .+  (
l  .x.  Z )
) )  e.  _V
3319, 31, 32fvmpt 5602 . . . 4  |-  ( X  e.  V  ->  ( G `  X )  =  ( iota_ l  e.  K E. z  e.  U  X  =  ( z  .+  ( l 
.x.  Z ) ) ) )
34 dfsbcq 2993 . . . 4  |-  ( ( G `  X )  =  ( iota_ l  e.  K E. z  e.  U  X  =  ( z  .+  ( l 
.x.  Z ) ) )  ->  ( [. ( G `  X )  /  l ]. E. z  e.  U  X  =  ( z  .+  ( l  .x.  Z
) )  <->  [. ( iota_ l  e.  K E. z  e.  U  X  =  ( z  .+  (
l  .x.  Z )
) )  /  l ]. E. z  e.  U  X  =  ( z  .+  ( l  .x.  Z
) ) ) )
359, 33, 343syl 18 . . 3  |-  ( ph  ->  ( [. ( G `
 X )  / 
l ]. E. z  e.  U  X  =  ( z  .+  ( l 
.x.  Z ) )  <->  [. ( iota_ l  e.  K E. z  e.  U  X  =  ( z  .+  ( l  .x.  Z
) ) )  / 
l ]. E. z  e.  U  X  =  ( z  .+  ( l 
.x.  Z ) ) ) )
3616, 35mpbird 223 . 2  |-  ( ph  ->  [. ( G `  X )  /  l ]. E. z  e.  U  X  =  ( z  .+  ( l  .x.  Z
) ) )
37 fvex 5539 . . 3  |-  ( G `
 X )  e. 
_V
38 oveq1 5865 . . . . . 6  |-  ( l  =  ( G `  X )  ->  (
l  .x.  Z )  =  ( ( G `
 X )  .x.  Z ) )
3938oveq2d 5874 . . . . 5  |-  ( l  =  ( G `  X )  ->  (
z  .+  ( l  .x.  Z ) )  =  ( z  .+  (
( G `  X
)  .x.  Z )
) )
4039eqeq2d 2294 . . . 4  |-  ( l  =  ( G `  X )  ->  ( X  =  ( z  .+  ( l  .x.  Z
) )  <->  X  =  ( z  .+  (
( G `  X
)  .x.  Z )
) ) )
4140rexbidv 2564 . . 3  |-  ( l  =  ( G `  X )  ->  ( E. z  e.  U  X  =  ( z  .+  ( l  .x.  Z
) )  <->  E. z  e.  U  X  =  ( z  .+  (
( G `  X
)  .x.  Z )
) ) )
4237, 41sbcie 3025 . 2  |-  ( [. ( G `  X )  /  l ]. E. z  e.  U  X  =  ( z  .+  ( l  .x.  Z
) )  <->  E. z  e.  U  X  =  ( z  .+  (
( G `  X
)  .x.  Z )
) )
4336, 42sylib 188 1  |-  ( ph  ->  E. z  e.  U  X  =  ( z  .+  ( ( G `  X )  .x.  Z
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1623    e. wcel 1684   E.wrex 2544   E!wreu 2545   [.wsbc 2991   {csn 3640    e. cmpt 4077   ` cfv 5255  (class class class)co 5858   iota_crio 6297   Basecbs 13148   +g cplusg 13208  Scalarcsca 13211   .scvsca 13212   0gc0g 13400   LSSumclsm 14945   LSpanclspn 15728   LVecclvec 15855  LSHypclsh 29165
This theorem is referenced by:  lshpkrlem6  29305
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-rmo 2551  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-int 3863  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-tpos 6234  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-3 9805  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-0g 13404  df-mnd 14367  df-submnd 14416  df-grp 14489  df-minusg 14490  df-sbg 14491  df-subg 14618  df-cntz 14793  df-lsm 14947  df-cmn 15091  df-abl 15092  df-mgp 15326  df-rng 15340  df-ur 15342  df-oppr 15405  df-dvdsr 15423  df-unit 15424  df-invr 15454  df-drng 15514  df-lmod 15629  df-lss 15690  df-lsp 15729  df-lvec 15856  df-lshyp 29167
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