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Theorem sspmlem 22231
Description: Lemma for sspm 22233 and others. (Contributed by NM, 1-Feb-2008.) (New usage is discouraged.)
Hypotheses
Ref Expression
sspmlem.y  |-  Y  =  ( BaseSet `  W )
sspmlem.h  |-  H  =  ( SubSp `  U )
sspmlem.1  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  H )  /\  ( x  e.  Y  /\  y  e.  Y
) )  ->  (
x F y )  =  ( x G y ) )
sspmlem.2  |-  ( W  e.  NrmCVec  ->  F : ( Y  X.  Y ) --> R )
sspmlem.3  |-  ( U  e.  NrmCVec  ->  G : ( ( BaseSet `  U )  X.  ( BaseSet `  U )
) --> S )
Assertion
Ref Expression
sspmlem  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  F  =  ( G  |`  ( Y  X.  Y
) ) )
Distinct variable groups:    x, y, F    x, G, y    x, H, y    x, U, y   
x, W, y    x, Y, y
Allowed substitution hints:    R( x, y)    S( x, y)

Proof of Theorem sspmlem
StepHypRef Expression
1 sspmlem.1 . . . . 5  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  H )  /\  ( x  e.  Y  /\  y  e.  Y
) )  ->  (
x F y )  =  ( x G y ) )
2 ovres 6213 . . . . . 6  |-  ( ( x  e.  Y  /\  y  e.  Y )  ->  ( x ( G  |`  ( Y  X.  Y
) ) y )  =  ( x G y ) )
32adantl 453 . . . . 5  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  H )  /\  ( x  e.  Y  /\  y  e.  Y
) )  ->  (
x ( G  |`  ( Y  X.  Y
) ) y )  =  ( x G y ) )
41, 3eqtr4d 2471 . . . 4  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  H )  /\  ( x  e.  Y  /\  y  e.  Y
) )  ->  (
x F y )  =  ( x ( G  |`  ( Y  X.  Y ) ) y ) )
54ralrimivva 2798 . . 3  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  A. x  e.  Y  A. y  e.  Y  ( x F y )  =  ( x ( G  |`  ( Y  X.  Y
) ) y ) )
6 eqid 2436 . . 3  |-  ( Y  X.  Y )  =  ( Y  X.  Y
)
75, 6jctil 524 . 2  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  (
( Y  X.  Y
)  =  ( Y  X.  Y )  /\  A. x  e.  Y  A. y  e.  Y  (
x F y )  =  ( x ( G  |`  ( Y  X.  Y ) ) y ) ) )
8 sspmlem.h . . . . 5  |-  H  =  ( SubSp `  U )
98sspnv 22225 . . . 4  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  W  e.  NrmCVec )
10 sspmlem.2 . . . 4  |-  ( W  e.  NrmCVec  ->  F : ( Y  X.  Y ) --> R )
11 ffn 5591 . . . 4  |-  ( F : ( Y  X.  Y ) --> R  ->  F  Fn  ( Y  X.  Y ) )
129, 10, 113syl 19 . . 3  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  F  Fn  ( Y  X.  Y
) )
13 sspmlem.3 . . . . . 6  |-  ( U  e.  NrmCVec  ->  G : ( ( BaseSet `  U )  X.  ( BaseSet `  U )
) --> S )
14 ffn 5591 . . . . . 6  |-  ( G : ( ( BaseSet `  U )  X.  ( BaseSet
`  U ) ) --> S  ->  G  Fn  ( ( BaseSet `  U
)  X.  ( BaseSet `  U ) ) )
1513, 14syl 16 . . . . 5  |-  ( U  e.  NrmCVec  ->  G  Fn  (
( BaseSet `  U )  X.  ( BaseSet `  U )
) )
1615adantr 452 . . . 4  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  G  Fn  ( ( BaseSet `  U
)  X.  ( BaseSet `  U ) ) )
17 eqid 2436 . . . . . 6  |-  ( BaseSet `  U )  =  (
BaseSet `  U )
18 sspmlem.y . . . . . 6  |-  Y  =  ( BaseSet `  W )
1917, 18, 8sspba 22226 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  Y  C_  ( BaseSet `  U )
)
20 xpss12 4981 . . . . 5  |-  ( ( Y  C_  ( BaseSet `  U )  /\  Y  C_  ( BaseSet `  U )
)  ->  ( Y  X.  Y )  C_  (
( BaseSet `  U )  X.  ( BaseSet `  U )
) )
2119, 19, 20syl2anc 643 . . . 4  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  ( Y  X.  Y )  C_  ( ( BaseSet `  U
)  X.  ( BaseSet `  U ) ) )
22 fnssres 5558 . . . 4  |-  ( ( G  Fn  ( (
BaseSet `  U )  X.  ( BaseSet `  U )
)  /\  ( Y  X.  Y )  C_  (
( BaseSet `  U )  X.  ( BaseSet `  U )
) )  ->  ( G  |`  ( Y  X.  Y ) )  Fn  ( Y  X.  Y
) )
2316, 21, 22syl2anc 643 . . 3  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  ( G  |`  ( Y  X.  Y ) )  Fn  ( Y  X.  Y
) )
24 eqfnov 6176 . . 3  |-  ( ( F  Fn  ( Y  X.  Y )  /\  ( G  |`  ( Y  X.  Y ) )  Fn  ( Y  X.  Y ) )  -> 
( F  =  ( G  |`  ( Y  X.  Y ) )  <->  ( ( Y  X.  Y )  =  ( Y  X.  Y
)  /\  A. x  e.  Y  A. y  e.  Y  ( x F y )  =  ( x ( G  |`  ( Y  X.  Y
) ) y ) ) ) )
2512, 23, 24syl2anc 643 . 2  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  ( F  =  ( G  |`  ( Y  X.  Y
) )  <->  ( ( Y  X.  Y )  =  ( Y  X.  Y
)  /\  A. x  e.  Y  A. y  e.  Y  ( x F y )  =  ( x ( G  |`  ( Y  X.  Y
) ) y ) ) ) )
267, 25mpbird 224 1  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  F  =  ( G  |`  ( Y  X.  Y
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2705    C_ wss 3320    X. cxp 4876    |` cres 4880    Fn wfn 5449   -->wf 5450   ` cfv 5454  (class class class)co 6081   NrmCVeccnv 22063   BaseSetcba 22065   SubSpcss 22220
This theorem is referenced by:  sspm  22233  sspi  22238  sspims  22240
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-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  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-ral 2710  df-rex 2711  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-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  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-fo 5460  df-fv 5462  df-ov 6084  df-oprab 6085  df-1st 6349  df-2nd 6350  df-vc 22025  df-nv 22071  df-va 22074  df-ba 22075  df-sm 22076  df-nmcv 22079  df-ssp 22221
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