MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sspmlem Structured version   Unicode version

Theorem sspmlem 25843
Description: Lemma for sspm 25845 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 6415 . . . . . 6  |-  ( ( x  e.  Y  /\  y  e.  Y )  ->  ( x ( G  |`  ( Y  X.  Y
) ) y )  =  ( x G y ) )
32adantl 464 . . . . 5  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  H )  /\  ( x  e.  Y  /\  y  e.  Y
) )  ->  (
x ( G  |`  ( Y  X.  Y
) ) y )  =  ( x G y ) )
41, 3eqtr4d 2498 . . . 4  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  H )  /\  ( x  e.  Y  /\  y  e.  Y
) )  ->  (
x F y )  =  ( x ( G  |`  ( Y  X.  Y ) ) y ) )
54ralrimivva 2875 . . 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 2454 . . 3  |-  ( Y  X.  Y )  =  ( Y  X.  Y
)
75, 6jctil 535 . 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 25837 . . . 4  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  W  e.  NrmCVec )
10 sspmlem.2 . . . 4  |-  ( W  e.  NrmCVec  ->  F : ( Y  X.  Y ) --> R )
11 ffn 5713 . . . 4  |-  ( F : ( Y  X.  Y ) --> R  ->  F  Fn  ( Y  X.  Y ) )
129, 10, 113syl 20 . . 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 5713 . . . . . 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 463 . . . 4  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  G  Fn  ( ( BaseSet `  U
)  X.  ( BaseSet `  U ) ) )
17 eqid 2454 . . . . . 6  |-  ( BaseSet `  U )  =  (
BaseSet `  U )
18 sspmlem.y . . . . . 6  |-  Y  =  ( BaseSet `  W )
1917, 18, 8sspba 25838 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  Y  C_  ( BaseSet `  U )
)
20 xpss12 5096 . . . . 5  |-  ( ( Y  C_  ( BaseSet `  U )  /\  Y  C_  ( BaseSet `  U )
)  ->  ( Y  X.  Y )  C_  (
( BaseSet `  U )  X.  ( BaseSet `  U )
) )
2119, 19, 20syl2anc 659 . . . 4  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  ( Y  X.  Y )  C_  ( ( BaseSet `  U
)  X.  ( BaseSet `  U ) ) )
22 fnssres 5676 . . . 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 659 . . 3  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  ( G  |`  ( Y  X.  Y ) )  Fn  ( Y  X.  Y
) )
24 eqfnov 6381 . . 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 659 . 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 232 1  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  F  =  ( G  |`  ( Y  X.  Y
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823   A.wral 2804    C_ wss 3461    X. cxp 4986    |` cres 4990    Fn wfn 5565   -->wf 5566   ` cfv 5570  (class class class)co 6270   NrmCVeccnv 25675   BaseSetcba 25677   SubSpcss 25832
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-fo 5576  df-fv 5578  df-ov 6273  df-oprab 6274  df-1st 6773  df-2nd 6774  df-vc 25637  df-nv 25683  df-va 25686  df-ba 25687  df-sm 25688  df-nmcv 25691  df-ssp 25833
This theorem is referenced by:  sspm  25845  sspi  25850  sspims  25852
  Copyright terms: Public domain W3C validator