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

Theorem rrxcph 22351
Description: Generalized Euclidean real spaces are pre-Hilbert spaces. (Contributed by Thierry Arnoux, 23-Jun-2019.) (Proof shortened by AV, 22-Jul-2019.)
Hypotheses
Ref Expression
rrxval.r  |-  H  =  (ℝ^ `  I )
rrxbase.b  |-  B  =  ( Base `  H
)
Assertion
Ref Expression
rrxcph  |-  ( I  e.  V  ->  H  e.  CPreHil )

Proof of Theorem rrxcph
Dummy variables  f  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rrxval.r . . 3  |-  H  =  (ℝ^ `  I )
21rrxval 22346 . 2  |-  ( I  e.  V  ->  H  =  (toCHil `  (RRfld freeLMod  I ) ) )
3 eqid 2451 . . 3  |-  (toCHil `  (RRfld freeLMod  I ) )  =  (toCHil `  (RRfld freeLMod  I ) )
4 eqid 2451 . . 3  |-  ( Base `  (RRfld freeLMod  I ) )  =  ( Base `  (RRfld freeLMod  I ) )
5 eqid 2451 . . 3  |-  (Scalar `  (RRfld freeLMod  I ) )  =  (Scalar `  (RRfld freeLMod  I ) )
6 eqid 2451 . . . 4  |-  (RRfld freeLMod  I )  =  (RRfld freeLMod  I )
7 rebase 19174 . . . 4  |-  RR  =  ( Base ` RRfld )
8 remulr 19179 . . . 4  |-  x.  =  ( .r ` RRfld )
9 eqid 2451 . . . 4  |-  ( .i
`  (RRfld freeLMod  I ) )  =  ( .i `  (RRfld freeLMod  I ) )
10 eqid 2451 . . . 4  |-  ( 0g
`  (RRfld freeLMod  I ) )  =  ( 0g `  (RRfld freeLMod  I ) )
11 re0g 19180 . . . 4  |-  0  =  ( 0g ` RRfld )
12 refldcj 19188 . . . 4  |-  *  =  ( *r ` RRfld )
13 refld 19187 . . . . 5  |- RRfld  e. Field
1413a1i 11 . . . 4  |-  ( I  e.  V  -> RRfld  e. Field )
15 fconstmpt 4878 . . . . 5  |-  ( I  X.  { 0 } )  =  ( x  e.  I  |->  0 )
166, 7, 4frlmbasf 19323 . . . . . . . 8  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  ->  f : I --> RR )
17 ffn 5728 . . . . . . . 8  |-  ( f : I --> RR  ->  f  Fn  I )
1816, 17syl 17 . . . . . . 7  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  ->  f  Fn  I
)
19183adant3 1028 . . . . . 6  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) )  /\  ( f ( .i
`  (RRfld freeLMod  I ) ) f )  =  0 )  ->  f  Fn  I )
20 simpl 459 . . . . . . . . . . . . . . . . 17  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  ->  I  e.  V
)
2113a1i 11 . . . . . . . . . . . . . . . . 17  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  -> RRfld  e. Field )
22 simpr 463 . . . . . . . . . . . . . . . . 17  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  ->  f  e.  (
Base `  (RRfld freeLMod  I ) ) )
236, 7, 8, 4, 9frlmipval 19337 . . . . . . . . . . . . . . . . 17  |-  ( ( ( I  e.  V  /\ RRfld  e. Field )  /\  (
f  e.  ( Base `  (RRfld freeLMod  I ) )  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) ) )  ->  ( f
( .i `  (RRfld freeLMod  I ) ) f )  =  (RRfld  gsumg  ( f  oF  x.  f ) ) )
2420, 21, 22, 22, 23syl22anc 1269 . . . . . . . . . . . . . . . 16  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  ->  ( f ( .i `  (RRfld freeLMod  I ) ) f )  =  (RRfld  gsumg  ( f  oF  x.  f ) ) )
25 ovex 6318 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( f `  x )  x.  ( f `  x ) )  e. 
_V
2625a1i 11 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  /\  x  e.  I )  ->  (
( f `  x
)  x.  ( f `
 x ) )  e.  _V )
27 inidm 3641 . . . . . . . . . . . . . . . . . . . 20  |-  ( I  i^i  I )  =  I
28 eqidd 2452 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  /\  x  e.  I )  ->  (
f `  x )  =  ( f `  x ) )
2918, 18, 20, 20, 27, 28, 28offval 6538 . . . . . . . . . . . . . . . . . . 19  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  ->  ( f  oF  x.  f )  =  ( x  e.  I  |->  ( ( f `
 x )  x.  ( f `  x
) ) ) )
3018, 18, 20, 20, 27, 28, 28ofval 6540 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  /\  x  e.  I )  ->  (
( f  oF  x.  f ) `  x )  =  ( ( f `  x
)  x.  ( f `
 x ) ) )
3116ffvelrnda 6022 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  /\  x  e.  I )  ->  (
f `  x )  e.  RR )
3231, 31remulcld 9671 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  /\  x  e.  I )  ->  (
( f `  x
)  x.  ( f `
 x ) )  e.  RR )
3330, 32eqeltrd 2529 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  /\  x  e.  I )  ->  (
( f  oF  x.  f ) `  x )  e.  RR )
3426, 29, 33fmpt2d 6053 . . . . . . . . . . . . . . . . . 18  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  ->  ( f  oF  x.  f ) : I --> RR )
35 ovex 6318 . . . . . . . . . . . . . . . . . . . 20  |-  ( f  oF  x.  f
)  e.  _V
3635a1i 11 . . . . . . . . . . . . . . . . . . 19  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  ->  ( f  oF  x.  f )  e.  _V )
37 ffun 5731 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( f  oF  x.  f ) : I --> RR  ->  Fun  ( f  oF  x.  f
) )
3834, 37syl 17 . . . . . . . . . . . . . . . . . . 19  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  ->  Fun  ( f  oF  x.  f
) )
396, 11, 4frlmbasfsupp 19321 . . . . . . . . . . . . . . . . . . 19  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  ->  f finSupp  0 )
40 0red 9644 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  ->  0  e.  RR )
41 simpr 463 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  /\  x  e.  RR )  ->  x  e.  RR )
4241recnd 9669 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  /\  x  e.  RR )  ->  x  e.  CC )
4342mul02d 9831 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  /\  x  e.  RR )  ->  (
0  x.  x )  =  0 )
4420, 40, 16, 16, 43suppofss1d 6952 . . . . . . . . . . . . . . . . . . 19  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  ->  ( ( f  oF  x.  f
) supp  0 )  C_  ( f supp  0 ) )
45 fsuppsssupp 7899 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( f  oF  x.  f )  e.  _V  /\  Fun  ( f  oF  x.  f ) )  /\  ( f finSupp  0  /\  ( ( f  oF  x.  f ) supp  0 )  C_  (
f supp  0 ) ) )  ->  ( f  oF  x.  f
) finSupp  0 )
4636, 38, 39, 44, 45syl22anc 1269 . . . . . . . . . . . . . . . . . 18  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  ->  ( f  oF  x.  f ) finSupp 
0 )
47 regsumsupp 19190 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( f  oF  x.  f ) : I --> RR  /\  (
f  oF  x.  f ) finSupp  0  /\  I  e.  V )  ->  (RRfld  gsumg  ( f  oF  x.  f ) )  =  sum_ x  e.  ( ( f  oF  x.  f ) supp  0
) ( ( f  oF  x.  f
) `  x )
)
4834, 46, 20, 47syl3anc 1268 . . . . . . . . . . . . . . . . 17  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  ->  (RRfld  gsumg  ( f  oF  x.  f ) )  =  sum_ x  e.  ( ( f  oF  x.  f ) supp  0
) ( ( f  oF  x.  f
) `  x )
)
49 suppssdm 6927 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( f supp  0 )  C_  dom  f
50 fdm 5733 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( f : I --> RR  ->  dom  f  =  I )
5116, 50syl 17 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  ->  dom  f  =  I )
5249, 51syl5sseq 3480 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  ->  ( f supp  0
)  C_  I )
5344, 52sstrd 3442 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  ->  ( ( f  oF  x.  f
) supp  0 )  C_  I )
5453sselda 3432 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  /\  x  e.  ( ( f  oF  x.  f ) supp  0 ) )  ->  x  e.  I )
5554, 30syldan 473 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  /\  x  e.  ( ( f  oF  x.  f ) supp  0 ) )  -> 
( ( f  oF  x.  f ) `
 x )  =  ( ( f `  x )  x.  (
f `  x )
) )
5655sumeq2dv 13769 . . . . . . . . . . . . . . . . 17  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  ->  sum_ x  e.  ( ( f  oF  x.  f ) supp  0
) ( ( f  oF  x.  f
) `  x )  =  sum_ x  e.  ( ( f  oF  x.  f ) supp  0
) ( ( f `
 x )  x.  ( f `  x
) ) )
5748, 56eqtrd 2485 . . . . . . . . . . . . . . . 16  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  ->  (RRfld  gsumg  ( f  oF  x.  f ) )  =  sum_ x  e.  ( ( f  oF  x.  f ) supp  0
) ( ( f `
 x )  x.  ( f `  x
) ) )
5824, 57eqtrd 2485 . . . . . . . . . . . . . . 15  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  ->  ( f ( .i `  (RRfld freeLMod  I ) ) f )  = 
sum_ x  e.  (
( f  oF  x.  f ) supp  0
) ( ( f `
 x )  x.  ( f `  x
) ) )
59583adant3 1028 . . . . . . . . . . . . . 14  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) )  /\  ( f ( .i
`  (RRfld freeLMod  I ) ) f )  =  0 )  ->  ( f
( .i `  (RRfld freeLMod  I ) ) f )  =  sum_ x  e.  ( ( f  oF  x.  f ) supp  0
) ( ( f `
 x )  x.  ( f `  x
) ) )
60 simp3 1010 . . . . . . . . . . . . . 14  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) )  /\  ( f ( .i
`  (RRfld freeLMod  I ) ) f )  =  0 )  ->  ( f
( .i `  (RRfld freeLMod  I ) ) f )  =  0 )
6159, 60eqtr3d 2487 . . . . . . . . . . . . 13  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) )  /\  ( f ( .i
`  (RRfld freeLMod  I ) ) f )  =  0 )  ->  sum_ x  e.  ( ( f  oF  x.  f ) supp  0 ) ( ( f `  x )  x.  ( f `  x ) )  =  0 )
6239fsuppimpd 7890 . . . . . . . . . . . . . . . 16  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  ->  ( f supp  0
)  e.  Fin )
63 ssfi 7792 . . . . . . . . . . . . . . . 16  |-  ( ( ( f supp  0 )  e.  Fin  /\  (
( f  oF  x.  f ) supp  0
)  C_  ( f supp  0 ) )  -> 
( ( f  oF  x.  f ) supp  0 )  e.  Fin )
6462, 44, 63syl2anc 667 . . . . . . . . . . . . . . 15  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  ->  ( ( f  oF  x.  f
) supp  0 )  e. 
Fin )
6554, 32syldan 473 . . . . . . . . . . . . . . 15  |-  ( ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  /\  x  e.  ( ( f  oF  x.  f ) supp  0 ) )  -> 
( ( f `  x )  x.  (
f `  x )
)  e.  RR )
6631msqge0d 10182 . . . . . . . . . . . . . . . 16  |-  ( ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  /\  x  e.  I )  ->  0  <_  ( ( f `  x )  x.  (
f `  x )
) )
6754, 66syldan 473 . . . . . . . . . . . . . . 15  |-  ( ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  /\  x  e.  ( ( f  oF  x.  f ) supp  0 ) )  -> 
0  <_  ( (
f `  x )  x.  ( f `  x
) ) )
6864, 65, 67fsum00 13858 . . . . . . . . . . . . . 14  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  ->  ( sum_ x  e.  ( ( f  oF  x.  f ) supp  0 ) ( ( f `  x )  x.  ( f `  x ) )  =  0  <->  A. x  e.  ( ( f  oF  x.  f ) supp  0
) ( ( f `
 x )  x.  ( f `  x
) )  =  0 ) )
69683adant3 1028 . . . . . . . . . . . . 13  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) )  /\  ( f ( .i
`  (RRfld freeLMod  I ) ) f )  =  0 )  ->  ( sum_ x  e.  ( ( f  oF  x.  f
) supp  0 ) ( ( f `  x
)  x.  ( f `
 x ) )  =  0  <->  A. x  e.  ( ( f  oF  x.  f ) supp  0 ) ( ( f `  x )  x.  ( f `  x ) )  =  0 ) )
7061, 69mpbid 214 . . . . . . . . . . . 12  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) )  /\  ( f ( .i
`  (RRfld freeLMod  I ) ) f )  =  0 )  ->  A. x  e.  ( ( f  oF  x.  f ) supp  0 ) ( ( f `  x )  x.  ( f `  x ) )  =  0 )
7170r19.21bi 2757 . . . . . . . . . . 11  |-  ( ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) )  /\  ( f ( .i `  (RRfld freeLMod  I ) ) f )  =  0 )  /\  x  e.  ( ( f  oF  x.  f ) supp  0 ) )  -> 
( ( f `  x )  x.  (
f `  x )
)  =  0 )
7271adantlr 721 . . . . . . . . . 10  |-  ( ( ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) )  /\  (
f ( .i `  (RRfld freeLMod  I ) ) f )  =  0 )  /\  x  e.  I
)  /\  x  e.  ( ( f  oF  x.  f ) supp  0 ) )  -> 
( ( f `  x )  x.  (
f `  x )
)  =  0 )
73313adantl3 1166 . . . . . . . . . . . . 13  |-  ( ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) )  /\  ( f ( .i `  (RRfld freeLMod  I ) ) f )  =  0 )  /\  x  e.  I )  ->  (
f `  x )  e.  RR )
7473recnd 9669 . . . . . . . . . . . 12  |-  ( ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) )  /\  ( f ( .i `  (RRfld freeLMod  I ) ) f )  =  0 )  /\  x  e.  I )  ->  (
f `  x )  e.  CC )
7574, 74mul0ord 10262 . . . . . . . . . . 11  |-  ( ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) )  /\  ( f ( .i `  (RRfld freeLMod  I ) ) f )  =  0 )  /\  x  e.  I )  ->  (
( ( f `  x )  x.  (
f `  x )
)  =  0  <->  (
( f `  x
)  =  0  \/  ( f `  x
)  =  0 ) ) )
7675adantr 467 . . . . . . . . . 10  |-  ( ( ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) )  /\  (
f ( .i `  (RRfld freeLMod  I ) ) f )  =  0 )  /\  x  e.  I
)  /\  x  e.  ( ( f  oF  x.  f ) supp  0 ) )  -> 
( ( ( f `
 x )  x.  ( f `  x
) )  =  0  <-> 
( ( f `  x )  =  0  \/  ( f `  x )  =  0 ) ) )
7772, 76mpbid 214 . . . . . . . . 9  |-  ( ( ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) )  /\  (
f ( .i `  (RRfld freeLMod  I ) ) f )  =  0 )  /\  x  e.  I
)  /\  x  e.  ( ( f  oF  x.  f ) supp  0 ) )  -> 
( ( f `  x )  =  0  \/  ( f `  x )  =  0 ) )
78 oridm 517 . . . . . . . . 9  |-  ( ( ( f `  x
)  =  0  \/  ( f `  x
)  =  0 )  <-> 
( f `  x
)  =  0 )
7977, 78sylib 200 . . . . . . . 8  |-  ( ( ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) )  /\  (
f ( .i `  (RRfld freeLMod  I ) ) f )  =  0 )  /\  x  e.  I
)  /\  x  e.  ( ( f  oF  x.  f ) supp  0 ) )  -> 
( f `  x
)  =  0 )
80343adant3 1028 . . . . . . . . . . 11  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) )  /\  ( f ( .i
`  (RRfld freeLMod  I ) ) f )  =  0 )  ->  ( f  oF  x.  f
) : I --> RR )
8180adantr 467 . . . . . . . . . 10  |-  ( ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) )  /\  ( f ( .i `  (RRfld freeLMod  I ) ) f )  =  0 )  /\  x  e.  I )  ->  (
f  oF  x.  f ) : I --> RR )
82 ssid 3451 . . . . . . . . . . 11  |-  ( ( f  oF  x.  f ) supp  0 ) 
C_  ( ( f  oF  x.  f
) supp  0 )
8382a1i 11 . . . . . . . . . 10  |-  ( ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) )  /\  ( f ( .i `  (RRfld freeLMod  I ) ) f )  =  0 )  /\  x  e.  I )  ->  (
( f  oF  x.  f ) supp  0
)  C_  ( (
f  oF  x.  f ) supp  0 ) )
84 simpl1 1011 . . . . . . . . . 10  |-  ( ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) )  /\  ( f ( .i `  (RRfld freeLMod  I ) ) f )  =  0 )  /\  x  e.  I )  ->  I  e.  V )
85 0red 9644 . . . . . . . . . 10  |-  ( ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) )  /\  ( f ( .i `  (RRfld freeLMod  I ) ) f )  =  0 )  /\  x  e.  I )  ->  0  e.  RR )
8681, 83, 84, 85suppssr 6946 . . . . . . . . 9  |-  ( ( ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) )  /\  (
f ( .i `  (RRfld freeLMod  I ) ) f )  =  0 )  /\  x  e.  I
)  /\  x  e.  ( I  \  (
( f  oF  x.  f ) supp  0
) ) )  -> 
( ( f  oF  x.  f ) `
 x )  =  0 )
87303adantl3 1166 . . . . . . . . . . . . 13  |-  ( ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) )  /\  ( f ( .i `  (RRfld freeLMod  I ) ) f )  =  0 )  /\  x  e.  I )  ->  (
( f  oF  x.  f ) `  x )  =  ( ( f `  x
)  x.  ( f `
 x ) ) )
8887eqeq1d 2453 . . . . . . . . . . . 12  |-  ( ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) )  /\  ( f ( .i `  (RRfld freeLMod  I ) ) f )  =  0 )  /\  x  e.  I )  ->  (
( ( f  oF  x.  f ) `
 x )  =  0  <->  ( ( f `
 x )  x.  ( f `  x
) )  =  0 ) )
8988, 75bitrd 257 . . . . . . . . . . 11  |-  ( ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) )  /\  ( f ( .i `  (RRfld freeLMod  I ) ) f )  =  0 )  /\  x  e.  I )  ->  (
( ( f  oF  x.  f ) `
 x )  =  0  <->  ( ( f `
 x )  =  0  \/  ( f `
 x )  =  0 ) ) )
9089, 78syl6bb 265 . . . . . . . . . 10  |-  ( ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) )  /\  ( f ( .i `  (RRfld freeLMod  I ) ) f )  =  0 )  /\  x  e.  I )  ->  (
( ( f  oF  x.  f ) `
 x )  =  0  <->  ( f `  x )  =  0 ) )
9190biimpa 487 . . . . . . . . 9  |-  ( ( ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) )  /\  (
f ( .i `  (RRfld freeLMod  I ) ) f )  =  0 )  /\  x  e.  I
)  /\  ( (
f  oF  x.  f ) `  x
)  =  0 )  ->  ( f `  x )  =  0 )
9286, 91syldan 473 . . . . . . . 8  |-  ( ( ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) )  /\  (
f ( .i `  (RRfld freeLMod  I ) ) f )  =  0 )  /\  x  e.  I
)  /\  x  e.  ( I  \  (
( f  oF  x.  f ) supp  0
) ) )  -> 
( f `  x
)  =  0 )
93 undif 3848 . . . . . . . . . . . . 13  |-  ( ( ( f  oF  x.  f ) supp  0
)  C_  I  <->  ( (
( f  oF  x.  f ) supp  0
)  u.  ( I 
\  ( ( f  oF  x.  f
) supp  0 ) ) )  =  I )
9453, 93sylib 200 . . . . . . . . . . . 12  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  ->  ( ( ( f  oF  x.  f ) supp  0 )  u.  ( I  \ 
( ( f  oF  x.  f ) supp  0 ) ) )  =  I )
9594eleq2d 2514 . . . . . . . . . . 11  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  ->  ( x  e.  ( ( ( f  oF  x.  f
) supp  0 )  u.  ( I  \  (
( f  oF  x.  f ) supp  0
) ) )  <->  x  e.  I ) )
96953adant3 1028 . . . . . . . . . 10  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) )  /\  ( f ( .i
`  (RRfld freeLMod  I ) ) f )  =  0 )  ->  ( x  e.  ( ( ( f  oF  x.  f
) supp  0 )  u.  ( I  \  (
( f  oF  x.  f ) supp  0
) ) )  <->  x  e.  I ) )
9796biimpar 488 . . . . . . . . 9  |-  ( ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) )  /\  ( f ( .i `  (RRfld freeLMod  I ) ) f )  =  0 )  /\  x  e.  I )  ->  x  e.  ( ( ( f  oF  x.  f
) supp  0 )  u.  ( I  \  (
( f  oF  x.  f ) supp  0
) ) ) )
98 elun 3574 . . . . . . . . 9  |-  ( x  e.  ( ( ( f  oF  x.  f ) supp  0 )  u.  ( I  \ 
( ( f  oF  x.  f ) supp  0 ) ) )  <-> 
( x  e.  ( ( f  oF  x.  f ) supp  0
)  \/  x  e.  ( I  \  (
( f  oF  x.  f ) supp  0
) ) ) )
9997, 98sylib 200 . . . . . . . 8  |-  ( ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) )  /\  ( f ( .i `  (RRfld freeLMod  I ) ) f )  =  0 )  /\  x  e.  I )  ->  (
x  e.  ( ( f  oF  x.  f ) supp  0 )  \/  x  e.  ( I  \  ( ( f  oF  x.  f ) supp  0 ) ) ) )
10079, 92, 99mpjaodan 795 . . . . . . 7  |-  ( ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) )  /\  ( f ( .i `  (RRfld freeLMod  I ) ) f )  =  0 )  /\  x  e.  I )  ->  (
f `  x )  =  0 )
101100ralrimiva 2802 . . . . . 6  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) )  /\  ( f ( .i
`  (RRfld freeLMod  I ) ) f )  =  0 )  ->  A. x  e.  I  ( f `  x )  =  0 )
102 fconstfv 6126 . . . . . . . 8  |-  ( f : I --> { 0 }  <->  ( f  Fn  I  /\  A. x  e.  I  ( f `  x )  =  0 ) )
103102biimpri 210 . . . . . . 7  |-  ( ( f  Fn  I  /\  A. x  e.  I  ( f `  x )  =  0 )  -> 
f : I --> { 0 } )
104 c0ex 9637 . . . . . . . 8  |-  0  e.  _V
105104fconst2 6121 . . . . . . 7  |-  ( f : I --> { 0 }  <->  f  =  ( I  X.  { 0 } ) )
106103, 105sylib 200 . . . . . 6  |-  ( ( f  Fn  I  /\  A. x  e.  I  ( f `  x )  =  0 )  -> 
f  =  ( I  X.  { 0 } ) )
10719, 101, 106syl2anc 667 . . . . 5  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) )  /\  ( f ( .i
`  (RRfld freeLMod  I ) ) f )  =  0 )  ->  f  =  ( I  X.  { 0 } ) )
108 isfld 17984 . . . . . . . . . . 11  |-  (RRfld  e. Field  <->  (RRfld  e.  DivRing  /\ RRfld  e.  CRing ) )
10913, 108mpbi 212 . . . . . . . . . 10  |-  (RRfld  e.  DivRing  /\ RRfld  e.  CRing )
110109simpli 460 . . . . . . . . 9  |- RRfld  e.  DivRing
111 drngring 17982 . . . . . . . . 9  |-  (RRfld  e.  DivRing  -> RRfld 
e.  Ring )
112110, 111ax-mp 5 . . . . . . . 8  |- RRfld  e.  Ring
1136, 11frlm0 19317 . . . . . . . 8  |-  ( (RRfld 
e.  Ring  /\  I  e.  V )  ->  (
I  X.  { 0 } )  =  ( 0g `  (RRfld freeLMod  I ) ) )
114112, 113mpan 676 . . . . . . 7  |-  ( I  e.  V  ->  (
I  X.  { 0 } )  =  ( 0g `  (RRfld freeLMod  I ) ) )
11515, 114syl5reqr 2500 . . . . . 6  |-  ( I  e.  V  ->  ( 0g `  (RRfld freeLMod  I ) )  =  ( x  e.  I  |->  0 ) )
1161153ad2ant1 1029 . . . . 5  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) )  /\  ( f ( .i
`  (RRfld freeLMod  I ) ) f )  =  0 )  ->  ( 0g `  (RRfld freeLMod  I ) )  =  ( x  e.  I  |->  0 ) )
11715, 107, 1163eqtr4a 2511 . . . 4  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) )  /\  ( f ( .i
`  (RRfld freeLMod  I ) ) f )  =  0 )  ->  f  =  ( 0g `  (RRfld freeLMod  I ) ) )
118 cjre 13202 . . . . 5  |-  ( x  e.  RR  ->  (
* `  x )  =  x )
119118adantl 468 . . . 4  |-  ( ( I  e.  V  /\  x  e.  RR )  ->  ( * `  x
)  =  x )
120 id 22 . . . 4  |-  ( I  e.  V  ->  I  e.  V )
1216, 7, 8, 4, 9, 10, 11, 12, 14, 117, 119, 120frlmphl 19339 . . 3  |-  ( I  e.  V  ->  (RRfld freeLMod  I )  e.  PreHil )
122 df-refld 19173 . . . 4  |- RRfld  =  (flds  RR )
1236frlmsca 19316 . . . . 5  |-  ( (RRfld 
e. Field  /\  I  e.  V
)  -> RRfld  =  (Scalar `  (RRfld freeLMod  I ) ) )
12413, 123mpan 676 . . . 4  |-  ( I  e.  V  -> RRfld  =  (Scalar `  (RRfld freeLMod  I ) ) )
125122, 124syl5reqr 2500 . . 3  |-  ( I  e.  V  ->  (Scalar `  (RRfld freeLMod  I ) )  =  (flds  RR ) )
126 simpr1 1014 . . . 4  |-  ( ( I  e.  V  /\  ( f  e.  RR  /\  f  e.  RR  /\  0  <_  f ) )  ->  f  e.  RR )
127 simpr3 1016 . . . 4  |-  ( ( I  e.  V  /\  ( f  e.  RR  /\  f  e.  RR  /\  0  <_  f ) )  ->  0  <_  f
)
128126, 127resqrtcld 13479 . . 3  |-  ( ( I  e.  V  /\  ( f  e.  RR  /\  f  e.  RR  /\  0  <_  f ) )  ->  ( sqr `  f
)  e.  RR )
12964, 65, 67fsumge0 13855 . . . . 5  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  ->  0  <_  sum_ x  e.  ( ( f  oF  x.  f ) supp  0 ) ( ( f `  x )  x.  ( f `  x ) ) )
130129, 57breqtrrd 4429 . . . 4  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  ->  0  <_  (RRfld  gsumg  (
f  oF  x.  f ) ) )
131130, 24breqtrrd 4429 . . 3  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  ->  0  <_  (
f ( .i `  (RRfld freeLMod  I ) ) f ) )
1323, 4, 5, 121, 125, 9, 128, 131tchcph 22211 . 2  |-  ( I  e.  V  ->  (toCHil `  (RRfld freeLMod  I ) )  e.  CPreHil )
1332, 132eqeltrd 2529 1  |-  ( I  e.  V  ->  H  e.  CPreHil )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    \/ wo 370    /\ wa 371    /\ w3a 985    = wceq 1444    e. wcel 1887   A.wral 2737   _Vcvv 3045    \ cdif 3401    u. cun 3402    C_ wss 3404   {csn 3968   class class class wbr 4402    |-> cmpt 4461    X. cxp 4832   dom cdm 4834   Fun wfun 5576    Fn wfn 5577   -->wf 5578   ` cfv 5582  (class class class)co 6290    oFcof 6529   supp csupp 6914   Fincfn 7569   finSupp cfsupp 7883   RRcr 9538   0cc0 9539    x. cmul 9544    <_ cle 9676   *ccj 13159   sum_csu 13752   Basecbs 15121   ↾s cress 15122  Scalarcsca 15193   .icip 15195   0gc0g 15338    gsumg cgsu 15339   Ringcrg 17780   CRingccrg 17781   DivRingcdr 17975  Fieldcfield 17976  ℂfldccnfld 18970  RRfldcrefld 19172   freeLMod cfrlm 19309   CPreHilccph 22144  toCHilctch 22145  ℝ^crrx 22342
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-inf2 8146  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617  ax-addf 9618  ax-mulf 9619
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-of 6531  df-om 6693  df-1st 6793  df-2nd 6794  df-supp 6915  df-tpos 6973  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-er 7363  df-map 7474  df-ixp 7523  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-fsupp 7884  df-sup 7956  df-inf 7957  df-oi 8025  df-card 8373  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-q 11265  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-ico 11641  df-fz 11785  df-fzo 11916  df-seq 12214  df-exp 12273  df-hash 12516  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-clim 13552  df-sum 13753  df-struct 15123  df-ndx 15124  df-slot 15125  df-base 15126  df-sets 15127  df-ress 15128  df-plusg 15203  df-mulr 15204  df-starv 15205  df-sca 15206  df-vsca 15207  df-ip 15208  df-tset 15209  df-ple 15210  df-ds 15212  df-unif 15213  df-hom 15214  df-cco 15215  df-rest 15321  df-topn 15322  df-0g 15340  df-gsum 15341  df-topgen 15342  df-prds 15346  df-pws 15348  df-mgm 16488  df-sgrp 16527  df-mnd 16537  df-mhm 16582  df-submnd 16583  df-grp 16673  df-minusg 16674  df-sbg 16675  df-subg 16814  df-ghm 16881  df-cntz 16971  df-cmn 17432  df-abl 17433  df-mgp 17724  df-ur 17736  df-ring 17782  df-cring 17783  df-oppr 17851  df-dvdsr 17869  df-unit 17870  df-invr 17900  df-dvr 17911  df-rnghom 17943  df-drng 17977  df-field 17978  df-subrg 18006  df-abv 18045  df-staf 18073  df-srng 18074  df-lmod 18093  df-lss 18156  df-lmhm 18245  df-lvec 18326  df-sra 18395  df-rgmod 18396  df-psmet 18962  df-xmet 18963  df-met 18964  df-bl 18965  df-mopn 18966  df-cnfld 18971  df-refld 19173  df-phl 19193  df-dsmm 19295  df-frlm 19310  df-top 19921  df-bases 19922  df-topon 19923  df-topsp 19924  df-xms 21335  df-ms 21336  df-nm 21597  df-ngp 21598  df-tng 21599  df-nrg 21600  df-nlm 21601  df-clm 22094  df-cph 22146  df-tch 22147  df-rrx 22344
This theorem is referenced by:  rrxngp  38151
  Copyright terms: Public domain W3C validator