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Theorem rrxcph 21587
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 21582 . 2  |-  ( I  e.  V  ->  H  =  (toCHil `  (RRfld freeLMod  I ) ) )
3 eqid 2467 . . 3  |-  (toCHil `  (RRfld freeLMod  I ) )  =  (toCHil `  (RRfld freeLMod  I ) )
4 eqid 2467 . . 3  |-  ( Base `  (RRfld freeLMod  I ) )  =  ( Base `  (RRfld freeLMod  I ) )
5 eqid 2467 . . 3  |-  (Scalar `  (RRfld freeLMod  I ) )  =  (Scalar `  (RRfld freeLMod  I ) )
6 eqid 2467 . . . 4  |-  (RRfld freeLMod  I )  =  (RRfld freeLMod  I )
7 rebase 18437 . . . 4  |-  RR  =  ( Base ` RRfld )
8 remulr 18442 . . . 4  |-  x.  =  ( .r ` RRfld )
9 eqid 2467 . . . 4  |-  ( .i
`  (RRfld freeLMod  I ) )  =  ( .i `  (RRfld freeLMod  I ) )
10 eqid 2467 . . . 4  |-  ( 0g
`  (RRfld freeLMod  I ) )  =  ( 0g `  (RRfld freeLMod  I ) )
11 re0g 18443 . . . 4  |-  0  =  ( 0g ` RRfld )
12 refldcj 18451 . . . 4  |-  *  =  ( *r ` RRfld )
13 refld 18450 . . . . 5  |- RRfld  e. Field
1413a1i 11 . . . 4  |-  ( I  e.  V  -> RRfld  e. Field )
15 fconstmpt 5043 . . . . 5  |-  ( I  X.  { 0 } )  =  ( x  e.  I  |->  0 )
166, 7, 4frlmbasf 18589 . . . . . . . 8  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  ->  f : I --> RR )
17 ffn 5731 . . . . . . . 8  |-  ( f : I --> RR  ->  f  Fn  I )
1816, 17syl 16 . . . . . . 7  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  ->  f  Fn  I
)
19183adant3 1016 . . . . . 6  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) )  /\  ( f ( .i
`  (RRfld freeLMod  I ) ) f )  =  0 )  ->  f  Fn  I )
20 simpl 457 . . . . . . . . . . . . . . . . 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 461 . . . . . . . . . . . . . . . . 17  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  ->  f  e.  (
Base `  (RRfld freeLMod  I ) ) )
236, 7, 8, 4, 9frlmipval 18605 . . . . . . . . . . . . . . . . 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 1229 . . . . . . . . . . . . . . . 16  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  ->  ( f ( .i `  (RRfld freeLMod  I ) ) f )  =  (RRfld  gsumg  ( f  oF  x.  f ) ) )
25 ovex 6309 . . . . . . . . . . . . . . . . . . . 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 3707 . . . . . . . . . . . . . . . . . . . 20  |-  ( I  i^i  I )  =  I
28 eqidd 2468 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  /\  x  e.  I )  ->  (
f `  x )  =  ( f `  x ) )
2918, 18, 20, 20, 27, 28, 28offval 6531 . . . . . . . . . . . . . . . . . . 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 6533 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  /\  x  e.  I )  ->  (
( f  oF  x.  f ) `  x )  =  ( ( f `  x
)  x.  ( f `
 x ) ) )
3116ffvelrnda 6021 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  /\  x  e.  I )  ->  (
f `  x )  e.  RR )
3231, 31remulcld 9624 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  /\  x  e.  I )  ->  (
( f `  x
)  x.  ( f `
 x ) )  e.  RR )
3330, 32eqeltrd 2555 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  /\  x  e.  I )  ->  (
( f  oF  x.  f ) `  x )  e.  RR )
3426, 29, 33fmpt2d 6051 . . . . . . . . . . . . . . . . . 18  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  ->  ( f  oF  x.  f ) : I --> RR )
35 ovex 6309 . . . . . . . . . . . . . . . . . . . 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 5733 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( f  oF  x.  f ) : I --> RR  ->  Fun  ( f  oF  x.  f
) )
3834, 37syl 16 . . . . . . . . . . . . . . . . . . 19  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  ->  Fun  ( f  oF  x.  f
) )
396, 11, 4frlmbasfsupp 18586 . . . . . . . . . . . . . . . . . . 19  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  ->  f finSupp  0 )
40 0red 9597 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  ->  0  e.  RR )
41 simpr 461 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  /\  x  e.  RR )  ->  x  e.  RR )
4241recnd 9622 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  /\  x  e.  RR )  ->  x  e.  CC )
4342mul02d 9777 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  /\  x  e.  RR )  ->  (
0  x.  x )  =  0 )
4420, 40, 16, 16, 43suppofss1d 6937 . . . . . . . . . . . . . . . . . . 19  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  ->  ( ( f  oF  x.  f
) supp  0 )  C_  ( f supp  0 ) )
45 fsuppsssupp 7845 . . . . . . . . . . . . . . . . . . 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 1229 . . . . . . . . . . . . . . . . . 18  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  ->  ( f  oF  x.  f ) finSupp 
0 )
47 regsumsupp 18453 . . . . . . . . . . . . . . . . . 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 1228 . . . . . . . . . . . . . . . . 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 6914 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( f supp  0 )  C_  dom  f
50 fdm 5735 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( f : I --> RR  ->  dom  f  =  I )
5116, 50syl 16 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  ->  dom  f  =  I )
5249, 51syl5sseq 3552 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  ->  ( f supp  0
)  C_  I )
5344, 52sstrd 3514 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  ->  ( ( f  oF  x.  f
) supp  0 )  C_  I )
5453sselda 3504 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  /\  x  e.  ( ( f  oF  x.  f ) supp  0 ) )  ->  x  e.  I )
5554, 30syldan 470 . . . . . . . . . . . . . . . . . 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 13488 . . . . . . . . . . . . . . . . 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 2508 . . . . . . . . . . . . . . . 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 2508 . . . . . . . . . . . . . . 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 1016 . . . . . . . . . . . . . 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 998 . . . . . . . . . . . . . 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 2510 . . . . . . . . . . . . 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 7836 . . . . . . . . . . . . . . . 16  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  ->  ( f supp  0
)  e.  Fin )
63 ssfi 7740 . . . . . . . . . . . . . . . 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 661 . . . . . . . . . . . . . . 15  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  ->  ( ( f  oF  x.  f
) supp  0 )  e. 
Fin )
6554, 32syldan 470 . . . . . . . . . . . . . . 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 10121 . . . . . . . . . . . . . . . 16  |-  ( ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  /\  x  e.  I )  ->  0  <_  ( ( f `  x )  x.  (
f `  x )
) )
6754, 66syldan 470 . . . . . . . . . . . . . . 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 13575 . . . . . . . . . . . . . 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 1016 . . . . . . . . . . . . 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 210 . . . . . . . . . . . 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 2833 . . . . . . . . . . 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 714 . . . . . . . . . 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 1154 . . . . . . . . . . . . 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 9622 . . . . . . . . . . . 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 10199 . . . . . . . . . . 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 465 . . . . . . . . . 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 210 . . . . . . . . 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 514 . . . . . . . . 9  |-  ( ( ( f `  x
)  =  0  \/  ( f `  x
)  =  0 )  <-> 
( f `  x
)  =  0 )
7977, 78sylib 196 . . . . . . . 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 1016 . . . . . . . . . . 11  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) )  /\  ( f ( .i
`  (RRfld freeLMod  I ) ) f )  =  0 )  ->  ( f  oF  x.  f
) : I --> RR )
8180adantr 465 . . . . . . . . . 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 3523 . . . . . . . . . . 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 999 . . . . . . . . . 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 9597 . . . . . . . . . 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 6931 . . . . . . . . 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 1154 . . . . . . . . . . . . 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 2469 . . . . . . . . . . . 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 253 . . . . . . . . . . 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 261 . . . . . . . . . 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 484 . . . . . . . . 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 470 . . . . . . . 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 3907 . . . . . . . . . . . . 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 196 . . . . . . . . . . . 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 2537 . . . . . . . . . . 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 1016 . . . . . . . . . 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 485 . . . . . . . . 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 3645 . . . . . . . . 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 196 . . . . . . . 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 784 . . . . . . 7  |-  ( ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) )  /\  ( f ( .i `  (RRfld freeLMod  I ) ) f )  =  0 )  /\  x  e.  I )  ->  (
f `  x )  =  0 )
101100ralrimiva 2878 . . . . . 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 6123 . . . . . . . 8  |-  ( f : I --> { 0 }  <->  ( f  Fn  I  /\  A. x  e.  I  ( f `  x )  =  0 ) )
103102biimpri 206 . . . . . . 7  |-  ( ( f  Fn  I  /\  A. x  e.  I  ( f `  x )  =  0 )  -> 
f : I --> { 0 } )
104 c0ex 9590 . . . . . . . 8  |-  0  e.  _V
105104fconst2 6117 . . . . . . 7  |-  ( f : I --> { 0 }  <->  f  =  ( I  X.  { 0 } ) )
106103, 105sylib 196 . . . . . 6  |-  ( ( f  Fn  I  /\  A. x  e.  I  ( f `  x )  =  0 )  -> 
f  =  ( I  X.  { 0 } ) )
10719, 101, 106syl2anc 661 . . . . 5  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) )  /\  ( f ( .i
`  (RRfld freeLMod  I ) ) f )  =  0 )  ->  f  =  ( I  X.  { 0 } ) )
108 isfld 17205 . . . . . . . . . . 11  |-  (RRfld  e. Field  <->  (RRfld  e.  DivRing  /\ RRfld  e.  CRing ) )
10913, 108mpbi 208 . . . . . . . . . 10  |-  (RRfld  e.  DivRing  /\ RRfld  e.  CRing )
110109simpli 458 . . . . . . . . 9  |- RRfld  e.  DivRing
111 drngrng 17203 . . . . . . . . 9  |-  (RRfld  e.  DivRing  -> RRfld 
e.  Ring )
112110, 111ax-mp 5 . . . . . . . 8  |- RRfld  e.  Ring
1136, 11frlm0 18580 . . . . . . . 8  |-  ( (RRfld 
e.  Ring  /\  I  e.  V )  ->  (
I  X.  { 0 } )  =  ( 0g `  (RRfld freeLMod  I ) ) )
114112, 113mpan 670 . . . . . . 7  |-  ( I  e.  V  ->  (
I  X.  { 0 } )  =  ( 0g `  (RRfld freeLMod  I ) ) )
11515, 114syl5reqr 2523 . . . . . 6  |-  ( I  e.  V  ->  ( 0g `  (RRfld freeLMod  I ) )  =  ( x  e.  I  |->  0 ) )
1161153ad2ant1 1017 . . . . 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 2534 . . . 4  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) )  /\  ( f ( .i
`  (RRfld freeLMod  I ) ) f )  =  0 )  ->  f  =  ( 0g `  (RRfld freeLMod  I ) ) )
118 cjre 12935 . . . . 5  |-  ( x  e.  RR  ->  (
* `  x )  =  x )
119118adantl 466 . . . 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 18607 . . 3  |-  ( I  e.  V  ->  (RRfld freeLMod  I )  e.  PreHil )
122 df-refld 18436 . . . 4  |- RRfld  =  (flds  RR )
1236frlmsca 18579 . . . . 5  |-  ( (RRfld 
e. Field  /\  I  e.  V
)  -> RRfld  =  (Scalar `  (RRfld freeLMod  I ) ) )
12413, 123mpan 670 . . . 4  |-  ( I  e.  V  -> RRfld  =  (Scalar `  (RRfld freeLMod  I ) ) )
125122, 124syl5reqr 2523 . . 3  |-  ( I  e.  V  ->  (Scalar `  (RRfld freeLMod  I ) )  =  (flds  RR ) )
126 simpr1 1002 . . . 4  |-  ( ( I  e.  V  /\  ( f  e.  RR  /\  f  e.  RR  /\  0  <_  f ) )  ->  f  e.  RR )
127 simpr3 1004 . . . 4  |-  ( ( I  e.  V  /\  ( f  e.  RR  /\  f  e.  RR  /\  0  <_  f ) )  ->  0  <_  f
)
128126, 127resqrtcld 13212 . . 3  |-  ( ( I  e.  V  /\  ( f  e.  RR  /\  f  e.  RR  /\  0  <_  f ) )  ->  ( sqr `  f
)  e.  RR )
12964, 65, 67fsumge0 13572 . . . . 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 4473 . . . 4  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  ->  0  <_  (RRfld  gsumg  (
f  oF  x.  f ) ) )
131130, 24breqtrrd 4473 . . 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 21443 . 2  |-  ( I  e.  V  ->  (toCHil `  (RRfld freeLMod  I ) )  e.  CPreHil )
1332, 132eqeltrd 2555 1  |-  ( I  e.  V  ->  H  e.  CPreHil )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2814   _Vcvv 3113    \ cdif 3473    u. cun 3474    C_ wss 3476   {csn 4027   class class class wbr 4447    |-> cmpt 4505    X. cxp 4997   dom cdm 4999   Fun wfun 5582    Fn wfn 5583   -->wf 5584   ` cfv 5588  (class class class)co 6284    oFcof 6522   supp csupp 6901   Fincfn 7516   finSupp cfsupp 7829   RRcr 9491   0cc0 9492    x. cmul 9497    <_ cle 9629   *ccj 12892   sum_csu 13471   Basecbs 14490   ↾s cress 14491  Scalarcsca 14558   .icip 14560   0gc0g 14695    gsumg cgsu 14696   Ringcrg 17000   CRingccrg 17001   DivRingcdr 17196  Fieldcfield 17197  ℂfldccnfld 18219  RRfldcrefld 18435   freeLMod cfrlm 18572   CPreHilccph 21376  toCHilctch 21377  ℝ^crrx 21578
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-inf2 8058  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569  ax-pre-sup 9570  ax-addf 9571  ax-mulf 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-of 6524  df-om 6685  df-1st 6784  df-2nd 6785  df-supp 6902  df-tpos 6955  df-recs 7042  df-rdg 7076  df-1o 7130  df-oadd 7134  df-er 7311  df-map 7422  df-ixp 7470  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-fsupp 7830  df-sup 7901  df-oi 7935  df-card 8320  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-div 10207  df-nn 10537  df-2 10594  df-3 10595  df-4 10596  df-5 10597  df-6 10598  df-7 10599  df-8 10600  df-9 10601  df-10 10602  df-n0 10796  df-z 10865  df-dec 10977  df-uz 11083  df-q 11183  df-rp 11221  df-xneg 11318  df-xadd 11319  df-xmul 11320  df-ico 11535  df-fz 11673  df-fzo 11793  df-seq 12076  df-exp 12135  df-hash 12374  df-cj 12895  df-re 12896  df-im 12897  df-sqrt 13031  df-abs 13032  df-clim 13274  df-sum 13472  df-struct 14492  df-ndx 14493  df-slot 14494  df-base 14495  df-sets 14496  df-ress 14497  df-plusg 14568  df-mulr 14569  df-starv 14570  df-sca 14571  df-vsca 14572  df-ip 14573  df-tset 14574  df-ple 14575  df-ds 14577  df-unif 14578  df-hom 14579  df-cco 14580  df-rest 14678  df-topn 14679  df-0g 14697  df-gsum 14698  df-topgen 14699  df-prds 14703  df-pws 14705  df-mnd 15732  df-mhm 15786  df-submnd 15787  df-grp 15867  df-minusg 15868  df-sbg 15869  df-subg 16003  df-ghm 16070  df-cntz 16160  df-cmn 16606  df-abl 16607  df-mgp 16944  df-ur 16956  df-rng 17002  df-cring 17003  df-oppr 17073  df-dvdsr 17091  df-unit 17092  df-invr 17122  df-dvr 17133  df-rnghom 17165  df-drng 17198  df-field 17199  df-subrg 17227  df-abv 17266  df-staf 17294  df-srng 17295  df-lmod 17314  df-lss 17379  df-lmhm 17468  df-lvec 17549  df-sra 17618  df-rgmod 17619  df-psmet 18210  df-xmet 18211  df-met 18212  df-bl 18213  df-mopn 18214  df-cnfld 18220  df-refld 18436  df-phl 18456  df-dsmm 18558  df-frlm 18573  df-top 19194  df-bases 19196  df-topon 19197  df-topsp 19198  df-xms 20586  df-ms 20587  df-nm 20866  df-ngp 20867  df-tng 20868  df-nrg 20869  df-nlm 20870  df-clm 21326  df-cph 21378  df-tch 21379  df-rrx 21580
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator