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Theorem rrxcph 22429
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 22424 . 2  |-  ( I  e.  V  ->  H  =  (toCHil `  (RRfld freeLMod  I ) ) )
3 eqid 2471 . . 3  |-  (toCHil `  (RRfld freeLMod  I ) )  =  (toCHil `  (RRfld freeLMod  I ) )
4 eqid 2471 . . 3  |-  ( Base `  (RRfld freeLMod  I ) )  =  ( Base `  (RRfld freeLMod  I ) )
5 eqid 2471 . . 3  |-  (Scalar `  (RRfld freeLMod  I ) )  =  (Scalar `  (RRfld freeLMod  I ) )
6 eqid 2471 . . . 4  |-  (RRfld freeLMod  I )  =  (RRfld freeLMod  I )
7 rebase 19251 . . . 4  |-  RR  =  ( Base ` RRfld )
8 remulr 19256 . . . 4  |-  x.  =  ( .r ` RRfld )
9 eqid 2471 . . . 4  |-  ( .i
`  (RRfld freeLMod  I ) )  =  ( .i `  (RRfld freeLMod  I ) )
10 eqid 2471 . . . 4  |-  ( 0g
`  (RRfld freeLMod  I ) )  =  ( 0g `  (RRfld freeLMod  I ) )
11 re0g 19257 . . . 4  |-  0  =  ( 0g ` RRfld )
12 refldcj 19265 . . . 4  |-  *  =  ( *r ` RRfld )
13 refld 19264 . . . . 5  |- RRfld  e. Field
1413a1i 11 . . . 4  |-  ( I  e.  V  -> RRfld  e. Field )
15 fconstmpt 4883 . . . . 5  |-  ( I  X.  { 0 } )  =  ( x  e.  I  |->  0 )
166, 7, 4frlmbasf 19400 . . . . . . . 8  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  ->  f : I --> RR )
17 ffn 5739 . . . . . . . 8  |-  ( f : I --> RR  ->  f  Fn  I )
1816, 17syl 17 . . . . . . 7  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  ->  f  Fn  I
)
19183adant3 1050 . . . . . 6  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) )  /\  ( f ( .i
`  (RRfld freeLMod  I ) ) f )  =  0 )  ->  f  Fn  I )
20 simpl 464 . . . . . . . . . . . . . . . . 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 468 . . . . . . . . . . . . . . . . 17  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  ->  f  e.  (
Base `  (RRfld freeLMod  I ) ) )
236, 7, 8, 4, 9frlmipval 19414 . . . . . . . . . . . . . . . . 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 1293 . . . . . . . . . . . . . . . 16  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  ->  ( f ( .i `  (RRfld freeLMod  I ) ) f )  =  (RRfld  gsumg  ( f  oF  x.  f ) ) )
25 ovex 6336 . . . . . . . . . . . . . . . . . . . 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 3632 . . . . . . . . . . . . . . . . . . . 20  |-  ( I  i^i  I )  =  I
28 eqidd 2472 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  /\  x  e.  I )  ->  (
f `  x )  =  ( f `  x ) )
2918, 18, 20, 20, 27, 28, 28offval 6557 . . . . . . . . . . . . . . . . . . 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 6559 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  /\  x  e.  I )  ->  (
( f  oF  x.  f ) `  x )  =  ( ( f `  x
)  x.  ( f `
 x ) ) )
3116ffvelrnda 6037 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  /\  x  e.  I )  ->  (
f `  x )  e.  RR )
3231, 31remulcld 9689 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  /\  x  e.  I )  ->  (
( f `  x
)  x.  ( f `
 x ) )  e.  RR )
3330, 32eqeltrd 2549 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  /\  x  e.  I )  ->  (
( f  oF  x.  f ) `  x )  e.  RR )
3426, 29, 33fmpt2d 6069 . . . . . . . . . . . . . . . . . 18  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  ->  ( f  oF  x.  f ) : I --> RR )
35 ovex 6336 . . . . . . . . . . . . . . . . . . . 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 5742 . . . . . . . . . . . . . . . . . . . 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 19398 . . . . . . . . . . . . . . . . . . 19  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  ->  f finSupp  0 )
40 0red 9662 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  ->  0  e.  RR )
41 simpr 468 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  /\  x  e.  RR )  ->  x  e.  RR )
4241recnd 9687 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  /\  x  e.  RR )  ->  x  e.  CC )
4342mul02d 9849 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  /\  x  e.  RR )  ->  (
0  x.  x )  =  0 )
4420, 40, 16, 16, 43suppofss1d 6971 . . . . . . . . . . . . . . . . . . 19  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  ->  ( ( f  oF  x.  f
) supp  0 )  C_  ( f supp  0 ) )
45 fsuppsssupp 7917 . . . . . . . . . . . . . . . . . . 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 1293 . . . . . . . . . . . . . . . . . 18  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  ->  ( f  oF  x.  f ) finSupp 
0 )
47 regsumsupp 19267 . . . . . . . . . . . . . . . . . 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 1292 . . . . . . . . . . . . . . . . 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 6946 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( f supp  0 )  C_  dom  f
50 fdm 5745 . . . . . . . . . . . . . . . . . . . . . . 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 3466 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  ->  ( f supp  0
)  C_  I )
5344, 52sstrd 3428 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  ->  ( ( f  oF  x.  f
) supp  0 )  C_  I )
5453sselda 3418 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  /\  x  e.  ( ( f  oF  x.  f ) supp  0 ) )  ->  x  e.  I )
5554, 30syldan 478 . . . . . . . . . . . . . . . . . 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 13846 . . . . . . . . . . . . . . . . 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 2505 . . . . . . . . . . . . . . . 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 2505 . . . . . . . . . . . . . . 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 1050 . . . . . . . . . . . . . 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 1032 . . . . . . . . . . . . . 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 2507 . . . . . . . . . . . . 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 7908 . . . . . . . . . . . . . . . 16  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  ->  ( f supp  0
)  e.  Fin )
63 ssfi 7810 . . . . . . . . . . . . . . . 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 673 . . . . . . . . . . . . . . 15  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  ->  ( ( f  oF  x.  f
) supp  0 )  e. 
Fin )
6554, 32syldan 478 . . . . . . . . . . . . . . 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 10203 . . . . . . . . . . . . . . . 16  |-  ( ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  /\  x  e.  I )  ->  0  <_  ( ( f `  x )  x.  (
f `  x )
) )
6754, 66syldan 478 . . . . . . . . . . . . . . 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 13935 . . . . . . . . . . . . . 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 1050 . . . . . . . . . . . . 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 215 . . . . . . . . . . . 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 2776 . . . . . . . . . . 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 729 . . . . . . . . . 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 1188 . . . . . . . . . . . . 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 9687 . . . . . . . . . . . 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 10284 . . . . . . . . . . 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 472 . . . . . . . . . 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 215 . . . . . . . . 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 523 . . . . . . . . 9  |-  ( ( ( f `  x
)  =  0  \/  ( f `  x
)  =  0 )  <-> 
( f `  x
)  =  0 )
7977, 78sylib 201 . . . . . . . 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 1050 . . . . . . . . . . 11  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) )  /\  ( f ( .i
`  (RRfld freeLMod  I ) ) f )  =  0 )  ->  ( f  oF  x.  f
) : I --> RR )
8180adantr 472 . . . . . . . . . 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 3437 . . . . . . . . . . 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 1033 . . . . . . . . . 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 9662 . . . . . . . . . 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 6965 . . . . . . . . 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 1188 . . . . . . . . . . . . 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 2473 . . . . . . . . . . . 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 261 . . . . . . . . . . 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 269 . . . . . . . . . 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 492 . . . . . . . . 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 478 . . . . . . . 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 3839 . . . . . . . . . . . . 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 201 . . . . . . . . . . . 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 2534 . . . . . . . . . . 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 1050 . . . . . . . . . 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 493 . . . . . . . . 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 3565 . . . . . . . . 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 201 . . . . . . . 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 803 . . . . . . 7  |-  ( ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) )  /\  ( f ( .i `  (RRfld freeLMod  I ) ) f )  =  0 )  /\  x  e.  I )  ->  (
f `  x )  =  0 )
101100ralrimiva 2809 . . . . . 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 6143 . . . . . . 7  |-  ( f : I --> { 0 }  <->  ( f  Fn  I  /\  A. x  e.  I  ( f `  x )  =  0 ) )
103 c0ex 9655 . . . . . . . 8  |-  0  e.  _V
104103fconst2 6137 . . . . . . 7  |-  ( f : I --> { 0 }  <->  f  =  ( I  X.  { 0 } ) )
105102, 104sylbb1 220 . . . . . 6  |-  ( ( f  Fn  I  /\  A. x  e.  I  ( f `  x )  =  0 )  -> 
f  =  ( I  X.  { 0 } ) )
10619, 101, 105syl2anc 673 . . . . 5  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) )  /\  ( f ( .i
`  (RRfld freeLMod  I ) ) f )  =  0 )  ->  f  =  ( I  X.  { 0 } ) )
107 isfld 18062 . . . . . . . . . . 11  |-  (RRfld  e. Field  <->  (RRfld  e.  DivRing  /\ RRfld  e.  CRing ) )
10813, 107mpbi 213 . . . . . . . . . 10  |-  (RRfld  e.  DivRing  /\ RRfld  e.  CRing )
109108simpli 465 . . . . . . . . 9  |- RRfld  e.  DivRing
110 drngring 18060 . . . . . . . . 9  |-  (RRfld  e.  DivRing  -> RRfld 
e.  Ring )
111109, 110ax-mp 5 . . . . . . . 8  |- RRfld  e.  Ring
1126, 11frlm0 19394 . . . . . . . 8  |-  ( (RRfld 
e.  Ring  /\  I  e.  V )  ->  (
I  X.  { 0 } )  =  ( 0g `  (RRfld freeLMod  I ) ) )
113111, 112mpan 684 . . . . . . 7  |-  ( I  e.  V  ->  (
I  X.  { 0 } )  =  ( 0g `  (RRfld freeLMod  I ) ) )
11415, 113syl5reqr 2520 . . . . . 6  |-  ( I  e.  V  ->  ( 0g `  (RRfld freeLMod  I ) )  =  ( x  e.  I  |->  0 ) )
1151143ad2ant1 1051 . . . . 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 ) )
11615, 106, 1153eqtr4a 2531 . . . 4  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) )  /\  ( f ( .i
`  (RRfld freeLMod  I ) ) f )  =  0 )  ->  f  =  ( 0g `  (RRfld freeLMod  I ) ) )
117 cjre 13279 . . . . 5  |-  ( x  e.  RR  ->  (
* `  x )  =  x )
118117adantl 473 . . . 4  |-  ( ( I  e.  V  /\  x  e.  RR )  ->  ( * `  x
)  =  x )
119 id 22 . . . 4  |-  ( I  e.  V  ->  I  e.  V )
1206, 7, 8, 4, 9, 10, 11, 12, 14, 116, 118, 119frlmphl 19416 . . 3  |-  ( I  e.  V  ->  (RRfld freeLMod  I )  e.  PreHil )
121 df-refld 19250 . . . 4  |- RRfld  =  (flds  RR )
1226frlmsca 19393 . . . . 5  |-  ( (RRfld 
e. Field  /\  I  e.  V
)  -> RRfld  =  (Scalar `  (RRfld freeLMod  I ) ) )
12313, 122mpan 684 . . . 4  |-  ( I  e.  V  -> RRfld  =  (Scalar `  (RRfld freeLMod  I ) ) )
124121, 123syl5reqr 2520 . . 3  |-  ( I  e.  V  ->  (Scalar `  (RRfld freeLMod  I ) )  =  (flds  RR ) )
125 simpr1 1036 . . . 4  |-  ( ( I  e.  V  /\  ( f  e.  RR  /\  f  e.  RR  /\  0  <_  f ) )  ->  f  e.  RR )
126 simpr3 1038 . . . 4  |-  ( ( I  e.  V  /\  ( f  e.  RR  /\  f  e.  RR  /\  0  <_  f ) )  ->  0  <_  f
)
127125, 126resqrtcld 13556 . . 3  |-  ( ( I  e.  V  /\  ( f  e.  RR  /\  f  e.  RR  /\  0  <_  f ) )  ->  ( sqr `  f
)  e.  RR )
12864, 65, 67fsumge0 13932 . . . . 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 ) ) )
129128, 57breqtrrd 4422 . . . 4  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  ->  0  <_  (RRfld  gsumg  (
f  oF  x.  f ) ) )
130129, 24breqtrrd 4422 . . 3  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  ->  0  <_  (
f ( .i `  (RRfld freeLMod  I ) ) f ) )
1313, 4, 5, 120, 124, 9, 127, 130tchcph 22289 . 2  |-  ( I  e.  V  ->  (toCHil `  (RRfld freeLMod  I ) )  e.  CPreHil )
1322, 131eqeltrd 2549 1  |-  ( I  e.  V  ->  H  e.  CPreHil )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    \/ wo 375    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904   A.wral 2756   _Vcvv 3031    \ cdif 3387    u. cun 3388    C_ wss 3390   {csn 3959   class class class wbr 4395    |-> cmpt 4454    X. cxp 4837   dom cdm 4839   Fun wfun 5583    Fn wfn 5584   -->wf 5585   ` cfv 5589  (class class class)co 6308    oFcof 6548   supp csupp 6933   Fincfn 7587   finSupp cfsupp 7901   RRcr 9556   0cc0 9557    x. cmul 9562    <_ cle 9694   *ccj 13236   sum_csu 13829   Basecbs 15199   ↾s cress 15200  Scalarcsca 15271   .icip 15273   0gc0g 15416    gsumg cgsu 15417   Ringcrg 17858   CRingccrg 17859   DivRingcdr 18053  Fieldcfield 18054  ℂfldccnfld 19047  RRfldcrefld 19249   freeLMod cfrlm 19386   CPreHilccph 22222  toCHilctch 22223  ℝ^crrx 22420
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635  ax-addf 9636  ax-mulf 9637
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-tpos 6991  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-map 7492  df-ixp 7541  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-q 11288  df-rp 11326  df-xneg 11432  df-xadd 11433  df-xmul 11434  df-ico 11666  df-fz 11811  df-fzo 11943  df-seq 12252  df-exp 12311  df-hash 12554  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-clim 13629  df-sum 13830  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-starv 15283  df-sca 15284  df-vsca 15285  df-ip 15286  df-tset 15287  df-ple 15288  df-ds 15290  df-unif 15291  df-hom 15292  df-cco 15293  df-rest 15399  df-topn 15400  df-0g 15418  df-gsum 15419  df-topgen 15420  df-prds 15424  df-pws 15426  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-mhm 16660  df-submnd 16661  df-grp 16751  df-minusg 16752  df-sbg 16753  df-subg 16892  df-ghm 16959  df-cntz 17049  df-cmn 17510  df-abl 17511  df-mgp 17802  df-ur 17814  df-ring 17860  df-cring 17861  df-oppr 17929  df-dvdsr 17947  df-unit 17948  df-invr 17978  df-dvr 17989  df-rnghom 18021  df-drng 18055  df-field 18056  df-subrg 18084  df-abv 18123  df-staf 18151  df-srng 18152  df-lmod 18171  df-lss 18234  df-lmhm 18323  df-lvec 18404  df-sra 18473  df-rgmod 18474  df-psmet 19039  df-xmet 19040  df-met 19041  df-bl 19042  df-mopn 19043  df-cnfld 19048  df-refld 19250  df-phl 19270  df-dsmm 19372  df-frlm 19387  df-top 19998  df-bases 19999  df-topon 20000  df-topsp 20001  df-xms 21413  df-ms 21414  df-nm 21675  df-ngp 21676  df-tng 21677  df-nrg 21678  df-nlm 21679  df-clm 22172  df-cph 22224  df-tch 22225  df-rrx 22422
This theorem is referenced by:  rrxngp  38263
  Copyright terms: Public domain W3C validator