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Theorem rrxcph 22338
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 22333 . 2  |-  ( I  e.  V  ->  H  =  (toCHil `  (RRfld freeLMod  I ) ) )
3 eqid 2422 . . 3  |-  (toCHil `  (RRfld freeLMod  I ) )  =  (toCHil `  (RRfld freeLMod  I ) )
4 eqid 2422 . . 3  |-  ( Base `  (RRfld freeLMod  I ) )  =  ( Base `  (RRfld freeLMod  I ) )
5 eqid 2422 . . 3  |-  (Scalar `  (RRfld freeLMod  I ) )  =  (Scalar `  (RRfld freeLMod  I ) )
6 eqid 2422 . . . 4  |-  (RRfld freeLMod  I )  =  (RRfld freeLMod  I )
7 rebase 19161 . . . 4  |-  RR  =  ( Base ` RRfld )
8 remulr 19166 . . . 4  |-  x.  =  ( .r ` RRfld )
9 eqid 2422 . . . 4  |-  ( .i
`  (RRfld freeLMod  I ) )  =  ( .i `  (RRfld freeLMod  I ) )
10 eqid 2422 . . . 4  |-  ( 0g
`  (RRfld freeLMod  I ) )  =  ( 0g `  (RRfld freeLMod  I ) )
11 re0g 19167 . . . 4  |-  0  =  ( 0g ` RRfld )
12 refldcj 19175 . . . 4  |-  *  =  ( *r ` RRfld )
13 refld 19174 . . . . 5  |- RRfld  e. Field
1413a1i 11 . . . 4  |-  ( I  e.  V  -> RRfld  e. Field )
15 fconstmpt 4894 . . . . 5  |-  ( I  X.  { 0 } )  =  ( x  e.  I  |->  0 )
166, 7, 4frlmbasf 19310 . . . . . . . 8  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  ->  f : I --> RR )
17 ffn 5743 . . . . . . . 8  |-  ( f : I --> RR  ->  f  Fn  I )
1816, 17syl 17 . . . . . . 7  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  ->  f  Fn  I
)
19183adant3 1025 . . . . . 6  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) )  /\  ( f ( .i
`  (RRfld freeLMod  I ) ) f )  =  0 )  ->  f  Fn  I )
20 simpl 458 . . . . . . . . . . . . . . . . 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 462 . . . . . . . . . . . . . . . . 17  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  ->  f  e.  (
Base `  (RRfld freeLMod  I ) ) )
236, 7, 8, 4, 9frlmipval 19324 . . . . . . . . . . . . . . . . 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 1265 . . . . . . . . . . . . . . . 16  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  ->  ( f ( .i `  (RRfld freeLMod  I ) ) f )  =  (RRfld  gsumg  ( f  oF  x.  f ) ) )
25 ovex 6330 . . . . . . . . . . . . . . . . . . . 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 3671 . . . . . . . . . . . . . . . . . . . 20  |-  ( I  i^i  I )  =  I
28 eqidd 2423 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  /\  x  e.  I )  ->  (
f `  x )  =  ( f `  x ) )
2918, 18, 20, 20, 27, 28, 28offval 6549 . . . . . . . . . . . . . . . . . . 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 6551 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  /\  x  e.  I )  ->  (
( f  oF  x.  f ) `  x )  =  ( ( f `  x
)  x.  ( f `
 x ) ) )
3116ffvelrnda 6034 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  /\  x  e.  I )  ->  (
f `  x )  e.  RR )
3231, 31remulcld 9672 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  /\  x  e.  I )  ->  (
( f `  x
)  x.  ( f `
 x ) )  e.  RR )
3330, 32eqeltrd 2510 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  /\  x  e.  I )  ->  (
( f  oF  x.  f ) `  x )  e.  RR )
3426, 29, 33fmpt2d 6065 . . . . . . . . . . . . . . . . . 18  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  ->  ( f  oF  x.  f ) : I --> RR )
35 ovex 6330 . . . . . . . . . . . . . . . . . . . 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 5745 . . . . . . . . . . . . . . . . . . . 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 19308 . . . . . . . . . . . . . . . . . . 19  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  ->  f finSupp  0 )
40 0red 9645 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  ->  0  e.  RR )
41 simpr 462 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  /\  x  e.  RR )  ->  x  e.  RR )
4241recnd 9670 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  /\  x  e.  RR )  ->  x  e.  CC )
4342mul02d 9832 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  /\  x  e.  RR )  ->  (
0  x.  x )  =  0 )
4420, 40, 16, 16, 43suppofss1d 6960 . . . . . . . . . . . . . . . . . . 19  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  ->  ( ( f  oF  x.  f
) supp  0 )  C_  ( f supp  0 ) )
45 fsuppsssupp 7902 . . . . . . . . . . . . . . . . . . 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 1265 . . . . . . . . . . . . . . . . . 18  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  ->  ( f  oF  x.  f ) finSupp 
0 )
47 regsumsupp 19177 . . . . . . . . . . . . . . . . . 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 1264 . . . . . . . . . . . . . . . . 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 6935 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( f supp  0 )  C_  dom  f
50 fdm 5747 . . . . . . . . . . . . . . . . . . . . . . 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 3512 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  ->  ( f supp  0
)  C_  I )
5344, 52sstrd 3474 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  ->  ( ( f  oF  x.  f
) supp  0 )  C_  I )
5453sselda 3464 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  /\  x  e.  ( ( f  oF  x.  f ) supp  0 ) )  ->  x  e.  I )
5554, 30syldan 472 . . . . . . . . . . . . . . . . . 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 13757 . . . . . . . . . . . . . . . . 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 2463 . . . . . . . . . . . . . . . 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 2463 . . . . . . . . . . . . . . 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 1025 . . . . . . . . . . . . . 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 1007 . . . . . . . . . . . . . 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 2465 . . . . . . . . . . . . 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 7893 . . . . . . . . . . . . . . . 16  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  ->  ( f supp  0
)  e.  Fin )
63 ssfi 7795 . . . . . . . . . . . . . . . 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 665 . . . . . . . . . . . . . . 15  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  ->  ( ( f  oF  x.  f
) supp  0 )  e. 
Fin )
6554, 32syldan 472 . . . . . . . . . . . . . . 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 10183 . . . . . . . . . . . . . . . 16  |-  ( ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  /\  x  e.  I )  ->  0  <_  ( ( f `  x )  x.  (
f `  x )
) )
6754, 66syldan 472 . . . . . . . . . . . . . . 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 13846 . . . . . . . . . . . . . 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 1025 . . . . . . . . . . . . 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 213 . . . . . . . . . . . 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 2794 . . . . . . . . . . 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 719 . . . . . . . . . 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 1163 . . . . . . . . . . . . 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 9670 . . . . . . . . . . . 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 10263 . . . . . . . . . . 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 466 . . . . . . . . . 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 213 . . . . . . . . 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 516 . . . . . . . . 9  |-  ( ( ( f `  x
)  =  0  \/  ( f `  x
)  =  0 )  <-> 
( f `  x
)  =  0 )
7977, 78sylib 199 . . . . . . . 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 1025 . . . . . . . . . . 11  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) )  /\  ( f ( .i
`  (RRfld freeLMod  I ) ) f )  =  0 )  ->  ( f  oF  x.  f
) : I --> RR )
8180adantr 466 . . . . . . . . . 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 3483 . . . . . . . . . . 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 1008 . . . . . . . . . 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 9645 . . . . . . . . . 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 6954 . . . . . . . . 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 1163 . . . . . . . . . . . . 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 2424 . . . . . . . . . . . 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 256 . . . . . . . . . . 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 264 . . . . . . . . . 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 486 . . . . . . . . 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 472 . . . . . . . 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 3876 . . . . . . . . . . . . 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 199 . . . . . . . . . . . 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 2492 . . . . . . . . . . 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 1025 . . . . . . . . . 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 487 . . . . . . . . 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 3606 . . . . . . . . 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 199 . . . . . . . 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 793 . . . . . . 7  |-  ( ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) )  /\  ( f ( .i `  (RRfld freeLMod  I ) ) f )  =  0 )  /\  x  e.  I )  ->  (
f `  x )  =  0 )
101100ralrimiva 2839 . . . . . 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 6138 . . . . . . . 8  |-  ( f : I --> { 0 }  <->  ( f  Fn  I  /\  A. x  e.  I  ( f `  x )  =  0 ) )
103102biimpri 209 . . . . . . 7  |-  ( ( f  Fn  I  /\  A. x  e.  I  ( f `  x )  =  0 )  -> 
f : I --> { 0 } )
104 c0ex 9638 . . . . . . . 8  |-  0  e.  _V
105104fconst2 6133 . . . . . . 7  |-  ( f : I --> { 0 }  <->  f  =  ( I  X.  { 0 } ) )
106103, 105sylib 199 . . . . . 6  |-  ( ( f  Fn  I  /\  A. x  e.  I  ( f `  x )  =  0 )  -> 
f  =  ( I  X.  { 0 } ) )
10719, 101, 106syl2anc 665 . . . . 5  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) )  /\  ( f ( .i
`  (RRfld freeLMod  I ) ) f )  =  0 )  ->  f  =  ( I  X.  { 0 } ) )
108 isfld 17972 . . . . . . . . . . 11  |-  (RRfld  e. Field  <->  (RRfld  e.  DivRing  /\ RRfld  e.  CRing ) )
10913, 108mpbi 211 . . . . . . . . . 10  |-  (RRfld  e.  DivRing  /\ RRfld  e.  CRing )
110109simpli 459 . . . . . . . . 9  |- RRfld  e.  DivRing
111 drngring 17970 . . . . . . . . 9  |-  (RRfld  e.  DivRing  -> RRfld 
e.  Ring )
112110, 111ax-mp 5 . . . . . . . 8  |- RRfld  e.  Ring
1136, 11frlm0 19304 . . . . . . . 8  |-  ( (RRfld 
e.  Ring  /\  I  e.  V )  ->  (
I  X.  { 0 } )  =  ( 0g `  (RRfld freeLMod  I ) ) )
114112, 113mpan 674 . . . . . . 7  |-  ( I  e.  V  ->  (
I  X.  { 0 } )  =  ( 0g `  (RRfld freeLMod  I ) ) )
11515, 114syl5reqr 2478 . . . . . 6  |-  ( I  e.  V  ->  ( 0g `  (RRfld freeLMod  I ) )  =  ( x  e.  I  |->  0 ) )
1161153ad2ant1 1026 . . . . 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 2489 . . . 4  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) )  /\  ( f ( .i
`  (RRfld freeLMod  I ) ) f )  =  0 )  ->  f  =  ( 0g `  (RRfld freeLMod  I ) ) )
118 cjre 13191 . . . . 5  |-  ( x  e.  RR  ->  (
* `  x )  =  x )
119118adantl 467 . . . 4  |-  ( ( I  e.  V  /\  x  e.  RR )  ->  ( * `  x
)  =  x )
120 id 23 . . . 4  |-  ( I  e.  V  ->  I  e.  V )
1216, 7, 8, 4, 9, 10, 11, 12, 14, 117, 119, 120frlmphl 19326 . . 3  |-  ( I  e.  V  ->  (RRfld freeLMod  I )  e.  PreHil )
122 df-refld 19160 . . . 4  |- RRfld  =  (flds  RR )
1236frlmsca 19303 . . . . 5  |-  ( (RRfld 
e. Field  /\  I  e.  V
)  -> RRfld  =  (Scalar `  (RRfld freeLMod  I ) ) )
12413, 123mpan 674 . . . 4  |-  ( I  e.  V  -> RRfld  =  (Scalar `  (RRfld freeLMod  I ) ) )
125122, 124syl5reqr 2478 . . 3  |-  ( I  e.  V  ->  (Scalar `  (RRfld freeLMod  I ) )  =  (flds  RR ) )
126 simpr1 1011 . . . 4  |-  ( ( I  e.  V  /\  ( f  e.  RR  /\  f  e.  RR  /\  0  <_  f ) )  ->  f  e.  RR )
127 simpr3 1013 . . . 4  |-  ( ( I  e.  V  /\  ( f  e.  RR  /\  f  e.  RR  /\  0  <_  f ) )  ->  0  <_  f
)
128126, 127resqrtcld 13468 . . 3  |-  ( ( I  e.  V  /\  ( f  e.  RR  /\  f  e.  RR  /\  0  <_  f ) )  ->  ( sqr `  f
)  e.  RR )
12964, 65, 67fsumge0 13843 . . . . 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 4447 . . . 4  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  ->  0  <_  (RRfld  gsumg  (
f  oF  x.  f ) ) )
131130, 24breqtrrd 4447 . . 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 22198 . 2  |-  ( I  e.  V  ->  (toCHil `  (RRfld freeLMod  I ) )  e.  CPreHil )
1332, 132eqeltrd 2510 1  |-  ( I  e.  V  ->  H  e.  CPreHil )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    \/ wo 369    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1868   A.wral 2775   _Vcvv 3081    \ cdif 3433    u. cun 3434    C_ wss 3436   {csn 3996   class class class wbr 4420    |-> cmpt 4479    X. cxp 4848   dom cdm 4850   Fun wfun 5592    Fn wfn 5593   -->wf 5594   ` cfv 5598  (class class class)co 6302    oFcof 6540   supp csupp 6922   Fincfn 7574   finSupp cfsupp 7886   RRcr 9539   0cc0 9540    x. cmul 9545    <_ cle 9677   *ccj 13148   sum_csu 13740   Basecbs 15109   ↾s cress 15110  Scalarcsca 15181   .icip 15183   0gc0g 15326    gsumg cgsu 15327   Ringcrg 17768   CRingccrg 17769   DivRingcdr 17963  Fieldcfield 17964  ℂfldccnfld 18958  RRfldcrefld 19159   freeLMod cfrlm 19296   CPreHilccph 22131  toCHilctch 22132  ℝ^crrx 22329
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4552  ax-pow 4599  ax-pr 4657  ax-un 6594  ax-inf2 8149  ax-cnex 9596  ax-resscn 9597  ax-1cn 9598  ax-icn 9599  ax-addcl 9600  ax-addrcl 9601  ax-mulcl 9602  ax-mulrcl 9603  ax-mulcom 9604  ax-addass 9605  ax-mulass 9606  ax-distr 9607  ax-i2m1 9608  ax-1ne0 9609  ax-1rid 9610  ax-rnegex 9611  ax-rrecex 9612  ax-cnre 9613  ax-pre-lttri 9614  ax-pre-lttrn 9615  ax-pre-ltadd 9616  ax-pre-mulgt0 9617  ax-pre-sup 9618  ax-addf 9619  ax-mulf 9620
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-int 4253  df-iun 4298  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4761  df-id 4765  df-po 4771  df-so 4772  df-fr 4809  df-se 4810  df-we 4811  df-xp 4856  df-rel 4857  df-cnv 4858  df-co 4859  df-dm 4860  df-rn 4861  df-res 4862  df-ima 4863  df-pred 5396  df-ord 5442  df-on 5443  df-lim 5444  df-suc 5445  df-iota 5562  df-fun 5600  df-fn 5601  df-f 5602  df-f1 5603  df-fo 5604  df-f1o 5605  df-fv 5606  df-isom 5607  df-riota 6264  df-ov 6305  df-oprab 6306  df-mpt2 6307  df-of 6542  df-om 6704  df-1st 6804  df-2nd 6805  df-supp 6923  df-tpos 6978  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-oadd 7191  df-er 7368  df-map 7479  df-ixp 7528  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-fsupp 7887  df-sup 7959  df-inf 7960  df-oi 8028  df-card 8375  df-pnf 9678  df-mnf 9679  df-xr 9680  df-ltxr 9681  df-le 9682  df-sub 9863  df-neg 9864  df-div 10271  df-nn 10611  df-2 10669  df-3 10670  df-4 10671  df-5 10672  df-6 10673  df-7 10674  df-8 10675  df-9 10676  df-10 10677  df-n0 10871  df-z 10939  df-dec 11053  df-uz 11161  df-q 11266  df-rp 11304  df-xneg 11410  df-xadd 11411  df-xmul 11412  df-ico 11642  df-fz 11786  df-fzo 11917  df-seq 12214  df-exp 12273  df-hash 12516  df-cj 13151  df-re 13152  df-im 13153  df-sqrt 13287  df-abs 13288  df-clim 13540  df-sum 13741  df-struct 15111  df-ndx 15112  df-slot 15113  df-base 15114  df-sets 15115  df-ress 15116  df-plusg 15191  df-mulr 15192  df-starv 15193  df-sca 15194  df-vsca 15195  df-ip 15196  df-tset 15197  df-ple 15198  df-ds 15200  df-unif 15201  df-hom 15202  df-cco 15203  df-rest 15309  df-topn 15310  df-0g 15328  df-gsum 15329  df-topgen 15330  df-prds 15334  df-pws 15336  df-mgm 16476  df-sgrp 16515  df-mnd 16525  df-mhm 16570  df-submnd 16571  df-grp 16661  df-minusg 16662  df-sbg 16663  df-subg 16802  df-ghm 16869  df-cntz 16959  df-cmn 17420  df-abl 17421  df-mgp 17712  df-ur 17724  df-ring 17770  df-cring 17771  df-oppr 17839  df-dvdsr 17857  df-unit 17858  df-invr 17888  df-dvr 17899  df-rnghom 17931  df-drng 17965  df-field 17966  df-subrg 17994  df-abv 18033  df-staf 18061  df-srng 18062  df-lmod 18081  df-lss 18144  df-lmhm 18233  df-lvec 18314  df-sra 18383  df-rgmod 18384  df-psmet 18950  df-xmet 18951  df-met 18952  df-bl 18953  df-mopn 18954  df-cnfld 18959  df-refld 19160  df-phl 19180  df-dsmm 19282  df-frlm 19297  df-top 19908  df-bases 19909  df-topon 19910  df-topsp 19911  df-xms 21322  df-ms 21323  df-nm 21584  df-ngp 21585  df-tng 21586  df-nrg 21587  df-nlm 21588  df-clm 22081  df-cph 22133  df-tch 22134  df-rrx 22331
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator