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Theorem rrxcph 21990
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 21985 . 2  |-  ( I  e.  V  ->  H  =  (toCHil `  (RRfld freeLMod  I ) ) )
3 eqid 2454 . . 3  |-  (toCHil `  (RRfld freeLMod  I ) )  =  (toCHil `  (RRfld freeLMod  I ) )
4 eqid 2454 . . 3  |-  ( Base `  (RRfld freeLMod  I ) )  =  ( Base `  (RRfld freeLMod  I ) )
5 eqid 2454 . . 3  |-  (Scalar `  (RRfld freeLMod  I ) )  =  (Scalar `  (RRfld freeLMod  I ) )
6 eqid 2454 . . . 4  |-  (RRfld freeLMod  I )  =  (RRfld freeLMod  I )
7 rebase 18815 . . . 4  |-  RR  =  ( Base ` RRfld )
8 remulr 18820 . . . 4  |-  x.  =  ( .r ` RRfld )
9 eqid 2454 . . . 4  |-  ( .i
`  (RRfld freeLMod  I ) )  =  ( .i `  (RRfld freeLMod  I ) )
10 eqid 2454 . . . 4  |-  ( 0g
`  (RRfld freeLMod  I ) )  =  ( 0g `  (RRfld freeLMod  I ) )
11 re0g 18821 . . . 4  |-  0  =  ( 0g ` RRfld )
12 refldcj 18829 . . . 4  |-  *  =  ( *r ` RRfld )
13 refld 18828 . . . . 5  |- RRfld  e. Field
1413a1i 11 . . . 4  |-  ( I  e.  V  -> RRfld  e. Field )
15 fconstmpt 5032 . . . . 5  |-  ( I  X.  { 0 } )  =  ( x  e.  I  |->  0 )
166, 7, 4frlmbasf 18965 . . . . . . . 8  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  ->  f : I --> RR )
17 ffn 5713 . . . . . . . 8  |-  ( f : I --> RR  ->  f  Fn  I )
1816, 17syl 16 . . . . . . 7  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  ->  f  Fn  I
)
19183adant3 1014 . . . . . 6  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) )  /\  ( f ( .i
`  (RRfld freeLMod  I ) ) f )  =  0 )  ->  f  Fn  I )
20 simpl 455 . . . . . . . . . . . . . . . . 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 459 . . . . . . . . . . . . . . . . 17  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  ->  f  e.  (
Base `  (RRfld freeLMod  I ) ) )
236, 7, 8, 4, 9frlmipval 18981 . . . . . . . . . . . . . . . . 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 1227 . . . . . . . . . . . . . . . 16  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  ->  ( f ( .i `  (RRfld freeLMod  I ) ) f )  =  (RRfld  gsumg  ( f  oF  x.  f ) ) )
25 ovex 6298 . . . . . . . . . . . . . . . . . . . 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 3693 . . . . . . . . . . . . . . . . . . . 20  |-  ( I  i^i  I )  =  I
28 eqidd 2455 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  /\  x  e.  I )  ->  (
f `  x )  =  ( f `  x ) )
2918, 18, 20, 20, 27, 28, 28offval 6520 . . . . . . . . . . . . . . . . . . 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 6522 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  /\  x  e.  I )  ->  (
( f  oF  x.  f ) `  x )  =  ( ( f `  x
)  x.  ( f `
 x ) ) )
3116ffvelrnda 6007 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  /\  x  e.  I )  ->  (
f `  x )  e.  RR )
3231, 31remulcld 9613 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  /\  x  e.  I )  ->  (
( f `  x
)  x.  ( f `
 x ) )  e.  RR )
3330, 32eqeltrd 2542 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  /\  x  e.  I )  ->  (
( f  oF  x.  f ) `  x )  e.  RR )
3426, 29, 33fmpt2d 6037 . . . . . . . . . . . . . . . . . 18  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  ->  ( f  oF  x.  f ) : I --> RR )
35 ovex 6298 . . . . . . . . . . . . . . . . . . . 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 5715 . . . . . . . . . . . . . . . . . . . 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 18963 . . . . . . . . . . . . . . . . . . 19  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  ->  f finSupp  0 )
40 0red 9586 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  ->  0  e.  RR )
41 simpr 459 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  /\  x  e.  RR )  ->  x  e.  RR )
4241recnd 9611 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  /\  x  e.  RR )  ->  x  e.  CC )
4342mul02d 9767 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  /\  x  e.  RR )  ->  (
0  x.  x )  =  0 )
4420, 40, 16, 16, 43suppofss1d 6929 . . . . . . . . . . . . . . . . . . 19  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  ->  ( ( f  oF  x.  f
) supp  0 )  C_  ( f supp  0 ) )
45 fsuppsssupp 7837 . . . . . . . . . . . . . . . . . . 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 1227 . . . . . . . . . . . . . . . . . 18  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  ->  ( f  oF  x.  f ) finSupp 
0 )
47 regsumsupp 18831 . . . . . . . . . . . . . . . . . 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 1226 . . . . . . . . . . . . . . . . 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 6904 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( f supp  0 )  C_  dom  f
50 fdm 5717 . . . . . . . . . . . . . . . . . . . . . . 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 3537 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  ->  ( f supp  0
)  C_  I )
5344, 52sstrd 3499 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  ->  ( ( f  oF  x.  f
) supp  0 )  C_  I )
5453sselda 3489 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  /\  x  e.  ( ( f  oF  x.  f ) supp  0 ) )  ->  x  e.  I )
5554, 30syldan 468 . . . . . . . . . . . . . . . . . 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 13607 . . . . . . . . . . . . . . . . 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 2495 . . . . . . . . . . . . . . . 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 2495 . . . . . . . . . . . . . . 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 1014 . . . . . . . . . . . . . 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 996 . . . . . . . . . . . . . 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 2497 . . . . . . . . . . . . 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 7828 . . . . . . . . . . . . . . . 16  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  ->  ( f supp  0
)  e.  Fin )
63 ssfi 7733 . . . . . . . . . . . . . . . 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 659 . . . . . . . . . . . . . . 15  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  ->  ( ( f  oF  x.  f
) supp  0 )  e. 
Fin )
6554, 32syldan 468 . . . . . . . . . . . . . . 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 10117 . . . . . . . . . . . . . . . 16  |-  ( ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  /\  x  e.  I )  ->  0  <_  ( ( f `  x )  x.  (
f `  x )
) )
6754, 66syldan 468 . . . . . . . . . . . . . . 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 13694 . . . . . . . . . . . . . 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 1014 . . . . . . . . . . . . 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 2823 . . . . . . . . . . 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 712 . . . . . . . . . 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 1152 . . . . . . . . . . . . 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 9611 . . . . . . . . . . . 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 10195 . . . . . . . . . . 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 463 . . . . . . . . . 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 512 . . . . . . . . 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 1014 . . . . . . . . . . 11  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) )  /\  ( f ( .i
`  (RRfld freeLMod  I ) ) f )  =  0 )  ->  ( f  oF  x.  f
) : I --> RR )
8180adantr 463 . . . . . . . . . 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 3508 . . . . . . . . . . 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 997 . . . . . . . . . 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 9586 . . . . . . . . . 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 6923 . . . . . . . . 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 1152 . . . . . . . . . . . . 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 2456 . . . . . . . . . . . 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 482 . . . . . . . . 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 468 . . . . . . . 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 3896 . . . . . . . . . . . . 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 2524 . . . . . . . . . . 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 1014 . . . . . . . . . 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 483 . . . . . . . . 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 3631 . . . . . . . . 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 2868 . . . . . 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 6108 . . . . . . . 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 9579 . . . . . . . 8  |-  0  e.  _V
105104fconst2 6104 . . . . . . 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 659 . . . . 5  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) )  /\  ( f ( .i
`  (RRfld freeLMod  I ) ) f )  =  0 )  ->  f  =  ( I  X.  { 0 } ) )
108 isfld 17600 . . . . . . . . . . 11  |-  (RRfld  e. Field  <->  (RRfld  e.  DivRing  /\ RRfld  e.  CRing ) )
10913, 108mpbi 208 . . . . . . . . . 10  |-  (RRfld  e.  DivRing  /\ RRfld  e.  CRing )
110109simpli 456 . . . . . . . . 9  |- RRfld  e.  DivRing
111 drngring 17598 . . . . . . . . 9  |-  (RRfld  e.  DivRing  -> RRfld 
e.  Ring )
112110, 111ax-mp 5 . . . . . . . 8  |- RRfld  e.  Ring
1136, 11frlm0 18958 . . . . . . . 8  |-  ( (RRfld 
e.  Ring  /\  I  e.  V )  ->  (
I  X.  { 0 } )  =  ( 0g `  (RRfld freeLMod  I ) ) )
114112, 113mpan 668 . . . . . . 7  |-  ( I  e.  V  ->  (
I  X.  { 0 } )  =  ( 0g `  (RRfld freeLMod  I ) ) )
11515, 114syl5reqr 2510 . . . . . 6  |-  ( I  e.  V  ->  ( 0g `  (RRfld freeLMod  I ) )  =  ( x  e.  I  |->  0 ) )
1161153ad2ant1 1015 . . . . 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 2521 . . . 4  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) )  /\  ( f ( .i
`  (RRfld freeLMod  I ) ) f )  =  0 )  ->  f  =  ( 0g `  (RRfld freeLMod  I ) ) )
118 cjre 13054 . . . . 5  |-  ( x  e.  RR  ->  (
* `  x )  =  x )
119118adantl 464 . . . 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 18983 . . 3  |-  ( I  e.  V  ->  (RRfld freeLMod  I )  e.  PreHil )
122 df-refld 18814 . . . 4  |- RRfld  =  (flds  RR )
1236frlmsca 18957 . . . . 5  |-  ( (RRfld 
e. Field  /\  I  e.  V
)  -> RRfld  =  (Scalar `  (RRfld freeLMod  I ) ) )
12413, 123mpan 668 . . . 4  |-  ( I  e.  V  -> RRfld  =  (Scalar `  (RRfld freeLMod  I ) ) )
125122, 124syl5reqr 2510 . . 3  |-  ( I  e.  V  ->  (Scalar `  (RRfld freeLMod  I ) )  =  (flds  RR ) )
126 simpr1 1000 . . . 4  |-  ( ( I  e.  V  /\  ( f  e.  RR  /\  f  e.  RR  /\  0  <_  f ) )  ->  f  e.  RR )
127 simpr3 1002 . . . 4  |-  ( ( I  e.  V  /\  ( f  e.  RR  /\  f  e.  RR  /\  0  <_  f ) )  ->  0  <_  f
)
128126, 127resqrtcld 13331 . . 3  |-  ( ( I  e.  V  /\  ( f  e.  RR  /\  f  e.  RR  /\  0  <_  f ) )  ->  ( sqr `  f
)  e.  RR )
12964, 65, 67fsumge0 13691 . . . . 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 4465 . . . 4  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  ->  0  <_  (RRfld  gsumg  (
f  oF  x.  f ) ) )
131130, 24breqtrrd 4465 . . 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 21846 . 2  |-  ( I  e.  V  ->  (toCHil `  (RRfld freeLMod  I ) )  e.  CPreHil )
1332, 132eqeltrd 2542 1  |-  ( I  e.  V  ->  H  e.  CPreHil )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 366    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823   A.wral 2804   _Vcvv 3106    \ cdif 3458    u. cun 3459    C_ wss 3461   {csn 4016   class class class wbr 4439    |-> cmpt 4497    X. cxp 4986   dom cdm 4988   Fun wfun 5564    Fn wfn 5565   -->wf 5566   ` cfv 5570  (class class class)co 6270    oFcof 6511   supp csupp 6891   Fincfn 7509   finSupp cfsupp 7821   RRcr 9480   0cc0 9481    x. cmul 9486    <_ cle 9618   *ccj 13011   sum_csu 13590   Basecbs 14716   ↾s cress 14717  Scalarcsca 14787   .icip 14789   0gc0g 14929    gsumg cgsu 14930   Ringcrg 17393   CRingccrg 17394   DivRingcdr 17591  Fieldcfield 17592  ℂfldccnfld 18615  RRfldcrefld 18813   freeLMod cfrlm 18950   CPreHilccph 21779  toCHilctch 21780  ℝ^crrx 21981
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559  ax-addf 9560  ax-mulf 9561
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-of 6513  df-om 6674  df-1st 6773  df-2nd 6774  df-supp 6892  df-tpos 6947  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-ixp 7463  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-fsupp 7822  df-sup 7893  df-oi 7927  df-card 8311  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10977  df-uz 11083  df-q 11184  df-rp 11222  df-xneg 11321  df-xadd 11322  df-xmul 11323  df-ico 11538  df-fz 11676  df-fzo 11800  df-seq 12090  df-exp 12149  df-hash 12388  df-cj 13014  df-re 13015  df-im 13016  df-sqrt 13150  df-abs 13151  df-clim 13393  df-sum 13591  df-struct 14718  df-ndx 14719  df-slot 14720  df-base 14721  df-sets 14722  df-ress 14723  df-plusg 14797  df-mulr 14798  df-starv 14799  df-sca 14800  df-vsca 14801  df-ip 14802  df-tset 14803  df-ple 14804  df-ds 14806  df-unif 14807  df-hom 14808  df-cco 14809  df-rest 14912  df-topn 14913  df-0g 14931  df-gsum 14932  df-topgen 14933  df-prds 14937  df-pws 14939  df-mgm 16071  df-sgrp 16110  df-mnd 16120  df-mhm 16165  df-submnd 16166  df-grp 16256  df-minusg 16257  df-sbg 16258  df-subg 16397  df-ghm 16464  df-cntz 16554  df-cmn 16999  df-abl 17000  df-mgp 17337  df-ur 17349  df-ring 17395  df-cring 17396  df-oppr 17467  df-dvdsr 17485  df-unit 17486  df-invr 17516  df-dvr 17527  df-rnghom 17559  df-drng 17593  df-field 17594  df-subrg 17622  df-abv 17661  df-staf 17689  df-srng 17690  df-lmod 17709  df-lss 17774  df-lmhm 17863  df-lvec 17944  df-sra 18013  df-rgmod 18014  df-psmet 18606  df-xmet 18607  df-met 18608  df-bl 18609  df-mopn 18610  df-cnfld 18616  df-refld 18814  df-phl 18834  df-dsmm 18936  df-frlm 18951  df-top 19566  df-bases 19568  df-topon 19569  df-topsp 19570  df-xms 20989  df-ms 20990  df-nm 21269  df-ngp 21270  df-tng 21271  df-nrg 21272  df-nlm 21273  df-clm 21729  df-cph 21781  df-tch 21782  df-rrx 21983
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator