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Theorem rrxmvallem 21559
Description: Support of the function used for building the distance . (Contributed by Thierry Arnoux, 30-Jun-2019.)
Hypothesis
Ref Expression
rrxmval.1  |-  X  =  { h  e.  ( RR  ^m  I )  |  h finSupp  0 }
Assertion
Ref Expression
rrxmvallem  |-  ( ( I  e.  V  /\  F  e.  X  /\  G  e.  X )  ->  ( ( k  e.  I  |->  ( ( ( F `  k )  -  ( G `  k ) ) ^
2 ) ) supp  0
)  C_  ( ( F supp  0 )  u.  ( G supp  0 ) ) )
Distinct variable groups:    h, F, k    h, G, k    h, I, k    h, V, k   
k, X
Allowed substitution hint:    X( h)

Proof of Theorem rrxmvallem
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 simprl 755 . . . . . . . . . 10  |-  ( ( ( ( I  e.  V  /\  F  e.  X  /\  G  e.  X )  /\  x  e.  I )  /\  (
( F `  x
)  =  0  /\  ( G `  x
)  =  0 ) )  ->  ( F `  x )  =  0 )
2 0cn 9577 . . . . . . . . . 10  |-  0  e.  CC
31, 2syl6eqel 2556 . . . . . . . . 9  |-  ( ( ( ( I  e.  V  /\  F  e.  X  /\  G  e.  X )  /\  x  e.  I )  /\  (
( F `  x
)  =  0  /\  ( G `  x
)  =  0 ) )  ->  ( F `  x )  e.  CC )
4 simprr 756 . . . . . . . . . 10  |-  ( ( ( ( I  e.  V  /\  F  e.  X  /\  G  e.  X )  /\  x  e.  I )  /\  (
( F `  x
)  =  0  /\  ( G `  x
)  =  0 ) )  ->  ( G `  x )  =  0 )
51, 4eqtr4d 2504 . . . . . . . . 9  |-  ( ( ( ( I  e.  V  /\  F  e.  X  /\  G  e.  X )  /\  x  e.  I )  /\  (
( F `  x
)  =  0  /\  ( G `  x
)  =  0 ) )  ->  ( F `  x )  =  ( G `  x ) )
63, 5subeq0bd 9974 . . . . . . . 8  |-  ( ( ( ( I  e.  V  /\  F  e.  X  /\  G  e.  X )  /\  x  e.  I )  /\  (
( F `  x
)  =  0  /\  ( G `  x
)  =  0 ) )  ->  ( ( F `  x )  -  ( G `  x ) )  =  0 )
76sq0id 12216 . . . . . . 7  |-  ( ( ( ( I  e.  V  /\  F  e.  X  /\  G  e.  X )  /\  x  e.  I )  /\  (
( F `  x
)  =  0  /\  ( G `  x
)  =  0 ) )  ->  ( (
( F `  x
)  -  ( G `
 x ) ) ^ 2 )  =  0 )
87ex 434 . . . . . 6  |-  ( ( ( I  e.  V  /\  F  e.  X  /\  G  e.  X
)  /\  x  e.  I )  ->  (
( ( F `  x )  =  0  /\  ( G `  x )  =  0 )  ->  ( (
( F `  x
)  -  ( G `
 x ) ) ^ 2 )  =  0 ) )
9 ioran 490 . . . . . . . . 9  |-  ( -.  ( ( F `  x )  =/=  0  \/  ( G `  x
)  =/=  0 )  <-> 
( -.  ( F `
 x )  =/=  0  /\  -.  ( G `  x )  =/=  0 ) )
10 nne 2661 . . . . . . . . . 10  |-  ( -.  ( F `  x
)  =/=  0  <->  ( F `  x )  =  0 )
11 nne 2661 . . . . . . . . . 10  |-  ( -.  ( G `  x
)  =/=  0  <->  ( G `  x )  =  0 )
1210, 11anbi12i 697 . . . . . . . . 9  |-  ( ( -.  ( F `  x )  =/=  0  /\  -.  ( G `  x )  =/=  0
)  <->  ( ( F `
 x )  =  0  /\  ( G `
 x )  =  0 ) )
139, 12bitri 249 . . . . . . . 8  |-  ( -.  ( ( F `  x )  =/=  0  \/  ( G `  x
)  =/=  0 )  <-> 
( ( F `  x )  =  0  /\  ( G `  x )  =  0 ) )
1413a1i 11 . . . . . . 7  |-  ( ( ( I  e.  V  /\  F  e.  X  /\  G  e.  X
)  /\  x  e.  I )  ->  ( -.  ( ( F `  x )  =/=  0  \/  ( G `  x
)  =/=  0 )  <-> 
( ( F `  x )  =  0  /\  ( G `  x )  =  0 ) ) )
15 eqid 2460 . . . . . . . . . . . 12  |-  ( k  e.  I  |->  ( ( ( F `  k
)  -  ( G `
 k ) ) ^ 2 ) )  =  ( k  e.  I  |->  ( ( ( F `  k )  -  ( G `  k ) ) ^
2 ) )
1615a1i 11 . . . . . . . . . . 11  |-  ( ( ( I  e.  V  /\  F  e.  X  /\  G  e.  X
)  /\  x  e.  I )  ->  (
k  e.  I  |->  ( ( ( F `  k )  -  ( G `  k )
) ^ 2 ) )  =  ( k  e.  I  |->  ( ( ( F `  k
)  -  ( G `
 k ) ) ^ 2 ) ) )
17 simpr 461 . . . . . . . . . . . . . 14  |-  ( ( ( ( I  e.  V  /\  F  e.  X  /\  G  e.  X )  /\  x  e.  I )  /\  k  =  x )  ->  k  =  x )
1817fveq2d 5861 . . . . . . . . . . . . 13  |-  ( ( ( ( I  e.  V  /\  F  e.  X  /\  G  e.  X )  /\  x  e.  I )  /\  k  =  x )  ->  ( F `  k )  =  ( F `  x ) )
1917fveq2d 5861 . . . . . . . . . . . . 13  |-  ( ( ( ( I  e.  V  /\  F  e.  X  /\  G  e.  X )  /\  x  e.  I )  /\  k  =  x )  ->  ( G `  k )  =  ( G `  x ) )
2018, 19oveq12d 6293 . . . . . . . . . . . 12  |-  ( ( ( ( I  e.  V  /\  F  e.  X  /\  G  e.  X )  /\  x  e.  I )  /\  k  =  x )  ->  (
( F `  k
)  -  ( G `
 k ) )  =  ( ( F `
 x )  -  ( G `  x ) ) )
2120oveq1d 6290 . . . . . . . . . . 11  |-  ( ( ( ( I  e.  V  /\  F  e.  X  /\  G  e.  X )  /\  x  e.  I )  /\  k  =  x )  ->  (
( ( F `  k )  -  ( G `  k )
) ^ 2 )  =  ( ( ( F `  x )  -  ( G `  x ) ) ^
2 ) )
22 simpr 461 . . . . . . . . . . 11  |-  ( ( ( I  e.  V  /\  F  e.  X  /\  G  e.  X
)  /\  x  e.  I )  ->  x  e.  I )
23 ovex 6300 . . . . . . . . . . . 12  |-  ( ( ( F `  x
)  -  ( G `
 x ) ) ^ 2 )  e. 
_V
2423a1i 11 . . . . . . . . . . 11  |-  ( ( ( I  e.  V  /\  F  e.  X  /\  G  e.  X
)  /\  x  e.  I )  ->  (
( ( F `  x )  -  ( G `  x )
) ^ 2 )  e.  _V )
2516, 21, 22, 24fvmptd 5946 . . . . . . . . . 10  |-  ( ( ( I  e.  V  /\  F  e.  X  /\  G  e.  X
)  /\  x  e.  I )  ->  (
( k  e.  I  |->  ( ( ( F `
 k )  -  ( G `  k ) ) ^ 2 ) ) `  x )  =  ( ( ( F `  x )  -  ( G `  x ) ) ^
2 ) )
2625neeq1d 2737 . . . . . . . . 9  |-  ( ( ( I  e.  V  /\  F  e.  X  /\  G  e.  X
)  /\  x  e.  I )  ->  (
( ( k  e.  I  |->  ( ( ( F `  k )  -  ( G `  k ) ) ^
2 ) ) `  x )  =/=  0  <->  ( ( ( F `  x )  -  ( G `  x )
) ^ 2 )  =/=  0 ) )
2726bicomd 201 . . . . . . . 8  |-  ( ( ( I  e.  V  /\  F  e.  X  /\  G  e.  X
)  /\  x  e.  I )  ->  (
( ( ( F `
 x )  -  ( G `  x ) ) ^ 2 )  =/=  0  <->  ( (
k  e.  I  |->  ( ( ( F `  k )  -  ( G `  k )
) ^ 2 ) ) `  x )  =/=  0 ) )
2827necon1bbid 2710 . . . . . . 7  |-  ( ( ( I  e.  V  /\  F  e.  X  /\  G  e.  X
)  /\  x  e.  I )  ->  ( -.  ( ( k  e.  I  |->  ( ( ( F `  k )  -  ( G `  k ) ) ^
2 ) ) `  x )  =/=  0  <->  ( ( ( F `  x )  -  ( G `  x )
) ^ 2 )  =  0 ) )
2914, 28imbi12d 320 . . . . . 6  |-  ( ( ( I  e.  V  /\  F  e.  X  /\  G  e.  X
)  /\  x  e.  I )  ->  (
( -.  ( ( F `  x )  =/=  0  \/  ( G `  x )  =/=  0 )  ->  -.  ( ( k  e.  I  |->  ( ( ( F `  k )  -  ( G `  k ) ) ^
2 ) ) `  x )  =/=  0
)  <->  ( ( ( F `  x )  =  0  /\  ( G `  x )  =  0 )  -> 
( ( ( F `
 x )  -  ( G `  x ) ) ^ 2 )  =  0 ) ) )
308, 29mpbird 232 . . . . 5  |-  ( ( ( I  e.  V  /\  F  e.  X  /\  G  e.  X
)  /\  x  e.  I )  ->  ( -.  ( ( F `  x )  =/=  0  \/  ( G `  x
)  =/=  0 )  ->  -.  ( (
k  e.  I  |->  ( ( ( F `  k )  -  ( G `  k )
) ^ 2 ) ) `  x )  =/=  0 ) )
3130con4d 105 . . . 4  |-  ( ( ( I  e.  V  /\  F  e.  X  /\  G  e.  X
)  /\  x  e.  I )  ->  (
( ( k  e.  I  |->  ( ( ( F `  k )  -  ( G `  k ) ) ^
2 ) ) `  x )  =/=  0  ->  ( ( F `  x )  =/=  0  \/  ( G `  x
)  =/=  0 ) ) )
3231ss2rabdv 3574 . . 3  |-  ( ( I  e.  V  /\  F  e.  X  /\  G  e.  X )  ->  { x  e.  I  |  ( ( k  e.  I  |->  ( ( ( F `  k
)  -  ( G `
 k ) ) ^ 2 ) ) `
 x )  =/=  0 }  C_  { x  e.  I  |  (
( F `  x
)  =/=  0  \/  ( G `  x
)  =/=  0 ) } )
33 unrab 3762 . . 3  |-  ( { x  e.  I  |  ( F `  x
)  =/=  0 }  u.  { x  e.  I  |  ( G `
 x )  =/=  0 } )  =  { x  e.  I  |  ( ( F `
 x )  =/=  0  \/  ( G `
 x )  =/=  0 ) }
3432, 33syl6sseqr 3544 . 2  |-  ( ( I  e.  V  /\  F  e.  X  /\  G  e.  X )  ->  { x  e.  I  |  ( ( k  e.  I  |->  ( ( ( F `  k
)  -  ( G `
 k ) ) ^ 2 ) ) `
 x )  =/=  0 }  C_  ( { x  e.  I  |  ( F `  x )  =/=  0 }  u.  { x  e.  I  |  ( G `  x )  =/=  0 } ) )
35 simp1 991 . . 3  |-  ( ( I  e.  V  /\  F  e.  X  /\  G  e.  X )  ->  I  e.  V )
36 ovex 6300 . . . . 5  |-  ( ( ( F `  k
)  -  ( G `
 k ) ) ^ 2 )  e. 
_V
3736, 15fnmpti 5700 . . . 4  |-  ( k  e.  I  |->  ( ( ( F `  k
)  -  ( G `
 k ) ) ^ 2 ) )  Fn  I
38 suppvalfn 6898 . . . 4  |-  ( ( ( k  e.  I  |->  ( ( ( F `
 k )  -  ( G `  k ) ) ^ 2 ) )  Fn  I  /\  I  e.  V  /\  0  e.  CC )  ->  ( ( k  e.  I  |->  ( ( ( F `  k )  -  ( G `  k ) ) ^
2 ) ) supp  0
)  =  { x  e.  I  |  (
( k  e.  I  |->  ( ( ( F `
 k )  -  ( G `  k ) ) ^ 2 ) ) `  x )  =/=  0 } )
3937, 2, 38mp3an13 1310 . . 3  |-  ( I  e.  V  ->  (
( k  e.  I  |->  ( ( ( F `
 k )  -  ( G `  k ) ) ^ 2 ) ) supp  0 )  =  { x  e.  I  |  ( ( k  e.  I  |->  ( ( ( F `  k
)  -  ( G `
 k ) ) ^ 2 ) ) `
 x )  =/=  0 } )
4035, 39syl 16 . 2  |-  ( ( I  e.  V  /\  F  e.  X  /\  G  e.  X )  ->  ( ( k  e.  I  |->  ( ( ( F `  k )  -  ( G `  k ) ) ^
2 ) ) supp  0
)  =  { x  e.  I  |  (
( k  e.  I  |->  ( ( ( F `
 k )  -  ( G `  k ) ) ^ 2 ) ) `  x )  =/=  0 } )
41 elrabi 3251 . . . . . . 7  |-  ( F  e.  { h  e.  ( RR  ^m  I
)  |  h finSupp  0 }  ->  F  e.  ( RR  ^m  I ) )
42 rrxmval.1 . . . . . . 7  |-  X  =  { h  e.  ( RR  ^m  I )  |  h finSupp  0 }
4341, 42eleq2s 2568 . . . . . 6  |-  ( F  e.  X  ->  F  e.  ( RR  ^m  I
) )
44 elmapi 7430 . . . . . 6  |-  ( F  e.  ( RR  ^m  I )  ->  F : I --> RR )
45 ffn 5722 . . . . . 6  |-  ( F : I --> RR  ->  F  Fn  I )
4643, 44, 453syl 20 . . . . 5  |-  ( F  e.  X  ->  F  Fn  I )
47463ad2ant2 1013 . . . 4  |-  ( ( I  e.  V  /\  F  e.  X  /\  G  e.  X )  ->  F  Fn  I )
482a1i 11 . . . 4  |-  ( ( I  e.  V  /\  F  e.  X  /\  G  e.  X )  ->  0  e.  CC )
49 suppvalfn 6898 . . . 4  |-  ( ( F  Fn  I  /\  I  e.  V  /\  0  e.  CC )  ->  ( F supp  0 )  =  { x  e.  I  |  ( F `
 x )  =/=  0 } )
5047, 35, 48, 49syl3anc 1223 . . 3  |-  ( ( I  e.  V  /\  F  e.  X  /\  G  e.  X )  ->  ( F supp  0 )  =  { x  e.  I  |  ( F `
 x )  =/=  0 } )
51 elrabi 3251 . . . . . . 7  |-  ( G  e.  { h  e.  ( RR  ^m  I
)  |  h finSupp  0 }  ->  G  e.  ( RR  ^m  I ) )
5251, 42eleq2s 2568 . . . . . 6  |-  ( G  e.  X  ->  G  e.  ( RR  ^m  I
) )
53 elmapi 7430 . . . . . 6  |-  ( G  e.  ( RR  ^m  I )  ->  G : I --> RR )
54 ffn 5722 . . . . . 6  |-  ( G : I --> RR  ->  G  Fn  I )
5552, 53, 543syl 20 . . . . 5  |-  ( G  e.  X  ->  G  Fn  I )
56553ad2ant3 1014 . . . 4  |-  ( ( I  e.  V  /\  F  e.  X  /\  G  e.  X )  ->  G  Fn  I )
57 suppvalfn 6898 . . . 4  |-  ( ( G  Fn  I  /\  I  e.  V  /\  0  e.  CC )  ->  ( G supp  0 )  =  { x  e.  I  |  ( G `
 x )  =/=  0 } )
5856, 35, 48, 57syl3anc 1223 . . 3  |-  ( ( I  e.  V  /\  F  e.  X  /\  G  e.  X )  ->  ( G supp  0 )  =  { x  e.  I  |  ( G `
 x )  =/=  0 } )
5950, 58uneq12d 3652 . 2  |-  ( ( I  e.  V  /\  F  e.  X  /\  G  e.  X )  ->  ( ( F supp  0
)  u.  ( G supp  0 ) )  =  ( { x  e.  I  |  ( F `
 x )  =/=  0 }  u.  {
x  e.  I  |  ( G `  x
)  =/=  0 } ) )
6034, 40, 593sstr4d 3540 1  |-  ( ( I  e.  V  /\  F  e.  X  /\  G  e.  X )  ->  ( ( k  e.  I  |->  ( ( ( F `  k )  -  ( G `  k ) ) ^
2 ) ) supp  0
)  C_  ( ( F supp  0 )  u.  ( G supp  0 ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762    =/= wne 2655   {crab 2811   _Vcvv 3106    u. cun 3467    C_ wss 3469   class class class wbr 4440    |-> cmpt 4498    Fn wfn 5574   -->wf 5575   ` cfv 5579  (class class class)co 6275   supp csupp 6891    ^m cmap 7410   finSupp cfsupp 7818   CCcc 9479   RRcr 9480   0cc0 9481    - cmin 9794   2c2 10574   ^cexp 12122
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  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
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-supp 6892  df-recs 7032  df-rdg 7066  df-er 7301  df-map 7412  df-en 7507  df-dom 7508  df-sdom 7509  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-2 10583  df-n0 10785  df-z 10854  df-uz 11072  df-seq 12064  df-exp 12123
This theorem is referenced by:  rrxmval  21560
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