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Theorem rrxmvallem 22000
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 754 . . . . . . . . . 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 2550 . . . . . . . . 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 755 . . . . . . . . . 10  |-  ( ( ( ( I  e.  V  /\  F  e.  X  /\  G  e.  X )  /\  x  e.  I )  /\  (
( F `  x
)  =  0  /\  ( G `  x
)  =  0 ) )  ->  ( G `  x )  =  0 )
51, 4eqtr4d 2498 . . . . . . . . 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 9981 . . . . . . . 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 12246 . . . . . . 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 432 . . . . . 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 488 . . . . . . . 8  |-  ( -.  ( ( F `  x )  =/=  0  \/  ( G `  x
)  =/=  0 )  <-> 
( -.  ( F `
 x )  =/=  0  /\  -.  ( G `  x )  =/=  0 ) )
10 nne 2655 . . . . . . . . 9  |-  ( -.  ( F `  x
)  =/=  0  <->  ( F `  x )  =  0 )
11 nne 2655 . . . . . . . . 9  |-  ( -.  ( G `  x
)  =/=  0  <->  ( G `  x )  =  0 )
1210, 11anbi12i 695 . . . . . . . 8  |-  ( ( -.  ( F `  x )  =/=  0  /\  -.  ( G `  x )  =/=  0
)  <->  ( ( F `
 x )  =  0  /\  ( G `
 x )  =  0 ) )
139, 12bitri 249 . . . . . . 7  |-  ( -.  ( ( F `  x )  =/=  0  \/  ( G `  x
)  =/=  0 )  <-> 
( ( F `  x )  =  0  /\  ( G `  x )  =  0 ) )
1413a1i 11 . . . . . 6  |-  ( ( ( 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 eqidd 2455 . . . . . . . . . 10  |-  ( ( ( 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 ) ) )
16 simpr 459 . . . . . . . . . . . . 13  |-  ( ( ( ( I  e.  V  /\  F  e.  X  /\  G  e.  X )  /\  x  e.  I )  /\  k  =  x )  ->  k  =  x )
1716fveq2d 5852 . . . . . . . . . . . 12  |-  ( ( ( ( I  e.  V  /\  F  e.  X  /\  G  e.  X )  /\  x  e.  I )  /\  k  =  x )  ->  ( F `  k )  =  ( F `  x ) )
1816fveq2d 5852 . . . . . . . . . . . 12  |-  ( ( ( ( I  e.  V  /\  F  e.  X  /\  G  e.  X )  /\  x  e.  I )  /\  k  =  x )  ->  ( G `  k )  =  ( G `  x ) )
1917, 18oveq12d 6288 . . . . . . . . . . 11  |-  ( ( ( ( I  e.  V  /\  F  e.  X  /\  G  e.  X )  /\  x  e.  I )  /\  k  =  x )  ->  (
( F `  k
)  -  ( G `
 k ) )  =  ( ( F `
 x )  -  ( G `  x ) ) )
2019oveq1d 6285 . . . . . . . . . 10  |-  ( ( ( ( I  e.  V  /\  F  e.  X  /\  G  e.  X )  /\  x  e.  I )  /\  k  =  x )  ->  (
( ( F `  k )  -  ( G `  k )
) ^ 2 )  =  ( ( ( F `  x )  -  ( G `  x ) ) ^
2 ) )
21 simpr 459 . . . . . . . . . 10  |-  ( ( ( I  e.  V  /\  F  e.  X  /\  G  e.  X
)  /\  x  e.  I )  ->  x  e.  I )
22 ovex 6298 . . . . . . . . . . 11  |-  ( ( ( F `  x
)  -  ( G `
 x ) ) ^ 2 )  e. 
_V
2322a1i 11 . . . . . . . . . 10  |-  ( ( ( I  e.  V  /\  F  e.  X  /\  G  e.  X
)  /\  x  e.  I )  ->  (
( ( F `  x )  -  ( G `  x )
) ^ 2 )  e.  _V )
2415, 20, 21, 23fvmptd 5936 . . . . . . . . 9  |-  ( ( ( 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 ) )
2524neeq1d 2731 . . . . . . . 8  |-  ( ( ( 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 ) )
2625bicomd 201 . . . . . . 7  |-  ( ( ( 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 ) )
2726necon1bbid 2704 . . . . . 6  |-  ( ( ( 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 ) )
288, 14, 273imtr4d 268 . . . . 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 ) )
2928con4d 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 ) ) )
3029ss2rabdv 3567 . . 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 ) } )
31 unrab 3766 . . 3  |-  ( { x  e.  I  |  ( F `  x
)  =/=  0 }  u.  { x  e.  I  |  ( G `
 x )  =/=  0 } )  =  { x  e.  I  |  ( ( F `
 x )  =/=  0  \/  ( G `
 x )  =/=  0 ) }
3230, 31syl6sseqr 3536 . 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 } ) )
33 simp1 994 . . 3  |-  ( ( I  e.  V  /\  F  e.  X  /\  G  e.  X )  ->  I  e.  V )
34 ovex 6298 . . . . 5  |-  ( ( ( F `  k
)  -  ( G `
 k ) ) ^ 2 )  e. 
_V
35 eqid 2454 . . . . 5  |-  ( k  e.  I  |->  ( ( ( F `  k
)  -  ( G `
 k ) ) ^ 2 ) )  =  ( k  e.  I  |->  ( ( ( F `  k )  -  ( G `  k ) ) ^
2 ) )
3634, 35fnmpti 5691 . . . 4  |-  ( k  e.  I  |->  ( ( ( F `  k
)  -  ( G `
 k ) ) ^ 2 ) )  Fn  I
37 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 } )
3836, 2, 37mp3an13 1313 . . 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 } )
3933, 38syl 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 } )
40 elrabi 3251 . . . . . . 7  |-  ( F  e.  { h  e.  ( RR  ^m  I
)  |  h finSupp  0 }  ->  F  e.  ( RR  ^m  I ) )
41 rrxmval.1 . . . . . . 7  |-  X  =  { h  e.  ( RR  ^m  I )  |  h finSupp  0 }
4240, 41eleq2s 2562 . . . . . 6  |-  ( F  e.  X  ->  F  e.  ( RR  ^m  I
) )
43 elmapi 7433 . . . . . 6  |-  ( F  e.  ( RR  ^m  I )  ->  F : I --> RR )
44 ffn 5713 . . . . . 6  |-  ( F : I --> RR  ->  F  Fn  I )
4542, 43, 443syl 20 . . . . 5  |-  ( F  e.  X  ->  F  Fn  I )
46453ad2ant2 1016 . . . 4  |-  ( ( I  e.  V  /\  F  e.  X  /\  G  e.  X )  ->  F  Fn  I )
472a1i 11 . . . 4  |-  ( ( I  e.  V  /\  F  e.  X  /\  G  e.  X )  ->  0  e.  CC )
48 suppvalfn 6898 . . . 4  |-  ( ( F  Fn  I  /\  I  e.  V  /\  0  e.  CC )  ->  ( F supp  0 )  =  { x  e.  I  |  ( F `
 x )  =/=  0 } )
4946, 33, 47, 48syl3anc 1226 . . 3  |-  ( ( I  e.  V  /\  F  e.  X  /\  G  e.  X )  ->  ( F supp  0 )  =  { x  e.  I  |  ( F `
 x )  =/=  0 } )
50 elrabi 3251 . . . . . . 7  |-  ( G  e.  { h  e.  ( RR  ^m  I
)  |  h finSupp  0 }  ->  G  e.  ( RR  ^m  I ) )
5150, 41eleq2s 2562 . . . . . 6  |-  ( G  e.  X  ->  G  e.  ( RR  ^m  I
) )
52 elmapi 7433 . . . . . 6  |-  ( G  e.  ( RR  ^m  I )  ->  G : I --> RR )
53 ffn 5713 . . . . . 6  |-  ( G : I --> RR  ->  G  Fn  I )
5451, 52, 533syl 20 . . . . 5  |-  ( G  e.  X  ->  G  Fn  I )
55543ad2ant3 1017 . . . 4  |-  ( ( I  e.  V  /\  F  e.  X  /\  G  e.  X )  ->  G  Fn  I )
56 suppvalfn 6898 . . . 4  |-  ( ( G  Fn  I  /\  I  e.  V  /\  0  e.  CC )  ->  ( G supp  0 )  =  { x  e.  I  |  ( G `
 x )  =/=  0 } )
5755, 33, 47, 56syl3anc 1226 . . 3  |-  ( ( I  e.  V  /\  F  e.  X  /\  G  e.  X )  ->  ( G supp  0 )  =  { x  e.  I  |  ( G `
 x )  =/=  0 } )
5849, 57uneq12d 3645 . 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 } ) )
5932, 39, 583sstr4d 3532 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 366    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649   {crab 2808   _Vcvv 3106    u. cun 3459    C_ wss 3461   class class class wbr 4439    |-> cmpt 4497    Fn wfn 5565   -->wf 5566   ` cfv 5570  (class class class)co 6270   supp csupp 6891    ^m cmap 7412   finSupp cfsupp 7821   CCcc 9479   RRcr 9480   0cc0 9481    - cmin 9796   2c2 10581   ^cexp 12151
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-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 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  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-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-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-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-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-supp 6892  df-recs 7034  df-rdg 7068  df-er 7303  df-map 7414  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-n0 10792  df-z 10861  df-uz 11083  df-seq 12093  df-exp 12152
This theorem is referenced by:  rrxmval  22001
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