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Theorem rrxmvallem 22300
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 762 . . . . . . . . . 10  |-  ( ( ( ( I  e.  V  /\  F  e.  X  /\  G  e.  X )  /\  x  e.  I )  /\  (
( F `  x
)  =  0  /\  ( G `  x
)  =  0 ) )  ->  ( F `  x )  =  0 )
2 0cn 9586 . . . . . . . . . 10  |-  0  e.  CC
31, 2syl6eqel 2514 . . . . . . . . 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 764 . . . . . . . . . 10  |-  ( ( ( ( I  e.  V  /\  F  e.  X  /\  G  e.  X )  /\  x  e.  I )  /\  (
( F `  x
)  =  0  /\  ( G `  x
)  =  0 ) )  ->  ( G `  x )  =  0 )
51, 4eqtr4d 2465 . . . . . . . . 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 9996 . . . . . . . 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 12318 . . . . . . 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 435 . . . . . 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 492 . . . . . . . 8  |-  ( -.  ( ( F `  x )  =/=  0  \/  ( G `  x
)  =/=  0 )  <-> 
( -.  ( F `
 x )  =/=  0  /\  -.  ( G `  x )  =/=  0 ) )
10 nne 2605 . . . . . . . . 9  |-  ( -.  ( F `  x
)  =/=  0  <->  ( F `  x )  =  0 )
11 nne 2605 . . . . . . . . 9  |-  ( -.  ( G `  x
)  =/=  0  <->  ( G `  x )  =  0 )
1210, 11anbi12i 701 . . . . . . . 8  |-  ( ( -.  ( F `  x )  =/=  0  /\  -.  ( G `  x )  =/=  0
)  <->  ( ( F `
 x )  =  0  /\  ( G `
 x )  =  0 ) )
139, 12bitri 252 . . . . . . 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 2429 . . . . . . . . . 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 462 . . . . . . . . . . . . 13  |-  ( ( ( ( I  e.  V  /\  F  e.  X  /\  G  e.  X )  /\  x  e.  I )  /\  k  =  x )  ->  k  =  x )
1716fveq2d 5829 . . . . . . . . . . . 12  |-  ( ( ( ( I  e.  V  /\  F  e.  X  /\  G  e.  X )  /\  x  e.  I )  /\  k  =  x )  ->  ( F `  k )  =  ( F `  x ) )
1816fveq2d 5829 . . . . . . . . . . . 12  |-  ( ( ( ( I  e.  V  /\  F  e.  X  /\  G  e.  X )  /\  x  e.  I )  /\  k  =  x )  ->  ( G `  k )  =  ( G `  x ) )
1917, 18oveq12d 6267 . . . . . . . . . . 11  |-  ( ( ( ( I  e.  V  /\  F  e.  X  /\  G  e.  X )  /\  x  e.  I )  /\  k  =  x )  ->  (
( F `  k
)  -  ( G `
 k ) )  =  ( ( F `
 x )  -  ( G `  x ) ) )
2019oveq1d 6264 . . . . . . . . . 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 462 . . . . . . . . . 10  |-  ( ( ( I  e.  V  /\  F  e.  X  /\  G  e.  X
)  /\  x  e.  I )  ->  x  e.  I )
22 ovex 6277 . . . . . . . . . . 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 5914 . . . . . . . . 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 2660 . . . . . . . 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 204 . . . . . . 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 2640 . . . . . 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 271 . . . . 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 108 . . . 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 3485 . . 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 3687 . . 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 3454 . 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 1005 . . 3  |-  ( ( I  e.  V  /\  F  e.  X  /\  G  e.  X )  ->  I  e.  V )
34 ovex 6277 . . . . 5  |-  ( ( ( F `  k
)  -  ( G `
 k ) ) ^ 2 )  e. 
_V
35 eqid 2428 . . . . 5  |-  ( k  e.  I  |->  ( ( ( F `  k
)  -  ( G `
 k ) ) ^ 2 ) )  =  ( k  e.  I  |->  ( ( ( F `  k )  -  ( G `  k ) ) ^
2 ) )
3634, 35fnmpti 5667 . . . 4  |-  ( k  e.  I  |->  ( ( ( F `  k
)  -  ( G `
 k ) ) ^ 2 ) )  Fn  I
37 suppvalfn 6876 . . . 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 1351 . . 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 17 . 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 3168 . . . . . . 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 2524 . . . . . 6  |-  ( F  e.  X  ->  F  e.  ( RR  ^m  I
) )
43 elmapi 7448 . . . . . 6  |-  ( F  e.  ( RR  ^m  I )  ->  F : I --> RR )
44 ffn 5689 . . . . . 6  |-  ( F : I --> RR  ->  F  Fn  I )
4542, 43, 443syl 18 . . . . 5  |-  ( F  e.  X  ->  F  Fn  I )
46453ad2ant2 1027 . . . 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 6876 . . . 4  |-  ( ( F  Fn  I  /\  I  e.  V  /\  0  e.  CC )  ->  ( F supp  0 )  =  { x  e.  I  |  ( F `
 x )  =/=  0 } )
4946, 33, 47, 48syl3anc 1264 . . 3  |-  ( ( I  e.  V  /\  F  e.  X  /\  G  e.  X )  ->  ( F supp  0 )  =  { x  e.  I  |  ( F `
 x )  =/=  0 } )
50 elrabi 3168 . . . . . . 7  |-  ( G  e.  { h  e.  ( RR  ^m  I
)  |  h finSupp  0 }  ->  G  e.  ( RR  ^m  I ) )
5150, 41eleq2s 2524 . . . . . 6  |-  ( G  e.  X  ->  G  e.  ( RR  ^m  I
) )
52 elmapi 7448 . . . . . 6  |-  ( G  e.  ( RR  ^m  I )  ->  G : I --> RR )
53 ffn 5689 . . . . . 6  |-  ( G : I --> RR  ->  G  Fn  I )
5451, 52, 533syl 18 . . . . 5  |-  ( G  e.  X  ->  G  Fn  I )
55543ad2ant3 1028 . . . 4  |-  ( ( I  e.  V  /\  F  e.  X  /\  G  e.  X )  ->  G  Fn  I )
56 suppvalfn 6876 . . . 4  |-  ( ( G  Fn  I  /\  I  e.  V  /\  0  e.  CC )  ->  ( G supp  0 )  =  { x  e.  I  |  ( G `
 x )  =/=  0 } )
5755, 33, 47, 56syl3anc 1264 . . 3  |-  ( ( I  e.  V  /\  F  e.  X  /\  G  e.  X )  ->  ( G supp  0 )  =  { x  e.  I  |  ( G `
 x )  =/=  0 } )
5849, 57uneq12d 3564 . 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 3450 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 187    \/ wo 369    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872    =/= wne 2599   {crab 2718   _Vcvv 3022    u. cun 3377    C_ wss 3379   class class class wbr 4366    |-> cmpt 4425    Fn wfn 5539   -->wf 5540   ` cfv 5544  (class class class)co 6249   supp csupp 6869    ^m cmap 7427   finSupp cfsupp 7836   CCcc 9488   RRcr 9489   0cc0 9490    - cmin 9811   2c2 10610   ^cexp 12222
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-rep 4479  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-nel 2602  df-ral 2719  df-rex 2720  df-reu 2721  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-pss 3395  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-tp 3946  df-op 3948  df-uni 4163  df-iun 4244  df-br 4367  df-opab 4426  df-mpt 4427  df-tr 4462  df-eprel 4707  df-id 4711  df-po 4717  df-so 4718  df-fr 4755  df-we 4757  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-pred 5342  df-ord 5388  df-on 5389  df-lim 5390  df-suc 5391  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-f1 5549  df-fo 5550  df-f1o 5551  df-fv 5552  df-riota 6211  df-ov 6252  df-oprab 6253  df-mpt2 6254  df-om 6651  df-1st 6751  df-2nd 6752  df-supp 6870  df-wrecs 6983  df-recs 7045  df-rdg 7083  df-er 7318  df-map 7429  df-en 7525  df-dom 7526  df-sdom 7527  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9813  df-neg 9814  df-nn 10561  df-2 10619  df-n0 10821  df-z 10889  df-uz 11111  df-seq 12164  df-exp 12223
This theorem is referenced by:  rrxmval  22301
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