MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rrxmvallem Structured version   Unicode version

Theorem rrxmvallem 20925
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 9399 . . . . . . . . . 10  |-  0  e.  CC
31, 2syl6eqel 2531 . . . . . . . . 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 2478 . . . . . . . . 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 9795 . . . . . . . 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 11980 . . . . . . 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 2626 . . . . . . . . . 10  |-  ( -.  ( F `  x
)  =/=  0  <->  ( F `  x )  =  0 )
11 nne 2626 . . . . . . . . . 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 2443 . . . . . . . . . . . 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 5716 . . . . . . . . . . . . 13  |-  ( ( ( ( I  e.  V  /\  F  e.  X  /\  G  e.  X )  /\  x  e.  I )  /\  k  =  x )  ->  ( F `  k )  =  ( F `  x ) )
1917fveq2d 5716 . . . . . . . . . . . . 13  |-  ( ( ( ( I  e.  V  /\  F  e.  X  /\  G  e.  X )  /\  x  e.  I )  /\  k  =  x )  ->  ( G `  k )  =  ( G `  x ) )
2018, 19oveq12d 6130 . . . . . . . . . . . 12  |-  ( ( ( ( I  e.  V  /\  F  e.  X  /\  G  e.  X )  /\  x  e.  I )  /\  k  =  x )  ->  (
( F `  k
)  -  ( G `
 k ) )  =  ( ( F `
 x )  -  ( G `  x ) ) )
2120oveq1d 6127 . . . . . . . . . . 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 6137 . . . . . . . . . . . 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 5800 . . . . . . . . . 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 2641 . . . . . . . . 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 2689 . . . . . . 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 3454 . . 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 3642 . . 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 3424 . 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 988 . . 3  |-  ( ( I  e.  V  /\  F  e.  X  /\  G  e.  X )  ->  I  e.  V )
36 ovex 6137 . . . . 5  |-  ( ( ( F `  k
)  -  ( G `
 k ) ) ^ 2 )  e. 
_V
3736, 15fnmpti 5560 . . . 4  |-  ( k  e.  I  |->  ( ( ( F `  k
)  -  ( G `
 k ) ) ^ 2 ) )  Fn  I
38 suppvalfn 6718 . . . 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 1305 . . 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 3135 . . . . . . 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 2535 . . . . . 6  |-  ( F  e.  X  ->  F  e.  ( RR  ^m  I
) )
44 elmapi 7255 . . . . . 6  |-  ( F  e.  ( RR  ^m  I )  ->  F : I --> RR )
45 ffn 5580 . . . . . 6  |-  ( F : I --> RR  ->  F  Fn  I )
4643, 44, 453syl 20 . . . . 5  |-  ( F  e.  X  ->  F  Fn  I )
47463ad2ant2 1010 . . . 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 6718 . . . 4  |-  ( ( F  Fn  I  /\  I  e.  V  /\  0  e.  CC )  ->  ( F supp  0 )  =  { x  e.  I  |  ( F `
 x )  =/=  0 } )
5047, 35, 48, 49syl3anc 1218 . . 3  |-  ( ( I  e.  V  /\  F  e.  X  /\  G  e.  X )  ->  ( F supp  0 )  =  { x  e.  I  |  ( F `
 x )  =/=  0 } )
51 elrabi 3135 . . . . . . 7  |-  ( G  e.  { h  e.  ( RR  ^m  I
)  |  h finSupp  0 }  ->  G  e.  ( RR  ^m  I ) )
5251, 42eleq2s 2535 . . . . . 6  |-  ( G  e.  X  ->  G  e.  ( RR  ^m  I
) )
53 elmapi 7255 . . . . . 6  |-  ( G  e.  ( RR  ^m  I )  ->  G : I --> RR )
54 ffn 5580 . . . . . 6  |-  ( G : I --> RR  ->  G  Fn  I )
5552, 53, 543syl 20 . . . . 5  |-  ( G  e.  X  ->  G  Fn  I )
56553ad2ant3 1011 . . . 4  |-  ( ( I  e.  V  /\  F  e.  X  /\  G  e.  X )  ->  G  Fn  I )
57 suppvalfn 6718 . . . 4  |-  ( ( G  Fn  I  /\  I  e.  V  /\  0  e.  CC )  ->  ( G supp  0 )  =  { x  e.  I  |  ( G `
 x )  =/=  0 } )
5856, 35, 48, 57syl3anc 1218 . . 3  |-  ( ( I  e.  V  /\  F  e.  X  /\  G  e.  X )  ->  ( G supp  0 )  =  { x  e.  I  |  ( G `
 x )  =/=  0 } )
5950, 58uneq12d 3532 . 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 3420 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 965    = wceq 1369    e. wcel 1756    =/= wne 2620   {crab 2740   _Vcvv 2993    u. cun 3347    C_ wss 3349   class class class wbr 4313    e. cmpt 4371    Fn wfn 5434   -->wf 5435   ` cfv 5439  (class class class)co 6112   supp csupp 6711    ^m cmap 7235   finSupp cfsupp 7641   CCcc 9301   RRcr 9302   0cc0 9303    - cmin 9616   2c2 10392   ^cexp 11886
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-om 6498  df-1st 6598  df-2nd 6599  df-supp 6712  df-recs 6853  df-rdg 6887  df-er 7122  df-map 7237  df-en 7332  df-dom 7333  df-sdom 7334  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-nn 10344  df-2 10401  df-n0 10601  df-z 10668  df-uz 10883  df-seq 11828  df-exp 11887
This theorem is referenced by:  rrxmval  20926
  Copyright terms: Public domain W3C validator