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Theorem rrxmetlem 21597
Description: Lemma for rrxmet 21598 (Contributed by Thierry Arnoux, 5-Jul-2019.)
Hypotheses
Ref Expression
rrxmval.1  |-  X  =  { h  e.  ( RR  ^m  I )  |  h finSupp  0 }
rrxmval.d  |-  D  =  ( dist `  (ℝ^ `  I ) )
rrxmetlem.1  |-  ( ph  ->  I  e.  V )
rrxmetlem.2  |-  ( ph  ->  F  e.  X )
rrxmetlem.3  |-  ( ph  ->  G  e.  X )
rrxmetlem.4  |-  ( ph  ->  A  C_  I )
rrxmetlem.5  |-  ( ph  ->  A  e.  Fin )
rrxmetlem.6  |-  ( ph  ->  ( ( F supp  0
)  u.  ( G supp  0 ) )  C_  A )
Assertion
Ref Expression
rrxmetlem  |-  ( ph  -> 
sum_ k  e.  ( ( F supp  0 )  u.  ( G supp  0
) ) ( ( ( F `  k
)  -  ( G `
 k ) ) ^ 2 )  = 
sum_ k  e.  A  ( ( ( F `
 k )  -  ( G `  k ) ) ^ 2 ) )
Distinct variable groups:    A, k    h, F, k    h, G, k    h, I, k   
h, V, k    k, X    ph, k
Allowed substitution hints:    ph( h)    A( h)    D( h, k)    X( h)

Proof of Theorem rrxmetlem
StepHypRef Expression
1 rrxmetlem.6 . 2  |-  ( ph  ->  ( ( F supp  0
)  u.  ( G supp  0 ) )  C_  A )
2 rrxmetlem.4 . . . . . . 7  |-  ( ph  ->  A  C_  I )
31, 2sstrd 3514 . . . . . 6  |-  ( ph  ->  ( ( F supp  0
)  u.  ( G supp  0 ) )  C_  I )
43sselda 3504 . . . . 5  |-  ( (
ph  /\  k  e.  ( ( F supp  0
)  u.  ( G supp  0 ) ) )  ->  k  e.  I
)
5 rrxmval.1 . . . . . . . 8  |-  X  =  { h  e.  ( RR  ^m  I )  |  h finSupp  0 }
6 rrxmetlem.2 . . . . . . . 8  |-  ( ph  ->  F  e.  X )
75, 6rrxf 21591 . . . . . . 7  |-  ( ph  ->  F : I --> RR )
87ffvelrnda 6021 . . . . . 6  |-  ( (
ph  /\  k  e.  I )  ->  ( F `  k )  e.  RR )
98recnd 9622 . . . . 5  |-  ( (
ph  /\  k  e.  I )  ->  ( F `  k )  e.  CC )
104, 9syldan 470 . . . 4  |-  ( (
ph  /\  k  e.  ( ( F supp  0
)  u.  ( G supp  0 ) ) )  ->  ( F `  k )  e.  CC )
11 rrxmetlem.3 . . . . . . . 8  |-  ( ph  ->  G  e.  X )
125, 11rrxf 21591 . . . . . . 7  |-  ( ph  ->  G : I --> RR )
1312ffvelrnda 6021 . . . . . 6  |-  ( (
ph  /\  k  e.  I )  ->  ( G `  k )  e.  RR )
1413recnd 9622 . . . . 5  |-  ( (
ph  /\  k  e.  I )  ->  ( G `  k )  e.  CC )
154, 14syldan 470 . . . 4  |-  ( (
ph  /\  k  e.  ( ( F supp  0
)  u.  ( G supp  0 ) ) )  ->  ( G `  k )  e.  CC )
1610, 15subcld 9930 . . 3  |-  ( (
ph  /\  k  e.  ( ( F supp  0
)  u.  ( G supp  0 ) ) )  ->  ( ( F `
 k )  -  ( G `  k ) )  e.  CC )
1716sqcld 12276 . 2  |-  ( (
ph  /\  k  e.  ( ( F supp  0
)  u.  ( G supp  0 ) ) )  ->  ( ( ( F `  k )  -  ( G `  k ) ) ^
2 )  e.  CC )
182ssdifd 3640 . . . 4  |-  ( ph  ->  ( A  \  (
( F supp  0 )  u.  ( G supp  0
) ) )  C_  ( I  \  (
( F supp  0 )  u.  ( G supp  0
) ) ) )
1918sselda 3504 . . 3  |-  ( (
ph  /\  k  e.  ( A  \  (
( F supp  0 )  u.  ( G supp  0
) ) ) )  ->  k  e.  ( I  \  ( ( F supp  0 )  u.  ( G supp  0 ) ) ) )
20 simpr 461 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( I  \  (
( F supp  0 )  u.  ( G supp  0
) ) ) )  ->  k  e.  ( I  \  ( ( F supp  0 )  u.  ( G supp  0 ) ) ) )
2120eldifad 3488 . . . . . 6  |-  ( (
ph  /\  k  e.  ( I  \  (
( F supp  0 )  u.  ( G supp  0
) ) ) )  ->  k  e.  I
)
2221, 9syldan 470 . . . . 5  |-  ( (
ph  /\  k  e.  ( I  \  (
( F supp  0 )  u.  ( G supp  0
) ) ) )  ->  ( F `  k )  e.  CC )
23 ssun1 3667 . . . . . . . 8  |-  ( F supp  0 )  C_  (
( F supp  0 )  u.  ( G supp  0
) )
2423a1i 11 . . . . . . 7  |-  ( ph  ->  ( F supp  0 ) 
C_  ( ( F supp  0 )  u.  ( G supp  0 ) ) )
25 rrxmetlem.1 . . . . . . 7  |-  ( ph  ->  I  e.  V )
26 0red 9597 . . . . . . 7  |-  ( ph  ->  0  e.  RR )
277, 24, 25, 26suppssr 6931 . . . . . 6  |-  ( (
ph  /\  k  e.  ( I  \  (
( F supp  0 )  u.  ( G supp  0
) ) ) )  ->  ( F `  k )  =  0 )
28 ssun2 3668 . . . . . . . 8  |-  ( G supp  0 )  C_  (
( F supp  0 )  u.  ( G supp  0
) )
2928a1i 11 . . . . . . 7  |-  ( ph  ->  ( G supp  0 ) 
C_  ( ( F supp  0 )  u.  ( G supp  0 ) ) )
3012, 29, 25, 26suppssr 6931 . . . . . 6  |-  ( (
ph  /\  k  e.  ( I  \  (
( F supp  0 )  u.  ( G supp  0
) ) ) )  ->  ( G `  k )  =  0 )
3127, 30eqtr4d 2511 . . . . 5  |-  ( (
ph  /\  k  e.  ( I  \  (
( F supp  0 )  u.  ( G supp  0
) ) ) )  ->  ( F `  k )  =  ( G `  k ) )
3222, 31subeq0bd 9985 . . . 4  |-  ( (
ph  /\  k  e.  ( I  \  (
( F supp  0 )  u.  ( G supp  0
) ) ) )  ->  ( ( F `
 k )  -  ( G `  k ) )  =  0 )
3332sq0id 12229 . . 3  |-  ( (
ph  /\  k  e.  ( I  \  (
( F supp  0 )  u.  ( G supp  0
) ) ) )  ->  ( ( ( F `  k )  -  ( G `  k ) ) ^
2 )  =  0 )
3419, 33syldan 470 . 2  |-  ( (
ph  /\  k  e.  ( A  \  (
( F supp  0 )  u.  ( G supp  0
) ) ) )  ->  ( ( ( F `  k )  -  ( G `  k ) ) ^
2 )  =  0 )
35 rrxmetlem.5 . 2  |-  ( ph  ->  A  e.  Fin )
361, 17, 34, 35fsumss 13510 1  |-  ( ph  -> 
sum_ k  e.  ( ( F supp  0 )  u.  ( G supp  0
) ) ( ( ( F `  k
)  -  ( G `
 k ) ) ^ 2 )  = 
sum_ k  e.  A  ( ( ( F `
 k )  -  ( G `  k ) ) ^ 2 ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   {crab 2818    \ cdif 3473    u. cun 3474    C_ wss 3476   class class class wbr 4447   ` cfv 5588  (class class class)co 6284   supp csupp 6901    ^m cmap 7420   Fincfn 7516   finSupp cfsupp 7829   CCcc 9490   RRcr 9491   0cc0 9492    - cmin 9805   2c2 10585   ^cexp 12134   sum_csu 13471   distcds 14564  ℝ^crrx 21578
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-inf2 8058  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569  ax-pre-sup 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-supp 6902  df-recs 7042  df-rdg 7076  df-1o 7130  df-oadd 7134  df-er 7311  df-map 7422  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-sup 7901  df-oi 7935  df-card 8320  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-div 10207  df-nn 10537  df-2 10594  df-3 10595  df-n0 10796  df-z 10865  df-uz 11083  df-rp 11221  df-fz 11673  df-fzo 11793  df-seq 12076  df-exp 12135  df-hash 12374  df-cj 12895  df-re 12896  df-im 12897  df-sqrt 13031  df-abs 13032  df-clim 13274  df-sum 13472
This theorem is referenced by:  rrxmet  21598
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