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Theorem rrxmetlem 20911
Description: Lemma for rrxmet 20912 (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 3371 . . . . . 6  |-  ( ph  ->  ( ( F supp  0
)  u.  ( G supp  0 ) )  C_  I )
43sselda 3361 . . . . 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 20905 . . . . . . 7  |-  ( ph  ->  F : I --> RR )
87ffvelrnda 5848 . . . . . 6  |-  ( (
ph  /\  k  e.  I )  ->  ( F `  k )  e.  RR )
98recnd 9417 . . . . 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 20905 . . . . . . 7  |-  ( ph  ->  G : I --> RR )
1312ffvelrnda 5848 . . . . . 6  |-  ( (
ph  /\  k  e.  I )  ->  ( G `  k )  e.  RR )
1413recnd 9417 . . . . 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 9724 . . 3  |-  ( (
ph  /\  k  e.  ( ( F supp  0
)  u.  ( G supp  0 ) ) )  ->  ( ( F `
 k )  -  ( G `  k ) )  e.  CC )
1716sqcld 12011 . 2  |-  ( (
ph  /\  k  e.  ( ( F supp  0
)  u.  ( G supp  0 ) ) )  ->  ( ( ( F `  k )  -  ( G `  k ) ) ^
2 )  e.  CC )
182ssdifd 3497 . . . 4  |-  ( ph  ->  ( A  \  (
( F supp  0 )  u.  ( G supp  0
) ) )  C_  ( I  \  (
( F supp  0 )  u.  ( G supp  0
) ) ) )
1918sselda 3361 . . 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 3345 . . . . . 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 3524 . . . . . . . 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 9392 . . . . . . 7  |-  ( ph  ->  0  e.  RR )
277, 24, 25, 26suppssr 6725 . . . . . 6  |-  ( (
ph  /\  k  e.  ( I  \  (
( F supp  0 )  u.  ( G supp  0
) ) ) )  ->  ( F `  k )  =  0 )
28 ssun2 3525 . . . . . . . 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 6725 . . . . . 6  |-  ( (
ph  /\  k  e.  ( I  \  (
( F supp  0 )  u.  ( G supp  0
) ) ) )  ->  ( G `  k )  =  0 )
3127, 30eqtr4d 2478 . . . . 5  |-  ( (
ph  /\  k  e.  ( I  \  (
( F supp  0 )  u.  ( G supp  0
) ) ) )  ->  ( F `  k )  =  ( G `  k ) )
3222, 31subeq0bd 9779 . . . 4  |-  ( (
ph  /\  k  e.  ( I  \  (
( F supp  0 )  u.  ( G supp  0
) ) ) )  ->  ( ( F `
 k )  -  ( G `  k ) )  =  0 )
3332sq0id 11964 . . 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 13207 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 1369    e. wcel 1756   {crab 2724    \ cdif 3330    u. cun 3331    C_ wss 3333   class class class wbr 4297   ` cfv 5423  (class class class)co 6096   supp csupp 6695    ^m cmap 7219   Fincfn 7315   finSupp cfsupp 7625   CCcc 9285   RRcr 9286   0cc0 9287    - cmin 9600   2c2 10376   ^cexp 11870   sum_csu 13168   distcds 14252  ℝ^crrx 20892
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 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-inf2 7852  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364  ax-pre-sup 9365
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  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 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-se 4685  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-isom 5432  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-supp 6696  df-recs 6837  df-rdg 6871  df-1o 6925  df-oadd 6929  df-er 7106  df-map 7221  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-sup 7696  df-oi 7729  df-card 8114  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-div 9999  df-nn 10328  df-2 10385  df-3 10386  df-n0 10585  df-z 10652  df-uz 10867  df-rp 10997  df-fz 11443  df-fzo 11554  df-seq 11812  df-exp 11871  df-hash 12109  df-cj 12593  df-re 12594  df-im 12595  df-sqr 12729  df-abs 12730  df-clim 12971  df-sum 13169
This theorem is referenced by:  rrxmet  20912
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