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Theorem axcgrid 23888
Description: If there is no distance between  A and  B, then  A  =  B. Axiom A3 of [Schwabhauser] p. 10. (Contributed by Scott Fenton, 3-Jun-2013.)
Assertion
Ref Expression
axcgrid  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  -> 
( <. A ,  B >.Cgr
<. C ,  C >.  ->  A  =  B )
)

Proof of Theorem axcgrid
Dummy variable  i is distinct from all other variables.
StepHypRef Expression
1 fveecn 23874 . . . . . . . . . 10  |-  ( ( C  e.  ( EE
`  N )  /\  i  e.  ( 1 ... N ) )  ->  ( C `  i )  e.  CC )
2 subid 9827 . . . . . . . . . . 11  |-  ( ( C `  i )  e.  CC  ->  (
( C `  i
)  -  ( C `
 i ) )  =  0 )
32sq0id 12216 . . . . . . . . . 10  |-  ( ( C `  i )  e.  CC  ->  (
( ( C `  i )  -  ( C `  i )
) ^ 2 )  =  0 )
41, 3syl 16 . . . . . . . . 9  |-  ( ( C  e.  ( EE
`  N )  /\  i  e.  ( 1 ... N ) )  ->  ( ( ( C `  i )  -  ( C `  i ) ) ^
2 )  =  0 )
54sumeq2dv 13474 . . . . . . . 8  |-  ( C  e.  ( EE `  N )  ->  sum_ i  e.  ( 1 ... N
) ( ( ( C `  i )  -  ( C `  i ) ) ^
2 )  =  sum_ i  e.  ( 1 ... N ) 0 )
6 fzfid 12039 . . . . . . . . 9  |-  ( C  e.  ( EE `  N )  ->  (
1 ... N )  e. 
Fin )
7 sumz 13493 . . . . . . . . . 10  |-  ( ( ( 1 ... N
)  C_  ( ZZ>= ` 
1 )  \/  (
1 ... N )  e. 
Fin )  ->  sum_ i  e.  ( 1 ... N
) 0  =  0 )
87olcs 395 . . . . . . . . 9  |-  ( ( 1 ... N )  e.  Fin  ->  sum_ i  e.  ( 1 ... N
) 0  =  0 )
96, 8syl 16 . . . . . . . 8  |-  ( C  e.  ( EE `  N )  ->  sum_ i  e.  ( 1 ... N
) 0  =  0 )
105, 9eqtrd 2501 . . . . . . 7  |-  ( C  e.  ( EE `  N )  ->  sum_ i  e.  ( 1 ... N
) ( ( ( C `  i )  -  ( C `  i ) ) ^
2 )  =  0 )
11103ad2ant3 1014 . . . . . 6  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  ->  sum_ i  e.  ( 1 ... N
) ( ( ( C `  i )  -  ( C `  i ) ) ^
2 )  =  0 )
1211eqeq2d 2474 . . . . 5  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  ->  ( sum_ i  e.  ( 1 ... N ) ( ( ( A `  i )  -  ( B `  i )
) ^ 2 )  =  sum_ i  e.  ( 1 ... N ) ( ( ( C `
 i )  -  ( C `  i ) ) ^ 2 )  <->  sum_ i  e.  ( 1 ... N ) ( ( ( A `  i )  -  ( B `  i )
) ^ 2 )  =  0 ) )
13 fzfid 12039 . . . . . . . 8  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  -> 
( 1 ... N
)  e.  Fin )
14 fveere 23873 . . . . . . . . . . 11  |-  ( ( A  e.  ( EE
`  N )  /\  i  e.  ( 1 ... N ) )  ->  ( A `  i )  e.  RR )
1514adantlr 714 . . . . . . . . . 10  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  i  e.  ( 1 ... N
) )  ->  ( A `  i )  e.  RR )
16 fveere 23873 . . . . . . . . . . 11  |-  ( ( B  e.  ( EE
`  N )  /\  i  e.  ( 1 ... N ) )  ->  ( B `  i )  e.  RR )
1716adantll 713 . . . . . . . . . 10  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  i  e.  ( 1 ... N
) )  ->  ( B `  i )  e.  RR )
1815, 17resubcld 9976 . . . . . . . . 9  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  i  e.  ( 1 ... N
) )  ->  (
( A `  i
)  -  ( B `
 i ) )  e.  RR )
1918resqcld 12291 . . . . . . . 8  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  i  e.  ( 1 ... N
) )  ->  (
( ( A `  i )  -  ( B `  i )
) ^ 2 )  e.  RR )
2018sqge0d 12292 . . . . . . . 8  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  i  e.  ( 1 ... N
) )  ->  0  <_  ( ( ( A `
 i )  -  ( B `  i ) ) ^ 2 ) )
2113, 19, 20fsum00 13561 . . . . . . 7  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  -> 
( sum_ i  e.  ( 1 ... N ) ( ( ( A `
 i )  -  ( B `  i ) ) ^ 2 )  =  0  <->  A. i  e.  ( 1 ... N
) ( ( ( A `  i )  -  ( B `  i ) ) ^
2 )  =  0 ) )
22 fveecn 23874 . . . . . . . . . 10  |-  ( ( A  e.  ( EE
`  N )  /\  i  e.  ( 1 ... N ) )  ->  ( A `  i )  e.  CC )
23 fveecn 23874 . . . . . . . . . 10  |-  ( ( B  e.  ( EE
`  N )  /\  i  e.  ( 1 ... N ) )  ->  ( B `  i )  e.  CC )
24 subcl 9808 . . . . . . . . . . . 12  |-  ( ( ( A `  i
)  e.  CC  /\  ( B `  i )  e.  CC )  -> 
( ( A `  i )  -  ( B `  i )
)  e.  CC )
25 sqeq0 12187 . . . . . . . . . . . 12  |-  ( ( ( A `  i
)  -  ( B `
 i ) )  e.  CC  ->  (
( ( ( A `
 i )  -  ( B `  i ) ) ^ 2 )  =  0  <->  ( ( A `  i )  -  ( B `  i ) )  =  0 ) )
2624, 25syl 16 . . . . . . . . . . 11  |-  ( ( ( A `  i
)  e.  CC  /\  ( B `  i )  e.  CC )  -> 
( ( ( ( A `  i )  -  ( B `  i ) ) ^
2 )  =  0  <-> 
( ( A `  i )  -  ( B `  i )
)  =  0 ) )
27 subeq0 9834 . . . . . . . . . . 11  |-  ( ( ( A `  i
)  e.  CC  /\  ( B `  i )  e.  CC )  -> 
( ( ( A `
 i )  -  ( B `  i ) )  =  0  <->  ( A `  i )  =  ( B `  i ) ) )
2826, 27bitrd 253 . . . . . . . . . 10  |-  ( ( ( A `  i
)  e.  CC  /\  ( B `  i )  e.  CC )  -> 
( ( ( ( A `  i )  -  ( B `  i ) ) ^
2 )  =  0  <-> 
( A `  i
)  =  ( B `
 i ) ) )
2922, 23, 28syl2an 477 . . . . . . . . 9  |-  ( ( ( A  e.  ( EE `  N )  /\  i  e.  ( 1 ... N ) )  /\  ( B  e.  ( EE `  N )  /\  i  e.  ( 1 ... N
) ) )  -> 
( ( ( ( A `  i )  -  ( B `  i ) ) ^
2 )  =  0  <-> 
( A `  i
)  =  ( B `
 i ) ) )
3029anandirs 828 . . . . . . . 8  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  i  e.  ( 1 ... N
) )  ->  (
( ( ( A `
 i )  -  ( B `  i ) ) ^ 2 )  =  0  <->  ( A `  i )  =  ( B `  i ) ) )
3130ralbidva 2893 . . . . . . 7  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  -> 
( A. i  e.  ( 1 ... N
) ( ( ( A `  i )  -  ( B `  i ) ) ^
2 )  =  0  <->  A. i  e.  (
1 ... N ) ( A `  i )  =  ( B `  i ) ) )
3221, 31bitrd 253 . . . . . 6  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  -> 
( sum_ i  e.  ( 1 ... N ) ( ( ( A `
 i )  -  ( B `  i ) ) ^ 2 )  =  0  <->  A. i  e.  ( 1 ... N
) ( A `  i )  =  ( B `  i ) ) )
33323adant3 1011 . . . . 5  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  ->  ( sum_ i  e.  ( 1 ... N ) ( ( ( A `  i )  -  ( B `  i )
) ^ 2 )  =  0  <->  A. i  e.  ( 1 ... N
) ( A `  i )  =  ( B `  i ) ) )
3412, 33bitrd 253 . . . 4  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  ->  ( sum_ i  e.  ( 1 ... N ) ( ( ( A `  i )  -  ( B `  i )
) ^ 2 )  =  sum_ i  e.  ( 1 ... N ) ( ( ( C `
 i )  -  ( C `  i ) ) ^ 2 )  <->  A. i  e.  (
1 ... N ) ( A `  i )  =  ( B `  i ) ) )
35 simp1 991 . . . . 5  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  ->  A  e.  ( EE `  N
) )
36 simp2 992 . . . . 5  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  ->  B  e.  ( EE `  N
) )
37 simp3 993 . . . . 5  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  ->  C  e.  ( EE `  N
) )
38 brcgr 23872 . . . . 5  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  -> 
( <. A ,  B >.Cgr
<. C ,  C >.  <->  sum_ i  e.  ( 1 ... N ) ( ( ( A `  i )  -  ( B `  i )
) ^ 2 )  =  sum_ i  e.  ( 1 ... N ) ( ( ( C `
 i )  -  ( C `  i ) ) ^ 2 ) ) )
3935, 36, 37, 37, 38syl22anc 1224 . . . 4  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  ->  ( <. A ,  B >.Cgr <. C ,  C >.  <->  sum_ i  e.  ( 1 ... N ) ( ( ( A `  i )  -  ( B `  i )
) ^ 2 )  =  sum_ i  e.  ( 1 ... N ) ( ( ( C `
 i )  -  ( C `  i ) ) ^ 2 ) ) )
40 eqeefv 23875 . . . . 5  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  -> 
( A  =  B  <->  A. i  e.  (
1 ... N ) ( A `  i )  =  ( B `  i ) ) )
41403adant3 1011 . . . 4  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  ->  ( A  =  B  <->  A. i  e.  ( 1 ... N
) ( A `  i )  =  ( B `  i ) ) )
4234, 39, 413bitr4d 285 . . 3  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  ->  ( <. A ,  B >.Cgr <. C ,  C >.  <->  A  =  B ) )
4342biimpd 207 . 2  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  ->  ( <. A ,  B >.Cgr <. C ,  C >.  ->  A  =  B )
)
4443adantl 466 1  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  -> 
( <. A ,  B >.Cgr
<. C ,  C >.  ->  A  =  B )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762   A.wral 2807    C_ wss 3469   <.cop 4026   class class class wbr 4440   ` cfv 5579  (class class class)co 6275   Fincfn 7506   CCcc 9479   RRcr 9480   0cc0 9481   1c1 9482    - cmin 9794   NNcn 10525   2c2 10574   ZZ>=cuz 11071   ...cfz 11661   ^cexp 12122   sum_csu 13457   EEcee 23860  Cgrccgr 23862
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-inf2 8047  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  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-fal 1380  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-se 4832  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-isom 5588  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-1o 7120  df-oadd 7124  df-er 7301  df-map 7412  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-sup 7890  df-oi 7924  df-card 8309  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196  df-nn 10526  df-2 10583  df-3 10584  df-n0 10785  df-z 10854  df-uz 11072  df-rp 11210  df-ico 11524  df-fz 11662  df-fzo 11782  df-seq 12064  df-exp 12123  df-hash 12361  df-cj 12882  df-re 12883  df-im 12884  df-sqr 13018  df-abs 13019  df-clim 13260  df-sum 13458  df-ee 23863  df-cgr 23865
This theorem is referenced by:  eengtrkg  23957  cgrtriv  29215  cgrid2  29216  cgrdegen  29217  segconeq  29223  btwntriv2  29225  btwnconn1lem7  29306  btwnconn1lem11  29310  btwnconn1lem12  29311
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