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Theorem axcgrid 25847
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 25833 . . . . . . . . . 10  |-  ( ( C  e.  ( EE
`  N )  /\  i  e.  ( 1 ... N ) )  ->  ( C `  i )  e.  CC )
2 subid 9313 . . . . . . . . . . 11  |-  ( ( C `  i )  e.  CC  ->  (
( C `  i
)  -  ( C `
 i ) )  =  0 )
32sq0id 11467 . . . . . . . . . 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 12489 . . . . . . . 8  |-  ( C  e.  ( EE `  N )  ->  sum_ i  e.  ( 1 ... N
) ( ( ( C `  i )  -  ( C `  i ) ) ^
2 )  =  sum_ i  e.  ( 1 ... N ) 0 )
6 fzfid 11304 . . . . . . . . 9  |-  ( C  e.  ( EE `  N )  ->  (
1 ... N )  e. 
Fin )
7 sumz 12508 . . . . . . . . . 10  |-  ( ( ( 1 ... N
)  C_  ( ZZ>= ` 
1 )  \/  (
1 ... N )  e. 
Fin )  ->  sum_ i  e.  ( 1 ... N
) 0  =  0 )
87olcs 385 . . . . . . . . 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 2467 . . . . . . 7  |-  ( C  e.  ( EE `  N )  ->  sum_ i  e.  ( 1 ... N
) ( ( ( C `  i )  -  ( C `  i ) ) ^
2 )  =  0 )
11103ad2ant3 980 . . . . . 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 2446 . . . . 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 11304 . . . . . . . 8  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  -> 
( 1 ... N
)  e.  Fin )
14 fveere 25832 . . . . . . . . . . 11  |-  ( ( A  e.  ( EE
`  N )  /\  i  e.  ( 1 ... N ) )  ->  ( A `  i )  e.  RR )
1514adantlr 696 . . . . . . . . . 10  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  i  e.  ( 1 ... N
) )  ->  ( A `  i )  e.  RR )
16 fveere 25832 . . . . . . . . . . 11  |-  ( ( B  e.  ( EE
`  N )  /\  i  e.  ( 1 ... N ) )  ->  ( B `  i )  e.  RR )
1716adantll 695 . . . . . . . . . 10  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  i  e.  ( 1 ... N
) )  ->  ( B `  i )  e.  RR )
1815, 17resubcld 9457 . . . . . . . . 9  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  i  e.  ( 1 ... N
) )  ->  (
( A `  i
)  -  ( B `
 i ) )  e.  RR )
1918resqcld 11541 . . . . . . . 8  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  i  e.  ( 1 ... N
) )  ->  (
( ( A `  i )  -  ( B `  i )
) ^ 2 )  e.  RR )
2018sqge0d 11542 . . . . . . . 8  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  i  e.  ( 1 ... N
) )  ->  0  <_  ( ( ( A `
 i )  -  ( B `  i ) ) ^ 2 ) )
2113, 19, 20fsum00 12569 . . . . . . 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 25833 . . . . . . . . . 10  |-  ( ( A  e.  ( EE
`  N )  /\  i  e.  ( 1 ... N ) )  ->  ( A `  i )  e.  CC )
23 fveecn 25833 . . . . . . . . . 10  |-  ( ( B  e.  ( EE
`  N )  /\  i  e.  ( 1 ... N ) )  ->  ( B `  i )  e.  CC )
24 subcl 9297 . . . . . . . . . . . 12  |-  ( ( ( A `  i
)  e.  CC  /\  ( B `  i )  e.  CC )  -> 
( ( A `  i )  -  ( B `  i )
)  e.  CC )
25 sqeq0 11438 . . . . . . . . . . . 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 9319 . . . . . . . . . . 11  |-  ( ( ( A `  i
)  e.  CC  /\  ( B `  i )  e.  CC )  -> 
( ( ( A `
 i )  -  ( B `  i ) )  =  0  <->  ( A `  i )  =  ( B `  i ) ) )
2826, 27bitrd 245 . . . . . . . . . 10  |-  ( ( ( A `  i
)  e.  CC  /\  ( B `  i )  e.  CC )  -> 
( ( ( ( A `  i )  -  ( B `  i ) ) ^
2 )  =  0  <-> 
( A `  i
)  =  ( B `
 i ) ) )
2922, 23, 28syl2an 464 . . . . . . . . 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 805 . . . . . . . 8  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  i  e.  ( 1 ... N
) )  ->  (
( ( ( A `
 i )  -  ( B `  i ) ) ^ 2 )  =  0  <->  ( A `  i )  =  ( B `  i ) ) )
3130ralbidva 2713 . . . . . . 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 245 . . . . . 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 977 . . . . 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 245 . . . 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 957 . . . . 5  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  ->  A  e.  ( EE `  N
) )
36 simp2 958 . . . . 5  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  ->  B  e.  ( EE `  N
) )
37 simp3 959 . . . . 5  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  ->  C  e.  ( EE `  N
) )
38 brcgr 25831 . . . . 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 1185 . . . 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 25834 . . . . 5  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  -> 
( A  =  B  <->  A. i  e.  (
1 ... N ) ( A `  i )  =  ( B `  i ) ) )
41403adant3 977 . . . 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 277 . . 3  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  ->  ( <. A ,  B >.Cgr <. C ,  C >.  <->  A  =  B ) )
4342biimpd 199 . 2  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  ->  ( <. A ,  B >.Cgr <. C ,  C >.  ->  A  =  B )
)
4443adantl 453 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 set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2697    C_ wss 3312   <.cop 3809   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   Fincfn 7101   CCcc 8980   RRcr 8981   0cc0 8982   1c1 8983    - cmin 9283   NNcn 9992   2c2 10041   ZZ>=cuz 10480   ...cfz 11035   ^cexp 11374   sum_csu 12471   EEcee 25819  Cgrccgr 25821
This theorem is referenced by:  cgrtriv  25928  cgrid2  25929  cgrdegen  25930  segconeq  25936  btwntriv2  25938  btwnconn1lem7  26019  btwnconn1lem11  26023  btwnconn1lem12  26024
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-oi 7471  df-card 7818  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-rp 10605  df-ico 10914  df-fz 11036  df-fzo 11128  df-seq 11316  df-exp 11375  df-hash 11611  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-clim 12274  df-sum 12472  df-ee 25822  df-cgr 25824
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