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Theorem axcgrid 23951
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 )
)
Dummy variable  i is distinct from all other variables.

Proof of Theorem axcgrid
StepHypRef Expression
1 fveecn 23937 . . . . . . . . . 10  |-  ( ( C  e.  ( EE
`  N )  /\  i  e.  ( 1 ... N ) )  ->  ( C `  i )  e.  CC )
2 subid 9062 . . . . . . . . . . . 12  |-  ( ( C `  i )  e.  CC  ->  (
( C `  i
)  -  ( C `
 i ) )  =  0 )
32oveq1d 5834 . . . . . . . . . . 11  |-  ( ( C `  i )  e.  CC  ->  (
( ( C `  i )  -  ( C `  i )
) ^ 2 )  =  ( 0 ^ 2 ) )
4 sq0 11189 . . . . . . . . . . 11  |-  ( 0 ^ 2 )  =  0
53, 4syl6eq 2332 . . . . . . . . . 10  |-  ( ( C `  i )  e.  CC  ->  (
( ( C `  i )  -  ( C `  i )
) ^ 2 )  =  0 )
61, 5syl 17 . . . . . . . . 9  |-  ( ( C  e.  ( EE
`  N )  /\  i  e.  ( 1 ... N ) )  ->  ( ( ( C `  i )  -  ( C `  i ) ) ^
2 )  =  0 )
76sumeq2dv 12170 . . . . . . . 8  |-  ( C  e.  ( EE `  N )  ->  sum_ i  e.  ( 1 ... N
) ( ( ( C `  i )  -  ( C `  i ) ) ^
2 )  =  sum_ i  e.  ( 1 ... N ) 0 )
8 fzfid 11029 . . . . . . . . 9  |-  ( C  e.  ( EE `  N )  ->  (
1 ... N )  e. 
Fin )
9 sumz 12189 . . . . . . . . . 10  |-  ( ( ( 1 ... N
)  C_  ( ZZ>= ` 
1 )  \/  (
1 ... N )  e. 
Fin )  ->  sum_ i  e.  ( 1 ... N
) 0  =  0 )
109olcs 386 . . . . . . . . 9  |-  ( ( 1 ... N )  e.  Fin  ->  sum_ i  e.  ( 1 ... N
) 0  =  0 )
118, 10syl 17 . . . . . . . 8  |-  ( C  e.  ( EE `  N )  ->  sum_ i  e.  ( 1 ... N
) 0  =  0 )
127, 11eqtrd 2316 . . . . . . 7  |-  ( C  e.  ( EE `  N )  ->  sum_ i  e.  ( 1 ... N
) ( ( ( C `  i )  -  ( C `  i ) ) ^
2 )  =  0 )
13123ad2ant3 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 )
1413eqeq2d 2295 . . . . 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 ) )
15 fzfid 11029 . . . . . . . 8  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  -> 
( 1 ... N
)  e.  Fin )
16 fveere 23936 . . . . . . . . . . 11  |-  ( ( A  e.  ( EE
`  N )  /\  i  e.  ( 1 ... N ) )  ->  ( A `  i )  e.  RR )
1716adantlr 697 . . . . . . . . . 10  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  i  e.  ( 1 ... N
) )  ->  ( A `  i )  e.  RR )
18 fveere 23936 . . . . . . . . . . 11  |-  ( ( B  e.  ( EE
`  N )  /\  i  e.  ( 1 ... N ) )  ->  ( B `  i )  e.  RR )
1918adantll 696 . . . . . . . . . 10  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  i  e.  ( 1 ... N
) )  ->  ( B `  i )  e.  RR )
2017, 19resubcld 9206 . . . . . . . . 9  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  i  e.  ( 1 ... N
) )  ->  (
( A `  i
)  -  ( B `
 i ) )  e.  RR )
2120resqcld 11265 . . . . . . . 8  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  i  e.  ( 1 ... N
) )  ->  (
( ( A `  i )  -  ( B `  i )
) ^ 2 )  e.  RR )
2220sqge0d 11266 . . . . . . . 8  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  i  e.  ( 1 ... N
) )  ->  0  <_  ( ( ( A `
 i )  -  ( B `  i ) ) ^ 2 ) )
2315, 21, 22fsum00 12250 . . . . . . 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 ) )
24 fveecn 23937 . . . . . . . . . 10  |-  ( ( A  e.  ( EE
`  N )  /\  i  e.  ( 1 ... N ) )  ->  ( A `  i )  e.  CC )
25 fveecn 23937 . . . . . . . . . 10  |-  ( ( B  e.  ( EE
`  N )  /\  i  e.  ( 1 ... N ) )  ->  ( B `  i )  e.  CC )
26 subcl 9046 . . . . . . . . . . . 12  |-  ( ( ( A `  i
)  e.  CC  /\  ( B `  i )  e.  CC )  -> 
( ( A `  i )  -  ( B `  i )
)  e.  CC )
27 sqeq0 11162 . . . . . . . . . . . 12  |-  ( ( ( A `  i
)  -  ( B `
 i ) )  e.  CC  ->  (
( ( ( A `
 i )  -  ( B `  i ) ) ^ 2 )  =  0  <->  ( ( A `  i )  -  ( B `  i ) )  =  0 ) )
2826, 27syl 17 . . . . . . . . . . 11  |-  ( ( ( A `  i
)  e.  CC  /\  ( B `  i )  e.  CC )  -> 
( ( ( ( A `  i )  -  ( B `  i ) ) ^
2 )  =  0  <-> 
( ( A `  i )  -  ( B `  i )
)  =  0 ) )
29 subeq0 9068 . . . . . . . . . . 11  |-  ( ( ( A `  i
)  e.  CC  /\  ( B `  i )  e.  CC )  -> 
( ( ( A `
 i )  -  ( B `  i ) )  =  0  <->  ( A `  i )  =  ( B `  i ) ) )
3028, 29bitrd 246 . . . . . . . . . 10  |-  ( ( ( A `  i
)  e.  CC  /\  ( B `  i )  e.  CC )  -> 
( ( ( ( A `  i )  -  ( B `  i ) ) ^
2 )  =  0  <-> 
( A `  i
)  =  ( B `
 i ) ) )
3124, 25, 30syl2an 465 . . . . . . . . 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 ) ) )
3231anandirs 806 . . . . . . . 8  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  i  e.  ( 1 ... N
) )  ->  (
( ( ( A `
 i )  -  ( B `  i ) ) ^ 2 )  =  0  <->  ( A `  i )  =  ( B `  i ) ) )
3332ralbidva 2560 . . . . . . 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 ) ) )
3423, 33bitrd 246 . . . . . 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 ) ) )
35343adant3 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 ) ) )
3614, 35bitrd 246 . . . 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 ) ) )
37 simp1 957 . . . . 5  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  ->  A  e.  ( EE `  N
) )
38 simp2 958 . . . . 5  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  ->  B  e.  ( EE `  N
) )
39 simp3 959 . . . . 5  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  ->  C  e.  ( EE `  N
) )
40 brcgr 23935 . . . . 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 ) ) )
4137, 38, 39, 39, 40syl22anc 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 ) ) )
42 eqeefv 23938 . . . . 5  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  -> 
( A  =  B  <->  A. i  e.  (
1 ... N ) ( A `  i )  =  ( B `  i ) ) )
43423adant3 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 ) ) )
4436, 41, 433bitr4d 278 . . 3  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  ->  ( <. A ,  B >.Cgr <. C ,  C >.  <->  A  =  B ) )
4544biimpd 200 . 2  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  ->  ( <. A ,  B >.Cgr <. C ,  C >.  ->  A  =  B )
)
4645adantl 454 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 6    <-> wb 178    /\ wa 360    /\ w3a 936    = wceq 1624    e. wcel 1685   A.wral 2544    C_ wss 3153   <.cop 3644   class class class wbr 4024   ` cfv 5221  (class class class)co 5819   Fincfn 6858   CCcc 8730   RRcr 8731   0cc0 8732   1c1 8733    - cmin 9032   NNcn 9741   2c2 9790   ZZ>=cuz 10225   ...cfz 10776   ^cexp 11098   sum_csu 12152   EEcee 23923  Cgrccgr 23925
This theorem is referenced by:  cgrtriv  24032  cgrid2  24033  cgrdegen  24034  segconeq  24040  btwntriv2  24042  btwnconn1lem7  24123  btwnconn1lem11  24127  btwnconn1lem12  24128
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-inf2 7337  ax-cnex 8788  ax-resscn 8789  ax-1cn 8790  ax-icn 8791  ax-addcl 8792  ax-addrcl 8793  ax-mulcl 8794  ax-mulrcl 8795  ax-mulcom 8796  ax-addass 8797  ax-mulass 8798  ax-distr 8799  ax-i2m1 8800  ax-1ne0 8801  ax-1rid 8802  ax-rnegex 8803  ax-rrecex 8804  ax-cnre 8805  ax-pre-lttri 8806  ax-pre-lttrn 8807  ax-pre-ltadd 8808  ax-pre-mulgt0 8809  ax-pre-sup 8810
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-int 3864  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-se 4352  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-isom 5230  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-1st 6083  df-2nd 6084  df-iota 6252  df-riota 6299  df-recs 6383  df-rdg 6418  df-1o 6474  df-oadd 6478  df-er 6655  df-map 6769  df-en 6859  df-dom 6860  df-sdom 6861  df-fin 6862  df-sup 7189  df-oi 7220  df-card 7567  df-pnf 8864  df-mnf 8865  df-xr 8866  df-ltxr 8867  df-le 8868  df-sub 9034  df-neg 9035  df-div 9419  df-nn 9742  df-2 9799  df-3 9800  df-n0 9961  df-z 10020  df-uz 10226  df-rp 10350  df-ico 10656  df-fz 10777  df-fzo 10865  df-seq 11041  df-exp 11099  df-hash 11332  df-cj 11578  df-re 11579  df-im 11580  df-sqr 11714  df-abs 11715  df-clim 11956  df-sum 12153  df-ee 23926  df-cgr 23928
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