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Theorem axcgrid 25103
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 25089 . . . . . . . . . 10  |-  ( ( C  e.  ( EE
`  N )  /\  i  e.  ( 1 ... N ) )  ->  ( C `  i )  e.  CC )
2 subid 9157 . . . . . . . . . . . 12  |-  ( ( C `  i )  e.  CC  ->  (
( C `  i
)  -  ( C `
 i ) )  =  0 )
32oveq1d 5960 . . . . . . . . . . 11  |-  ( ( C `  i )  e.  CC  ->  (
( ( C `  i )  -  ( C `  i )
) ^ 2 )  =  ( 0 ^ 2 ) )
4 sq0 11288 . . . . . . . . . . 11  |-  ( 0 ^ 2 )  =  0
53, 4syl6eq 2406 . . . . . . . . . 10  |-  ( ( C `  i )  e.  CC  ->  (
( ( C `  i )  -  ( C `  i )
) ^ 2 )  =  0 )
61, 5syl 15 . . . . . . . . 9  |-  ( ( C  e.  ( EE
`  N )  /\  i  e.  ( 1 ... N ) )  ->  ( ( ( C `  i )  -  ( C `  i ) ) ^
2 )  =  0 )
76sumeq2dv 12273 . . . . . . . 8  |-  ( C  e.  ( EE `  N )  ->  sum_ i  e.  ( 1 ... N
) ( ( ( C `  i )  -  ( C `  i ) ) ^
2 )  =  sum_ i  e.  ( 1 ... N ) 0 )
8 fzfid 11127 . . . . . . . . 9  |-  ( C  e.  ( EE `  N )  ->  (
1 ... N )  e. 
Fin )
9 sumz 12292 . . . . . . . . . 10  |-  ( ( ( 1 ... N
)  C_  ( ZZ>= ` 
1 )  \/  (
1 ... N )  e. 
Fin )  ->  sum_ i  e.  ( 1 ... N
) 0  =  0 )
109olcs 384 . . . . . . . . 9  |-  ( ( 1 ... N )  e.  Fin  ->  sum_ i  e.  ( 1 ... N
) 0  =  0 )
118, 10syl 15 . . . . . . . 8  |-  ( C  e.  ( EE `  N )  ->  sum_ i  e.  ( 1 ... N
) 0  =  0 )
127, 11eqtrd 2390 . . . . . . 7  |-  ( C  e.  ( EE `  N )  ->  sum_ i  e.  ( 1 ... N
) ( ( ( C `  i )  -  ( C `  i ) ) ^
2 )  =  0 )
13123ad2ant3 978 . . . . . 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 2369 . . . . 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 11127 . . . . . . . 8  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  -> 
( 1 ... N
)  e.  Fin )
16 fveere 25088 . . . . . . . . . . 11  |-  ( ( A  e.  ( EE
`  N )  /\  i  e.  ( 1 ... N ) )  ->  ( A `  i )  e.  RR )
1716adantlr 695 . . . . . . . . . 10  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  i  e.  ( 1 ... N
) )  ->  ( A `  i )  e.  RR )
18 fveere 25088 . . . . . . . . . . 11  |-  ( ( B  e.  ( EE
`  N )  /\  i  e.  ( 1 ... N ) )  ->  ( B `  i )  e.  RR )
1918adantll 694 . . . . . . . . . 10  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  i  e.  ( 1 ... N
) )  ->  ( B `  i )  e.  RR )
2017, 19resubcld 9301 . . . . . . . . 9  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  i  e.  ( 1 ... N
) )  ->  (
( A `  i
)  -  ( B `
 i ) )  e.  RR )
2120resqcld 11364 . . . . . . . 8  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  i  e.  ( 1 ... N
) )  ->  (
( ( A `  i )  -  ( B `  i )
) ^ 2 )  e.  RR )
2220sqge0d 11365 . . . . . . . 8  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  i  e.  ( 1 ... N
) )  ->  0  <_  ( ( ( A `
 i )  -  ( B `  i ) ) ^ 2 ) )
2315, 21, 22fsum00 12353 . . . . . . 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 25089 . . . . . . . . . 10  |-  ( ( A  e.  ( EE
`  N )  /\  i  e.  ( 1 ... N ) )  ->  ( A `  i )  e.  CC )
25 fveecn 25089 . . . . . . . . . 10  |-  ( ( B  e.  ( EE
`  N )  /\  i  e.  ( 1 ... N ) )  ->  ( B `  i )  e.  CC )
26 subcl 9141 . . . . . . . . . . . 12  |-  ( ( ( A `  i
)  e.  CC  /\  ( B `  i )  e.  CC )  -> 
( ( A `  i )  -  ( B `  i )
)  e.  CC )
27 sqeq0 11261 . . . . . . . . . . . 12  |-  ( ( ( A `  i
)  -  ( B `
 i ) )  e.  CC  ->  (
( ( ( A `
 i )  -  ( B `  i ) ) ^ 2 )  =  0  <->  ( ( A `  i )  -  ( B `  i ) )  =  0 ) )
2826, 27syl 15 . . . . . . . . . . 11  |-  ( ( ( A `  i
)  e.  CC  /\  ( B `  i )  e.  CC )  -> 
( ( ( ( A `  i )  -  ( B `  i ) ) ^
2 )  =  0  <-> 
( ( A `  i )  -  ( B `  i )
)  =  0 ) )
29 subeq0 9163 . . . . . . . . . . 11  |-  ( ( ( A `  i
)  e.  CC  /\  ( B `  i )  e.  CC )  -> 
( ( ( A `
 i )  -  ( B `  i ) )  =  0  <->  ( A `  i )  =  ( B `  i ) ) )
3028, 29bitrd 244 . . . . . . . . . 10  |-  ( ( ( A `  i
)  e.  CC  /\  ( B `  i )  e.  CC )  -> 
( ( ( ( A `  i )  -  ( B `  i ) ) ^
2 )  =  0  <-> 
( A `  i
)  =  ( B `
 i ) ) )
3124, 25, 30syl2an 463 . . . . . . . . 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 804 . . . . . . . 8  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  i  e.  ( 1 ... N
) )  ->  (
( ( ( A `
 i )  -  ( B `  i ) ) ^ 2 )  =  0  <->  ( A `  i )  =  ( B `  i ) ) )
3332ralbidva 2635 . . . . . . 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 244 . . . . . 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 975 . . . . 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 244 . . . 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 955 . . . . 5  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  ->  A  e.  ( EE `  N
) )
38 simp2 956 . . . . 5  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  ->  B  e.  ( EE `  N
) )
39 simp3 957 . . . . 5  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  ->  C  e.  ( EE `  N
) )
40 brcgr 25087 . . . . 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 1183 . . . 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 25090 . . . . 5  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  -> 
( A  =  B  <->  A. i  e.  (
1 ... N ) ( A `  i )  =  ( B `  i ) ) )
43423adant3 975 . . . 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 276 . . 3  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  ->  ( <. A ,  B >.Cgr <. C ,  C >.  <->  A  =  B ) )
4544biimpd 198 . 2  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  ->  ( <. A ,  B >.Cgr <. C ,  C >.  ->  A  =  B )
)
4645adantl 452 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 176    /\ wa 358    /\ w3a 934    = wceq 1642    e. wcel 1710   A.wral 2619    C_ wss 3228   <.cop 3719   class class class wbr 4104   ` cfv 5337  (class class class)co 5945   Fincfn 6951   CCcc 8825   RRcr 8826   0cc0 8827   1c1 8828    - cmin 9127   NNcn 9836   2c2 9885   ZZ>=cuz 10322   ...cfz 10874   ^cexp 11197   sum_csu 12255   EEcee 25075  Cgrccgr 25077
This theorem is referenced by:  cgrtriv  25184  cgrid2  25185  cgrdegen  25186  segconeq  25192  btwntriv2  25194  btwnconn1lem7  25275  btwnconn1lem11  25279  btwnconn1lem12  25280
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-inf2 7432  ax-cnex 8883  ax-resscn 8884  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-addrcl 8888  ax-mulcl 8889  ax-mulrcl 8890  ax-mulcom 8891  ax-addass 8892  ax-mulass 8893  ax-distr 8894  ax-i2m1 8895  ax-1ne0 8896  ax-1rid 8897  ax-rnegex 8898  ax-rrecex 8899  ax-cnre 8900  ax-pre-lttri 8901  ax-pre-lttrn 8902  ax-pre-ltadd 8903  ax-pre-mulgt0 8904  ax-pre-sup 8905
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-int 3944  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-se 4435  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-isom 5346  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-1st 6209  df-2nd 6210  df-riota 6391  df-recs 6475  df-rdg 6510  df-1o 6566  df-oadd 6570  df-er 6747  df-map 6862  df-en 6952  df-dom 6953  df-sdom 6954  df-fin 6955  df-sup 7284  df-oi 7315  df-card 7662  df-pnf 8959  df-mnf 8960  df-xr 8961  df-ltxr 8962  df-le 8963  df-sub 9129  df-neg 9130  df-div 9514  df-nn 9837  df-2 9894  df-3 9895  df-n0 10058  df-z 10117  df-uz 10323  df-rp 10447  df-ico 10754  df-fz 10875  df-fzo 10963  df-seq 11139  df-exp 11198  df-hash 11431  df-cj 11680  df-re 11681  df-im 11682  df-sqr 11816  df-abs 11817  df-clim 12058  df-sum 12256  df-ee 25078  df-cgr 25080
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