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Theorem axcgrid 23905
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
StepHypRef Expression
1 fveecn 23891 . . . . . . . . . 10  |-  ( ( C  e.  ( EE
`  N )  /\  i  e.  ( 1 ... N ) )  ->  ( C `  i )  e.  CC )
2 subid 9021 . . . . . . . . . . . 12  |-  ( ( C `  i )  e.  CC  ->  (
( C `  i
)  -  ( C `
 i ) )  =  0 )
32oveq1d 5793 . . . . . . . . . . 11  |-  ( ( C `  i )  e.  CC  ->  (
( ( C `  i )  -  ( C `  i )
) ^ 2 )  =  ( 0 ^ 2 ) )
4 sq0 11147 . . . . . . . . . . 11  |-  ( 0 ^ 2 )  =  0
53, 4syl6eq 2304 . . . . . . . . . 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 12127 . . . . . . . 8  |-  ( C  e.  ( EE `  N )  ->  sum_ i  e.  ( 1 ... N
) ( ( ( C `  i )  -  ( C `  i ) ) ^
2 )  =  sum_ i  e.  ( 1 ... N ) 0 )
8 fzfid 10987 . . . . . . . . 9  |-  ( C  e.  ( EE `  N )  ->  (
1 ... N )  e. 
Fin )
9 sumz 12146 . . . . . . . . . 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 2288 . . . . . . 7  |-  ( C  e.  ( EE `  N )  ->  sum_ i  e.  ( 1 ... N
) ( ( ( C `  i )  -  ( C `  i ) ) ^
2 )  =  0 )
13123ad2ant3 983 . . . . . 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 2267 . . . . 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 10987 . . . . . . . 8  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  -> 
( 1 ... N
)  e.  Fin )
16 fveere 23890 . . . . . . . . . . 11  |-  ( ( A  e.  ( EE
`  N )  /\  i  e.  ( 1 ... N ) )  ->  ( A `  i )  e.  RR )
1716adantlr 698 . . . . . . . . . 10  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  i  e.  ( 1 ... N
) )  ->  ( A `  i )  e.  RR )
18 fveere 23890 . . . . . . . . . . 11  |-  ( ( B  e.  ( EE
`  N )  /\  i  e.  ( 1 ... N ) )  ->  ( B `  i )  e.  RR )
1918adantll 697 . . . . . . . . . 10  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  i  e.  ( 1 ... N
) )  ->  ( B `  i )  e.  RR )
2017, 19resubcld 9165 . . . . . . . . 9  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  i  e.  ( 1 ... N
) )  ->  (
( A `  i
)  -  ( B `
 i ) )  e.  RR )
2120resqcld 11223 . . . . . . . 8  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  i  e.  ( 1 ... N
) )  ->  (
( ( A `  i )  -  ( B `  i )
) ^ 2 )  e.  RR )
2220sqge0d 11224 . . . . . . . 8  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  i  e.  ( 1 ... N
) )  ->  0  <_  ( ( ( A `
 i )  -  ( B `  i ) ) ^ 2 ) )
2315, 21, 22fsum00 12207 . . . . . . 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 23891 . . . . . . . . . 10  |-  ( ( A  e.  ( EE
`  N )  /\  i  e.  ( 1 ... N ) )  ->  ( A `  i )  e.  CC )
25 fveecn 23891 . . . . . . . . . 10  |-  ( ( B  e.  ( EE
`  N )  /\  i  e.  ( 1 ... N ) )  ->  ( B `  i )  e.  CC )
26 subcl 9005 . . . . . . . . . . . 12  |-  ( ( ( A `  i
)  e.  CC  /\  ( B `  i )  e.  CC )  -> 
( ( A `  i )  -  ( B `  i )
)  e.  CC )
27 sqeq0 11120 . . . . . . . . . . . 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 9027 . . . . . . . . . . 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 807 . . . . . . . 8  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  i  e.  ( 1 ... N
) )  ->  (
( ( ( A `
 i )  -  ( B `  i ) ) ^ 2 )  =  0  <->  ( A `  i )  =  ( B `  i ) ) )
3332ralbidva 2532 . . . . . . 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 980 . . . . 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 960 . . . . 5  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  ->  A  e.  ( EE `  N
) )
38 simp2 961 . . . . 5  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  ->  B  e.  ( EE `  N
) )
39 simp3 962 . . . . 5  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  ->  C  e.  ( EE `  N
) )
40 brcgr 23889 . . . . 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 1188 . . . 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 23892 . . . . 5  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  -> 
( A  =  B  <->  A. i  e.  (
1 ... N ) ( A `  i )  =  ( B `  i ) ) )
43423adant3 980 . . . 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 939    = wceq 1619    e. wcel 1621   A.wral 2516    C_ wss 3113   <.cop 3603   class class class wbr 3983   ` cfv 4659  (class class class)co 5778   Fincfn 6817   CCcc 8689   RRcr 8690   0cc0 8691   1c1 8692    - cmin 8991   NNcn 9700   2c2 9749   ZZ>=cuz 10183   ...cfz 10734   ^cexp 11056   sum_csu 12109   EEcee 23877  Cgrccgr 23879
This theorem is referenced by:  cgrtriv  23986  cgrid2  23987  cgrdegen  23988  segconeq  23994  btwntriv2  23996  btwnconn1lem7  24077  btwnconn1lem11  24081  btwnconn1lem12  24082
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4091  ax-sep 4101  ax-nul 4109  ax-pow 4146  ax-pr 4172  ax-un 4470  ax-inf2 7296  ax-cnex 8747  ax-resscn 8748  ax-1cn 8749  ax-icn 8750  ax-addcl 8751  ax-addrcl 8752  ax-mulcl 8753  ax-mulrcl 8754  ax-mulcom 8755  ax-addass 8756  ax-mulass 8757  ax-distr 8758  ax-i2m1 8759  ax-1ne0 8760  ax-1rid 8761  ax-rnegex 8762  ax-rrecex 8763  ax-cnre 8764  ax-pre-lttri 8765  ax-pre-lttrn 8766  ax-pre-ltadd 8767  ax-pre-mulgt0 8768  ax-pre-sup 8769
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2521  df-rex 2522  df-reu 2523  df-rmo 2524  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-pss 3129  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-tp 3608  df-op 3609  df-uni 3788  df-int 3823  df-iun 3867  df-br 3984  df-opab 4038  df-mpt 4039  df-tr 4074  df-eprel 4263  df-id 4267  df-po 4272  df-so 4273  df-fr 4310  df-se 4311  df-we 4312  df-ord 4353  df-on 4354  df-lim 4355  df-suc 4356  df-om 4615  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-isom 4676  df-ov 5781  df-oprab 5782  df-mpt2 5783  df-1st 6042  df-2nd 6043  df-iota 6211  df-riota 6258  df-recs 6342  df-rdg 6377  df-1o 6433  df-oadd 6437  df-er 6614  df-map 6728  df-en 6818  df-dom 6819  df-sdom 6820  df-fin 6821  df-sup 7148  df-oi 7179  df-card 7526  df-pnf 8823  df-mnf 8824  df-xr 8825  df-ltxr 8826  df-le 8827  df-sub 8993  df-neg 8994  df-div 9378  df-n 9701  df-2 9758  df-3 9759  df-n0 9919  df-z 9978  df-uz 10184  df-rp 10308  df-ico 10614  df-fz 10735  df-fzo 10823  df-seq 10999  df-exp 11057  df-hash 11290  df-cj 11535  df-re 11536  df-im 11537  df-sqr 11671  df-abs 11672  df-clim 11913  df-sum 12110  df-ee 23880  df-cgr 23882
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