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Theorem axcgrrflx 24544
Description:  A is as far from  B as  B is from  A. Axiom A1 of [Schwabhauser] p. 10. (Contributed by Scott Fenton, 3-Jun-2013.)
Assertion
Ref Expression
axcgrrflx  |-  ( ( N  e.  NN  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  ->  <. A ,  B >.Cgr <. B ,  A >. )

Proof of Theorem axcgrrflx
Dummy variable  i is distinct from all other variables.
StepHypRef Expression
1 fveecn 24532 . . . . . 6  |-  ( ( A  e.  ( EE
`  N )  /\  i  e.  ( 1 ... N ) )  ->  ( A `  i )  e.  CC )
2 fveecn 24532 . . . . . 6  |-  ( ( B  e.  ( EE
`  N )  /\  i  e.  ( 1 ... N ) )  ->  ( B `  i )  e.  CC )
3 sqsubswap 11167 . . . . . 6  |-  ( ( ( A `  i
)  e.  CC  /\  ( B `  i )  e.  CC )  -> 
( ( ( A `
 i )  -  ( B `  i ) ) ^ 2 )  =  ( ( ( B `  i )  -  ( A `  i ) ) ^
2 ) )
41, 2, 3syl2an 463 . . . . 5  |-  ( ( ( A  e.  ( EE `  N )  /\  i  e.  ( 1 ... N ) )  /\  ( B  e.  ( EE `  N )  /\  i  e.  ( 1 ... N
) ) )  -> 
( ( ( A `
 i )  -  ( B `  i ) ) ^ 2 )  =  ( ( ( B `  i )  -  ( A `  i ) ) ^
2 ) )
54anandirs 804 . . . 4  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  i  e.  ( 1 ... N
) )  ->  (
( ( A `  i )  -  ( B `  i )
) ^ 2 )  =  ( ( ( B `  i )  -  ( A `  i ) ) ^
2 ) )
65sumeq2dv 12178 . . 3  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  ->  sum_ i  e.  ( 1 ... N ) ( ( ( A `  i )  -  ( B `  i )
) ^ 2 )  =  sum_ i  e.  ( 1 ... N ) ( ( ( B `
 i )  -  ( A `  i ) ) ^ 2 ) )
7 id 19 . . . 4  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  -> 
( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) )
8 simpr 447 . . . 4  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  ->  B  e.  ( EE `  N ) )
9 simpl 443 . . . 4  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  ->  A  e.  ( EE `  N ) )
10 brcgr 24530 . . . 4  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( B  e.  ( EE `  N )  /\  A  e.  ( EE `  N
) ) )  -> 
( <. A ,  B >.Cgr
<. B ,  A >.  <->  sum_ i  e.  ( 1 ... N ) ( ( ( A `  i )  -  ( B `  i )
) ^ 2 )  =  sum_ i  e.  ( 1 ... N ) ( ( ( B `
 i )  -  ( A `  i ) ) ^ 2 ) ) )
117, 8, 9, 10syl12anc 1180 . . 3  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  -> 
( <. A ,  B >.Cgr
<. B ,  A >.  <->  sum_ i  e.  ( 1 ... N ) ( ( ( A `  i )  -  ( B `  i )
) ^ 2 )  =  sum_ i  e.  ( 1 ... N ) ( ( ( B `
 i )  -  ( A `  i ) ) ^ 2 ) ) )
126, 11mpbird 223 . 2  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  ->  <. A ,  B >.Cgr <. B ,  A >. )
13123adant1 973 1  |-  ( ( N  e.  NN  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  ->  <. A ,  B >.Cgr <. B ,  A >. )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1625    e. wcel 1686   <.cop 3645   class class class wbr 4025   ` cfv 5257  (class class class)co 5860   CCcc 8737   1c1 8740    - cmin 9039   NNcn 9748   2c2 9797   ...cfz 10784   ^cexp 11106   sum_csu 12160   EEcee 24518  Cgrccgr 24520
This theorem is referenced by:  cgrrflx2d  24609  cgrrflx  24612  endofsegid  24710
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-cnex 8795  ax-resscn 8796  ax-1cn 8797  ax-icn 8798  ax-addcl 8799  ax-addrcl 8800  ax-mulcl 8801  ax-mulrcl 8802  ax-mulcom 8803  ax-addass 8804  ax-mulass 8805  ax-distr 8806  ax-i2m1 8807  ax-1ne0 8808  ax-1rid 8809  ax-rnegex 8810  ax-rrecex 8811  ax-cnre 8812  ax-pre-lttri 8813  ax-pre-lttrn 8814  ax-pre-ltadd 8815  ax-pre-mulgt0 8816
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-1st 6124  df-2nd 6125  df-riota 6306  df-recs 6390  df-rdg 6425  df-er 6662  df-map 6776  df-en 6866  df-dom 6867  df-sdom 6868  df-pnf 8871  df-mnf 8872  df-xr 8873  df-ltxr 8874  df-le 8875  df-sub 9041  df-neg 9042  df-nn 9749  df-2 9806  df-n0 9968  df-z 10027  df-uz 10233  df-fz 10785  df-seq 11049  df-exp 11107  df-sum 12161  df-ee 24521  df-cgr 24523
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