Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  axcgrrflx Structured version   Unicode version

Theorem axcgrrflx 25846
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 25834 . . . . . 6  |-  ( ( A  e.  ( EE
`  N )  /\  i  e.  ( 1 ... N ) )  ->  ( A `  i )  e.  CC )
2 fveecn 25834 . . . . . 6  |-  ( ( B  e.  ( EE
`  N )  /\  i  e.  ( 1 ... N ) )  ->  ( B `  i )  e.  CC )
3 sqsubswap 11436 . . . . . 6  |-  ( ( ( A `  i
)  e.  CC  /\  ( B `  i )  e.  CC )  -> 
( ( ( A `
 i )  -  ( B `  i ) ) ^ 2 )  =  ( ( ( B `  i )  -  ( A `  i ) ) ^
2 ) )
41, 2, 3syl2an 464 . . . . 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 805 . . . 4  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  i  e.  ( 1 ... N
) )  ->  (
( ( A `  i )  -  ( B `  i )
) ^ 2 )  =  ( ( ( B `  i )  -  ( A `  i ) ) ^
2 ) )
65sumeq2dv 12490 . . 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 20 . . . 4  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  -> 
( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) )
8 simpr 448 . . . 4  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  ->  B  e.  ( EE `  N ) )
9 simpl 444 . . . 4  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  ->  A  e.  ( EE `  N ) )
10 brcgr 25832 . . . 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 1182 . . 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 224 . 2  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  ->  <. A ,  B >.Cgr <. B ,  A >. )
13123adant1 975 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 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   <.cop 3810   class class class wbr 4205   ` cfv 5447  (class class class)co 6074   CCcc 8981   1c1 8984    - cmin 9284   NNcn 9993   2c2 10042   ...cfz 11036   ^cexp 11375   sum_csu 12472   EEcee 25820  Cgrccgr 25822
This theorem is referenced by:  cgrrflx2d  25911  cgrrflx  25914  endofsegid  26012
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 2417  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396  ax-un 4694  ax-cnex 9039  ax-resscn 9040  ax-1cn 9041  ax-icn 9042  ax-addcl 9043  ax-addrcl 9044  ax-mulcl 9045  ax-mulrcl 9046  ax-mulcom 9047  ax-addass 9048  ax-mulass 9049  ax-distr 9050  ax-i2m1 9051  ax-1ne0 9052  ax-1rid 9053  ax-rnegex 9054  ax-rrecex 9055  ax-cnre 9056  ax-pre-lttri 9057  ax-pre-lttrn 9058  ax-pre-ltadd 9059  ax-pre-mulgt0 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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2703  df-rex 2704  df-reu 2705  df-rab 2707  df-v 2951  df-sbc 3155  df-csb 3245  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-pss 3329  df-nul 3622  df-if 3733  df-pw 3794  df-sn 3813  df-pr 3814  df-tp 3815  df-op 3816  df-uni 4009  df-iun 4088  df-br 4206  df-opab 4260  df-mpt 4261  df-tr 4296  df-eprel 4487  df-id 4491  df-po 4496  df-so 4497  df-fr 4534  df-we 4536  df-ord 4577  df-on 4578  df-lim 4579  df-suc 4580  df-om 4839  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-res 4883  df-ima 4884  df-iota 5411  df-fun 5449  df-fn 5450  df-f 5451  df-f1 5452  df-fo 5453  df-f1o 5454  df-fv 5455  df-ov 6077  df-oprab 6078  df-mpt2 6079  df-1st 6342  df-2nd 6343  df-riota 6542  df-recs 6626  df-rdg 6661  df-er 6898  df-map 7013  df-en 7103  df-dom 7104  df-sdom 7105  df-pnf 9115  df-mnf 9116  df-xr 9117  df-ltxr 9118  df-le 9119  df-sub 9286  df-neg 9287  df-nn 9994  df-2 10051  df-n0 10215  df-z 10276  df-uz 10482  df-fz 11037  df-seq 11317  df-exp 11376  df-sum 12473  df-ee 25823  df-cgr 25825
  Copyright terms: Public domain W3C validator