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Theorem axbtwnid 25870
Description: Points are indivisible. That is, if  A lies between  B and  B, then  A  =  B. Axiom A6 of [Schwabhauser] p. 11. (Contributed by Scott Fenton, 3-Jun-2013.)
Assertion
Ref Expression
axbtwnid  |-  ( ( N  e.  NN  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  ->  ( A  Btwn  <. B ,  B >.  ->  A  =  B ) )

Proof of Theorem axbtwnid
Dummy variables  t 
i are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp2 958 . . 3  |-  ( ( N  e.  NN  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  ->  A  e.  ( EE `  N
) )
2 simp3 959 . . 3  |-  ( ( N  e.  NN  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  ->  B  e.  ( EE `  N
) )
3 brbtwn 25830 . . 3  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  ->  ( A  Btwn  <. B ,  B >.  <->  E. t  e.  (
0 [,] 1 ) A. i  e.  ( 1 ... N ) ( A `  i
)  =  ( ( ( 1  -  t
)  x.  ( B `
 i ) )  +  ( t  x.  ( B `  i
) ) ) ) )
41, 2, 2, 3syl3anc 1184 . 2  |-  ( ( N  e.  NN  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  ->  ( A  Btwn  <. B ,  B >.  <->  E. t  e.  (
0 [,] 1 ) A. i  e.  ( 1 ... N ) ( A `  i
)  =  ( ( ( 1  -  t
)  x.  ( B `
 i ) )  +  ( t  x.  ( B `  i
) ) ) ) )
5 0re 9083 . . . . . . 7  |-  0  e.  RR
6 1re 9082 . . . . . . 7  |-  1  e.  RR
75, 6elicc2i 10968 . . . . . 6  |-  ( t  e.  ( 0 [,] 1 )  <->  ( t  e.  RR  /\  0  <_ 
t  /\  t  <_  1 ) )
87simp1bi 972 . . . . 5  |-  ( t  e.  ( 0 [,] 1 )  ->  t  e.  RR )
98recnd 9106 . . . 4  |-  ( t  e.  ( 0 [,] 1 )  ->  t  e.  CC )
10 eqeefv 25834 . . . . . . . 8  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  -> 
( A  =  B  <->  A. i  e.  (
1 ... N ) ( A `  i )  =  ( B `  i ) ) )
11103adant1 975 . . . . . . 7  |-  ( ( N  e.  NN  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  ->  ( A  =  B  <->  A. i  e.  ( 1 ... N
) ( A `  i )  =  ( B `  i ) ) )
1211adantr 452 . . . . . 6  |-  ( ( ( N  e.  NN  /\  A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  /\  t  e.  CC )  ->  ( A  =  B  <->  A. i  e.  (
1 ... N ) ( A `  i )  =  ( B `  i ) ) )
13 ax-1cn 9040 . . . . . . . . . . . 12  |-  1  e.  CC
14 npcan 9306 . . . . . . . . . . . 12  |-  ( ( 1  e.  CC  /\  t  e.  CC )  ->  ( ( 1  -  t )  +  t )  =  1 )
1513, 14mpan 652 . . . . . . . . . . 11  |-  ( t  e.  CC  ->  (
( 1  -  t
)  +  t )  =  1 )
1615ad2antlr 708 . . . . . . . . . 10  |-  ( ( ( ( N  e.  NN  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) )  /\  t  e.  CC )  /\  i  e.  ( 1 ... N
) )  ->  (
( 1  -  t
)  +  t )  =  1 )
1716oveq1d 6088 . . . . . . . . 9  |-  ( ( ( ( N  e.  NN  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) )  /\  t  e.  CC )  /\  i  e.  ( 1 ... N
) )  ->  (
( ( 1  -  t )  +  t )  x.  ( B `
 i ) )  =  ( 1  x.  ( B `  i
) ) )
18 subcl 9297 . . . . . . . . . . . 12  |-  ( ( 1  e.  CC  /\  t  e.  CC )  ->  ( 1  -  t
)  e.  CC )
1913, 18mpan 652 . . . . . . . . . . 11  |-  ( t  e.  CC  ->  (
1  -  t )  e.  CC )
2019ad2antlr 708 . . . . . . . . . 10  |-  ( ( ( ( N  e.  NN  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) )  /\  t  e.  CC )  /\  i  e.  ( 1 ... N
) )  ->  (
1  -  t )  e.  CC )
21 simplr 732 . . . . . . . . . 10  |-  ( ( ( ( N  e.  NN  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) )  /\  t  e.  CC )  /\  i  e.  ( 1 ... N
) )  ->  t  e.  CC )
22 simpll3 998 . . . . . . . . . . 11  |-  ( ( ( ( N  e.  NN  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) )  /\  t  e.  CC )  /\  i  e.  ( 1 ... N
) )  ->  B  e.  ( EE `  N
) )
23 fveecn 25833 . . . . . . . . . . 11  |-  ( ( B  e.  ( EE
`  N )  /\  i  e.  ( 1 ... N ) )  ->  ( B `  i )  e.  CC )
2422, 23sylancom 649 . . . . . . . . . 10  |-  ( ( ( ( N  e.  NN  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) )  /\  t  e.  CC )  /\  i  e.  ( 1 ... N
) )  ->  ( B `  i )  e.  CC )
2520, 21, 24adddird 9105 . . . . . . . . 9  |-  ( ( ( ( N  e.  NN  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) )  /\  t  e.  CC )  /\  i  e.  ( 1 ... N
) )  ->  (
( ( 1  -  t )  +  t )  x.  ( B `
 i ) )  =  ( ( ( 1  -  t )  x.  ( B `  i ) )  +  ( t  x.  ( B `  i )
) ) )
2624mulid2d 9098 . . . . . . . . 9  |-  ( ( ( ( N  e.  NN  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) )  /\  t  e.  CC )  /\  i  e.  ( 1 ... N
) )  ->  (
1  x.  ( B `
 i ) )  =  ( B `  i ) )
2717, 25, 263eqtr3rd 2476 . . . . . . . 8  |-  ( ( ( ( N  e.  NN  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) )  /\  t  e.  CC )  /\  i  e.  ( 1 ... N
) )  ->  ( B `  i )  =  ( ( ( 1  -  t )  x.  ( B `  i ) )  +  ( t  x.  ( B `  i )
) ) )
2827eqeq2d 2446 . . . . . . 7  |-  ( ( ( ( N  e.  NN  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) )  /\  t  e.  CC )  /\  i  e.  ( 1 ... N
) )  ->  (
( A `  i
)  =  ( B `
 i )  <->  ( A `  i )  =  ( ( ( 1  -  t )  x.  ( B `  i )
)  +  ( t  x.  ( B `  i ) ) ) ) )
2928ralbidva 2713 . . . . . 6  |-  ( ( ( N  e.  NN  /\  A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  /\  t  e.  CC )  ->  ( A. i  e.  ( 1 ... N
) ( A `  i )  =  ( B `  i )  <->  A. i  e.  (
1 ... N ) ( A `  i )  =  ( ( ( 1  -  t )  x.  ( B `  i ) )  +  ( t  x.  ( B `  i )
) ) ) )
3012, 29bitrd 245 . . . . 5  |-  ( ( ( N  e.  NN  /\  A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  /\  t  e.  CC )  ->  ( A  =  B  <->  A. i  e.  (
1 ... N ) ( A `  i )  =  ( ( ( 1  -  t )  x.  ( B `  i ) )  +  ( t  x.  ( B `  i )
) ) ) )
3130biimprd 215 . . . 4  |-  ( ( ( N  e.  NN  /\  A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  /\  t  e.  CC )  ->  ( A. i  e.  ( 1 ... N
) ( A `  i )  =  ( ( ( 1  -  t )  x.  ( B `  i )
)  +  ( t  x.  ( B `  i ) ) )  ->  A  =  B ) )
329, 31sylan2 461 . . 3  |-  ( ( ( N  e.  NN  /\  A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  /\  t  e.  ( 0 [,] 1 ) )  ->  ( A. i  e.  ( 1 ... N
) ( A `  i )  =  ( ( ( 1  -  t )  x.  ( B `  i )
)  +  ( t  x.  ( B `  i ) ) )  ->  A  =  B ) )
3332rexlimdva 2822 . 2  |-  ( ( N  e.  NN  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  ->  ( E. t  e.  (
0 [,] 1 ) A. i  e.  ( 1 ... N ) ( A `  i
)  =  ( ( ( 1  -  t
)  x.  ( B `
 i ) )  +  ( t  x.  ( B `  i
) ) )  ->  A  =  B )
)
344, 33sylbid 207 1  |-  ( ( N  e.  NN  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  ->  ( A  Btwn  <. B ,  B >.  ->  A  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2697   E.wrex 2698   <.cop 3809   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   CCcc 8980   RRcr 8981   0cc0 8982   1c1 8983    + caddc 8985    x. cmul 8987    <_ cle 9113    - cmin 9283   NNcn 9992   [,]cicc 10911   ...cfz 11035   EEcee 25819    Btwn cbtwn 25820
This theorem is referenced by:  btwncomim  25939  btwnswapid  25943  btwnintr  25945  btwnexch3  25946  ifscgr  25970  idinside  26010  btwnconn1lem12  26024  outsideofrflx  26053
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 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-z 10275  df-uz 10481  df-icc 10915  df-fz 11036  df-ee 25822  df-btwn 25823
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