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Theorem axbtwnid 24567
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 956 . . 3  |-  ( ( N  e.  NN  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  ->  A  e.  ( EE `  N
) )
2 simp3 957 . . 3  |-  ( ( N  e.  NN  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  ->  B  e.  ( EE `  N
) )
3 brbtwn 24527 . . 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 1182 . 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 8838 . . . . . . 7  |-  0  e.  RR
6 1re 8837 . . . . . . 7  |-  1  e.  RR
75, 6elicc2i 10716 . . . . . 6  |-  ( t  e.  ( 0 [,] 1 )  <->  ( t  e.  RR  /\  0  <_ 
t  /\  t  <_  1 ) )
87simp1bi 970 . . . . 5  |-  ( t  e.  ( 0 [,] 1 )  ->  t  e.  RR )
98recnd 8861 . . . 4  |-  ( t  e.  ( 0 [,] 1 )  ->  t  e.  CC )
10 eqeefv 24531 . . . . . . . 8  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  -> 
( A  =  B  <->  A. i  e.  (
1 ... N ) ( A `  i )  =  ( B `  i ) ) )
11103adant1 973 . . . . . . 7  |-  ( ( N  e.  NN  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  ->  ( A  =  B  <->  A. i  e.  ( 1 ... N
) ( A `  i )  =  ( B `  i ) ) )
1211adantr 451 . . . . . 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 8795 . . . . . . . . . . . 12  |-  1  e.  CC
14 npcan 9060 . . . . . . . . . . . 12  |-  ( ( 1  e.  CC  /\  t  e.  CC )  ->  ( ( 1  -  t )  +  t )  =  1 )
1513, 14mpan 651 . . . . . . . . . . 11  |-  ( t  e.  CC  ->  (
( 1  -  t
)  +  t )  =  1 )
1615ad2antlr 707 . . . . . . . . . 10  |-  ( ( ( ( N  e.  NN  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) )  /\  t  e.  CC )  /\  i  e.  ( 1 ... N
) )  ->  (
( 1  -  t
)  +  t )  =  1 )
1716oveq1d 5873 . . . . . . . . 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 9051 . . . . . . . . . . . 12  |-  ( ( 1  e.  CC  /\  t  e.  CC )  ->  ( 1  -  t
)  e.  CC )
1913, 18mpan 651 . . . . . . . . . . 11  |-  ( t  e.  CC  ->  (
1  -  t )  e.  CC )
2019ad2antlr 707 . . . . . . . . . 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 731 . . . . . . . . . 10  |-  ( ( ( ( N  e.  NN  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) )  /\  t  e.  CC )  /\  i  e.  ( 1 ... N
) )  ->  t  e.  CC )
22 simpll3 996 . . . . . . . . . . 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 24530 . . . . . . . . . . 11  |-  ( ( B  e.  ( EE
`  N )  /\  i  e.  ( 1 ... N ) )  ->  ( B `  i )  e.  CC )
2422, 23sylancom 648 . . . . . . . . . 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 8860 . . . . . . . . 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 8853 . . . . . . . . 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 2324 . . . . . . . 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 2294 . . . . . . 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 2559 . . . . . 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 244 . . . . 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 214 . . . 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 460 . . 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 2667 . 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 206 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   <.cop 3643   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742    <_ cle 8868    - cmin 9037   NNcn 9746   [,]cicc 10659   ...cfz 10782   EEcee 24516    Btwn cbtwn 24517
This theorem is referenced by:  btwncomim  24636  btwnswapid  24640  btwnintr  24642  btwnexch3  24643  ifscgr  24667  idinside  24707  btwnconn1lem12  24721  outsideofrflx  24750
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-z 10025  df-uz 10231  df-icc 10663  df-fz 10783  df-ee 24519  df-btwn 24520
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