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Theorem axbtwnid 23153
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 989 . . 3  |-  ( ( N  e.  NN  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  ->  A  e.  ( EE `  N
) )
2 simp3 990 . . 3  |-  ( ( N  e.  NN  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  ->  B  e.  ( EE `  N
) )
3 brbtwn 23113 . . 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 1218 . 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 9378 . . . . . . 7  |-  0  e.  RR
6 1re 9377 . . . . . . 7  |-  1  e.  RR
75, 6elicc2i 11353 . . . . . 6  |-  ( t  e.  ( 0 [,] 1 )  <->  ( t  e.  RR  /\  0  <_ 
t  /\  t  <_  1 ) )
87simp1bi 1003 . . . . 5  |-  ( t  e.  ( 0 [,] 1 )  ->  t  e.  RR )
98recnd 9404 . . . 4  |-  ( t  e.  ( 0 [,] 1 )  ->  t  e.  CC )
10 eqeefv 23117 . . . . . . . 8  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  -> 
( A  =  B  <->  A. i  e.  (
1 ... N ) ( A `  i )  =  ( B `  i ) ) )
11103adant1 1006 . . . . . . 7  |-  ( ( N  e.  NN  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  ->  ( A  =  B  <->  A. i  e.  ( 1 ... N
) ( A `  i )  =  ( B `  i ) ) )
1211adantr 465 . . . . . 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 9332 . . . . . . . . . . . 12  |-  1  e.  CC
14 npcan 9611 . . . . . . . . . . . 12  |-  ( ( 1  e.  CC  /\  t  e.  CC )  ->  ( ( 1  -  t )  +  t )  =  1 )
1513, 14mpan 670 . . . . . . . . . . 11  |-  ( t  e.  CC  ->  (
( 1  -  t
)  +  t )  =  1 )
1615ad2antlr 726 . . . . . . . . . 10  |-  ( ( ( ( N  e.  NN  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) )  /\  t  e.  CC )  /\  i  e.  ( 1 ... N
) )  ->  (
( 1  -  t
)  +  t )  =  1 )
1716oveq1d 6101 . . . . . . . . 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 9601 . . . . . . . . . . . 12  |-  ( ( 1  e.  CC  /\  t  e.  CC )  ->  ( 1  -  t
)  e.  CC )
1913, 18mpan 670 . . . . . . . . . . 11  |-  ( t  e.  CC  ->  (
1  -  t )  e.  CC )
2019ad2antlr 726 . . . . . . . . . 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 754 . . . . . . . . . 10  |-  ( ( ( ( N  e.  NN  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) )  /\  t  e.  CC )  /\  i  e.  ( 1 ... N
) )  ->  t  e.  CC )
22 simpll3 1029 . . . . . . . . . . 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 23116 . . . . . . . . . . 11  |-  ( ( B  e.  ( EE
`  N )  /\  i  e.  ( 1 ... N ) )  ->  ( B `  i )  e.  CC )
2422, 23sylancom 667 . . . . . . . . . 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 9403 . . . . . . . . 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 9396 . . . . . . . . 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 2479 . . . . . . . 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 2449 . . . . . . 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 2726 . . . . . 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 253 . . . . 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 223 . . . 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 474 . . 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 2836 . 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 215 1  |-  ( ( N  e.  NN  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  ->  ( A  Btwn  <. B ,  B >.  ->  A  =  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2710   E.wrex 2711   <.cop 3878   class class class wbr 4287   ` cfv 5413  (class class class)co 6086   CCcc 9272   RRcr 9273   0cc0 9274   1c1 9275    + caddc 9277    x. cmul 9279    <_ cle 9411    - cmin 9587   NNcn 10314   [,]cicc 11295   ...cfz 11429   EEcee 23102    Btwn cbtwn 23103
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-er 7093  df-map 7208  df-en 7303  df-dom 7304  df-sdom 7305  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-nn 10315  df-z 10639  df-uz 10854  df-icc 11299  df-fz 11430  df-ee 23105  df-btwn 23106
This theorem is referenced by:  eengtrkg  23199  btwncomim  28013  btwnswapid  28017  btwnintr  28019  btwnexch3  28020  ifscgr  28044  idinside  28084  btwnconn1lem12  28098  outsideofrflx  28127
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