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Theorem axbtwnid 23336
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 23296 . . 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 1219 . 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 9496 . . . . . . 7  |-  0  e.  RR
6 1re 9495 . . . . . . 7  |-  1  e.  RR
75, 6elicc2i 11471 . . . . . 6  |-  ( t  e.  ( 0 [,] 1 )  <->  ( t  e.  RR  /\  0  <_ 
t  /\  t  <_  1 ) )
87simp1bi 1003 . . . . 5  |-  ( t  e.  ( 0 [,] 1 )  ->  t  e.  RR )
98recnd 9522 . . . 4  |-  ( t  e.  ( 0 [,] 1 )  ->  t  e.  CC )
10 eqeefv 23300 . . . . . . . 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 9450 . . . . . . . . . . . 12  |-  1  e.  CC
14 npcan 9729 . . . . . . . . . . . 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 6214 . . . . . . . . 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 9719 . . . . . . . . . . . 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 23299 . . . . . . . . . . 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 9521 . . . . . . . . 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 9514 . . . . . . . . 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 2504 . . . . . . . 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 2468 . . . . . . 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 2843 . . . . . 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 2945 . 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 1370    e. wcel 1758   A.wral 2798   E.wrex 2799   <.cop 3990   class class class wbr 4399   ` cfv 5525  (class class class)co 6199   CCcc 9390   RRcr 9391   0cc0 9392   1c1 9393    + caddc 9395    x. cmul 9397    <_ cle 9529    - cmin 9705   NNcn 10432   [,]cicc 11413   ...cfz 11553   EEcee 23285    Btwn cbtwn 23286
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481  ax-cnex 9448  ax-resscn 9449  ax-1cn 9450  ax-icn 9451  ax-addcl 9452  ax-addrcl 9453  ax-mulcl 9454  ax-mulrcl 9455  ax-mulcom 9456  ax-addass 9457  ax-mulass 9458  ax-distr 9459  ax-i2m1 9460  ax-1ne0 9461  ax-1rid 9462  ax-rnegex 9463  ax-rrecex 9464  ax-cnre 9465  ax-pre-lttri 9466  ax-pre-lttrn 9467  ax-pre-ltadd 9468  ax-pre-mulgt0 9469
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-nel 2650  df-ral 2803  df-rex 2804  df-reu 2805  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-pss 3451  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-tp 3989  df-op 3991  df-uni 4199  df-iun 4280  df-br 4400  df-opab 4458  df-mpt 4459  df-tr 4493  df-eprel 4739  df-id 4743  df-po 4748  df-so 4749  df-fr 4786  df-we 4788  df-ord 4829  df-on 4830  df-lim 4831  df-suc 4832  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6160  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-om 6586  df-1st 6686  df-2nd 6687  df-recs 6941  df-rdg 6975  df-er 7210  df-map 7325  df-en 7420  df-dom 7421  df-sdom 7422  df-pnf 9530  df-mnf 9531  df-xr 9532  df-ltxr 9533  df-le 9534  df-sub 9707  df-neg 9708  df-nn 10433  df-z 10757  df-uz 10972  df-icc 11417  df-fz 11554  df-ee 23288  df-btwn 23289
This theorem is referenced by:  eengtrkg  23382  btwncomim  28187  btwnswapid  28191  btwnintr  28193  btwnexch3  28194  ifscgr  28218  idinside  28258  btwnconn1lem12  28272  outsideofrflx  28301
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