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Theorem axbtwnid 23977
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 23937 . . 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 8834 . . . . . . 7  |-  0  e.  RR
6 1re 8833 . . . . . . 7  |-  1  e.  RR
75, 6elicc2i 10712 . . . . . 6  |-  ( t  e.  ( 0 [,] 1 )  <->  ( t  e.  RR  /\  0  <_ 
t  /\  t  <_  1 ) )
87simp1bi 970 . . . . 5  |-  ( t  e.  ( 0 [,] 1 )  ->  t  e.  RR )
98recnd 8857 . . . 4  |-  ( t  e.  ( 0 [,] 1 )  ->  t  e.  CC )
10 eqeefv 23941 . . . . . . . 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 8791 . . . . . . . . . . . 12  |-  1  e.  CC
14 npcan 9056 . . . . . . . . . . . 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 5835 . . . . . . . . 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 9047 . . . . . . . . . . . 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 23940 . . . . . . . . . . 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 8856 . . . . . . . . 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 8849 . . . . . . . . 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 2325 . . . . . . . 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 2295 . . . . . . 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 2560 . . . . . 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 2668 . 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 1685   A.wral 2544   E.wrex 2545   <.cop 3644   class class class wbr 4024   ` cfv 5221  (class class class)co 5820   CCcc 8731   RRcr 8732   0cc0 8733   1c1 8734    + caddc 8736    x. cmul 8738    <_ cle 8864    - cmin 9033   NNcn 9742   [,]cicc 10655   ...cfz 10778   EEcee 23926    Btwn cbtwn 23927
This theorem is referenced by:  btwncomim  24046  btwnswapid  24050  btwnintr  24052  btwnexch3  24053  ifscgr  24077  idinside  24117  btwnconn1lem12  24131  outsideofrflx  24160
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 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-cnex 8789  ax-resscn 8790  ax-1cn 8791  ax-icn 8792  ax-addcl 8793  ax-addrcl 8794  ax-mulcl 8795  ax-mulrcl 8796  ax-mulcom 8797  ax-addass 8798  ax-mulass 8799  ax-distr 8800  ax-i2m1 8801  ax-1ne0 8802  ax-1rid 8803  ax-rnegex 8804  ax-rrecex 8805  ax-cnre 8806  ax-pre-lttri 8807  ax-pre-lttrn 8808  ax-pre-ltadd 8809  ax-pre-mulgt0 8810
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 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-1st 6084  df-2nd 6085  df-iota 6253  df-riota 6300  df-recs 6384  df-rdg 6419  df-er 6656  df-map 6770  df-en 6860  df-dom 6861  df-sdom 6862  df-pnf 8865  df-mnf 8866  df-xr 8867  df-ltxr 8868  df-le 8869  df-sub 9035  df-neg 9036  df-nn 9743  df-z 10021  df-uz 10227  df-icc 10659  df-fz 10779  df-ee 23929  df-btwn 23930
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