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Theorem ax5seglem5 23905
Description: Lemma for ax5seg 23910. If  B is between  A and  C, and  A is distinct from  B, then  A is distinct from  C. (Contributed by Scott Fenton, 11-Jun-2013.)
Assertion
Ref Expression
ax5seglem5  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  /\  ( A  =/=  B  /\  T  e.  ( 0 [,] 1
)  /\  A. i  e.  ( 1 ... N
) ( B `  i )  =  ( ( ( 1  -  T )  x.  ( A `  i )
)  +  ( T  x.  ( C `  i ) ) ) ) )  ->  sum_ j  e.  ( 1 ... N
) ( ( ( A `  j )  -  ( C `  j ) ) ^
2 )  =/=  0
)
Distinct variable groups:    A, i,
j    B, i, j    C, i, j    T, i    i, N, j
Allowed substitution hint:    T( j)

Proof of Theorem ax5seglem5
StepHypRef Expression
1 fveq1 5856 . . . . . . . . . . . . . . 15  |-  ( A  =  C  ->  ( A `  i )  =  ( C `  i ) )
21oveq2d 6291 . . . . . . . . . . . . . 14  |-  ( A  =  C  ->  ( T  x.  ( A `  i ) )  =  ( T  x.  ( C `  i )
) )
32oveq2d 6291 . . . . . . . . . . . . 13  |-  ( A  =  C  ->  (
( ( 1  -  T )  x.  ( A `  i )
)  +  ( T  x.  ( A `  i ) ) )  =  ( ( ( 1  -  T )  x.  ( A `  i ) )  +  ( T  x.  ( C `  i )
) ) )
43eqeq2d 2474 . . . . . . . . . . . 12  |-  ( A  =  C  ->  (
( B `  i
)  =  ( ( ( 1  -  T
)  x.  ( A `
 i ) )  +  ( T  x.  ( A `  i ) ) )  <->  ( B `  i )  =  ( ( ( 1  -  T )  x.  ( A `  i )
)  +  ( T  x.  ( C `  i ) ) ) ) )
54ralbidv 2896 . . . . . . . . . . 11  |-  ( A  =  C  ->  ( A. i  e.  (
1 ... N ) ( B `  i )  =  ( ( ( 1  -  T )  x.  ( A `  i ) )  +  ( T  x.  ( A `  i )
) )  <->  A. i  e.  ( 1 ... N
) ( B `  i )  =  ( ( ( 1  -  T )  x.  ( A `  i )
)  +  ( T  x.  ( C `  i ) ) ) ) )
65biimparc 487 . . . . . . . . . 10  |-  ( ( A. i  e.  ( 1 ... N ) ( B `  i
)  =  ( ( ( 1  -  T
)  x.  ( A `
 i ) )  +  ( T  x.  ( C `  i ) ) )  /\  A  =  C )  ->  A. i  e.  ( 1 ... N
) ( B `  i )  =  ( ( ( 1  -  T )  x.  ( A `  i )
)  +  ( T  x.  ( A `  i ) ) ) )
7 simplr1 1033 . . . . . . . . . . . 12  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  /\  T  e.  ( 0 [,] 1
) )  ->  A  e.  ( EE `  N
) )
8 simplr2 1034 . . . . . . . . . . . 12  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  /\  T  e.  ( 0 [,] 1
) )  ->  B  e.  ( EE `  N
) )
9 eqeefv 23875 . . . . . . . . . . . 12  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  -> 
( A  =  B  <->  A. i  e.  (
1 ... N ) ( A `  i )  =  ( B `  i ) ) )
107, 8, 9syl2anc 661 . . . . . . . . . . 11  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  /\  T  e.  ( 0 [,] 1
) )  ->  ( A  =  B  <->  A. i  e.  ( 1 ... N
) ( A `  i )  =  ( B `  i ) ) )
11 fveecn 23874 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  ( EE
`  N )  /\  i  e.  ( 1 ... N ) )  ->  ( A `  i )  e.  CC )
127, 11sylan 471 . . . . . . . . . . . . . 14  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) ) )  /\  T  e.  ( 0 [,] 1
) )  /\  i  e.  ( 1 ... N
) )  ->  ( A `  i )  e.  CC )
13 0re 9585 . . . . . . . . . . . . . . . . . 18  |-  0  e.  RR
14 1re 9584 . . . . . . . . . . . . . . . . . 18  |-  1  e.  RR
1513, 14elicc2i 11579 . . . . . . . . . . . . . . . . 17  |-  ( T  e.  ( 0 [,] 1 )  <->  ( T  e.  RR  /\  0  <_  T  /\  T  <_  1
) )
1615simp1bi 1006 . . . . . . . . . . . . . . . 16  |-  ( T  e.  ( 0 [,] 1 )  ->  T  e.  RR )
1716recnd 9611 . . . . . . . . . . . . . . 15  |-  ( T  e.  ( 0 [,] 1 )  ->  T  e.  CC )
1817ad2antlr 726 . . . . . . . . . . . . . 14  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) ) )  /\  T  e.  ( 0 [,] 1
) )  /\  i  e.  ( 1 ... N
) )  ->  T  e.  CC )
19 ax-1cn 9539 . . . . . . . . . . . . . . . . . . 19  |-  1  e.  CC
20 npcan 9818 . . . . . . . . . . . . . . . . . . 19  |-  ( ( 1  e.  CC  /\  T  e.  CC )  ->  ( ( 1  -  T )  +  T
)  =  1 )
2119, 20mpan 670 . . . . . . . . . . . . . . . . . 18  |-  ( T  e.  CC  ->  (
( 1  -  T
)  +  T )  =  1 )
2221oveq1d 6290 . . . . . . . . . . . . . . . . 17  |-  ( T  e.  CC  ->  (
( ( 1  -  T )  +  T
)  x.  ( A `
 i ) )  =  ( 1  x.  ( A `  i
) ) )
23 mulid2 9583 . . . . . . . . . . . . . . . . 17  |-  ( ( A `  i )  e.  CC  ->  (
1  x.  ( A `
 i ) )  =  ( A `  i ) )
2422, 23sylan9eqr 2523 . . . . . . . . . . . . . . . 16  |-  ( ( ( A `  i
)  e.  CC  /\  T  e.  CC )  ->  ( ( ( 1  -  T )  +  T )  x.  ( A `  i )
)  =  ( A `
 i ) )
25 subcl 9808 . . . . . . . . . . . . . . . . . . 19  |-  ( ( 1  e.  CC  /\  T  e.  CC )  ->  ( 1  -  T
)  e.  CC )
2619, 25mpan 670 . . . . . . . . . . . . . . . . . 18  |-  ( T  e.  CC  ->  (
1  -  T )  e.  CC )
2726adantl 466 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A `  i
)  e.  CC  /\  T  e.  CC )  ->  ( 1  -  T
)  e.  CC )
28 simpr 461 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A `  i
)  e.  CC  /\  T  e.  CC )  ->  T  e.  CC )
29 simpl 457 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A `  i
)  e.  CC  /\  T  e.  CC )  ->  ( A `  i
)  e.  CC )
3027, 28, 29adddird 9610 . . . . . . . . . . . . . . . 16  |-  ( ( ( A `  i
)  e.  CC  /\  T  e.  CC )  ->  ( ( ( 1  -  T )  +  T )  x.  ( A `  i )
)  =  ( ( ( 1  -  T
)  x.  ( A `
 i ) )  +  ( T  x.  ( A `  i ) ) ) )
3124, 30eqtr3d 2503 . . . . . . . . . . . . . . 15  |-  ( ( ( A `  i
)  e.  CC  /\  T  e.  CC )  ->  ( A `  i
)  =  ( ( ( 1  -  T
)  x.  ( A `
 i ) )  +  ( T  x.  ( A `  i ) ) ) )
3231eqeq1d 2462 . . . . . . . . . . . . . 14  |-  ( ( ( A `  i
)  e.  CC  /\  T  e.  CC )  ->  ( ( A `  i )  =  ( B `  i )  <-> 
( ( ( 1  -  T )  x.  ( A `  i
) )  +  ( T  x.  ( A `
 i ) ) )  =  ( B `
 i ) ) )
3312, 18, 32syl2anc 661 . . . . . . . . . . . . 13  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) ) )  /\  T  e.  ( 0 [,] 1
) )  /\  i  e.  ( 1 ... N
) )  ->  (
( A `  i
)  =  ( B `
 i )  <->  ( (
( 1  -  T
)  x.  ( A `
 i ) )  +  ( T  x.  ( A `  i ) ) )  =  ( B `  i ) ) )
34 eqcom 2469 . . . . . . . . . . . . 13  |-  ( ( ( ( 1  -  T )  x.  ( A `  i )
)  +  ( T  x.  ( A `  i ) ) )  =  ( B `  i )  <->  ( B `  i )  =  ( ( ( 1  -  T )  x.  ( A `  i )
)  +  ( T  x.  ( A `  i ) ) ) )
3533, 34syl6bb 261 . . . . . . . . . . . 12  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) ) )  /\  T  e.  ( 0 [,] 1
) )  /\  i  e.  ( 1 ... N
) )  ->  (
( A `  i
)  =  ( B `
 i )  <->  ( B `  i )  =  ( ( ( 1  -  T )  x.  ( A `  i )
)  +  ( T  x.  ( A `  i ) ) ) ) )
3635ralbidva 2893 . . . . . . . . . . 11  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  /\  T  e.  ( 0 [,] 1
) )  ->  ( A. i  e.  (
1 ... N ) ( A `  i )  =  ( B `  i )  <->  A. i  e.  ( 1 ... N
) ( B `  i )  =  ( ( ( 1  -  T )  x.  ( A `  i )
)  +  ( T  x.  ( A `  i ) ) ) ) )
3710, 36bitrd 253 . . . . . . . . . 10  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  /\  T  e.  ( 0 [,] 1
) )  ->  ( A  =  B  <->  A. i  e.  ( 1 ... N
) ( B `  i )  =  ( ( ( 1  -  T )  x.  ( A `  i )
)  +  ( T  x.  ( A `  i ) ) ) ) )
386, 37syl5ibr 221 . . . . . . . . 9  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  /\  T  e.  ( 0 [,] 1
) )  ->  (
( A. i  e.  ( 1 ... N
) ( B `  i )  =  ( ( ( 1  -  T )  x.  ( A `  i )
)  +  ( T  x.  ( C `  i ) ) )  /\  A  =  C )  ->  A  =  B ) )
3938expd 436 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  /\  T  e.  ( 0 [,] 1
) )  ->  ( A. i  e.  (
1 ... N ) ( B `  i )  =  ( ( ( 1  -  T )  x.  ( A `  i ) )  +  ( T  x.  ( C `  i )
) )  ->  ( A  =  C  ->  A  =  B ) ) )
4039impr 619 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  /\  ( T  e.  ( 0 [,] 1 )  /\  A. i  e.  ( 1 ... N ) ( B `  i )  =  ( ( ( 1  -  T )  x.  ( A `  i ) )  +  ( T  x.  ( C `  i )
) ) ) )  ->  ( A  =  C  ->  A  =  B ) )
4140necon3d 2684 . . . . . 6  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  /\  ( T  e.  ( 0 [,] 1 )  /\  A. i  e.  ( 1 ... N ) ( B `  i )  =  ( ( ( 1  -  T )  x.  ( A `  i ) )  +  ( T  x.  ( C `  i )
) ) ) )  ->  ( A  =/= 
B  ->  A  =/=  C ) )
4241ex 434 . . . . 5  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  -> 
( ( T  e.  ( 0 [,] 1
)  /\  A. i  e.  ( 1 ... N
) ( B `  i )  =  ( ( ( 1  -  T )  x.  ( A `  i )
)  +  ( T  x.  ( C `  i ) ) ) )  ->  ( A  =/=  B  ->  A  =/=  C ) ) )
4342com23 78 . . . 4  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  -> 
( A  =/=  B  ->  ( ( T  e.  ( 0 [,] 1
)  /\  A. i  e.  ( 1 ... N
) ( B `  i )  =  ( ( ( 1  -  T )  x.  ( A `  i )
)  +  ( T  x.  ( C `  i ) ) ) )  ->  A  =/=  C ) ) )
4443exp4a 606 . . 3  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  -> 
( A  =/=  B  ->  ( T  e.  ( 0 [,] 1 )  ->  ( A. i  e.  ( 1 ... N
) ( B `  i )  =  ( ( ( 1  -  T )  x.  ( A `  i )
)  +  ( T  x.  ( C `  i ) ) )  ->  A  =/=  C
) ) ) )
45443imp2 1206 . 2  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  /\  ( A  =/=  B  /\  T  e.  ( 0 [,] 1
)  /\  A. i  e.  ( 1 ... N
) ( B `  i )  =  ( ( ( 1  -  T )  x.  ( A `  i )
)  +  ( T  x.  ( C `  i ) ) ) ) )  ->  A  =/=  C )
46 simplr1 1033 . . . 4  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  /\  ( A  =/=  B  /\  T  e.  ( 0 [,] 1
)  /\  A. i  e.  ( 1 ... N
) ( B `  i )  =  ( ( ( 1  -  T )  x.  ( A `  i )
)  +  ( T  x.  ( C `  i ) ) ) ) )  ->  A  e.  ( EE `  N
) )
47 simplr3 1035 . . . 4  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  /\  ( A  =/=  B  /\  T  e.  ( 0 [,] 1
)  /\  A. i  e.  ( 1 ... N
) ( B `  i )  =  ( ( ( 1  -  T )  x.  ( A `  i )
)  +  ( T  x.  ( C `  i ) ) ) ) )  ->  C  e.  ( EE `  N
) )
48 eqeelen 23876 . . . 4  |-  ( ( A  e.  ( EE
`  N )  /\  C  e.  ( EE `  N ) )  -> 
( A  =  C  <->  sum_ j  e.  ( 1 ... N ) ( ( ( A `  j )  -  ( C `  j )
) ^ 2 )  =  0 ) )
4946, 47, 48syl2anc 661 . . 3  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  /\  ( A  =/=  B  /\  T  e.  ( 0 [,] 1
)  /\  A. i  e.  ( 1 ... N
) ( B `  i )  =  ( ( ( 1  -  T )  x.  ( A `  i )
)  +  ( T  x.  ( C `  i ) ) ) ) )  ->  ( A  =  C  <->  sum_ j  e.  ( 1 ... N
) ( ( ( A `  j )  -  ( C `  j ) ) ^
2 )  =  0 ) )
5049necon3bid 2718 . 2  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  /\  ( A  =/=  B  /\  T  e.  ( 0 [,] 1
)  /\  A. i  e.  ( 1 ... N
) ( B `  i )  =  ( ( ( 1  -  T )  x.  ( A `  i )
)  +  ( T  x.  ( C `  i ) ) ) ) )  ->  ( A  =/=  C  <->  sum_ j  e.  ( 1 ... N
) ( ( ( A `  j )  -  ( C `  j ) ) ^
2 )  =/=  0
) )
5145, 50mpbid 210 1  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  /\  ( A  =/=  B  /\  T  e.  ( 0 [,] 1
)  /\  A. i  e.  ( 1 ... N
) ( B `  i )  =  ( ( ( 1  -  T )  x.  ( A `  i )
)  +  ( T  x.  ( C `  i ) ) ) ) )  ->  sum_ j  e.  ( 1 ... N
) ( ( ( A `  j )  -  ( C `  j ) ) ^
2 )  =/=  0
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762    =/= wne 2655   A.wral 2807   class class class wbr 4440   ` cfv 5579  (class class class)co 6275   CCcc 9479   RRcr 9480   0cc0 9481   1c1 9482    + caddc 9484    x. cmul 9486    <_ cle 9618    - cmin 9794   NNcn 10525   2c2 10574   [,]cicc 11521   ...cfz 11661   ^cexp 12122   sum_csu 13457   EEcee 23860
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-inf2 8047  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-fal 1380  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-se 4832  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-isom 5588  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-1o 7120  df-oadd 7124  df-er 7301  df-map 7412  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-sup 7890  df-oi 7924  df-card 8309  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196  df-nn 10526  df-2 10583  df-3 10584  df-n0 10785  df-z 10854  df-uz 11072  df-rp 11210  df-ico 11524  df-icc 11525  df-fz 11662  df-fzo 11782  df-seq 12064  df-exp 12123  df-hash 12361  df-cj 12882  df-re 12883  df-im 12884  df-sqr 13018  df-abs 13019  df-clim 13260  df-sum 13458  df-ee 23863
This theorem is referenced by:  ax5seglem6  23906
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