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Theorem ax5seglem5 24101
Description: Lemma for ax5seg 24106. 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 5851 . . . . . . . . . . . . . . 15  |-  ( A  =  C  ->  ( A `  i )  =  ( C `  i ) )
21oveq2d 6293 . . . . . . . . . . . . . 14  |-  ( A  =  C  ->  ( T  x.  ( A `  i ) )  =  ( T  x.  ( C `  i )
) )
32oveq2d 6293 . . . . . . . . . . . . 13  |-  ( A  =  C  ->  (
( ( 1  -  T )  x.  ( A `  i )
)  +  ( T  x.  ( A `  i ) ) )  =  ( ( ( 1  -  T )  x.  ( A `  i ) )  +  ( T  x.  ( C `  i )
) ) )
43eqeq2d 2455 . . . . . . . . . . . 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 2880 . . . . . . . . . . 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 1037 . . . . . . . . . . . 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 1038 . . . . . . . . . . . 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 24071 . . . . . . . . . . . 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 24070 . . . . . . . . . . . . . . 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 9594 . . . . . . . . . . . . . . . . . 18  |-  0  e.  RR
14 1re 9593 . . . . . . . . . . . . . . . . . 18  |-  1  e.  RR
1513, 14elicc2i 11594 . . . . . . . . . . . . . . . . 17  |-  ( T  e.  ( 0 [,] 1 )  <->  ( T  e.  RR  /\  0  <_  T  /\  T  <_  1
) )
1615simp1bi 1010 . . . . . . . . . . . . . . . 16  |-  ( T  e.  ( 0 [,] 1 )  ->  T  e.  RR )
1716recnd 9620 . . . . . . . . . . . . . . 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 9548 . . . . . . . . . . . . . . . . . . 19  |-  1  e.  CC
20 npcan 9829 . . . . . . . . . . . . . . . . . . 19  |-  ( ( 1  e.  CC  /\  T  e.  CC )  ->  ( ( 1  -  T )  +  T
)  =  1 )
2119, 20mpan 670 . . . . . . . . . . . . . . . . . 18  |-  ( T  e.  CC  ->  (
( 1  -  T
)  +  T )  =  1 )
2221oveq1d 6292 . . . . . . . . . . . . . . . . 17  |-  ( T  e.  CC  ->  (
( ( 1  -  T )  +  T
)  x.  ( A `
 i ) )  =  ( 1  x.  ( A `  i
) ) )
23 mulid2 9592 . . . . . . . . . . . . . . . . 17  |-  ( ( A `  i )  e.  CC  ->  (
1  x.  ( A `
 i ) )  =  ( A `  i ) )
2422, 23sylan9eqr 2504 . . . . . . . . . . . . . . . 16  |-  ( ( ( A `  i
)  e.  CC  /\  T  e.  CC )  ->  ( ( ( 1  -  T )  +  T )  x.  ( A `  i )
)  =  ( A `
 i ) )
25 subcl 9819 . . . . . . . . . . . . . . . . . . 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 9619 . . . . . . . . . . . . . . . 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 2484 . . . . . . . . . . . . . . 15  |-  ( ( ( A `  i
)  e.  CC  /\  T  e.  CC )  ->  ( A `  i
)  =  ( ( ( 1  -  T
)  x.  ( A `
 i ) )  +  ( T  x.  ( A `  i ) ) ) )
3231eqeq1d 2443 . . . . . . . . . . . . . 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 2450 . . . . . . . . . . . . 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 2877 . . . . . . . . . . 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 2665 . . . . . 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 1210 . 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 1037 . . . 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 1039 . . . 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 24072 . . . 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 2699 . 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 972    = wceq 1381    e. wcel 1802    =/= wne 2636   A.wral 2791   class class class wbr 4433   ` cfv 5574  (class class class)co 6277   CCcc 9488   RRcr 9489   0cc0 9490   1c1 9491    + caddc 9493    x. cmul 9495    <_ cle 9627    - cmin 9805   NNcn 10537   2c2 10586   [,]cicc 11536   ...cfz 11676   ^cexp 12140   sum_csu 13482   EEcee 24056
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-inf2 8056  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567  ax-pre-sup 9568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-fal 1387  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-int 4268  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-se 4825  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-isom 5583  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6682  df-1st 6781  df-2nd 6782  df-recs 7040  df-rdg 7074  df-1o 7128  df-oadd 7132  df-er 7309  df-map 7420  df-en 7515  df-dom 7516  df-sdom 7517  df-fin 7518  df-sup 7899  df-oi 7933  df-card 8318  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-div 10208  df-nn 10538  df-2 10595  df-3 10596  df-n0 10797  df-z 10866  df-uz 11086  df-rp 11225  df-ico 11539  df-icc 11540  df-fz 11677  df-fzo 11799  df-seq 12082  df-exp 12141  df-hash 12380  df-cj 12906  df-re 12907  df-im 12908  df-sqrt 13042  df-abs 13043  df-clim 13285  df-sum 13483  df-ee 24059
This theorem is referenced by:  ax5seglem6  24102
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