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Theorem ax5seglem5 25864
Description: Lemma for ax5seg 25869. 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 5719 . . . . . . . . . . . . . . 15  |-  ( A  =  C  ->  ( A `  i )  =  ( C `  i ) )
21oveq2d 6089 . . . . . . . . . . . . . 14  |-  ( A  =  C  ->  ( T  x.  ( A `  i ) )  =  ( T  x.  ( C `  i )
) )
32oveq2d 6089 . . . . . . . . . . . . 13  |-  ( A  =  C  ->  (
( ( 1  -  T )  x.  ( A `  i )
)  +  ( T  x.  ( A `  i ) ) )  =  ( ( ( 1  -  T )  x.  ( A `  i ) )  +  ( T  x.  ( C `  i )
) ) )
43eqeq2d 2446 . . . . . . . . . . . 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 2717 . . . . . . . . . . 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 474 . . . . . . . . . 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 999 . . . . . . . . . . . 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 1000 . . . . . . . . . . . 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 25834 . . . . . . . . . . . 12  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  -> 
( A  =  B  <->  A. i  e.  (
1 ... N ) ( A `  i )  =  ( B `  i ) ) )
107, 8, 9syl2anc 643 . . . . . . . . . . 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 25833 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  ( EE
`  N )  /\  i  e.  ( 1 ... N ) )  ->  ( A `  i )  e.  CC )
127, 11sylan 458 . . . . . . . . . . . . . 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 9083 . . . . . . . . . . . . . . . . . 18  |-  0  e.  RR
14 1re 9082 . . . . . . . . . . . . . . . . . 18  |-  1  e.  RR
1513, 14elicc2i 10968 . . . . . . . . . . . . . . . . 17  |-  ( T  e.  ( 0 [,] 1 )  <->  ( T  e.  RR  /\  0  <_  T  /\  T  <_  1
) )
1615simp1bi 972 . . . . . . . . . . . . . . . 16  |-  ( T  e.  ( 0 [,] 1 )  ->  T  e.  RR )
1716recnd 9106 . . . . . . . . . . . . . . 15  |-  ( T  e.  ( 0 [,] 1 )  ->  T  e.  CC )
1817ad2antlr 708 . . . . . . . . . . . . . 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 9040 . . . . . . . . . . . . . . . . . . 19  |-  1  e.  CC
20 npcan 9306 . . . . . . . . . . . . . . . . . . 19  |-  ( ( 1  e.  CC  /\  T  e.  CC )  ->  ( ( 1  -  T )  +  T
)  =  1 )
2119, 20mpan 652 . . . . . . . . . . . . . . . . . 18  |-  ( T  e.  CC  ->  (
( 1  -  T
)  +  T )  =  1 )
2221oveq1d 6088 . . . . . . . . . . . . . . . . 17  |-  ( T  e.  CC  ->  (
( ( 1  -  T )  +  T
)  x.  ( A `
 i ) )  =  ( 1  x.  ( A `  i
) ) )
23 mulid2 9081 . . . . . . . . . . . . . . . . 17  |-  ( ( A `  i )  e.  CC  ->  (
1  x.  ( A `
 i ) )  =  ( A `  i ) )
2422, 23sylan9eqr 2489 . . . . . . . . . . . . . . . 16  |-  ( ( ( A `  i
)  e.  CC  /\  T  e.  CC )  ->  ( ( ( 1  -  T )  +  T )  x.  ( A `  i )
)  =  ( A `
 i ) )
25 subcl 9297 . . . . . . . . . . . . . . . . . . 19  |-  ( ( 1  e.  CC  /\  T  e.  CC )  ->  ( 1  -  T
)  e.  CC )
2619, 25mpan 652 . . . . . . . . . . . . . . . . . 18  |-  ( T  e.  CC  ->  (
1  -  T )  e.  CC )
2726adantl 453 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A `  i
)  e.  CC  /\  T  e.  CC )  ->  ( 1  -  T
)  e.  CC )
28 simpr 448 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A `  i
)  e.  CC  /\  T  e.  CC )  ->  T  e.  CC )
29 simpl 444 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A `  i
)  e.  CC  /\  T  e.  CC )  ->  ( A `  i
)  e.  CC )
3027, 28, 29adddird 9105 . . . . . . . . . . . . . . . 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 2469 . . . . . . . . . . . . . . 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 643 . . . . . . . . . . . . 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 2437 . . . . . . . . . . . . 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 253 . . . . . . . . . . . 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 2713 . . . . . . . . . . 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 245 . . . . . . . . . 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 213 . . . . . . . . 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 ) )
3938exp3a 426 . . . . . . . 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 603 . . . . . . 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 2636 . . . . . 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 424 . . . . 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 74 . . . 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 590 . . 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 1168 . 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 999 . . . 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 1001 . . . 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 25835 . . . 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 643 . . 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 2633 . 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 202 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 set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   CCcc 8980   RRcr 8981   0cc0 8982   1c1 8983    + caddc 8985    x. cmul 8987    <_ cle 9113    - cmin 9283   NNcn 9992   2c2 10041   [,]cicc 10911   ...cfz 11035   ^cexp 11374   sum_csu 12471   EEcee 25819
This theorem is referenced by:  ax5seglem6  25865
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-oi 7471  df-card 7818  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-rp 10605  df-ico 10914  df-icc 10915  df-fz 11036  df-fzo 11128  df-seq 11316  df-exp 11375  df-hash 11611  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-clim 12274  df-sum 12472  df-ee 25822
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