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Theorem ax5seglem5 25588
Description: Lemma for ax5seg 25593. 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 5669 . . . . . . . . . . . . . . 15  |-  ( A  =  C  ->  ( A `  i )  =  ( C `  i ) )
21oveq2d 6038 . . . . . . . . . . . . . 14  |-  ( A  =  C  ->  ( T  x.  ( A `  i ) )  =  ( T  x.  ( C `  i )
) )
32oveq2d 6038 . . . . . . . . . . . . 13  |-  ( A  =  C  ->  (
( ( 1  -  T )  x.  ( A `  i )
)  +  ( T  x.  ( A `  i ) ) )  =  ( ( ( 1  -  T )  x.  ( A `  i ) )  +  ( T  x.  ( C `  i )
) ) )
43eqeq2d 2400 . . . . . . . . . . . 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 2671 . . . . . . . . . . 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 25558 . . . . . . . . . . . 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 25557 . . . . . . . . . . . . . . 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 9026 . . . . . . . . . . . . . . . . . 18  |-  0  e.  RR
14 1re 9025 . . . . . . . . . . . . . . . . . 18  |-  1  e.  RR
1513, 14elicc2i 10910 . . . . . . . . . . . . . . . . 17  |-  ( T  e.  ( 0 [,] 1 )  <->  ( T  e.  RR  /\  0  <_  T  /\  T  <_  1
) )
1615simp1bi 972 . . . . . . . . . . . . . . . 16  |-  ( T  e.  ( 0 [,] 1 )  ->  T  e.  RR )
1716recnd 9049 . . . . . . . . . . . . . . 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 8983 . . . . . . . . . . . . . . . . . . 19  |-  1  e.  CC
20 npcan 9248 . . . . . . . . . . . . . . . . . . 19  |-  ( ( 1  e.  CC  /\  T  e.  CC )  ->  ( ( 1  -  T )  +  T
)  =  1 )
2119, 20mpan 652 . . . . . . . . . . . . . . . . . 18  |-  ( T  e.  CC  ->  (
( 1  -  T
)  +  T )  =  1 )
2221oveq1d 6037 . . . . . . . . . . . . . . . . 17  |-  ( T  e.  CC  ->  (
( ( 1  -  T )  +  T
)  x.  ( A `
 i ) )  =  ( 1  x.  ( A `  i
) ) )
23 mulid2 9024 . . . . . . . . . . . . . . . . 17  |-  ( ( A `  i )  e.  CC  ->  (
1  x.  ( A `
 i ) )  =  ( A `  i ) )
2422, 23sylan9eqr 2443 . . . . . . . . . . . . . . . 16  |-  ( ( ( A `  i
)  e.  CC  /\  T  e.  CC )  ->  ( ( ( 1  -  T )  +  T )  x.  ( A `  i )
)  =  ( A `
 i ) )
25 subcl 9239 . . . . . . . . . . . . . . . . . . 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 9048 . . . . . . . . . . . . . . . 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 2423 . . . . . . . . . . . . . . 15  |-  ( ( ( A `  i
)  e.  CC  /\  T  e.  CC )  ->  ( A `  i
)  =  ( ( ( 1  -  T
)  x.  ( A `
 i ) )  +  ( T  x.  ( A `  i ) ) ) )
3231eqeq1d 2397 . . . . . . . . . . . . . 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 2391 . . . . . . . . . . . . 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 2667 . . . . . . . . . . 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 2590 . . . . . 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 25559 . . . 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 2587 . 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 1649    e. wcel 1717    =/= wne 2552   A.wral 2651   class class class wbr 4155   ` cfv 5396  (class class class)co 6022   CCcc 8923   RRcr 8924   0cc0 8925   1c1 8926    + caddc 8928    x. cmul 8930    <_ cle 9056    - cmin 9225   NNcn 9934   2c2 9983   [,]cicc 10853   ...cfz 10977   ^cexp 11311   sum_csu 12408   EEcee 25543
This theorem is referenced by:  ax5seglem6  25589
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-inf2 7531  ax-cnex 8981  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002  ax-pre-sup 9003
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-int 3995  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-se 4485  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-isom 5405  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-riota 6487  df-recs 6571  df-rdg 6606  df-1o 6662  df-oadd 6666  df-er 6843  df-map 6958  df-en 7048  df-dom 7049  df-sdom 7050  df-fin 7051  df-sup 7383  df-oi 7414  df-card 7761  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-div 9612  df-nn 9935  df-2 9992  df-3 9993  df-n0 10156  df-z 10217  df-uz 10423  df-rp 10547  df-ico 10856  df-icc 10857  df-fz 10978  df-fzo 11068  df-seq 11253  df-exp 11312  df-hash 11548  df-cj 11833  df-re 11834  df-im 11835  df-sqr 11969  df-abs 11970  df-clim 12211  df-sum 12409  df-ee 25546
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