MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ax5seglem5 Structured version   Unicode version

Theorem ax5seglem5 23179
Description: Lemma for ax5seg 23184. 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 5690 . . . . . . . . . . . . . . 15  |-  ( A  =  C  ->  ( A `  i )  =  ( C `  i ) )
21oveq2d 6107 . . . . . . . . . . . . . 14  |-  ( A  =  C  ->  ( T  x.  ( A `  i ) )  =  ( T  x.  ( C `  i )
) )
32oveq2d 6107 . . . . . . . . . . . . 13  |-  ( A  =  C  ->  (
( ( 1  -  T )  x.  ( A `  i )
)  +  ( T  x.  ( A `  i ) ) )  =  ( ( ( 1  -  T )  x.  ( A `  i ) )  +  ( T  x.  ( C `  i )
) ) )
43eqeq2d 2454 . . . . . . . . . . . 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 2735 . . . . . . . . . . 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 1030 . . . . . . . . . . . 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 1031 . . . . . . . . . . . 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 23149 . . . . . . . . . . . 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 23148 . . . . . . . . . . . . . . 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 9386 . . . . . . . . . . . . . . . . . 18  |-  0  e.  RR
14 1re 9385 . . . . . . . . . . . . . . . . . 18  |-  1  e.  RR
1513, 14elicc2i 11361 . . . . . . . . . . . . . . . . 17  |-  ( T  e.  ( 0 [,] 1 )  <->  ( T  e.  RR  /\  0  <_  T  /\  T  <_  1
) )
1615simp1bi 1003 . . . . . . . . . . . . . . . 16  |-  ( T  e.  ( 0 [,] 1 )  ->  T  e.  RR )
1716recnd 9412 . . . . . . . . . . . . . . 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 9340 . . . . . . . . . . . . . . . . . . 19  |-  1  e.  CC
20 npcan 9619 . . . . . . . . . . . . . . . . . . 19  |-  ( ( 1  e.  CC  /\  T  e.  CC )  ->  ( ( 1  -  T )  +  T
)  =  1 )
2119, 20mpan 670 . . . . . . . . . . . . . . . . . 18  |-  ( T  e.  CC  ->  (
( 1  -  T
)  +  T )  =  1 )
2221oveq1d 6106 . . . . . . . . . . . . . . . . 17  |-  ( T  e.  CC  ->  (
( ( 1  -  T )  +  T
)  x.  ( A `
 i ) )  =  ( 1  x.  ( A `  i
) ) )
23 mulid2 9384 . . . . . . . . . . . . . . . . 17  |-  ( ( A `  i )  e.  CC  ->  (
1  x.  ( A `
 i ) )  =  ( A `  i ) )
2422, 23sylan9eqr 2497 . . . . . . . . . . . . . . . 16  |-  ( ( ( A `  i
)  e.  CC  /\  T  e.  CC )  ->  ( ( ( 1  -  T )  +  T )  x.  ( A `  i )
)  =  ( A `
 i ) )
25 subcl 9609 . . . . . . . . . . . . . . . . . . 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 9411 . . . . . . . . . . . . . . . 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 2477 . . . . . . . . . . . . . . 15  |-  ( ( ( A `  i
)  e.  CC  /\  T  e.  CC )  ->  ( A `  i
)  =  ( ( ( 1  -  T
)  x.  ( A `
 i ) )  +  ( T  x.  ( A `  i ) ) ) )
3231eqeq1d 2451 . . . . . . . . . . . . . 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 2445 . . . . . . . . . . . . 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 2731 . . . . . . . . . . 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 2646 . . . . . 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 1202 . 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 1030 . . . 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 1032 . . . 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 23150 . . . 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 2643 . 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 965    = wceq 1369    e. wcel 1756    =/= wne 2606   A.wral 2715   class class class wbr 4292   ` cfv 5418  (class class class)co 6091   CCcc 9280   RRcr 9281   0cc0 9282   1c1 9283    + caddc 9285    x. cmul 9287    <_ cle 9419    - cmin 9595   NNcn 10322   2c2 10371   [,]cicc 11303   ...cfz 11437   ^cexp 11865   sum_csu 13163   EEcee 23134
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-inf2 7847  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-pre-sup 9360
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-se 4680  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-isom 5427  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-1o 6920  df-oadd 6924  df-er 7101  df-map 7216  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-sup 7691  df-oi 7724  df-card 8109  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-2 10380  df-3 10381  df-n0 10580  df-z 10647  df-uz 10862  df-rp 10992  df-ico 11306  df-icc 11307  df-fz 11438  df-fzo 11549  df-seq 11807  df-exp 11866  df-hash 12104  df-cj 12588  df-re 12589  df-im 12590  df-sqr 12724  df-abs 12725  df-clim 12966  df-sum 13164  df-ee 23137
This theorem is referenced by:  ax5seglem6  23180
  Copyright terms: Public domain W3C validator