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Theorem ax5seglem4 24213
Description: Lemma for ax5seg 24219. Given two distinct points, the scaling constant in a betweenness statement is non-zero. (Contributed by Scott Fenton, 11-Jun-2013.)
Assertion
Ref Expression
ax5seglem4  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  /\  A. i  e.  ( 1 ... N ) ( B `  i )  =  ( ( ( 1  -  T )  x.  ( A `  i ) )  +  ( T  x.  ( C `  i )
) )  /\  A  =/=  B )  ->  T  =/=  0 )
Distinct variable groups:    A, i    B, i    C, i    i, N    T, i

Proof of Theorem ax5seglem4
StepHypRef Expression
1 oveq2 6289 . . . . . . . . . . 11  |-  ( T  =  0  ->  (
1  -  T )  =  ( 1  -  0 ) )
2 1m0e1 10653 . . . . . . . . . . 11  |-  ( 1  -  0 )  =  1
31, 2syl6eq 2500 . . . . . . . . . 10  |-  ( T  =  0  ->  (
1  -  T )  =  1 )
43oveq1d 6296 . . . . . . . . 9  |-  ( T  =  0  ->  (
( 1  -  T
)  x.  ( A `
 i ) )  =  ( 1  x.  ( A `  i
) ) )
5 oveq1 6288 . . . . . . . . 9  |-  ( T  =  0  ->  ( T  x.  ( C `  i ) )  =  ( 0  x.  ( C `  i )
) )
64, 5oveq12d 6299 . . . . . . . 8  |-  ( T  =  0  ->  (
( ( 1  -  T )  x.  ( A `  i )
)  +  ( T  x.  ( C `  i ) ) )  =  ( ( 1  x.  ( A `  i ) )  +  ( 0  x.  ( C `  i )
) ) )
76eqeq2d 2457 . . . . . . 7  |-  ( T  =  0  ->  (
( B `  i
)  =  ( ( ( 1  -  T
)  x.  ( A `
 i ) )  +  ( T  x.  ( C `  i ) ) )  <->  ( B `  i )  =  ( ( 1  x.  ( A `  i )
)  +  ( 0  x.  ( C `  i ) ) ) ) )
87ralbidv 2882 . . . . . 6  |-  ( T  =  0  ->  ( A. i  e.  (
1 ... N ) ( B `  i )  =  ( ( ( 1  -  T )  x.  ( A `  i ) )  +  ( T  x.  ( C `  i )
) )  <->  A. i  e.  ( 1 ... N
) ( B `  i )  =  ( ( 1  x.  ( A `  i )
)  +  ( 0  x.  ( C `  i ) ) ) ) )
98biimpac 486 . . . . 5  |-  ( ( A. i  e.  ( 1 ... N ) ( B `  i
)  =  ( ( ( 1  -  T
)  x.  ( A `
 i ) )  +  ( T  x.  ( C `  i ) ) )  /\  T  =  0 )  ->  A. i  e.  (
1 ... N ) ( B `  i )  =  ( ( 1  x.  ( A `  i ) )  +  ( 0  x.  ( C `  i )
) ) )
10 eqeefv 24184 . . . . . . . 8  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  -> 
( A  =  B  <->  A. i  e.  (
1 ... N ) ( A `  i )  =  ( B `  i ) ) )
11103adant1 1015 . . . . . . 7  |-  ( ( N  e.  NN  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  ->  ( A  =  B  <->  A. i  e.  ( 1 ... N
) ( A `  i )  =  ( B `  i ) ) )
12113adant3r3 1208 . . . . . 6  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  -> 
( A  =  B  <->  A. i  e.  (
1 ... N ) ( A `  i )  =  ( B `  i ) ) )
13 eqcom 2452 . . . . . . . 8  |-  ( ( ( 1  x.  ( A `  i )
)  +  ( 0  x.  ( C `  i ) ) )  =  ( B `  i )  <->  ( B `  i )  =  ( ( 1  x.  ( A `  i )
)  +  ( 0  x.  ( C `  i ) ) ) )
14 simplr1 1039 . . . . . . . . . . 11  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  /\  i  e.  ( 1 ... N
) )  ->  A  e.  ( EE `  N
) )
15 fveecn 24183 . . . . . . . . . . 11  |-  ( ( A  e.  ( EE
`  N )  /\  i  e.  ( 1 ... N ) )  ->  ( A `  i )  e.  CC )
1614, 15sylancom 667 . . . . . . . . . 10  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  /\  i  e.  ( 1 ... N
) )  ->  ( A `  i )  e.  CC )
17 simplr3 1041 . . . . . . . . . . 11  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  /\  i  e.  ( 1 ... N
) )  ->  C  e.  ( EE `  N
) )
18 fveecn 24183 . . . . . . . . . . 11  |-  ( ( C  e.  ( EE
`  N )  /\  i  e.  ( 1 ... N ) )  ->  ( C `  i )  e.  CC )
1917, 18sylancom 667 . . . . . . . . . 10  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  /\  i  e.  ( 1 ... N
) )  ->  ( C `  i )  e.  CC )
20 mulid2 9597 . . . . . . . . . . . 12  |-  ( ( A `  i )  e.  CC  ->  (
1  x.  ( A `
 i ) )  =  ( A `  i ) )
21 mul02 9761 . . . . . . . . . . . 12  |-  ( ( C `  i )  e.  CC  ->  (
0  x.  ( C `
 i ) )  =  0 )
2220, 21oveqan12d 6300 . . . . . . . . . . 11  |-  ( ( ( A `  i
)  e.  CC  /\  ( C `  i )  e.  CC )  -> 
( ( 1  x.  ( A `  i
) )  +  ( 0  x.  ( C `
 i ) ) )  =  ( ( A `  i )  +  0 ) )
23 addid1 9763 . . . . . . . . . . . 12  |-  ( ( A `  i )  e.  CC  ->  (
( A `  i
)  +  0 )  =  ( A `  i ) )
2423adantr 465 . . . . . . . . . . 11  |-  ( ( ( A `  i
)  e.  CC  /\  ( C `  i )  e.  CC )  -> 
( ( A `  i )  +  0 )  =  ( A `
 i ) )
2522, 24eqtrd 2484 . . . . . . . . . 10  |-  ( ( ( A `  i
)  e.  CC  /\  ( C `  i )  e.  CC )  -> 
( ( 1  x.  ( A `  i
) )  +  ( 0  x.  ( C `
 i ) ) )  =  ( A `
 i ) )
2616, 19, 25syl2anc 661 . . . . . . . . 9  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  /\  i  e.  ( 1 ... N
) )  ->  (
( 1  x.  ( A `  i )
)  +  ( 0  x.  ( C `  i ) ) )  =  ( A `  i ) )
2726eqeq1d 2445 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  /\  i  e.  ( 1 ... N
) )  ->  (
( ( 1  x.  ( A `  i
) )  +  ( 0  x.  ( C `
 i ) ) )  =  ( B `
 i )  <->  ( A `  i )  =  ( B `  i ) ) )
2813, 27syl5rbbr 260 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  /\  i  e.  ( 1 ... N
) )  ->  (
( A `  i
)  =  ( B `
 i )  <->  ( B `  i )  =  ( ( 1  x.  ( A `  i )
)  +  ( 0  x.  ( C `  i ) ) ) ) )
2928ralbidva 2879 . . . . . 6  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  -> 
( A. i  e.  ( 1 ... N
) ( A `  i )  =  ( B `  i )  <->  A. i  e.  (
1 ... N ) ( B `  i )  =  ( ( 1  x.  ( A `  i ) )  +  ( 0  x.  ( C `  i )
) ) ) )
3012, 29bitrd 253 . . . . 5  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  -> 
( A  =  B  <->  A. i  e.  (
1 ... N ) ( B `  i )  =  ( ( 1  x.  ( A `  i ) )  +  ( 0  x.  ( C `  i )
) ) ) )
319, 30syl5ibr 221 . . . 4  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  -> 
( ( A. i  e.  ( 1 ... N
) ( B `  i )  =  ( ( ( 1  -  T )  x.  ( A `  i )
)  +  ( T  x.  ( C `  i ) ) )  /\  T  =  0 )  ->  A  =  B ) )
3231expdimp 437 . . 3  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  /\  A. i  e.  ( 1 ... N ) ( B `  i )  =  ( ( ( 1  -  T )  x.  ( A `  i ) )  +  ( T  x.  ( C `  i )
) ) )  -> 
( T  =  0  ->  A  =  B ) )
3332necon3d 2667 . 2  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  /\  A. i  e.  ( 1 ... N ) ( B `  i )  =  ( ( ( 1  -  T )  x.  ( A `  i ) )  +  ( T  x.  ( C `  i )
) ) )  -> 
( A  =/=  B  ->  T  =/=  0 ) )
34333impia 1194 1  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  /\  A. i  e.  ( 1 ... N ) ( B `  i )  =  ( ( ( 1  -  T )  x.  ( A `  i ) )  +  ( T  x.  ( C `  i )
) )  /\  A  =/=  B )  ->  T  =/=  0 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 974    = wceq 1383    e. wcel 1804    =/= wne 2638   A.wral 2793   ` cfv 5578  (class class class)co 6281   CCcc 9493   0cc0 9495   1c1 9496    + caddc 9498    x. cmul 9500    - cmin 9810   NNcn 10543   ...cfz 11683   EEcee 24169
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-po 4790  df-so 4791  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-er 7313  df-map 7424  df-en 7519  df-dom 7520  df-sdom 7521  df-pnf 9633  df-mnf 9634  df-ltxr 9636  df-sub 9812  df-ee 24172
This theorem is referenced by:  ax5seg  24219
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