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Theorem ax5seglem4 23178
Description: Lemma for ax5seg 23184. 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 6099 . . . . . . . . . . 11  |-  ( T  =  0  ->  (
1  -  T )  =  ( 1  -  0 ) )
2 1m0e1 10432 . . . . . . . . . . 11  |-  ( 1  -  0 )  =  1
31, 2syl6eq 2491 . . . . . . . . . 10  |-  ( T  =  0  ->  (
1  -  T )  =  1 )
43oveq1d 6106 . . . . . . . . 9  |-  ( T  =  0  ->  (
( 1  -  T
)  x.  ( A `
 i ) )  =  ( 1  x.  ( A `  i
) ) )
5 oveq1 6098 . . . . . . . . 9  |-  ( T  =  0  ->  ( T  x.  ( C `  i ) )  =  ( 0  x.  ( C `  i )
) )
64, 5oveq12d 6109 . . . . . . . 8  |-  ( T  =  0  ->  (
( ( 1  -  T )  x.  ( A `  i )
)  +  ( T  x.  ( C `  i ) ) )  =  ( ( 1  x.  ( A `  i ) )  +  ( 0  x.  ( C `  i )
) ) )
76eqeq2d 2454 . . . . . . 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 2735 . . . . . 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 23149 . . . . . . . 8  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  -> 
( A  =  B  <->  A. i  e.  (
1 ... N ) ( A `  i )  =  ( B `  i ) ) )
11103adant1 1006 . . . . . . 7  |-  ( ( N  e.  NN  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  ->  ( A  =  B  <->  A. i  e.  ( 1 ... N
) ( A `  i )  =  ( B `  i ) ) )
12113adant3r3 1198 . . . . . 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 2445 . . . . . . . 8  |-  ( ( ( 1  x.  ( A `  i )
)  +  ( 0  x.  ( C `  i ) ) )  =  ( B `  i )  <->  ( B `  i )  =  ( ( 1  x.  ( A `  i )
)  +  ( 0  x.  ( C `  i ) ) ) )
14 simplr1 1030 . . . . . . . . . . 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 23148 . . . . . . . . . . 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 1032 . . . . . . . . . . 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 23148 . . . . . . . . . . 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 9384 . . . . . . . . . . . 12  |-  ( ( A `  i )  e.  CC  ->  (
1  x.  ( A `
 i ) )  =  ( A `  i ) )
21 mul02 9547 . . . . . . . . . . . 12  |-  ( ( C `  i )  e.  CC  ->  (
0  x.  ( C `
 i ) )  =  0 )
2220, 21oveqan12d 6110 . . . . . . . . . . 11  |-  ( ( ( A `  i
)  e.  CC  /\  ( C `  i )  e.  CC )  -> 
( ( 1  x.  ( A `  i
) )  +  ( 0  x.  ( C `
 i ) ) )  =  ( ( A `  i )  +  0 ) )
23 addid1 9549 . . . . . . . . . . . 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 2475 . . . . . . . . . 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 2451 . . . . . . . 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 2731 . . . . . 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 2646 . 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 1184 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 965    = wceq 1369    e. wcel 1756    =/= wne 2606   A.wral 2715   ` cfv 5418  (class class class)co 6091   CCcc 9280   0cc0 9282   1c1 9283    + caddc 9285    x. cmul 9287    - cmin 9595   NNcn 10322   ...cfz 11437   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-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  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
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  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-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-po 4641  df-so 4642  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-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-er 7101  df-map 7216  df-en 7311  df-dom 7312  df-sdom 7313  df-pnf 9420  df-mnf 9421  df-ltxr 9423  df-sub 9597  df-ee 23137
This theorem is referenced by:  ax5seg  23184
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