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Theorem ax5seglem4 25863
Description: Lemma for ax5seg 25869. 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 6081 . . . . . . . . . . 11  |-  ( T  =  0  ->  (
1  -  T )  =  ( 1  -  0 ) )
2 ax-1cn 9040 . . . . . . . . . . . 12  |-  1  e.  CC
32subid1i 9364 . . . . . . . . . . 11  |-  ( 1  -  0 )  =  1
41, 3syl6eq 2483 . . . . . . . . . 10  |-  ( T  =  0  ->  (
1  -  T )  =  1 )
54oveq1d 6088 . . . . . . . . 9  |-  ( T  =  0  ->  (
( 1  -  T
)  x.  ( A `
 i ) )  =  ( 1  x.  ( A `  i
) ) )
6 oveq1 6080 . . . . . . . . 9  |-  ( T  =  0  ->  ( T  x.  ( C `  i ) )  =  ( 0  x.  ( C `  i )
) )
75, 6oveq12d 6091 . . . . . . . 8  |-  ( T  =  0  ->  (
( ( 1  -  T )  x.  ( A `  i )
)  +  ( T  x.  ( C `  i ) ) )  =  ( ( 1  x.  ( A `  i ) )  +  ( 0  x.  ( C `  i )
) ) )
87eqeq2d 2446 . . . . . . 7  |-  ( T  =  0  ->  (
( B `  i
)  =  ( ( ( 1  -  T
)  x.  ( A `
 i ) )  +  ( T  x.  ( C `  i ) ) )  <->  ( B `  i )  =  ( ( 1  x.  ( A `  i )
)  +  ( 0  x.  ( C `  i ) ) ) ) )
98ralbidv 2717 . . . . . 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 ) ) ) ) )
109biimpac 473 . . . . 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 )
) ) )
11 eqeefv 25834 . . . . . . . 8  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  -> 
( A  =  B  <->  A. i  e.  (
1 ... N ) ( A `  i )  =  ( B `  i ) ) )
12113adant1 975 . . . . . . 7  |-  ( ( N  e.  NN  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  ->  ( A  =  B  <->  A. i  e.  ( 1 ... N
) ( A `  i )  =  ( B `  i ) ) )
13123adant3r3 1164 . . . . . 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 ) ) )
14 eqcom 2437 . . . . . . . 8  |-  ( ( ( 1  x.  ( A `  i )
)  +  ( 0  x.  ( C `  i ) ) )  =  ( B `  i )  <->  ( B `  i )  =  ( ( 1  x.  ( A `  i )
)  +  ( 0  x.  ( C `  i ) ) ) )
15 simplr1 999 . . . . . . . . . . 11  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  /\  i  e.  ( 1 ... N
) )  ->  A  e.  ( EE `  N
) )
16 fveecn 25833 . . . . . . . . . . 11  |-  ( ( A  e.  ( EE
`  N )  /\  i  e.  ( 1 ... N ) )  ->  ( A `  i )  e.  CC )
1715, 16sylancom 649 . . . . . . . . . 10  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  /\  i  e.  ( 1 ... N
) )  ->  ( A `  i )  e.  CC )
18 simplr3 1001 . . . . . . . . . . 11  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  /\  i  e.  ( 1 ... N
) )  ->  C  e.  ( EE `  N
) )
19 fveecn 25833 . . . . . . . . . . 11  |-  ( ( C  e.  ( EE
`  N )  /\  i  e.  ( 1 ... N ) )  ->  ( C `  i )  e.  CC )
2018, 19sylancom 649 . . . . . . . . . 10  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  /\  i  e.  ( 1 ... N
) )  ->  ( C `  i )  e.  CC )
21 mulid2 9081 . . . . . . . . . . . 12  |-  ( ( A `  i )  e.  CC  ->  (
1  x.  ( A `
 i ) )  =  ( A `  i ) )
22 mul02 9236 . . . . . . . . . . . 12  |-  ( ( C `  i )  e.  CC  ->  (
0  x.  ( C `
 i ) )  =  0 )
2321, 22oveqan12d 6092 . . . . . . . . . . 11  |-  ( ( ( A `  i
)  e.  CC  /\  ( C `  i )  e.  CC )  -> 
( ( 1  x.  ( A `  i
) )  +  ( 0  x.  ( C `
 i ) ) )  =  ( ( A `  i )  +  0 ) )
24 addid1 9238 . . . . . . . . . . . 12  |-  ( ( A `  i )  e.  CC  ->  (
( A `  i
)  +  0 )  =  ( A `  i ) )
2524adantr 452 . . . . . . . . . . 11  |-  ( ( ( A `  i
)  e.  CC  /\  ( C `  i )  e.  CC )  -> 
( ( A `  i )  +  0 )  =  ( A `
 i ) )
2623, 25eqtrd 2467 . . . . . . . . . 10  |-  ( ( ( A `  i
)  e.  CC  /\  ( C `  i )  e.  CC )  -> 
( ( 1  x.  ( A `  i
) )  +  ( 0  x.  ( C `
 i ) ) )  =  ( A `
 i ) )
2717, 20, 26syl2anc 643 . . . . . . . . 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 ) )
2827eqeq1d 2443 . . . . . . . 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 ) ) )
2914, 28syl5rbbr 252 . . . . . . 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 ) ) ) ) )
3029ralbidva 2713 . . . . . 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 )
) ) ) )
3113, 30bitrd 245 . . . . 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 )
) ) ) )
3210, 31syl5ibr 213 . . . 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 ) )
3332expdimp 427 . . 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 ) )
3433necon3d 2636 . 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 ) )
35343impia 1150 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 set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   ` cfv 5446  (class class class)co 6073   CCcc 8980   0cc0 8982   1c1 8983    + caddc 8985    x. cmul 8987    - cmin 9283   NNcn 9992   ...cfz 11035   EEcee 25819
This theorem is referenced by:  ax5seg  25869
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-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  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
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-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-po 4495  df-so 4496  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-ov 6076  df-oprab 6077  df-mpt2 6078  df-riota 6541  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-ltxr 9117  df-sub 9285  df-ee 25822
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