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Theorem ax5seglem4 25871
Description: Lemma for ax5seg 25877. 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 6089 . . . . . . . . . . 11  |-  ( T  =  0  ->  (
1  -  T )  =  ( 1  -  0 ) )
2 ax-1cn 9048 . . . . . . . . . . . 12  |-  1  e.  CC
32subid1i 9372 . . . . . . . . . . 11  |-  ( 1  -  0 )  =  1
41, 3syl6eq 2484 . . . . . . . . . 10  |-  ( T  =  0  ->  (
1  -  T )  =  1 )
54oveq1d 6096 . . . . . . . . 9  |-  ( T  =  0  ->  (
( 1  -  T
)  x.  ( A `
 i ) )  =  ( 1  x.  ( A `  i
) ) )
6 oveq1 6088 . . . . . . . . 9  |-  ( T  =  0  ->  ( T  x.  ( C `  i ) )  =  ( 0  x.  ( C `  i )
) )
75, 6oveq12d 6099 . . . . . . . 8  |-  ( T  =  0  ->  (
( ( 1  -  T )  x.  ( A `  i )
)  +  ( T  x.  ( C `  i ) ) )  =  ( ( 1  x.  ( A `  i ) )  +  ( 0  x.  ( C `  i )
) ) )
87eqeq2d 2447 . . . . . . 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 2725 . . . . . 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 25842 . . . . . . . 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 2438 . . . . . . . 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 25841 . . . . . . . . . . 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 25841 . . . . . . . . . . 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 9089 . . . . . . . . . . . 12  |-  ( ( A `  i )  e.  CC  ->  (
1  x.  ( A `
 i ) )  =  ( A `  i ) )
22 mul02 9244 . . . . . . . . . . . 12  |-  ( ( C `  i )  e.  CC  ->  (
0  x.  ( C `
 i ) )  =  0 )
2321, 22oveqan12d 6100 . . . . . . . . . . 11  |-  ( ( ( A `  i
)  e.  CC  /\  ( C `  i )  e.  CC )  -> 
( ( 1  x.  ( A `  i
) )  +  ( 0  x.  ( C `
 i ) ) )  =  ( ( A `  i )  +  0 ) )
24 addid1 9246 . . . . . . . . . . . 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 2468 . . . . . . . . . 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 2444 . . . . . . . 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 2721 . . . . . 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 2639 . 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 2599   A.wral 2705   ` cfv 5454  (class class class)co 6081   CCcc 8988   0cc0 8990   1c1 8991    + caddc 8993    x. cmul 8995    - cmin 9291   NNcn 10000   ...cfz 11043   EEcee 25827
This theorem is referenced by:  ax5seg  25877
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 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-po 4503  df-so 4504  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-riota 6549  df-er 6905  df-map 7020  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-ltxr 9125  df-sub 9293  df-ee 25830
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