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Theorem ax5seglem4 24560
Description: Lemma for ax5seg 24566. 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 5866 . . . . . . . . . . 11  |-  ( T  =  0  ->  (
1  -  T )  =  ( 1  -  0 ) )
2 ax-1cn 8795 . . . . . . . . . . . 12  |-  1  e.  CC
32subid1i 9118 . . . . . . . . . . 11  |-  ( 1  -  0 )  =  1
41, 3syl6eq 2331 . . . . . . . . . 10  |-  ( T  =  0  ->  (
1  -  T )  =  1 )
54oveq1d 5873 . . . . . . . . 9  |-  ( T  =  0  ->  (
( 1  -  T
)  x.  ( A `
 i ) )  =  ( 1  x.  ( A `  i
) ) )
6 oveq1 5865 . . . . . . . . 9  |-  ( T  =  0  ->  ( T  x.  ( C `  i ) )  =  ( 0  x.  ( C `  i )
) )
75, 6oveq12d 5876 . . . . . . . 8  |-  ( T  =  0  ->  (
( ( 1  -  T )  x.  ( A `  i )
)  +  ( T  x.  ( C `  i ) ) )  =  ( ( 1  x.  ( A `  i ) )  +  ( 0  x.  ( C `  i )
) ) )
87eqeq2d 2294 . . . . . . 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 2563 . . . . . 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 472 . . . . 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 24531 . . . . . . . 8  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  -> 
( A  =  B  <->  A. i  e.  (
1 ... N ) ( A `  i )  =  ( B `  i ) ) )
12113adant1 973 . . . . . . 7  |-  ( ( N  e.  NN  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  ->  ( A  =  B  <->  A. i  e.  ( 1 ... N
) ( A `  i )  =  ( B `  i ) ) )
13123adant3r3 1162 . . . . . 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 2285 . . . . . . . 8  |-  ( ( ( 1  x.  ( A `  i )
)  +  ( 0  x.  ( C `  i ) ) )  =  ( B `  i )  <->  ( B `  i )  =  ( ( 1  x.  ( A `  i )
)  +  ( 0  x.  ( C `  i ) ) ) )
15 simplr1 997 . . . . . . . . . . 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 24530 . . . . . . . . . . 11  |-  ( ( A  e.  ( EE
`  N )  /\  i  e.  ( 1 ... N ) )  ->  ( A `  i )  e.  CC )
1715, 16sylancom 648 . . . . . . . . . 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 999 . . . . . . . . . . 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 24530 . . . . . . . . . . 11  |-  ( ( C  e.  ( EE
`  N )  /\  i  e.  ( 1 ... N ) )  ->  ( C `  i )  e.  CC )
2018, 19sylancom 648 . . . . . . . . . 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 8836 . . . . . . . . . . . 12  |-  ( ( A `  i )  e.  CC  ->  (
1  x.  ( A `
 i ) )  =  ( A `  i ) )
22 mul02 8990 . . . . . . . . . . . 12  |-  ( ( C `  i )  e.  CC  ->  (
0  x.  ( C `
 i ) )  =  0 )
2321, 22oveqan12d 5877 . . . . . . . . . . 11  |-  ( ( ( A `  i
)  e.  CC  /\  ( C `  i )  e.  CC )  -> 
( ( 1  x.  ( A `  i
) )  +  ( 0  x.  ( C `
 i ) ) )  =  ( ( A `  i )  +  0 ) )
24 addid1 8992 . . . . . . . . . . . 12  |-  ( ( A `  i )  e.  CC  ->  (
( A `  i
)  +  0 )  =  ( A `  i ) )
2524adantr 451 . . . . . . . . . . 11  |-  ( ( ( A `  i
)  e.  CC  /\  ( C `  i )  e.  CC )  -> 
( ( A `  i )  +  0 )  =  ( A `
 i ) )
2623, 25eqtrd 2315 . . . . . . . . . 10  |-  ( ( ( A `  i
)  e.  CC  /\  ( C `  i )  e.  CC )  -> 
( ( 1  x.  ( A `  i
) )  +  ( 0  x.  ( C `
 i ) ) )  =  ( A `
 i ) )
2717, 20, 26syl2anc 642 . . . . . . . . 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 2291 . . . . . . . 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 251 . . . . . . 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 2559 . . . . . 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 244 . . . . 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 212 . . . 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 426 . . 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 2484 . 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 1148 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   ` cfv 5255  (class class class)co 5858   CCcc 8735   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742    - cmin 9037   NNcn 9746   ...cfz 10782   EEcee 24516
This theorem is referenced by:  ax5seg  24566
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-ltxr 8872  df-sub 9039  df-ee 24519
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