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Theorem idinside 24707
Description: Law for finding a point inside a segment. Theorem 4.19 of [Schwabhauser] p. 38. (Contributed by Scott Fenton, 7-Oct-2013.)
Assertion
Ref Expression
idinside  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) )  ->  ( ( C 
Btwn  <. A ,  B >.  /\  <. A ,  C >.Cgr
<. A ,  D >.  /\ 
<. B ,  C >.Cgr <. B ,  D >. )  ->  C  =  D ) )

Proof of Theorem idinside
StepHypRef Expression
1 simp1 955 . . . . . 6  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) )  ->  N  e.  NN )
2 simp3l 983 . . . . . 6  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) )  ->  C  e.  ( EE `  N ) )
3 simp3r 984 . . . . . 6  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) )  ->  D  e.  ( EE `  N ) )
4 cgrid2 24626 . . . . . 6  |-  ( ( N  e.  NN  /\  ( C  e.  ( EE `  N )  /\  C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  -> 
( <. C ,  C >.Cgr
<. C ,  D >.  ->  C  =  D )
)
51, 2, 2, 3, 4syl13anc 1184 . . . . 5  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) )  ->  ( <. C ,  C >.Cgr <. C ,  D >.  ->  C  =  D ) )
6 simp2l 981 . . . . . 6  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) )  ->  A  e.  ( EE `  N ) )
7 axbtwnid 24567 . . . . . 6  |-  ( ( N  e.  NN  /\  C  e.  ( EE `  N )  /\  A  e.  ( EE `  N
) )  ->  ( C  Btwn  <. A ,  A >.  ->  C  =  A ) )
81, 2, 6, 7syl3anc 1182 . . . . 5  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) )  ->  ( C  Btwn  <. A ,  A >.  ->  C  =  A )
)
9 opeq1 3796 . . . . . . . . 9  |-  ( C  =  A  ->  <. C ,  C >.  =  <. A ,  C >. )
10 opeq1 3796 . . . . . . . . 9  |-  ( C  =  A  ->  <. C ,  D >.  =  <. A ,  D >. )
119, 10breq12d 4036 . . . . . . . 8  |-  ( C  =  A  ->  ( <. C ,  C >.Cgr <. C ,  D >.  <->  <. A ,  C >.Cgr <. A ,  D >. ) )
1211imbi1d 308 . . . . . . 7  |-  ( C  =  A  ->  (
( <. C ,  C >.Cgr
<. C ,  D >.  ->  C  =  D )  <->  (
<. A ,  C >.Cgr <. A ,  D >.  ->  C  =  D )
) )
1312biimpcd 215 . . . . . 6  |-  ( (
<. C ,  C >.Cgr <. C ,  D >.  ->  C  =  D )  ->  ( C  =  A  ->  ( <. A ,  C >.Cgr <. A ,  D >.  ->  C  =  D ) ) )
14 ax-1 5 . . . . . 6  |-  ( C  =  D  ->  ( <. B ,  C >.Cgr <. B ,  D >.  ->  C  =  D )
)
1513, 14syl8 65 . . . . 5  |-  ( (
<. C ,  C >.Cgr <. C ,  D >.  ->  C  =  D )  ->  ( C  =  A  ->  ( <. A ,  C >.Cgr <. A ,  D >.  ->  ( <. B ,  C >.Cgr <. B ,  D >.  ->  C  =  D ) ) ) )
165, 8, 15sylsyld 52 . . . 4  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) )  ->  ( C  Btwn  <. A ,  A >.  -> 
( <. A ,  C >.Cgr
<. A ,  D >.  -> 
( <. B ,  C >.Cgr
<. B ,  D >.  ->  C  =  D )
) ) )
17163impd 1165 . . 3  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) )  ->  ( ( C 
Btwn  <. A ,  A >.  /\  <. A ,  C >.Cgr
<. A ,  D >.  /\ 
<. B ,  C >.Cgr <. B ,  D >. )  ->  C  =  D ) )
18 opeq2 3797 . . . . . 6  |-  ( A  =  B  ->  <. A ,  A >.  =  <. A ,  B >. )
1918breq2d 4035 . . . . 5  |-  ( A  =  B  ->  ( C  Btwn  <. A ,  A >.  <-> 
C  Btwn  <. A ,  B >. ) )
20193anbi1d 1256 . . . 4  |-  ( A  =  B  ->  (
( C  Btwn  <. A ,  A >.  /\  <. A ,  C >.Cgr <. A ,  D >.  /\  <. B ,  C >.Cgr
<. B ,  D >. )  <-> 
( C  Btwn  <. A ,  B >.  /\  <. A ,  C >.Cgr <. A ,  D >.  /\  <. B ,  C >.Cgr
<. B ,  D >. ) ) )
2120imbi1d 308 . . 3  |-  ( A  =  B  ->  (
( ( C  Btwn  <. A ,  A >.  /\ 
<. A ,  C >.Cgr <. A ,  D >.  /\ 
<. B ,  C >.Cgr <. B ,  D >. )  ->  C  =  D )  <->  ( ( C 
Btwn  <. A ,  B >.  /\  <. A ,  C >.Cgr
<. A ,  D >.  /\ 
<. B ,  C >.Cgr <. B ,  D >. )  ->  C  =  D ) ) )
2217, 21syl5ib 210 . 2  |-  ( A  =  B  ->  (
( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  -> 
( ( C  Btwn  <. A ,  B >.  /\ 
<. A ,  C >.Cgr <. A ,  D >.  /\ 
<. B ,  C >.Cgr <. B ,  D >. )  ->  C  =  D ) ) )
23 simpr1 961 . . . . . 6  |-  ( ( A  =/=  B  /\  ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) ) )  ->  N  e.  NN )
24 simpr2l 1014 . . . . . 6  |-  ( ( A  =/=  B  /\  ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) ) )  ->  A  e.  ( EE `  N ) )
25 simpr2r 1015 . . . . . 6  |-  ( ( A  =/=  B  /\  ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) ) )  ->  B  e.  ( EE `  N ) )
26 simpr3l 1016 . . . . . 6  |-  ( ( A  =/=  B  /\  ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) ) )  ->  C  e.  ( EE `  N ) )
27 btwncolinear1 24692 . . . . . 6  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  -> 
( C  Btwn  <. A ,  B >.  ->  A  Colinear  <. B ,  C >. ) )
2823, 24, 25, 26, 27syl13anc 1184 . . . . 5  |-  ( ( A  =/=  B  /\  ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) ) )  ->  ( C  Btwn  <. A ,  B >.  ->  A  Colinear  <. B ,  C >. ) )
29 idd 21 . . . . 5  |-  ( ( A  =/=  B  /\  ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) ) )  ->  ( <. A ,  C >.Cgr <. A ,  D >.  ->  <. A ,  C >.Cgr
<. A ,  D >. ) )
30 idd 21 . . . . 5  |-  ( ( A  =/=  B  /\  ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) ) )  ->  ( <. B ,  C >.Cgr <. B ,  D >.  ->  <. B ,  C >.Cgr
<. B ,  D >. ) )
3128, 29, 303anim123d 1259 . . . 4  |-  ( ( A  =/=  B  /\  ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) ) )  ->  ( ( C 
Btwn  <. A ,  B >.  /\  <. A ,  C >.Cgr
<. A ,  D >.  /\ 
<. B ,  C >.Cgr <. B ,  D >. )  ->  ( A  Colinear  <. B ,  C >.  /\  <. A ,  C >.Cgr <. A ,  D >.  /\  <. B ,  C >.Cgr <. B ,  D >. ) ) )
32 simp1 955 . . . . . . . . 9  |-  ( ( A  Colinear  <. B ,  C >.  /\  <. A ,  C >.Cgr
<. A ,  D >.  /\ 
<. B ,  C >.Cgr <. B ,  D >. )  ->  A  Colinear  <. B ,  C >. )
3332anim2i 552 . . . . . . . 8  |-  ( ( A  =/=  B  /\  ( A  Colinear  <. B ,  C >.  /\  <. A ,  C >.Cgr <. A ,  D >.  /\  <. B ,  C >.Cgr
<. B ,  D >. ) )  ->  ( A  =/=  B  /\  A  Colinear  <. B ,  C >. )
)
34 3simpc 954 . . . . . . . . 9  |-  ( ( A  Colinear  <. B ,  C >.  /\  <. A ,  C >.Cgr
<. A ,  D >.  /\ 
<. B ,  C >.Cgr <. B ,  D >. )  ->  ( <. A ,  C >.Cgr <. A ,  D >.  /\  <. B ,  C >.Cgr
<. B ,  D >. ) )
3534adantl 452 . . . . . . . 8  |-  ( ( A  =/=  B  /\  ( A  Colinear  <. B ,  C >.  /\  <. A ,  C >.Cgr <. A ,  D >.  /\  <. B ,  C >.Cgr
<. B ,  D >. ) )  ->  ( <. A ,  C >.Cgr <. A ,  D >.  /\  <. B ,  C >.Cgr <. B ,  D >. ) )
3633, 35jca 518 . . . . . . 7  |-  ( ( A  =/=  B  /\  ( A  Colinear  <. B ,  C >.  /\  <. A ,  C >.Cgr <. A ,  D >.  /\  <. B ,  C >.Cgr
<. B ,  D >. ) )  ->  ( ( A  =/=  B  /\  A  Colinear  <. B ,  C >. )  /\  ( <. A ,  C >.Cgr <. A ,  D >.  /\  <. B ,  C >.Cgr
<. B ,  D >. ) ) )
37 lineid 24706 . . . . . . 7  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) )  ->  ( ( ( A  =/=  B  /\  A  Colinear  <. B ,  C >. )  /\  ( <. A ,  C >.Cgr <. A ,  D >.  /\ 
<. B ,  C >.Cgr <. B ,  D >. ) )  ->  C  =  D ) )
3836, 37syl5 28 . . . . . 6  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) )  ->  ( ( A  =/=  B  /\  ( A  Colinear  <. B ,  C >.  /\  <. A ,  C >.Cgr
<. A ,  D >.  /\ 
<. B ,  C >.Cgr <. B ,  D >. ) )  ->  C  =  D ) )
3938exp3a 425 . . . . 5  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) )  ->  ( A  =/= 
B  ->  ( ( A  Colinear  <. B ,  C >.  /\  <. A ,  C >.Cgr
<. A ,  D >.  /\ 
<. B ,  C >.Cgr <. B ,  D >. )  ->  C  =  D ) ) )
4039impcom 419 . . . 4  |-  ( ( A  =/=  B  /\  ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) ) )  ->  ( ( A 
Colinear 
<. B ,  C >.  /\ 
<. A ,  C >.Cgr <. A ,  D >.  /\ 
<. B ,  C >.Cgr <. B ,  D >. )  ->  C  =  D ) )
4131, 40syld 40 . . 3  |-  ( ( A  =/=  B  /\  ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) ) )  ->  ( ( C 
Btwn  <. A ,  B >.  /\  <. A ,  C >.Cgr
<. A ,  D >.  /\ 
<. B ,  C >.Cgr <. B ,  D >. )  ->  C  =  D ) )
4241ex 423 . 2  |-  ( A  =/=  B  ->  (
( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  -> 
( ( C  Btwn  <. A ,  B >.  /\ 
<. A ,  C >.Cgr <. A ,  D >.  /\ 
<. B ,  C >.Cgr <. B ,  D >. )  ->  C  =  D ) ) )
4322, 42pm2.61ine 2522 1  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) )  ->  ( ( C 
Btwn  <. A ,  B >.  /\  <. A ,  C >.Cgr
<. A ,  D >.  /\ 
<. B ,  C >.Cgr <. B ,  D >. )  ->  C  =  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   <.cop 3643   class class class wbr 4023   ` cfv 5255   NNcn 9746   EEcee 24516    Btwn cbtwn 24517  Cgrccgr 24518    Colinear ccolin 24660
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-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  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  ax-pre-mulgt0 8814  ax-pre-sup 8815
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-rmo 2551  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-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  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-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-sum 12159  df-ee 24519  df-btwn 24520  df-cgr 24521  df-ofs 24606  df-ifs 24662  df-cgr3 24663  df-colinear 24664  df-fs 24665
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