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Theorem idinside 27962
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 981 . . . . . 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 1009 . . . . . 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 1010 . . . . . 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 27881 . . . . . 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 1213 . . . . 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 1007 . . . . . 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 23008 . . . . . 6  |-  ( ( N  e.  NN  /\  C  e.  ( EE `  N )  /\  A  e.  ( EE `  N
) )  ->  ( C  Btwn  <. A ,  A >.  ->  C  =  A ) )
81, 2, 6, 7syl3anc 1211 . . . . 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 4047 . . . . . . . . 9  |-  ( C  =  A  ->  <. C ,  C >.  =  <. A ,  C >. )
10 opeq1 4047 . . . . . . . . 9  |-  ( C  =  A  ->  <. C ,  D >.  =  <. A ,  D >. )
119, 10breq12d 4293 . . . . . . . 8  |-  ( C  =  A  ->  ( <. C ,  C >.Cgr <. C ,  D >.  <->  <. A ,  C >.Cgr <. A ,  D >. ) )
1211imbi1d 317 . . . . . . 7  |-  ( C  =  A  ->  (
( <. C ,  C >.Cgr
<. C ,  D >.  ->  C  =  D )  <->  (
<. A ,  C >.Cgr <. A ,  D >.  ->  C  =  D )
) )
1312biimpcd 224 . . . . . 6  |-  ( (
<. C ,  C >.Cgr <. C ,  D >.  ->  C  =  D )  ->  ( C  =  A  ->  ( <. A ,  C >.Cgr <. A ,  D >.  ->  C  =  D ) ) )
14 ax-1 6 . . . . . 6  |-  ( C  =  D  ->  ( <. B ,  C >.Cgr <. B ,  D >.  ->  C  =  D )
)
1513, 14syl8 70 . . . . 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 56 . . . 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 1194 . . 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 4048 . . . . . 6  |-  ( A  =  B  ->  <. A ,  A >.  =  <. A ,  B >. )
1918breq2d 4292 . . . . 5  |-  ( A  =  B  ->  ( C  Btwn  <. A ,  A >.  <-> 
C  Btwn  <. A ,  B >. ) )
20193anbi1d 1286 . . . 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 317 . . 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 219 . 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 987 . . . . . 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 1040 . . . . . 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 1041 . . . . . 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 1042 . . . . . 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 27947 . . . . . 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 1213 . . . . 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 24 . . . . 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 24 . . . . 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 1289 . . . 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 981 . . . . . . . . 9  |-  ( ( A  Colinear  <. B ,  C >.  /\  <. A ,  C >.Cgr
<. A ,  D >.  /\ 
<. B ,  C >.Cgr <. B ,  D >. )  ->  A  Colinear  <. B ,  C >. )
3332anim2i 564 . . . . . . . 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 980 . . . . . . . . 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 463 . . . . . . . 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 529 . . . . . . 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 27961 . . . . . . 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 32 . . . . . 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 436 . . . . 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 430 . . . 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 44 . . 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 434 . 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 2677 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 958    = wceq 1362    e. wcel 1755    =/= wne 2596   <.cop 3871   class class class wbr 4280   ` cfv 5406   NNcn 10310   EEcee 22957    Btwn cbtwn 22958  Cgrccgr 22959    Colinear ccolin 27915
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-inf2 7835  ax-cnex 9326  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347  ax-pre-sup 9348
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-fal 1368  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-se 4667  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-isom 5415  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-om 6466  df-1st 6566  df-2nd 6567  df-recs 6818  df-rdg 6852  df-1o 6908  df-oadd 6912  df-er 7089  df-map 7204  df-en 7299  df-dom 7300  df-sdom 7301  df-fin 7302  df-sup 7679  df-oi 7712  df-card 8097  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-div 9982  df-nn 10311  df-2 10368  df-3 10369  df-n0 10568  df-z 10635  df-uz 10850  df-rp 10980  df-ico 11294  df-icc 11295  df-fz 11425  df-fzo 11533  df-seq 11791  df-exp 11850  df-hash 12088  df-cj 12572  df-re 12573  df-im 12574  df-sqr 12708  df-abs 12709  df-clim 12950  df-sum 13148  df-ee 22960  df-btwn 22961  df-cgr 22962  df-ofs 27861  df-colinear 27917  df-ifs 27918  df-cgr3 27919  df-fs 27920
This theorem is referenced by: (None)
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