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Theorem idinside 24083
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 960 . . . . . 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 988 . . . . . 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 989 . . . . . 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 24002 . . . . . 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 1189 . . . . 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 986 . . . . . 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 23943 . . . . . 6  |-  ( ( N  e.  NN  /\  C  e.  ( EE `  N )  /\  A  e.  ( EE `  N
) )  ->  ( C  Btwn  <. A ,  A >.  ->  C  =  A ) )
81, 2, 6, 7syl3anc 1187 . . . . 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 3770 . . . . . . . . 9  |-  ( C  =  A  ->  <. C ,  C >.  =  <. A ,  C >. )
10 opeq1 3770 . . . . . . . . 9  |-  ( C  =  A  ->  <. C ,  D >.  =  <. A ,  D >. )
119, 10breq12d 4010 . . . . . . . 8  |-  ( C  =  A  ->  ( <. C ,  C >.Cgr <. C ,  D >.  <->  <. A ,  C >.Cgr <. A ,  D >. ) )
1211imbi1d 310 . . . . . . 7  |-  ( C  =  A  ->  (
( <. C ,  C >.Cgr
<. C ,  D >.  ->  C  =  D )  <->  (
<. A ,  C >.Cgr <. A ,  D >.  ->  C  =  D )
) )
1312biimpcd 217 . . . . . 6  |-  ( (
<. C ,  C >.Cgr <. C ,  D >.  ->  C  =  D )  ->  ( C  =  A  ->  ( <. A ,  C >.Cgr <. A ,  D >.  ->  C  =  D ) ) )
14 ax-1 7 . . . . . 6  |-  ( C  =  D  ->  ( <. B ,  C >.Cgr <. B ,  D >.  ->  C  =  D )
)
1513, 14syl8 67 . . . . 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 54 . . . 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 1170 . . 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 3771 . . . . . 6  |-  ( A  =  B  ->  <. A ,  A >.  =  <. A ,  B >. )
1918breq2d 4009 . . . . 5  |-  ( A  =  B  ->  ( C  Btwn  <. A ,  A >.  <-> 
C  Btwn  <. A ,  B >. ) )
20193anbi1d 1261 . . . 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 310 . . 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 212 . 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 966 . . . . . 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 1019 . . . . . 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 1020 . . . . . 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 1021 . . . . . 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 24068 . . . . . 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 1189 . . . . 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 23 . . . . 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 23 . . . . 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 1264 . . . 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 960 . . . . . . . . 9  |-  ( ( A  Colinear  <. B ,  C >.  /\  <. A ,  C >.Cgr
<. A ,  D >.  /\ 
<. B ,  C >.Cgr <. B ,  D >. )  ->  A  Colinear  <. B ,  C >. )
3332anim2i 555 . . . . . . . 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 959 . . . . . . . . 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 454 . . . . . . . 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 520 . . . . . . 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 24082 . . . . . . 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 30 . . . . . 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 427 . . . . 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 421 . . . 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 42 . . 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 425 . 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 2497 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 6    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    =/= wne 2421   <.cop 3617   class class class wbr 3997   ` cfv 4673   NNcn 9714   EEcee 23892    Btwn cbtwn 23893  Cgrccgr 23894    Colinear ccolin 24036
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484  ax-inf2 7310  ax-cnex 8761  ax-resscn 8762  ax-1cn 8763  ax-icn 8764  ax-addcl 8765  ax-addrcl 8766  ax-mulcl 8767  ax-mulrcl 8768  ax-mulcom 8769  ax-addass 8770  ax-mulass 8771  ax-distr 8772  ax-i2m1 8773  ax-1ne0 8774  ax-1rid 8775  ax-rnegex 8776  ax-rrecex 8777  ax-cnre 8778  ax-pre-lttri 8779  ax-pre-lttrn 8780  ax-pre-ltadd 8781  ax-pre-mulgt0 8782  ax-pre-sup 8783
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-int 3837  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-se 4325  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-isom 4690  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-1st 6056  df-2nd 6057  df-iota 6225  df-riota 6272  df-recs 6356  df-rdg 6391  df-1o 6447  df-oadd 6451  df-er 6628  df-map 6742  df-en 6832  df-dom 6833  df-sdom 6834  df-fin 6835  df-sup 7162  df-oi 7193  df-card 7540  df-pnf 8837  df-mnf 8838  df-xr 8839  df-ltxr 8840  df-le 8841  df-sub 9007  df-neg 9008  df-div 9392  df-n 9715  df-2 9772  df-3 9773  df-n0 9934  df-z 9993  df-uz 10199  df-rp 10323  df-ico 10629  df-icc 10630  df-fz 10750  df-fzo 10838  df-seq 11014  df-exp 11072  df-hash 11305  df-cj 11550  df-re 11551  df-im 11552  df-sqr 11686  df-abs 11687  df-clim 11928  df-sum 12125  df-ee 23895  df-btwn 23896  df-cgr 23897  df-ofs 23982  df-ifs 24038  df-cgr3 24039  df-colinear 24040  df-fs 24041
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