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Theorem idinside 24047
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 23966 . . . . . 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 23907 . . . . . 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 3737 . . . . . . . . 9  |-  ( C  =  A  ->  <. C ,  C >.  =  <. A ,  C >. )
10 opeq1 3737 . . . . . . . . 9  |-  ( C  =  A  ->  <. C ,  D >.  =  <. A ,  D >. )
119, 10breq12d 3976 . . . . . . . 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 3738 . . . . . 6  |-  ( A  =  B  ->  <. A ,  A >.  =  <. A ,  B >. )
1918breq2d 3975 . . . . 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 24032 . . . . . 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 24046 . . . . . . 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 2495 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 2419   <.cop 3584   class class class wbr 3963   ` cfv 4638   NNcn 9679   EEcee 23856    Btwn cbtwn 23857  Cgrccgr 23858    Colinear ccolin 24000
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 2237  ax-rep 4071  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449  ax-inf2 7275  ax-cnex 8726  ax-resscn 8727  ax-1cn 8728  ax-icn 8729  ax-addcl 8730  ax-addrcl 8731  ax-mulcl 8732  ax-mulrcl 8733  ax-mulcom 8734  ax-addass 8735  ax-mulass 8736  ax-distr 8737  ax-i2m1 8738  ax-1ne0 8739  ax-1rid 8740  ax-rnegex 8741  ax-rrecex 8742  ax-cnre 8743  ax-pre-lttri 8744  ax-pre-lttrn 8745  ax-pre-ltadd 8746  ax-pre-mulgt0 8747  ax-pre-sup 8748
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 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-pss 3110  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-tp 3589  df-op 3590  df-uni 3769  df-int 3804  df-iun 3848  df-br 3964  df-opab 4018  df-mpt 4019  df-tr 4054  df-eprel 4242  df-id 4246  df-po 4251  df-so 4252  df-fr 4289  df-se 4290  df-we 4291  df-ord 4332  df-on 4333  df-lim 4334  df-suc 4335  df-om 4594  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-isom 4655  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-1st 6021  df-2nd 6022  df-iota 6190  df-riota 6237  df-recs 6321  df-rdg 6356  df-1o 6412  df-oadd 6416  df-er 6593  df-map 6707  df-en 6797  df-dom 6798  df-sdom 6799  df-fin 6800  df-sup 7127  df-oi 7158  df-card 7505  df-pnf 8802  df-mnf 8803  df-xr 8804  df-ltxr 8805  df-le 8806  df-sub 8972  df-neg 8973  df-div 9357  df-n 9680  df-2 9737  df-3 9738  df-n0 9898  df-z 9957  df-uz 10163  df-rp 10287  df-ico 10593  df-icc 10594  df-fz 10714  df-fzo 10802  df-seq 10978  df-exp 11036  df-hash 11269  df-cj 11514  df-re 11515  df-im 11516  df-sqr 11650  df-abs 11651  df-clim 11892  df-sum 12089  df-ee 23859  df-btwn 23860  df-cgr 23861  df-ofs 23946  df-ifs 24002  df-cgr3 24003  df-colinear 24004  df-fs 24005
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