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Theorem idinside 24117
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 24036 . . . . . 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 23977 . . . . . 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 3797 . . . . . . . . 9  |-  ( C  =  A  ->  <. C ,  C >.  =  <. A ,  C >. )
10 opeq1 3797 . . . . . . . . 9  |-  ( C  =  A  ->  <. C ,  D >.  =  <. A ,  D >. )
119, 10breq12d 4037 . . . . . . . 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 3798 . . . . . 6  |-  ( A  =  B  ->  <. A ,  A >.  =  <. A ,  B >. )
1918breq2d 4036 . . . . 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 24102 . . . . . 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 24116 . . . . . . 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 2523 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 1685    =/= wne 2447   <.cop 3644   class class class wbr 4024   ` cfv 5221   NNcn 9742   EEcee 23926    Btwn cbtwn 23927  Cgrccgr 23928    Colinear ccolin 24070
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 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-inf2 7338  ax-cnex 8789  ax-resscn 8790  ax-1cn 8791  ax-icn 8792  ax-addcl 8793  ax-addrcl 8794  ax-mulcl 8795  ax-mulrcl 8796  ax-mulcom 8797  ax-addass 8798  ax-mulass 8799  ax-distr 8800  ax-i2m1 8801  ax-1ne0 8802  ax-1rid 8803  ax-rnegex 8804  ax-rrecex 8805  ax-cnre 8806  ax-pre-lttri 8807  ax-pre-lttrn 8808  ax-pre-ltadd 8809  ax-pre-mulgt0 8810  ax-pre-sup 8811
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 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-int 3864  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-se 4352  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-isom 5230  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-1st 6084  df-2nd 6085  df-iota 6253  df-riota 6300  df-recs 6384  df-rdg 6419  df-1o 6475  df-oadd 6479  df-er 6656  df-map 6770  df-en 6860  df-dom 6861  df-sdom 6862  df-fin 6863  df-sup 7190  df-oi 7221  df-card 7568  df-pnf 8865  df-mnf 8866  df-xr 8867  df-ltxr 8868  df-le 8869  df-sub 9035  df-neg 9036  df-div 9420  df-nn 9743  df-2 9800  df-3 9801  df-n0 9962  df-z 10021  df-uz 10227  df-rp 10351  df-ico 10658  df-icc 10659  df-fz 10779  df-fzo 10867  df-seq 11043  df-exp 11101  df-hash 11334  df-cj 11580  df-re 11581  df-im 11582  df-sqr 11716  df-abs 11717  df-clim 11958  df-sum 12155  df-ee 23929  df-btwn 23930  df-cgr 23931  df-ofs 24016  df-ifs 24072  df-cgr3 24073  df-colinear 24074  df-fs 24075
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