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Theorem colinearxfr 24074
Description: Transfer law for colinearity. Theorem 4.13 of [Schwabhauser] p. 37. (Contributed by Scott Fenton, 5-Oct-2013.)
Assertion
Ref Expression
colinearxfr  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  ->  (
( B  Colinear  <. A ,  C >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. E ,  F >. >. )  ->  E  Colinear  <. D ,  F >. ) )

Proof of Theorem colinearxfr
StepHypRef Expression
1 btwnxfr 24055 . . . . . . . 8  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  ->  (
( B  Btwn  <. A ,  C >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. E ,  F >. >. )  ->  E  Btwn  <. D ,  F >. ) )
21exp3acom23 1368 . . . . . . 7  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  ->  ( <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. E ,  F >. >.  ->  ( B  Btwn  <. A ,  C >.  ->  E  Btwn  <. D ,  F >. ) ) )
32imp 420 . . . . . 6  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  /\  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. E ,  F >. >. )  ->  ( B  Btwn  <. A ,  C >.  ->  E  Btwn  <. D ,  F >. ) )
4 cgr3permute4 24049 . . . . . . . 8  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  ->  ( <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. E ,  F >. >.  <->  <. C ,  <. A ,  B >. >.Cgr3 <. F ,  <. D ,  E >. >. ) )
5 biid 229 . . . . . . . . . 10  |-  ( N  e.  NN  <->  N  e.  NN )
6 3anrot 944 . . . . . . . . . 10  |-  ( ( C  e.  ( EE
`  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  <->  ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )
7 3anrot 944 . . . . . . . . . 10  |-  ( ( F  e.  ( EE
`  N )  /\  D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
) )  <->  ( D  e.  ( EE `  N
)  /\  E  e.  ( EE `  N )  /\  F  e.  ( EE `  N ) ) )
8 btwnxfr 24055 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  ( C  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( F  e.  ( EE `  N )  /\  D  e.  ( EE `  N
)  /\  E  e.  ( EE `  N ) ) )  ->  (
( A  Btwn  <. C ,  B >.  /\  <. C ,  <. A ,  B >. >.Cgr3 <. F ,  <. D ,  E >. >. )  ->  D  Btwn  <. F ,  E >. ) )
95, 6, 7, 8syl3anbr 1231 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  ->  (
( A  Btwn  <. C ,  B >.  /\  <. C ,  <. A ,  B >. >.Cgr3 <. F ,  <. D ,  E >. >. )  ->  D  Btwn  <. F ,  E >. ) )
109exp3acom23 1368 . . . . . . . 8  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  ->  ( <. C ,  <. A ,  B >. >.Cgr3 <. F ,  <. D ,  E >. >.  ->  ( A  Btwn  <. C ,  B >.  ->  D  Btwn  <. F ,  E >. ) ) )
114, 10sylbid 208 . . . . . . 7  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  ->  ( <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. E ,  F >. >.  ->  ( A  Btwn  <. C ,  B >.  ->  D  Btwn  <. F ,  E >. ) ) )
1211imp 420 . . . . . 6  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  /\  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. E ,  F >. >. )  ->  ( A  Btwn  <. C ,  B >.  ->  D  Btwn  <. F ,  E >. ) )
13 cgr3permute3 24046 . . . . . . . 8  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  ->  ( <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. E ,  F >. >.  <->  <. B ,  <. C ,  A >. >.Cgr3 <. E ,  <. F ,  D >. >. ) )
14 3anrot 944 . . . . . . . . . 10  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  <->  ( B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N )  /\  A  e.  ( EE `  N ) ) )
15 3anrot 944 . . . . . . . . . 10  |-  ( ( D  e.  ( EE
`  N )  /\  E  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) )  <->  ( E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) )
16 btwnxfr 24055 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N )  /\  A  e.  ( EE `  N
) )  /\  ( E  e.  ( EE `  N )  /\  F  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  ->  (
( C  Btwn  <. B ,  A >.  /\  <. B ,  <. C ,  A >. >.Cgr3 <. E ,  <. F ,  D >. >. )  ->  F  Btwn  <. E ,  D >. ) )
175, 14, 15, 16syl3anb 1230 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  ->  (
( C  Btwn  <. B ,  A >.  /\  <. B ,  <. C ,  A >. >.Cgr3 <. E ,  <. F ,  D >. >. )  ->  F  Btwn  <. E ,  D >. ) )
1817exp3acom23 1368 . . . . . . . 8  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  ->  ( <. B ,  <. C ,  A >. >.Cgr3 <. E ,  <. F ,  D >. >.  ->  ( C  Btwn  <. B ,  A >.  ->  F  Btwn  <. E ,  D >. ) ) )
1913, 18sylbid 208 . . . . . . 7  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  ->  ( <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. E ,  F >. >.  ->  ( C  Btwn  <. B ,  A >.  ->  F  Btwn  <. E ,  D >. ) ) )
2019imp 420 . . . . . 6  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  /\  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. E ,  F >. >. )  ->  ( C  Btwn  <. B ,  A >.  ->  F  Btwn  <. E ,  D >. ) )
213, 12, 203orim123d 1265 . . . . 5  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  /\  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. E ,  F >. >. )  ->  ( ( B  Btwn  <. A ,  C >.  \/  A  Btwn  <. C ,  B >.  \/  C  Btwn  <. B ,  A >. )  ->  ( E  Btwn  <. D ,  F >.  \/  D  Btwn  <. F ,  E >.  \/  F  Btwn  <. E ,  D >. ) ) )
22 simp1 960 . . . . . . 7  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  ->  N  e.  NN )
23 simp22 994 . . . . . . 7  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  ->  B  e.  ( EE `  N
) )
24 simp21 993 . . . . . . 7  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  ->  A  e.  ( EE `  N
) )
25 simp23 995 . . . . . . 7  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  ->  C  e.  ( EE `  N
) )
26 brcolinear 24058 . . . . . . 7  |-  ( ( N  e.  NN  /\  ( B  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  -> 
( B  Colinear  <. A ,  C >. 
<->  ( B  Btwn  <. A ,  C >.  \/  A  Btwn  <. C ,  B >.  \/  C  Btwn  <. B ,  A >. ) ) )
2722, 23, 24, 25, 26syl13anc 1189 . . . . . 6  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  ->  ( B  Colinear  <. A ,  C >.  <-> 
( B  Btwn  <. A ,  C >.  \/  A  Btwn  <. C ,  B >.  \/  C  Btwn  <. B ,  A >. ) ) )
2827adantr 453 . . . . 5  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  /\  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. E ,  F >. >. )  ->  ( B  Colinear  <. A ,  C >. 
<->  ( B  Btwn  <. A ,  C >.  \/  A  Btwn  <. C ,  B >.  \/  C  Btwn  <. B ,  A >. ) ) )
29 simp32 997 . . . . . . 7  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  ->  E  e.  ( EE `  N
) )
30 simp31 996 . . . . . . 7  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  ->  D  e.  ( EE `  N
) )
31 simp33 998 . . . . . . 7  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  ->  F  e.  ( EE `  N
) )
32 brcolinear 24058 . . . . . . 7  |-  ( ( N  e.  NN  /\  ( E  e.  ( EE `  N )  /\  D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  -> 
( E  Colinear  <. D ,  F >. 
<->  ( E  Btwn  <. D ,  F >.  \/  D  Btwn  <. F ,  E >.  \/  F  Btwn  <. E ,  D >. ) ) )
3322, 29, 30, 31, 32syl13anc 1189 . . . . . 6  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  ->  ( E  Colinear  <. D ,  F >.  <-> 
( E  Btwn  <. D ,  F >.  \/  D  Btwn  <. F ,  E >.  \/  F  Btwn  <. E ,  D >. ) ) )
3433adantr 453 . . . . 5  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  /\  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. E ,  F >. >. )  ->  ( E  Colinear  <. D ,  F >. 
<->  ( E  Btwn  <. D ,  F >.  \/  D  Btwn  <. F ,  E >.  \/  F  Btwn  <. E ,  D >. ) ) )
3521, 28, 343imtr4d 261 . . . 4  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  /\  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. E ,  F >. >. )  ->  ( B  Colinear  <. A ,  C >.  ->  E  Colinear  <. D ,  F >. ) )
3635ex 425 . . 3  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  ->  ( <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. E ,  F >. >.  ->  ( B  Colinear  <. A ,  C >.  ->  E  Colinear  <. D ,  F >. ) ) )
3736com23 74 . 2  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  ->  ( B  Colinear  <. A ,  C >.  ->  ( <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. E ,  F >. >.  ->  E  Colinear  <. D ,  F >. ) ) )
3837imp3a 422 1  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  ->  (
( B  Colinear  <. A ,  C >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. E ,  F >. >. )  ->  E  Colinear  <. D ,  F >. ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    \/ w3o 938    /\ w3a 939    e. wcel 1621   <.cop 3617   class class class wbr 3997   ` cfv 4673   NNcn 9714   EEcee 23892    Btwn cbtwn 23893  Cgr3ccgr3 24035    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
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