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Theorem btwnouttr2 24020
Description: Outer transitivity law for betweenness. Left hand side of Theorem 3.1 of [Schwabhauser] p. 30. (Contributed by Scott Fenton, 12-Jun-2013.)
Assertion
Ref Expression
btwnouttr2  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) )  ->  ( ( B  =/=  C  /\  B  Btwn  <. A ,  C >.  /\  C  Btwn  <. B ,  D >. )  ->  C  Btwn  <. A ,  D >. ) )

Proof of Theorem btwnouttr2
StepHypRef Expression
1 simp1 960 . . . . 5  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) )  ->  N  e.  NN )
2 simp2l 986 . . . . 5  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) )  ->  A  e.  ( EE `  N ) )
3 simp3l 988 . . . . 5  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) )  ->  C  e.  ( EE `  N ) )
4 simp3r 989 . . . . 5  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) )  ->  D  e.  ( EE `  N ) )
5 axsegcon 23930 . . . . 5  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) )  ->  E. x  e.  ( EE `  N ) ( C  Btwn  <. A ,  x >.  /\  <. C ,  x >.Cgr <. C ,  D >. ) )
61, 2, 3, 3, 4, 5syl122anc 1196 . . . 4  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) )  ->  E. x  e.  ( EE `  N ) ( C  Btwn  <. A ,  x >.  /\  <. C ,  x >.Cgr <. C ,  D >. ) )
76adantr 453 . . 3  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  ( B  =/=  C  /\  B  Btwn  <. A ,  C >.  /\  C  Btwn  <. B ,  D >. ) )  ->  E. x  e.  ( EE `  N
) ( C  Btwn  <. A ,  x >.  /\ 
<. C ,  x >.Cgr <. C ,  D >. ) )
8 simprrl 743 . . . . . . 7  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  x  e.  ( EE `  N ) )  /\  ( ( B  =/= 
C  /\  B  Btwn  <. A ,  C >.  /\  C  Btwn  <. B ,  D >. )  /\  ( C  Btwn  <. A ,  x >.  /\  <. C ,  x >.Cgr
<. C ,  D >. ) ) )  ->  C  Btwn  <. A ,  x >. )
9 simprl1 1005 . . . . . . . . 9  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  x  e.  ( EE `  N ) )  /\  ( ( B  =/= 
C  /\  B  Btwn  <. A ,  C >.  /\  C  Btwn  <. B ,  D >. )  /\  ( C  Btwn  <. A ,  x >.  /\  <. C ,  x >.Cgr
<. C ,  D >. ) ) )  ->  B  =/=  C )
10 simpl2 964 . . . . . . . . . . . . 13  |-  ( ( ( B  =/=  C  /\  B  Btwn  <. A ,  C >.  /\  C  Btwn  <. B ,  D >. )  /\  ( C  Btwn  <. A ,  x >.  /\ 
<. C ,  x >.Cgr <. C ,  D >. ) )  ->  B  Btwn  <. A ,  C >. )
11 simprl 735 . . . . . . . . . . . . 13  |-  ( ( ( B  =/=  C  /\  B  Btwn  <. A ,  C >.  /\  C  Btwn  <. B ,  D >. )  /\  ( C  Btwn  <. A ,  x >.  /\ 
<. C ,  x >.Cgr <. C ,  D >. ) )  ->  C  Btwn  <. A ,  x >. )
1210, 11jca 520 . . . . . . . . . . . 12  |-  ( ( ( B  =/=  C  /\  B  Btwn  <. A ,  C >.  /\  C  Btwn  <. B ,  D >. )  /\  ( C  Btwn  <. A ,  x >.  /\ 
<. C ,  x >.Cgr <. C ,  D >. ) )  ->  ( B  Btwn  <. A ,  C >.  /\  C  Btwn  <. A ,  x >. ) )
1312adantl 454 . . . . . . . . . . 11  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  x  e.  ( EE `  N ) )  /\  ( ( B  =/= 
C  /\  B  Btwn  <. A ,  C >.  /\  C  Btwn  <. B ,  D >. )  /\  ( C  Btwn  <. A ,  x >.  /\  <. C ,  x >.Cgr
<. C ,  D >. ) ) )  ->  ( B  Btwn  <. A ,  C >.  /\  C  Btwn  <. A ,  x >. ) )
14 simpl1 963 . . . . . . . . . . . . 13  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  x  e.  ( EE `  N ) )  ->  N  e.  NN )
15 simpl2l 1013 . . . . . . . . . . . . 13  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  x  e.  ( EE `  N ) )  ->  A  e.  ( EE `  N ) )
16 simpl2r 1014 . . . . . . . . . . . . 13  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  x  e.  ( EE `  N ) )  ->  B  e.  ( EE `  N ) )
17 simpl3l 1015 . . . . . . . . . . . . 13  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  x  e.  ( EE `  N ) )  ->  C  e.  ( EE `  N ) )
18 simpr 449 . . . . . . . . . . . . 13  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  x  e.  ( EE `  N ) )  ->  x  e.  ( EE `  N ) )
19 btwnexch3 24018 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  x  e.  ( EE `  N ) ) )  ->  ( ( B 
Btwn  <. A ,  C >.  /\  C  Btwn  <. A ,  x >. )  ->  C  Btwn  <. B ,  x >. ) )
2014, 15, 16, 17, 18, 19syl122anc 1196 . . . . . . . . . . . 12  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  x  e.  ( EE `  N ) )  -> 
( ( B  Btwn  <. A ,  C >.  /\  C  Btwn  <. A ,  x >. )  ->  C  Btwn  <. B ,  x >. ) )
2120adantr 453 . . . . . . . . . . 11  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  x  e.  ( EE `  N ) )  /\  ( ( B  =/= 
C  /\  B  Btwn  <. A ,  C >.  /\  C  Btwn  <. B ,  D >. )  /\  ( C  Btwn  <. A ,  x >.  /\  <. C ,  x >.Cgr
<. C ,  D >. ) ) )  ->  (
( B  Btwn  <. A ,  C >.  /\  C  Btwn  <. A ,  x >. )  ->  C  Btwn  <. B ,  x >. ) )
2213, 21mpd 16 . . . . . . . . . 10  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  x  e.  ( EE `  N ) )  /\  ( ( B  =/= 
C  /\  B  Btwn  <. A ,  C >.  /\  C  Btwn  <. B ,  D >. )  /\  ( C  Btwn  <. A ,  x >.  /\  <. C ,  x >.Cgr
<. C ,  D >. ) ) )  ->  C  Btwn  <. B ,  x >. )
23 simprrr 744 . . . . . . . . . 10  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  x  e.  ( EE `  N ) )  /\  ( ( B  =/= 
C  /\  B  Btwn  <. A ,  C >.  /\  C  Btwn  <. B ,  D >. )  /\  ( C  Btwn  <. A ,  x >.  /\  <. C ,  x >.Cgr
<. C ,  D >. ) ) )  ->  <. C ,  x >.Cgr <. C ,  D >. )
2422, 23jca 520 . . . . . . . . 9  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  x  e.  ( EE `  N ) )  /\  ( ( B  =/= 
C  /\  B  Btwn  <. A ,  C >.  /\  C  Btwn  <. B ,  D >. )  /\  ( C  Btwn  <. A ,  x >.  /\  <. C ,  x >.Cgr
<. C ,  D >. ) ) )  ->  ( C  Btwn  <. B ,  x >.  /\  <. C ,  x >.Cgr
<. C ,  D >. ) )
25 simprl3 1007 . . . . . . . . . 10  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  x  e.  ( EE `  N ) )  /\  ( ( B  =/= 
C  /\  B  Btwn  <. A ,  C >.  /\  C  Btwn  <. B ,  D >. )  /\  ( C  Btwn  <. A ,  x >.  /\  <. C ,  x >.Cgr
<. C ,  D >. ) ) )  ->  C  Btwn  <. B ,  D >. )
26 simpl3r 1016 . . . . . . . . . . . 12  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  x  e.  ( EE `  N ) )  ->  D  e.  ( EE `  N ) )
2714, 17, 26cgrrflxd 23986 . . . . . . . . . . 11  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  x  e.  ( EE `  N ) )  ->  <. C ,  D >.Cgr <. C ,  D >. )
2827adantr 453 . . . . . . . . . 10  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  x  e.  ( EE `  N ) )  /\  ( ( B  =/= 
C  /\  B  Btwn  <. A ,  C >.  /\  C  Btwn  <. B ,  D >. )  /\  ( C  Btwn  <. A ,  x >.  /\  <. C ,  x >.Cgr
<. C ,  D >. ) ) )  ->  <. C ,  D >.Cgr <. C ,  D >. )
2925, 28jca 520 . . . . . . . . 9  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  x  e.  ( EE `  N ) )  /\  ( ( B  =/= 
C  /\  B  Btwn  <. A ,  C >.  /\  C  Btwn  <. B ,  D >. )  /\  ( C  Btwn  <. A ,  x >.  /\  <. C ,  x >.Cgr
<. C ,  D >. ) ) )  ->  ( C  Btwn  <. B ,  D >.  /\  <. C ,  D >.Cgr
<. C ,  D >. ) )
30 segconeq 24008 . . . . . . . . . . 11  |-  ( ( N  e.  NN  /\  ( C  e.  ( EE `  N )  /\  C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) )  /\  ( B  e.  ( EE `  N )  /\  x  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  ->  (
( B  =/=  C  /\  ( C  Btwn  <. B ,  x >.  /\  <. C ,  x >.Cgr <. C ,  D >. )  /\  ( C 
Btwn  <. B ,  D >.  /\  <. C ,  D >.Cgr
<. C ,  D >. ) )  ->  x  =  D ) )
3114, 17, 17, 26, 16, 18, 26, 30syl133anc 1210 . . . . . . . . . 10  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  x  e.  ( EE `  N ) )  -> 
( ( B  =/= 
C  /\  ( C  Btwn  <. B ,  x >.  /\  <. C ,  x >.Cgr
<. C ,  D >. )  /\  ( C  Btwn  <. B ,  D >.  /\ 
<. C ,  D >.Cgr <. C ,  D >. ) )  ->  x  =  D ) )
3231adantr 453 . . . . . . . . 9  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  x  e.  ( EE `  N ) )  /\  ( ( B  =/= 
C  /\  B  Btwn  <. A ,  C >.  /\  C  Btwn  <. B ,  D >. )  /\  ( C  Btwn  <. A ,  x >.  /\  <. C ,  x >.Cgr
<. C ,  D >. ) ) )  ->  (
( B  =/=  C  /\  ( C  Btwn  <. B ,  x >.  /\  <. C ,  x >.Cgr <. C ,  D >. )  /\  ( C 
Btwn  <. B ,  D >.  /\  <. C ,  D >.Cgr
<. C ,  D >. ) )  ->  x  =  D ) )
339, 24, 29, 32mp3and 1285 . . . . . . . 8  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  x  e.  ( EE `  N ) )  /\  ( ( B  =/= 
C  /\  B  Btwn  <. A ,  C >.  /\  C  Btwn  <. B ,  D >. )  /\  ( C  Btwn  <. A ,  x >.  /\  <. C ,  x >.Cgr
<. C ,  D >. ) ) )  ->  x  =  D )
3433opeq2d 3777 . . . . . . 7  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  x  e.  ( EE `  N ) )  /\  ( ( B  =/= 
C  /\  B  Btwn  <. A ,  C >.  /\  C  Btwn  <. B ,  D >. )  /\  ( C  Btwn  <. A ,  x >.  /\  <. C ,  x >.Cgr
<. C ,  D >. ) ) )  ->  <. A ,  x >.  =  <. A ,  D >. )
358, 34breqtrd 4021 . . . . . 6  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  x  e.  ( EE `  N ) )  /\  ( ( B  =/= 
C  /\  B  Btwn  <. A ,  C >.  /\  C  Btwn  <. B ,  D >. )  /\  ( C  Btwn  <. A ,  x >.  /\  <. C ,  x >.Cgr
<. C ,  D >. ) ) )  ->  C  Btwn  <. A ,  D >. )
3635expr 601 . . . . 5  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  x  e.  ( EE `  N ) )  /\  ( B  =/=  C  /\  B  Btwn  <. A ,  C >.  /\  C  Btwn  <. B ,  D >. ) )  ->  ( ( C  Btwn  <. A ,  x >.  /\  <. C ,  x >.Cgr
<. C ,  D >. )  ->  C  Btwn  <. A ,  D >. ) )
3736an32s 782 . . . 4  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  ( B  =/=  C  /\  B  Btwn  <. A ,  C >.  /\  C  Btwn  <. B ,  D >. ) )  /\  x  e.  ( EE `  N
) )  ->  (
( C  Btwn  <. A ,  x >.  /\  <. C ,  x >.Cgr <. C ,  D >. )  ->  C  Btwn  <. A ,  D >. ) )
3837rexlimdva 2642 . . 3  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  ( B  =/=  C  /\  B  Btwn  <. A ,  C >.  /\  C  Btwn  <. B ,  D >. ) )  ->  ( E. x  e.  ( EE `  N ) ( C 
Btwn  <. A ,  x >.  /\  <. C ,  x >.Cgr
<. C ,  D >. )  ->  C  Btwn  <. A ,  D >. ) )
397, 38mpd 16 . 2  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  ( B  =/=  C  /\  B  Btwn  <. A ,  C >.  /\  C  Btwn  <. B ,  D >. ) )  ->  C  Btwn  <. A ,  D >. )
4039ex 425 1  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) )  ->  ( ( B  =/=  C  /\  B  Btwn  <. A ,  C >.  /\  C  Btwn  <. B ,  D >. )  ->  C  Btwn  <. A ,  D >. ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    =/= wne 2421   E.wrex 2519   <.cop 3617   class class class wbr 3997   ` cfv 4673   NNcn 9714   EEcee 23891    Btwn cbtwn 23892  Cgrccgr 23893
This theorem is referenced by:  btwnexch2  24021  btwnouttr  24022  btwnoutside  24123  lineelsb2  24146
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 9933  df-z 9992  df-uz 10198  df-rp 10322  df-ico 10628  df-icc 10629  df-fz 10749  df-fzo 10837  df-seq 11013  df-exp 11071  df-hash 11304  df-cj 11549  df-re 11550  df-im 11551  df-sqr 11685  df-abs 11686  df-clim 11927  df-sum 12124  df-ee 23894  df-btwn 23895  df-cgr 23896  df-ofs 23981
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