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Theorem ivthicc 19224
Description: The interval between any two points of a continuous real function is contained in the range of the function. Equivalently, the range of a continuous real function is convex. (Contributed by Mario Carneiro, 12-Aug-2014.)
Hypotheses
Ref Expression
ivthicc.1  |-  ( ph  ->  A  e.  RR )
ivthicc.2  |-  ( ph  ->  B  e.  RR )
ivthicc.3  |-  ( ph  ->  M  e.  ( A [,] B ) )
ivthicc.4  |-  ( ph  ->  N  e.  ( A [,] B ) )
ivthicc.5  |-  ( ph  ->  ( A [,] B
)  C_  D )
ivthicc.7  |-  ( ph  ->  F  e.  ( D
-cn-> CC ) )
ivthicc.8  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( F `  x )  e.  RR )
Assertion
Ref Expression
ivthicc  |-  ( ph  ->  ( ( F `  M ) [,] ( F `  N )
)  C_  ran  F )
Distinct variable groups:    x, D    x, F    x, M    x, N    ph, x    x, A    x, B

Proof of Theorem ivthicc
Dummy variables  z 
y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ivthicc.3 . . . . . . . 8  |-  ( ph  ->  M  e.  ( A [,] B ) )
2 ivthicc.1 . . . . . . . . 9  |-  ( ph  ->  A  e.  RR )
3 ivthicc.2 . . . . . . . . 9  |-  ( ph  ->  B  e.  RR )
4 elicc2 10909 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( M  e.  ( A [,] B )  <-> 
( M  e.  RR  /\  A  <_  M  /\  M  <_  B ) ) )
52, 3, 4syl2anc 643 . . . . . . . 8  |-  ( ph  ->  ( M  e.  ( A [,] B )  <-> 
( M  e.  RR  /\  A  <_  M  /\  M  <_  B ) ) )
61, 5mpbid 202 . . . . . . 7  |-  ( ph  ->  ( M  e.  RR  /\  A  <_  M  /\  M  <_  B ) )
76simp1d 969 . . . . . 6  |-  ( ph  ->  M  e.  RR )
8 ivthicc.4 . . . . . . . 8  |-  ( ph  ->  N  e.  ( A [,] B ) )
9 elicc2 10909 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( N  e.  ( A [,] B )  <-> 
( N  e.  RR  /\  A  <_  N  /\  N  <_  B ) ) )
102, 3, 9syl2anc 643 . . . . . . . 8  |-  ( ph  ->  ( N  e.  ( A [,] B )  <-> 
( N  e.  RR  /\  A  <_  N  /\  N  <_  B ) ) )
118, 10mpbid 202 . . . . . . 7  |-  ( ph  ->  ( N  e.  RR  /\  A  <_  N  /\  N  <_  B ) )
1211simp1d 969 . . . . . 6  |-  ( ph  ->  N  e.  RR )
137, 12lttri4d 9148 . . . . 5  |-  ( ph  ->  ( M  <  N  \/  M  =  N  \/  N  <  M ) )
1413adantr 452 . . . 4  |-  ( (
ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  ->  ( M  <  N  \/  M  =  N  \/  N  <  M ) )
15 simpll 731 . . . . . 6  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  M  <  N )  ->  ph )
167ad2antrr 707 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  M  <  N )  ->  M  e.  RR )
1712ad2antrr 707 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  M  <  N )  ->  N  e.  RR )
18 ivthicc.8 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( F `  x )  e.  RR )
1918ralrimiva 2734 . . . . . . . . . . 11  |-  ( ph  ->  A. x  e.  ( A [,] B ) ( F `  x
)  e.  RR )
20 fveq2 5670 . . . . . . . . . . . . 13  |-  ( x  =  M  ->  ( F `  x )  =  ( F `  M ) )
2120eleq1d 2455 . . . . . . . . . . . 12  |-  ( x  =  M  ->  (
( F `  x
)  e.  RR  <->  ( F `  M )  e.  RR ) )
2221rspcv 2993 . . . . . . . . . . 11  |-  ( M  e.  ( A [,] B )  ->  ( A. x  e.  ( A [,] B ) ( F `  x )  e.  RR  ->  ( F `  M )  e.  RR ) )
231, 19, 22sylc 58 . . . . . . . . . 10  |-  ( ph  ->  ( F `  M
)  e.  RR )
24 fveq2 5670 . . . . . . . . . . . . 13  |-  ( x  =  N  ->  ( F `  x )  =  ( F `  N ) )
2524eleq1d 2455 . . . . . . . . . . . 12  |-  ( x  =  N  ->  (
( F `  x
)  e.  RR  <->  ( F `  N )  e.  RR ) )
2625rspcv 2993 . . . . . . . . . . 11  |-  ( N  e.  ( A [,] B )  ->  ( A. x  e.  ( A [,] B ) ( F `  x )  e.  RR  ->  ( F `  N )  e.  RR ) )
278, 19, 26sylc 58 . . . . . . . . . 10  |-  ( ph  ->  ( F `  N
)  e.  RR )
28 iccssre 10926 . . . . . . . . . 10  |-  ( ( ( F `  M
)  e.  RR  /\  ( F `  N )  e.  RR )  -> 
( ( F `  M ) [,] ( F `  N )
)  C_  RR )
2923, 27, 28syl2anc 643 . . . . . . . . 9  |-  ( ph  ->  ( ( F `  M ) [,] ( F `  N )
)  C_  RR )
3029sselda 3293 . . . . . . . 8  |-  ( (
ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  ->  y  e.  RR )
3130adantr 452 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  M  <  N )  ->  y  e.  RR )
32 simpr 448 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  M  <  N )  ->  M  <  N )
336simp2d 970 . . . . . . . . . 10  |-  ( ph  ->  A  <_  M )
3411simp3d 971 . . . . . . . . . 10  |-  ( ph  ->  N  <_  B )
35 iccss 10912 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  <_  M  /\  N  <_  B
) )  ->  ( M [,] N )  C_  ( A [,] B ) )
362, 3, 33, 34, 35syl22anc 1185 . . . . . . . . 9  |-  ( ph  ->  ( M [,] N
)  C_  ( A [,] B ) )
37 ivthicc.5 . . . . . . . . 9  |-  ( ph  ->  ( A [,] B
)  C_  D )
3836, 37sstrd 3303 . . . . . . . 8  |-  ( ph  ->  ( M [,] N
)  C_  D )
3938ad2antrr 707 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  M  <  N )  ->  ( M [,] N )  C_  D )
40 ivthicc.7 . . . . . . . 8  |-  ( ph  ->  F  e.  ( D
-cn-> CC ) )
4140ad2antrr 707 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  M  <  N )  ->  F  e.  ( D -cn-> CC ) )
4236sselda 3293 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( M [,] N ) )  ->  x  e.  ( A [,] B ) )
4342, 18syldan 457 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( M [,] N ) )  ->  ( F `  x )  e.  RR )
4415, 43sylan 458 . . . . . . 7  |-  ( ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N
) ) )  /\  M  <  N )  /\  x  e.  ( M [,] N ) )  -> 
( F `  x
)  e.  RR )
45 elicc2 10909 . . . . . . . . . . 11  |-  ( ( ( F `  M
)  e.  RR  /\  ( F `  N )  e.  RR )  -> 
( y  e.  ( ( F `  M
) [,] ( F `
 N ) )  <-> 
( y  e.  RR  /\  ( F `  M
)  <_  y  /\  y  <_  ( F `  N ) ) ) )
4623, 27, 45syl2anc 643 . . . . . . . . . 10  |-  ( ph  ->  ( y  e.  ( ( F `  M
) [,] ( F `
 N ) )  <-> 
( y  e.  RR  /\  ( F `  M
)  <_  y  /\  y  <_  ( F `  N ) ) ) )
4746biimpa 471 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  ->  (
y  e.  RR  /\  ( F `  M )  <_  y  /\  y  <_  ( F `  N
) ) )
48 3simpc 956 . . . . . . . . 9  |-  ( ( y  e.  RR  /\  ( F `  M )  <_  y  /\  y  <_  ( F `  N
) )  ->  (
( F `  M
)  <_  y  /\  y  <_  ( F `  N ) ) )
4947, 48syl 16 . . . . . . . 8  |-  ( (
ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  ->  (
( F `  M
)  <_  y  /\  y  <_  ( F `  N ) ) )
5049adantr 452 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  M  <  N )  ->  (
( F `  M
)  <_  y  /\  y  <_  ( F `  N ) ) )
5116, 17, 31, 32, 39, 41, 44, 50ivthle 19222 . . . . . 6  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  M  <  N )  ->  E. z  e.  ( M [,] N
) ( F `  z )  =  y )
5238sselda 3293 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ( M [,] N ) )  ->  z  e.  D )
53 cncff 18796 . . . . . . . . . . 11  |-  ( F  e.  ( D -cn-> CC )  ->  F : D
--> CC )
54 ffn 5533 . . . . . . . . . . 11  |-  ( F : D --> CC  ->  F  Fn  D )
5540, 53, 543syl 19 . . . . . . . . . 10  |-  ( ph  ->  F  Fn  D )
56 fnfvelrn 5808 . . . . . . . . . 10  |-  ( ( F  Fn  D  /\  z  e.  D )  ->  ( F `  z
)  e.  ran  F
)
5755, 56sylan 458 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  D )  ->  ( F `  z )  e.  ran  F )
58 eleq1 2449 . . . . . . . . 9  |-  ( ( F `  z )  =  y  ->  (
( F `  z
)  e.  ran  F  <->  y  e.  ran  F ) )
5957, 58syl5ibcom 212 . . . . . . . 8  |-  ( (
ph  /\  z  e.  D )  ->  (
( F `  z
)  =  y  -> 
y  e.  ran  F
) )
6052, 59syldan 457 . . . . . . 7  |-  ( (
ph  /\  z  e.  ( M [,] N ) )  ->  ( ( F `  z )  =  y  ->  y  e. 
ran  F ) )
6160rexlimdva 2775 . . . . . 6  |-  ( ph  ->  ( E. z  e.  ( M [,] N
) ( F `  z )  =  y  ->  y  e.  ran  F ) )
6215, 51, 61sylc 58 . . . . 5  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  M  <  N )  ->  y  e.  ran  F )
63 simplr 732 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  M  =  N )  ->  y  e.  ( ( F `  M ) [,] ( F `  N )
) )
64 simpr 448 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  M  =  N )  ->  M  =  N )
6564fveq2d 5674 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  M  =  N )  ->  ( F `  M )  =  ( F `  N ) )
6665oveq2d 6038 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  M  =  N )  ->  (
( F `  M
) [,] ( F `
 M ) )  =  ( ( F `
 M ) [,] ( F `  N
) ) )
6723rexrd 9069 . . . . . . . . . . 11  |-  ( ph  ->  ( F `  M
)  e.  RR* )
6867ad2antrr 707 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  M  =  N )  ->  ( F `  M )  e.  RR* )
69 iccid 10895 . . . . . . . . . 10  |-  ( ( F `  M )  e.  RR*  ->  ( ( F `  M ) [,] ( F `  M ) )  =  { ( F `  M ) } )
7068, 69syl 16 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  M  =  N )  ->  (
( F `  M
) [,] ( F `
 M ) )  =  { ( F `
 M ) } )
7166, 70eqtr3d 2423 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  M  =  N )  ->  (
( F `  M
) [,] ( F `
 N ) )  =  { ( F `
 M ) } )
7263, 71eleqtrd 2465 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  M  =  N )  ->  y  e.  { ( F `  M ) } )
73 elsni 3783 . . . . . . 7  |-  ( y  e.  { ( F `
 M ) }  ->  y  =  ( F `  M ) )
7472, 73syl 16 . . . . . 6  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  M  =  N )  ->  y  =  ( F `  M ) )
7537, 1sseldd 3294 . . . . . . . 8  |-  ( ph  ->  M  e.  D )
76 fnfvelrn 5808 . . . . . . . 8  |-  ( ( F  Fn  D  /\  M  e.  D )  ->  ( F `  M
)  e.  ran  F
)
7755, 75, 76syl2anc 643 . . . . . . 7  |-  ( ph  ->  ( F `  M
)  e.  ran  F
)
7877ad2antrr 707 . . . . . 6  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  M  =  N )  ->  ( F `  M )  e.  ran  F )
7974, 78eqeltrd 2463 . . . . 5  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  M  =  N )  ->  y  e.  ran  F )
80 simpll 731 . . . . . 6  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  N  <  M )  ->  ph )
8112ad2antrr 707 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  N  <  M )  ->  N  e.  RR )
827ad2antrr 707 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  N  <  M )  ->  M  e.  RR )
8330adantr 452 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  N  <  M )  ->  y  e.  RR )
84 simpr 448 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  N  <  M )  ->  N  <  M )
8511simp2d 970 . . . . . . . . . 10  |-  ( ph  ->  A  <_  N )
866simp3d 971 . . . . . . . . . 10  |-  ( ph  ->  M  <_  B )
87 iccss 10912 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  <_  N  /\  M  <_  B
) )  ->  ( N [,] M )  C_  ( A [,] B ) )
882, 3, 85, 86, 87syl22anc 1185 . . . . . . . . 9  |-  ( ph  ->  ( N [,] M
)  C_  ( A [,] B ) )
8988, 37sstrd 3303 . . . . . . . 8  |-  ( ph  ->  ( N [,] M
)  C_  D )
9089ad2antrr 707 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  N  <  M )  ->  ( N [,] M )  C_  D )
9140ad2antrr 707 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  N  <  M )  ->  F  e.  ( D -cn-> CC ) )
9288sselda 3293 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( N [,] M ) )  ->  x  e.  ( A [,] B ) )
9392, 18syldan 457 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( N [,] M ) )  ->  ( F `  x )  e.  RR )
9480, 93sylan 458 . . . . . . 7  |-  ( ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N
) ) )  /\  N  <  M )  /\  x  e.  ( N [,] M ) )  -> 
( F `  x
)  e.  RR )
9549adantr 452 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  N  <  M )  ->  (
( F `  M
)  <_  y  /\  y  <_  ( F `  N ) ) )
9681, 82, 83, 84, 90, 91, 94, 95ivthle2 19223 . . . . . 6  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  N  <  M )  ->  E. z  e.  ( N [,] M
) ( F `  z )  =  y )
9789sselda 3293 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ( N [,] M ) )  ->  z  e.  D )
9897, 59syldan 457 . . . . . . 7  |-  ( (
ph  /\  z  e.  ( N [,] M ) )  ->  ( ( F `  z )  =  y  ->  y  e. 
ran  F ) )
9998rexlimdva 2775 . . . . . 6  |-  ( ph  ->  ( E. z  e.  ( N [,] M
) ( F `  z )  =  y  ->  y  e.  ran  F ) )
10080, 96, 99sylc 58 . . . . 5  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  N  <  M )  ->  y  e.  ran  F )
10162, 79, 1003jaodan 1250 . . . 4  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  ( M  <  N  \/  M  =  N  \/  N  <  M ) )  -> 
y  e.  ran  F
)
10214, 101mpdan 650 . . 3  |-  ( (
ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  ->  y  e.  ran  F )
103102ex 424 . 2  |-  ( ph  ->  ( y  e.  ( ( F `  M
) [,] ( F `
 N ) )  ->  y  e.  ran  F ) )
104103ssrdv 3299 1  |-  ( ph  ->  ( ( F `  M ) [,] ( F `  N )
)  C_  ran  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    \/ w3o 935    /\ w3a 936    = wceq 1649    e. wcel 1717   A.wral 2651   E.wrex 2652    C_ wss 3265   {csn 3759   class class class wbr 4155   ran crn 4821    Fn wfn 5391   -->wf 5392   ` cfv 5396  (class class class)co 6022   CCcc 8923   RRcr 8924   RR*cxr 9054    < clt 9055    <_ cle 9056   [,]cicc 10853   -cn->ccncf 18779
This theorem is referenced by:  evthicc2  19226
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-inf2 7531  ax-cnex 8981  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002  ax-pre-sup 9003  ax-mulf 9005
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-int 3995  df-iun 4039  df-iin 4040  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-se 4485  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-isom 5405  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-of 6246  df-1st 6290  df-2nd 6291  df-riota 6487  df-recs 6571  df-rdg 6606  df-1o 6662  df-2o 6663  df-oadd 6666  df-er 6843  df-map 6958  df-ixp 7002  df-en 7048  df-dom 7049  df-sdom 7050  df-fin 7051  df-fi 7353  df-sup 7383  df-oi 7414  df-card 7761  df-cda 7983  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-div 9612  df-nn 9935  df-2 9992  df-3 9993  df-4 9994  df-5 9995  df-6 9996  df-7 9997  df-8 9998  df-9 9999  df-10 10000  df-n0 10156  df-z 10217  df-dec 10317  df-uz 10423  df-q 10509  df-rp 10547  df-xneg 10644  df-xadd 10645  df-xmul 10646  df-ioo 10854  df-icc 10857  df-fz 10978  df-fzo 11068  df-seq 11253  df-exp 11312  df-hash 11548  df-cj 11833  df-re 11834  df-im 11835  df-sqr 11969  df-abs 11970  df-struct 13400  df-ndx 13401  df-slot 13402  df-base 13403  df-sets 13404  df-ress 13405  df-plusg 13471  df-mulr 13472  df-starv 13473  df-sca 13474  df-vsca 13475  df-tset 13477  df-ple 13478  df-ds 13480  df-unif 13481  df-hom 13482  df-cco 13483  df-rest 13579  df-topn 13580  df-topgen 13596  df-pt 13597  df-prds 13600  df-xrs 13655  df-0g 13656  df-gsum 13657  df-qtop 13662  df-imas 13663  df-xps 13665  df-mre 13740  df-mrc 13741  df-acs 13743  df-mnd 14619  df-submnd 14668  df-mulg 14744  df-cntz 15045  df-cmn 15343  df-xmet 16621  df-met 16622  df-bl 16623  df-mopn 16624  df-cnfld 16629  df-top 16888  df-bases 16890  df-topon 16891  df-topsp 16892  df-cn 17215  df-cnp 17216  df-tx 17517  df-hmeo 17710  df-xms 18261  df-ms 18262  df-tms 18263  df-cncf 18781
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