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Theorem ivthicc 19308
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 10931 . . . . . . . . 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 10931 . . . . . . . . 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 9170 . . . . 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 2749 . . . . . . . . . . 11  |-  ( ph  ->  A. x  e.  ( A [,] B ) ( F `  x
)  e.  RR )
20 fveq2 5687 . . . . . . . . . . . . 13  |-  ( x  =  M  ->  ( F `  x )  =  ( F `  M ) )
2120eleq1d 2470 . . . . . . . . . . . 12  |-  ( x  =  M  ->  (
( F `  x
)  e.  RR  <->  ( F `  M )  e.  RR ) )
2221rspcv 3008 . . . . . . . . . . 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 5687 . . . . . . . . . . . . 13  |-  ( x  =  N  ->  ( F `  x )  =  ( F `  N ) )
2524eleq1d 2470 . . . . . . . . . . . 12  |-  ( x  =  N  ->  (
( F `  x
)  e.  RR  <->  ( F `  N )  e.  RR ) )
2625rspcv 3008 . . . . . . . . . . 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 10948 . . . . . . . . . 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 3308 . . . . . . . 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 10934 . . . . . . . . . 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 3318 . . . . . . . 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 3308 . . . . . . . . 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 10931 . . . . . . . . . . 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 19306 . . . . . 6  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  M  <  N )  ->  E. z  e.  ( M [,] N
) ( F `  z )  =  y )
5238sselda 3308 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ( M [,] N ) )  ->  z  e.  D )
53 cncff 18876 . . . . . . . . . . 11  |-  ( F  e.  ( D -cn-> CC )  ->  F : D
--> CC )
54 ffn 5550 . . . . . . . . . . 11  |-  ( F : D --> CC  ->  F  Fn  D )
5540, 53, 543syl 19 . . . . . . . . . 10  |-  ( ph  ->  F  Fn  D )
56 fnfvelrn 5826 . . . . . . . . . 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 2464 . . . . . . . . 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 2790 . . . . . 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 5691 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  M  =  N )  ->  ( F `  M )  =  ( F `  N ) )
6665oveq2d 6056 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  M  =  N )  ->  (
( F `  M
) [,] ( F `
 M ) )  =  ( ( F `
 M ) [,] ( F `  N
) ) )
6723rexrd 9090 . . . . . . . . . . 11  |-  ( ph  ->  ( F `  M
)  e.  RR* )
6867ad2antrr 707 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  M  =  N )  ->  ( F `  M )  e.  RR* )
69 iccid 10917 . . . . . . . . . 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 2438 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  M  =  N )  ->  (
( F `  M
) [,] ( F `
 N ) )  =  { ( F `
 M ) } )
7263, 71eleqtrd 2480 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  M  =  N )  ->  y  e.  { ( F `  M ) } )
73 elsni 3798 . . . . . . 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 3309 . . . . . . . 8  |-  ( ph  ->  M  e.  D )
76 fnfvelrn 5826 . . . . . . . 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 2478 . . . . 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 10934 . . . . . . . . . 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 3318 . . . . . . . 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 3308 . . . . . . . . 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 19307 . . . . . 6  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  N  <  M )  ->  E. z  e.  ( N [,] M
) ( F `  z )  =  y )
9789sselda 3308 . . . . . . . 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 2790 . . . . . 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 3314 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 1721   A.wral 2666   E.wrex 2667    C_ wss 3280   {csn 3774   class class class wbr 4172   ran crn 4838    Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6040   CCcc 8944   RRcr 8945   RR*cxr 9075    < clt 9076    <_ cle 9077   [,]cicc 10875   -cn->ccncf 18859
This theorem is referenced by:  evthicc2  19310
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-mulf 9026
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-sup 7404  df-oi 7435  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-ioo 10876  df-icc 10879  df-fz 11000  df-fzo 11091  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-hom 13508  df-cco 13509  df-rest 13605  df-topn 13606  df-topgen 13622  df-pt 13623  df-prds 13626  df-xrs 13681  df-0g 13682  df-gsum 13683  df-qtop 13688  df-imas 13689  df-xps 13691  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-submnd 14694  df-mulg 14770  df-cntz 15071  df-cmn 15369  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-cnfld 16659  df-top 16918  df-bases 16920  df-topon 16921  df-topsp 16922  df-cn 17245  df-cnp 17246  df-tx 17547  df-hmeo 17740  df-xms 18303  df-ms 18304  df-tms 18305  df-cncf 18861
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