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Theorem ivthicc 19345
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 10965 . . . . . . . . 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 10965 . . . . . . . . 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 9204 . . . . 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 2781 . . . . . . . . . . 11  |-  ( ph  ->  A. x  e.  ( A [,] B ) ( F `  x
)  e.  RR )
20 fveq2 5720 . . . . . . . . . . . . 13  |-  ( x  =  M  ->  ( F `  x )  =  ( F `  M ) )
2120eleq1d 2501 . . . . . . . . . . . 12  |-  ( x  =  M  ->  (
( F `  x
)  e.  RR  <->  ( F `  M )  e.  RR ) )
2221rspcv 3040 . . . . . . . . . . 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 5720 . . . . . . . . . . . . 13  |-  ( x  =  N  ->  ( F `  x )  =  ( F `  N ) )
2524eleq1d 2501 . . . . . . . . . . . 12  |-  ( x  =  N  ->  (
( F `  x
)  e.  RR  <->  ( F `  N )  e.  RR ) )
2625rspcv 3040 . . . . . . . . . . 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 10982 . . . . . . . . . 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 3340 . . . . . . . 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 10968 . . . . . . . . . 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 3350 . . . . . . . 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 3340 . . . . . . . . 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 10965 . . . . . . . . . . 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 19343 . . . . . 6  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  M  <  N )  ->  E. z  e.  ( M [,] N
) ( F `  z )  =  y )
5238sselda 3340 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ( M [,] N ) )  ->  z  e.  D )
53 cncff 18913 . . . . . . . . . . 11  |-  ( F  e.  ( D -cn-> CC )  ->  F : D
--> CC )
54 ffn 5583 . . . . . . . . . . 11  |-  ( F : D --> CC  ->  F  Fn  D )
5540, 53, 543syl 19 . . . . . . . . . 10  |-  ( ph  ->  F  Fn  D )
56 fnfvelrn 5859 . . . . . . . . . 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 2495 . . . . . . . . 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 2822 . . . . . 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 5724 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  M  =  N )  ->  ( F `  M )  =  ( F `  N ) )
6665oveq2d 6089 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  M  =  N )  ->  (
( F `  M
) [,] ( F `
 M ) )  =  ( ( F `
 M ) [,] ( F `  N
) ) )
6723rexrd 9124 . . . . . . . . . . 11  |-  ( ph  ->  ( F `  M
)  e.  RR* )
6867ad2antrr 707 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  M  =  N )  ->  ( F `  M )  e.  RR* )
69 iccid 10951 . . . . . . . . . 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 2469 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  M  =  N )  ->  (
( F `  M
) [,] ( F `
 N ) )  =  { ( F `
 M ) } )
7263, 71eleqtrd 2511 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  M  =  N )  ->  y  e.  { ( F `  M ) } )
73 elsni 3830 . . . . . . 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 3341 . . . . . . . 8  |-  ( ph  ->  M  e.  D )
76 fnfvelrn 5859 . . . . . . . 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 2509 . . . . 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 10968 . . . . . . . . . 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 3350 . . . . . . . 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 3340 . . . . . . . . 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 19344 . . . . . 6  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  N  <  M )  ->  E. z  e.  ( N [,] M
) ( F `  z )  =  y )
9789sselda 3340 . . . . . . . 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 2822 . . . . . 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 3346 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 1652    e. wcel 1725   A.wral 2697   E.wrex 2698    C_ wss 3312   {csn 3806   class class class wbr 4204   ran crn 4871    Fn wfn 5441   -->wf 5442   ` cfv 5446  (class class class)co 6073   CCcc 8978   RRcr 8979   RR*cxr 9109    < clt 9110    <_ cle 9111   [,]cicc 10909   -cn->ccncf 18896
This theorem is referenced by:  evthicc2  19347
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7586  ax-cnex 9036  ax-resscn 9037  ax-1cn 9038  ax-icn 9039  ax-addcl 9040  ax-addrcl 9041  ax-mulcl 9042  ax-mulrcl 9043  ax-mulcom 9044  ax-addass 9045  ax-mulass 9046  ax-distr 9047  ax-i2m1 9048  ax-1ne0 9049  ax-1rid 9050  ax-rnegex 9051  ax-rrecex 9052  ax-cnre 9053  ax-pre-lttri 9054  ax-pre-lttrn 9055  ax-pre-ltadd 9056  ax-pre-mulgt0 9057  ax-pre-sup 9058  ax-mulf 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-2o 6717  df-oadd 6720  df-er 6897  df-map 7012  df-ixp 7056  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-fi 7408  df-sup 7438  df-oi 7469  df-card 7816  df-cda 8038  df-pnf 9112  df-mnf 9113  df-xr 9114  df-ltxr 9115  df-le 9116  df-sub 9283  df-neg 9284  df-div 9668  df-nn 9991  df-2 10048  df-3 10049  df-4 10050  df-5 10051  df-6 10052  df-7 10053  df-8 10054  df-9 10055  df-10 10056  df-n0 10212  df-z 10273  df-dec 10373  df-uz 10479  df-q 10565  df-rp 10603  df-xneg 10700  df-xadd 10701  df-xmul 10702  df-ioo 10910  df-icc 10913  df-fz 11034  df-fzo 11126  df-seq 11314  df-exp 11373  df-hash 11609  df-cj 11894  df-re 11895  df-im 11896  df-sqr 12030  df-abs 12031  df-struct 13461  df-ndx 13462  df-slot 13463  df-base 13464  df-sets 13465  df-ress 13466  df-plusg 13532  df-mulr 13533  df-starv 13534  df-sca 13535  df-vsca 13536  df-tset 13538  df-ple 13539  df-ds 13541  df-unif 13542  df-hom 13543  df-cco 13544  df-rest 13640  df-topn 13641  df-topgen 13657  df-pt 13658  df-prds 13661  df-xrs 13716  df-0g 13717  df-gsum 13718  df-qtop 13723  df-imas 13724  df-xps 13726  df-mre 13801  df-mrc 13802  df-acs 13804  df-mnd 14680  df-submnd 14729  df-mulg 14805  df-cntz 15106  df-cmn 15404  df-psmet 16684  df-xmet 16685  df-met 16686  df-bl 16687  df-mopn 16688  df-cnfld 16694  df-top 16953  df-bases 16955  df-topon 16956  df-topsp 16957  df-cn 17281  df-cnp 17282  df-tx 17584  df-hmeo 17777  df-xms 18340  df-ms 18341  df-tms 18342  df-cncf 18898
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