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Theorem ivthicc 21738
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 11601 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( M  e.  ( A [,] B )  <-> 
( M  e.  RR  /\  A  <_  M  /\  M  <_  B ) ) )
52, 3, 4syl2anc 661 . . . . . . . 8  |-  ( ph  ->  ( M  e.  ( A [,] B )  <-> 
( M  e.  RR  /\  A  <_  M  /\  M  <_  B ) ) )
61, 5mpbid 210 . . . . . . 7  |-  ( ph  ->  ( M  e.  RR  /\  A  <_  M  /\  M  <_  B ) )
76simp1d 1008 . . . . . 6  |-  ( ph  ->  M  e.  RR )
8 ivthicc.4 . . . . . . . 8  |-  ( ph  ->  N  e.  ( A [,] B ) )
9 elicc2 11601 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( N  e.  ( A [,] B )  <-> 
( N  e.  RR  /\  A  <_  N  /\  N  <_  B ) ) )
102, 3, 9syl2anc 661 . . . . . . . 8  |-  ( ph  ->  ( N  e.  ( A [,] B )  <-> 
( N  e.  RR  /\  A  <_  N  /\  N  <_  B ) ) )
118, 10mpbid 210 . . . . . . 7  |-  ( ph  ->  ( N  e.  RR  /\  A  <_  N  /\  N  <_  B ) )
1211simp1d 1008 . . . . . 6  |-  ( ph  ->  N  e.  RR )
137, 12lttri4d 9737 . . . . 5  |-  ( ph  ->  ( M  <  N  \/  M  =  N  \/  N  <  M ) )
1413adantr 465 . . . 4  |-  ( (
ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  ->  ( M  <  N  \/  M  =  N  \/  N  <  M ) )
15 simpll 753 . . . . . 6  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  M  <  N )  ->  ph )
167ad2antrr 725 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  M  <  N )  ->  M  e.  RR )
1712ad2antrr 725 . . . . . . 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 2881 . . . . . . . . . . 11  |-  ( ph  ->  A. x  e.  ( A [,] B ) ( F `  x
)  e.  RR )
20 fveq2 5872 . . . . . . . . . . . . 13  |-  ( x  =  M  ->  ( F `  x )  =  ( F `  M ) )
2120eleq1d 2536 . . . . . . . . . . . 12  |-  ( x  =  M  ->  (
( F `  x
)  e.  RR  <->  ( F `  M )  e.  RR ) )
2221rspcv 3215 . . . . . . . . . . 11  |-  ( M  e.  ( A [,] B )  ->  ( A. x  e.  ( A [,] B ) ( F `  x )  e.  RR  ->  ( F `  M )  e.  RR ) )
231, 19, 22sylc 60 . . . . . . . . . 10  |-  ( ph  ->  ( F `  M
)  e.  RR )
24 fveq2 5872 . . . . . . . . . . . . 13  |-  ( x  =  N  ->  ( F `  x )  =  ( F `  N ) )
2524eleq1d 2536 . . . . . . . . . . . 12  |-  ( x  =  N  ->  (
( F `  x
)  e.  RR  <->  ( F `  N )  e.  RR ) )
2625rspcv 3215 . . . . . . . . . . 11  |-  ( N  e.  ( A [,] B )  ->  ( A. x  e.  ( A [,] B ) ( F `  x )  e.  RR  ->  ( F `  N )  e.  RR ) )
278, 19, 26sylc 60 . . . . . . . . . 10  |-  ( ph  ->  ( F `  N
)  e.  RR )
28 iccssre 11618 . . . . . . . . . 10  |-  ( ( ( F `  M
)  e.  RR  /\  ( F `  N )  e.  RR )  -> 
( ( F `  M ) [,] ( F `  N )
)  C_  RR )
2923, 27, 28syl2anc 661 . . . . . . . . 9  |-  ( ph  ->  ( ( F `  M ) [,] ( F `  N )
)  C_  RR )
3029sselda 3509 . . . . . . . 8  |-  ( (
ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  ->  y  e.  RR )
3130adantr 465 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  M  <  N )  ->  y  e.  RR )
32 simpr 461 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  M  <  N )  ->  M  <  N )
336simp2d 1009 . . . . . . . . . 10  |-  ( ph  ->  A  <_  M )
3411simp3d 1010 . . . . . . . . . 10  |-  ( ph  ->  N  <_  B )
35 iccss 11604 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  <_  M  /\  N  <_  B
) )  ->  ( M [,] N )  C_  ( A [,] B ) )
362, 3, 33, 34, 35syl22anc 1229 . . . . . . . . 9  |-  ( ph  ->  ( M [,] N
)  C_  ( A [,] B ) )
37 ivthicc.5 . . . . . . . . 9  |-  ( ph  ->  ( A [,] B
)  C_  D )
3836, 37sstrd 3519 . . . . . . . 8  |-  ( ph  ->  ( M [,] N
)  C_  D )
3938ad2antrr 725 . . . . . . 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 725 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  M  <  N )  ->  F  e.  ( D -cn-> CC ) )
4236sselda 3509 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( M [,] N ) )  ->  x  e.  ( A [,] B ) )
4342, 18syldan 470 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( M [,] N ) )  ->  ( F `  x )  e.  RR )
4415, 43sylan 471 . . . . . . 7  |-  ( ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N
) ) )  /\  M  <  N )  /\  x  e.  ( M [,] N ) )  -> 
( F `  x
)  e.  RR )
45 elicc2 11601 . . . . . . . . . . 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 661 . . . . . . . . . 10  |-  ( ph  ->  ( y  e.  ( ( F `  M
) [,] ( F `
 N ) )  <-> 
( y  e.  RR  /\  ( F `  M
)  <_  y  /\  y  <_  ( F `  N ) ) ) )
4746biimpa 484 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  ->  (
y  e.  RR  /\  ( F `  M )  <_  y  /\  y  <_  ( F `  N
) ) )
48 3simpc 995 . . . . . . . . 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 465 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  M  <  N )  ->  (
( F `  M
)  <_  y  /\  y  <_  ( F `  N ) ) )
5116, 17, 31, 32, 39, 41, 44, 50ivthle 21736 . . . . . 6  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  M  <  N )  ->  E. z  e.  ( M [,] N
) ( F `  z )  =  y )
5238sselda 3509 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ( M [,] N ) )  ->  z  e.  D )
53 cncff 21265 . . . . . . . . . . 11  |-  ( F  e.  ( D -cn-> CC )  ->  F : D
--> CC )
54 ffn 5737 . . . . . . . . . . 11  |-  ( F : D --> CC  ->  F  Fn  D )
5540, 53, 543syl 20 . . . . . . . . . 10  |-  ( ph  ->  F  Fn  D )
56 fnfvelrn 6029 . . . . . . . . . 10  |-  ( ( F  Fn  D  /\  z  e.  D )  ->  ( F `  z
)  e.  ran  F
)
5755, 56sylan 471 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  D )  ->  ( F `  z )  e.  ran  F )
58 eleq1 2539 . . . . . . . . 9  |-  ( ( F `  z )  =  y  ->  (
( F `  z
)  e.  ran  F  <->  y  e.  ran  F ) )
5957, 58syl5ibcom 220 . . . . . . . 8  |-  ( (
ph  /\  z  e.  D )  ->  (
( F `  z
)  =  y  -> 
y  e.  ran  F
) )
6052, 59syldan 470 . . . . . . 7  |-  ( (
ph  /\  z  e.  ( M [,] N ) )  ->  ( ( F `  z )  =  y  ->  y  e. 
ran  F ) )
6160rexlimdva 2959 . . . . . 6  |-  ( ph  ->  ( E. z  e.  ( M [,] N
) ( F `  z )  =  y  ->  y  e.  ran  F ) )
6215, 51, 61sylc 60 . . . . 5  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  M  <  N )  ->  y  e.  ran  F )
63 simplr 754 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  M  =  N )  ->  y  e.  ( ( F `  M ) [,] ( F `  N )
) )
64 simpr 461 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  M  =  N )  ->  M  =  N )
6564fveq2d 5876 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  M  =  N )  ->  ( F `  M )  =  ( F `  N ) )
6665oveq2d 6311 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  M  =  N )  ->  (
( F `  M
) [,] ( F `
 M ) )  =  ( ( F `
 M ) [,] ( F `  N
) ) )
6723rexrd 9655 . . . . . . . . . . 11  |-  ( ph  ->  ( F `  M
)  e.  RR* )
6867ad2antrr 725 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  M  =  N )  ->  ( F `  M )  e.  RR* )
69 iccid 11586 . . . . . . . . . 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 2510 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  M  =  N )  ->  (
( F `  M
) [,] ( F `
 N ) )  =  { ( F `
 M ) } )
7263, 71eleqtrd 2557 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  M  =  N )  ->  y  e.  { ( F `  M ) } )
73 elsni 4058 . . . . . . 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 3510 . . . . . . . 8  |-  ( ph  ->  M  e.  D )
76 fnfvelrn 6029 . . . . . . . 8  |-  ( ( F  Fn  D  /\  M  e.  D )  ->  ( F `  M
)  e.  ran  F
)
7755, 75, 76syl2anc 661 . . . . . . 7  |-  ( ph  ->  ( F `  M
)  e.  ran  F
)
7877ad2antrr 725 . . . . . 6  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  M  =  N )  ->  ( F `  M )  e.  ran  F )
7974, 78eqeltrd 2555 . . . . 5  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  M  =  N )  ->  y  e.  ran  F )
80 simpll 753 . . . . . 6  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  N  <  M )  ->  ph )
8112ad2antrr 725 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  N  <  M )  ->  N  e.  RR )
827ad2antrr 725 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  N  <  M )  ->  M  e.  RR )
8330adantr 465 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  N  <  M )  ->  y  e.  RR )
84 simpr 461 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  N  <  M )  ->  N  <  M )
8511simp2d 1009 . . . . . . . . . 10  |-  ( ph  ->  A  <_  N )
866simp3d 1010 . . . . . . . . . 10  |-  ( ph  ->  M  <_  B )
87 iccss 11604 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  <_  N  /\  M  <_  B
) )  ->  ( N [,] M )  C_  ( A [,] B ) )
882, 3, 85, 86, 87syl22anc 1229 . . . . . . . . 9  |-  ( ph  ->  ( N [,] M
)  C_  ( A [,] B ) )
8988, 37sstrd 3519 . . . . . . . 8  |-  ( ph  ->  ( N [,] M
)  C_  D )
9089ad2antrr 725 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  N  <  M )  ->  ( N [,] M )  C_  D )
9140ad2antrr 725 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  N  <  M )  ->  F  e.  ( D -cn-> CC ) )
9288sselda 3509 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( N [,] M ) )  ->  x  e.  ( A [,] B ) )
9392, 18syldan 470 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( N [,] M ) )  ->  ( F `  x )  e.  RR )
9480, 93sylan 471 . . . . . . 7  |-  ( ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N
) ) )  /\  N  <  M )  /\  x  e.  ( N [,] M ) )  -> 
( F `  x
)  e.  RR )
9549adantr 465 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  N  <  M )  ->  (
( F `  M
)  <_  y  /\  y  <_  ( F `  N ) ) )
9681, 82, 83, 84, 90, 91, 94, 95ivthle2 21737 . . . . . 6  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  N  <  M )  ->  E. z  e.  ( N [,] M
) ( F `  z )  =  y )
9789sselda 3509 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ( N [,] M ) )  ->  z  e.  D )
9897, 59syldan 470 . . . . . . 7  |-  ( (
ph  /\  z  e.  ( N [,] M ) )  ->  ( ( F `  z )  =  y  ->  y  e. 
ran  F ) )
9998rexlimdva 2959 . . . . . 6  |-  ( ph  ->  ( E. z  e.  ( N [,] M
) ( F `  z )  =  y  ->  y  e.  ran  F ) )
10080, 96, 99sylc 60 . . . . 5  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  N  <  M )  ->  y  e.  ran  F )
10162, 79, 1003jaodan 1294 . . . 4  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  ( M  <  N  \/  M  =  N  \/  N  <  M ) )  -> 
y  e.  ran  F
)
10214, 101mpdan 668 . . 3  |-  ( (
ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  ->  y  e.  ran  F )
103102ex 434 . 2  |-  ( ph  ->  ( y  e.  ( ( F `  M
) [,] ( F `
 N ) )  ->  y  e.  ran  F ) )
104103ssrdv 3515 1  |-  ( ph  ->  ( ( F `  M ) [,] ( F `  N )
)  C_  ran  F )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    \/ w3o 972    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2817   E.wrex 2818    C_ wss 3481   {csn 4033   class class class wbr 4453   ran crn 5006    Fn wfn 5589   -->wf 5590   ` cfv 5594  (class class class)co 6295   CCcc 9502   RRcr 9503   RR*cxr 9639    < clt 9640    <_ cle 9641   [,]cicc 11544   -cn->ccncf 21248
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582  ax-mulf 9584
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-iin 4334  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-of 6535  df-om 6696  df-1st 6795  df-2nd 6796  df-supp 6914  df-recs 7054  df-rdg 7088  df-1o 7142  df-2o 7143  df-oadd 7146  df-er 7323  df-map 7434  df-ixp 7482  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-fsupp 7842  df-fi 7883  df-sup 7913  df-oi 7947  df-card 8332  df-cda 8560  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-9 10613  df-10 10614  df-n0 10808  df-z 10877  df-dec 10989  df-uz 11095  df-q 11195  df-rp 11233  df-xneg 11330  df-xadd 11331  df-xmul 11332  df-ioo 11545  df-icc 11548  df-fz 11685  df-fzo 11805  df-seq 12088  df-exp 12147  df-hash 12386  df-cj 12912  df-re 12913  df-im 12914  df-sqrt 13048  df-abs 13049  df-struct 14509  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-ress 14514  df-plusg 14585  df-mulr 14586  df-starv 14587  df-sca 14588  df-vsca 14589  df-ip 14590  df-tset 14591  df-ple 14592  df-ds 14594  df-unif 14595  df-hom 14596  df-cco 14597  df-rest 14695  df-topn 14696  df-0g 14714  df-gsum 14715  df-topgen 14716  df-pt 14717  df-prds 14720  df-xrs 14774  df-qtop 14779  df-imas 14780  df-xps 14782  df-mre 14858  df-mrc 14859  df-acs 14861  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-submnd 15840  df-mulg 15932  df-cntz 16227  df-cmn 16673  df-psmet 18281  df-xmet 18282  df-met 18283  df-bl 18284  df-mopn 18285  df-cnfld 18291  df-top 19268  df-bases 19270  df-topon 19271  df-topsp 19272  df-cn 19596  df-cnp 19597  df-tx 19931  df-hmeo 20124  df-xms 20691  df-ms 20692  df-tms 20693  df-cncf 21250
This theorem is referenced by:  evthicc2  21740
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