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Theorem ivthicc 20784
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 11348 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( M  e.  ( A [,] B )  <-> 
( M  e.  RR  /\  A  <_  M  /\  M  <_  B ) ) )
52, 3, 4syl2anc 654 . . . . . . . 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 993 . . . . . 6  |-  ( ph  ->  M  e.  RR )
8 ivthicc.4 . . . . . . . 8  |-  ( ph  ->  N  e.  ( A [,] B ) )
9 elicc2 11348 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( N  e.  ( A [,] B )  <-> 
( N  e.  RR  /\  A  <_  N  /\  N  <_  B ) ) )
102, 3, 9syl2anc 654 . . . . . . . 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 993 . . . . . 6  |-  ( ph  ->  N  e.  RR )
137, 12lttri4d 9503 . . . . 5  |-  ( ph  ->  ( M  <  N  \/  M  =  N  \/  N  <  M ) )
1413adantr 462 . . . 4  |-  ( (
ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  ->  ( M  <  N  \/  M  =  N  \/  N  <  M ) )
15 simpll 746 . . . . . 6  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  M  <  N )  ->  ph )
167ad2antrr 718 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  M  <  N )  ->  M  e.  RR )
1712ad2antrr 718 . . . . . . 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 2789 . . . . . . . . . . 11  |-  ( ph  ->  A. x  e.  ( A [,] B ) ( F `  x
)  e.  RR )
20 fveq2 5679 . . . . . . . . . . . . 13  |-  ( x  =  M  ->  ( F `  x )  =  ( F `  M ) )
2120eleq1d 2499 . . . . . . . . . . . 12  |-  ( x  =  M  ->  (
( F `  x
)  e.  RR  <->  ( F `  M )  e.  RR ) )
2221rspcv 3058 . . . . . . . . . . 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 5679 . . . . . . . . . . . . 13  |-  ( x  =  N  ->  ( F `  x )  =  ( F `  N ) )
2524eleq1d 2499 . . . . . . . . . . . 12  |-  ( x  =  N  ->  (
( F `  x
)  e.  RR  <->  ( F `  N )  e.  RR ) )
2625rspcv 3058 . . . . . . . . . . 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 11365 . . . . . . . . . 10  |-  ( ( ( F `  M
)  e.  RR  /\  ( F `  N )  e.  RR )  -> 
( ( F `  M ) [,] ( F `  N )
)  C_  RR )
2923, 27, 28syl2anc 654 . . . . . . . . 9  |-  ( ph  ->  ( ( F `  M ) [,] ( F `  N )
)  C_  RR )
3029sselda 3344 . . . . . . . 8  |-  ( (
ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  ->  y  e.  RR )
3130adantr 462 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  M  <  N )  ->  y  e.  RR )
32 simpr 458 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  M  <  N )  ->  M  <  N )
336simp2d 994 . . . . . . . . . 10  |-  ( ph  ->  A  <_  M )
3411simp3d 995 . . . . . . . . . 10  |-  ( ph  ->  N  <_  B )
35 iccss 11351 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  <_  M  /\  N  <_  B
) )  ->  ( M [,] N )  C_  ( A [,] B ) )
362, 3, 33, 34, 35syl22anc 1212 . . . . . . . . 9  |-  ( ph  ->  ( M [,] N
)  C_  ( A [,] B ) )
37 ivthicc.5 . . . . . . . . 9  |-  ( ph  ->  ( A [,] B
)  C_  D )
3836, 37sstrd 3354 . . . . . . . 8  |-  ( ph  ->  ( M [,] N
)  C_  D )
3938ad2antrr 718 . . . . . . 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 718 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  M  <  N )  ->  F  e.  ( D -cn-> CC ) )
4236sselda 3344 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( M [,] N ) )  ->  x  e.  ( A [,] B ) )
4342, 18syldan 467 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( M [,] N ) )  ->  ( F `  x )  e.  RR )
4415, 43sylan 468 . . . . . . 7  |-  ( ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N
) ) )  /\  M  <  N )  /\  x  e.  ( M [,] N ) )  -> 
( F `  x
)  e.  RR )
45 elicc2 11348 . . . . . . . . . . 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 654 . . . . . . . . . 10  |-  ( ph  ->  ( y  e.  ( ( F `  M
) [,] ( F `
 N ) )  <-> 
( y  e.  RR  /\  ( F `  M
)  <_  y  /\  y  <_  ( F `  N ) ) ) )
4746biimpa 481 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  ->  (
y  e.  RR  /\  ( F `  M )  <_  y  /\  y  <_  ( F `  N
) ) )
48 3simpc 980 . . . . . . . . 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 462 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  M  <  N )  ->  (
( F `  M
)  <_  y  /\  y  <_  ( F `  N ) ) )
5116, 17, 31, 32, 39, 41, 44, 50ivthle 20782 . . . . . 6  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  M  <  N )  ->  E. z  e.  ( M [,] N
) ( F `  z )  =  y )
5238sselda 3344 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ( M [,] N ) )  ->  z  e.  D )
53 cncff 20311 . . . . . . . . . . 11  |-  ( F  e.  ( D -cn-> CC )  ->  F : D
--> CC )
54 ffn 5547 . . . . . . . . . . 11  |-  ( F : D --> CC  ->  F  Fn  D )
5540, 53, 543syl 20 . . . . . . . . . 10  |-  ( ph  ->  F  Fn  D )
56 fnfvelrn 5828 . . . . . . . . . 10  |-  ( ( F  Fn  D  /\  z  e.  D )  ->  ( F `  z
)  e.  ran  F
)
5755, 56sylan 468 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  D )  ->  ( F `  z )  e.  ran  F )
58 eleq1 2493 . . . . . . . . 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 467 . . . . . . 7  |-  ( (
ph  /\  z  e.  ( M [,] N ) )  ->  ( ( F `  z )  =  y  ->  y  e. 
ran  F ) )
6160rexlimdva 2831 . . . . . 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 747 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  M  =  N )  ->  y  e.  ( ( F `  M ) [,] ( F `  N )
) )
64 simpr 458 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  M  =  N )  ->  M  =  N )
6564fveq2d 5683 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  M  =  N )  ->  ( F `  M )  =  ( F `  N ) )
6665oveq2d 6096 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  M  =  N )  ->  (
( F `  M
) [,] ( F `
 M ) )  =  ( ( F `
 M ) [,] ( F `  N
) ) )
6723rexrd 9421 . . . . . . . . . . 11  |-  ( ph  ->  ( F `  M
)  e.  RR* )
6867ad2antrr 718 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  M  =  N )  ->  ( F `  M )  e.  RR* )
69 iccid 11333 . . . . . . . . . 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 2467 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  M  =  N )  ->  (
( F `  M
) [,] ( F `
 N ) )  =  { ( F `
 M ) } )
7263, 71eleqtrd 2509 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  M  =  N )  ->  y  e.  { ( F `  M ) } )
73 elsni 3890 . . . . . . 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 3345 . . . . . . . 8  |-  ( ph  ->  M  e.  D )
76 fnfvelrn 5828 . . . . . . . 8  |-  ( ( F  Fn  D  /\  M  e.  D )  ->  ( F `  M
)  e.  ran  F
)
7755, 75, 76syl2anc 654 . . . . . . 7  |-  ( ph  ->  ( F `  M
)  e.  ran  F
)
7877ad2antrr 718 . . . . . 6  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  M  =  N )  ->  ( F `  M )  e.  ran  F )
7974, 78eqeltrd 2507 . . . . 5  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  M  =  N )  ->  y  e.  ran  F )
80 simpll 746 . . . . . 6  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  N  <  M )  ->  ph )
8112ad2antrr 718 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  N  <  M )  ->  N  e.  RR )
827ad2antrr 718 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  N  <  M )  ->  M  e.  RR )
8330adantr 462 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  N  <  M )  ->  y  e.  RR )
84 simpr 458 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  N  <  M )  ->  N  <  M )
8511simp2d 994 . . . . . . . . . 10  |-  ( ph  ->  A  <_  N )
866simp3d 995 . . . . . . . . . 10  |-  ( ph  ->  M  <_  B )
87 iccss 11351 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  <_  N  /\  M  <_  B
) )  ->  ( N [,] M )  C_  ( A [,] B ) )
882, 3, 85, 86, 87syl22anc 1212 . . . . . . . . 9  |-  ( ph  ->  ( N [,] M
)  C_  ( A [,] B ) )
8988, 37sstrd 3354 . . . . . . . 8  |-  ( ph  ->  ( N [,] M
)  C_  D )
9089ad2antrr 718 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  N  <  M )  ->  ( N [,] M )  C_  D )
9140ad2antrr 718 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  N  <  M )  ->  F  e.  ( D -cn-> CC ) )
9288sselda 3344 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( N [,] M ) )  ->  x  e.  ( A [,] B ) )
9392, 18syldan 467 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( N [,] M ) )  ->  ( F `  x )  e.  RR )
9480, 93sylan 468 . . . . . . 7  |-  ( ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N
) ) )  /\  N  <  M )  /\  x  e.  ( N [,] M ) )  -> 
( F `  x
)  e.  RR )
9549adantr 462 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  N  <  M )  ->  (
( F `  M
)  <_  y  /\  y  <_  ( F `  N ) ) )
9681, 82, 83, 84, 90, 91, 94, 95ivthle2 20783 . . . . . 6  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  N  <  M )  ->  E. z  e.  ( N [,] M
) ( F `  z )  =  y )
9789sselda 3344 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ( N [,] M ) )  ->  z  e.  D )
9897, 59syldan 467 . . . . . . 7  |-  ( (
ph  /\  z  e.  ( N [,] M ) )  ->  ( ( F `  z )  =  y  ->  y  e. 
ran  F ) )
9998rexlimdva 2831 . . . . . 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 1277 . . . 4  |-  ( ( ( ph  /\  y  e.  ( ( F `  M ) [,] ( F `  N )
) )  /\  ( M  <  N  \/  M  =  N  \/  N  <  M ) )  -> 
y  e.  ran  F
)
10214, 101mpdan 661 . . 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 3350 1  |-  ( ph  ->  ( ( F `  M ) [,] ( F `  N )
)  C_  ran  F )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    \/ w3o 957    /\ w3a 958    = wceq 1362    e. wcel 1755   A.wral 2705   E.wrex 2706    C_ wss 3316   {csn 3865   class class class wbr 4280   ran crn 4828    Fn wfn 5401   -->wf 5402   ` cfv 5406  (class class class)co 6080   CCcc 9268   RRcr 9269   RR*cxr 9405    < clt 9406    <_ cle 9407   [,]cicc 11291   -cn->ccncf 20294
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-inf2 7835  ax-cnex 9326  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347  ax-pre-sup 9348  ax-mulf 9350
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-iin 4162  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-se 4667  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-isom 5415  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-of 6309  df-om 6466  df-1st 6566  df-2nd 6567  df-supp 6680  df-recs 6818  df-rdg 6852  df-1o 6908  df-2o 6909  df-oadd 6912  df-er 7089  df-map 7204  df-ixp 7252  df-en 7299  df-dom 7300  df-sdom 7301  df-fin 7302  df-fsupp 7609  df-fi 7649  df-sup 7679  df-oi 7712  df-card 8097  df-cda 8325  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-div 9982  df-nn 10311  df-2 10368  df-3 10369  df-4 10370  df-5 10371  df-6 10372  df-7 10373  df-8 10374  df-9 10375  df-10 10376  df-n0 10568  df-z 10635  df-dec 10744  df-uz 10850  df-q 10942  df-rp 10980  df-xneg 11077  df-xadd 11078  df-xmul 11079  df-ioo 11292  df-icc 11295  df-fz 11425  df-fzo 11533  df-seq 11791  df-exp 11850  df-hash 12088  df-cj 12572  df-re 12573  df-im 12574  df-sqr 12708  df-abs 12709  df-struct 14159  df-ndx 14160  df-slot 14161  df-base 14162  df-sets 14163  df-ress 14164  df-plusg 14234  df-mulr 14235  df-starv 14236  df-sca 14237  df-vsca 14238  df-ip 14239  df-tset 14240  df-ple 14241  df-ds 14243  df-unif 14244  df-hom 14245  df-cco 14246  df-rest 14344  df-topn 14345  df-0g 14363  df-gsum 14364  df-topgen 14365  df-pt 14366  df-prds 14369  df-xrs 14423  df-qtop 14428  df-imas 14429  df-xps 14431  df-mre 14507  df-mrc 14508  df-acs 14510  df-mnd 15398  df-submnd 15448  df-mulg 15528  df-cntz 15815  df-cmn 16259  df-psmet 17653  df-xmet 17654  df-met 17655  df-bl 17656  df-mopn 17657  df-cnfld 17663  df-top 18345  df-bases 18347  df-topon 18348  df-topsp 18349  df-cn 18673  df-cnp 18674  df-tx 18977  df-hmeo 19170  df-xms 19737  df-ms 19738  df-tms 19739  df-cncf 20296
This theorem is referenced by:  evthicc2  20786
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