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Theorem evth 22065
Description: The Extreme Value Theorem. A continuous function from a nonempty compact topological space to the reals attains its maximum at some point in the domain. (Contributed by Mario Carneiro, 12-Aug-2014.)
Hypotheses
Ref Expression
bndth.1  |-  X  = 
U. J
bndth.2  |-  K  =  ( topGen `  ran  (,) )
bndth.3  |-  ( ph  ->  J  e.  Comp )
bndth.4  |-  ( ph  ->  F  e.  ( J  Cn  K ) )
evth.5  |-  ( ph  ->  X  =/=  (/) )
Assertion
Ref Expression
evth  |-  ( ph  ->  E. x  e.  X  A. y  e.  X  ( F `  y )  <_  ( F `  x ) )
Distinct variable groups:    x, y, F    y, K    ph, x, y   
x, X, y    x, J, y
Allowed substitution hint:    K( x)

Proof of Theorem evth
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 bndth.1 . . . . 5  |-  X  = 
U. J
2 bndth.2 . . . . 5  |-  K  =  ( topGen `  ran  (,) )
3 bndth.3 . . . . . 6  |-  ( ph  ->  J  e.  Comp )
43adantr 472 . . . . 5  |-  ( (
ph  /\  F : X
--> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  ->  J  e.  Comp )
5 cmptop 20487 . . . . . . . . . 10  |-  ( J  e.  Comp  ->  J  e. 
Top )
64, 5syl 17 . . . . . . . . 9  |-  ( (
ph  /\  F : X
--> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  ->  J  e.  Top )
71toptopon 20025 . . . . . . . . 9  |-  ( J  e.  Top  <->  J  e.  (TopOn `  X ) )
86, 7sylib 201 . . . . . . . 8  |-  ( (
ph  /\  F : X
--> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  ->  J  e.  (TopOn `  X )
)
9 eqid 2471 . . . . . . . . . . 11  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
109cnfldtopon 21881 . . . . . . . . . 10  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
1110a1i 11 . . . . . . . . 9  |-  ( (
ph  /\  F : X
--> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  ->  ( TopOpen
` fld
)  e.  (TopOn `  CC ) )
12 1cnd 9677 . . . . . . . . 9  |-  ( (
ph  /\  F : X
--> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  ->  1  e.  CC )
138, 11, 12cnmptc 20754 . . . . . . . 8  |-  ( (
ph  /\  F : X
--> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  ->  (
z  e.  X  |->  1 )  e.  ( J  Cn  ( TopOpen ` fld ) ) )
14 bndth.4 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  F  e.  ( J  Cn  K ) )
15 uniretop 21861 . . . . . . . . . . . . . . . . . . 19  |-  RR  =  U. ( topGen `  ran  (,) )
162unieqi 4199 . . . . . . . . . . . . . . . . . . 19  |-  U. K  =  U. ( topGen `  ran  (,) )
1715, 16eqtr4i 2496 . . . . . . . . . . . . . . . . . 18  |-  RR  =  U. K
181, 17cnf 20339 . . . . . . . . . . . . . . . . 17  |-  ( F  e.  ( J  Cn  K )  ->  F : X --> RR )
1914, 18syl 17 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  F : X --> RR )
20 frn 5747 . . . . . . . . . . . . . . . 16  |-  ( F : X --> RR  ->  ran 
F  C_  RR )
2119, 20syl 17 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ran  F  C_  RR )
22 fdm 5745 . . . . . . . . . . . . . . . . . 18  |-  ( F : X --> RR  ->  dom 
F  =  X )
2319, 22syl 17 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  dom  F  =  X )
24 evth.5 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  X  =/=  (/) )
2523, 24eqnetrd 2710 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  dom  F  =/=  (/) )
26 dm0rn0 5057 . . . . . . . . . . . . . . . . 17  |-  ( dom 
F  =  (/)  <->  ran  F  =  (/) )
2726necon3bii 2695 . . . . . . . . . . . . . . . 16  |-  ( dom 
F  =/=  (/)  <->  ran  F  =/=  (/) )
2825, 27sylib 201 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ran  F  =/=  (/) )
291, 2, 3, 14bndth 22064 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  E. x  e.  RR  A. y  e.  X  ( F `  y )  <_  x )
30 ffn 5739 . . . . . . . . . . . . . . . . . . 19  |-  ( F : X --> RR  ->  F  Fn  X )
3119, 30syl 17 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  F  Fn  X )
32 breq1 4398 . . . . . . . . . . . . . . . . . . 19  |-  ( z  =  ( F `  y )  ->  (
z  <_  x  <->  ( F `  y )  <_  x
) )
3332ralrn 6040 . . . . . . . . . . . . . . . . . 18  |-  ( F  Fn  X  ->  ( A. z  e.  ran  F  z  <_  x  <->  A. y  e.  X  ( F `  y )  <_  x
) )
3431, 33syl 17 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( A. z  e. 
ran  F  z  <_  x  <->  A. y  e.  X  ( F `  y )  <_  x ) )
3534rexbidv 2892 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( E. x  e.  RR  A. z  e. 
ran  F  z  <_  x  <->  E. x  e.  RR  A. y  e.  X  ( F `  y )  <_  x ) )
3629, 35mpbird 240 . . . . . . . . . . . . . . 15  |-  ( ph  ->  E. x  e.  RR  A. z  e.  ran  F  z  <_  x )
3721, 28, 363jca 1210 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ran  F  C_  RR  /\  ran  F  =/=  (/)  /\  E. x  e.  RR  A. z  e. 
ran  F  z  <_  x ) )
38 suprcl 10591 . . . . . . . . . . . . . 14  |-  ( ( ran  F  C_  RR  /\ 
ran  F  =/=  (/)  /\  E. x  e.  RR  A. z  e.  ran  F  z  <_  x )  ->  sup ( ran  F ,  RR ,  <  )  e.  RR )
3937, 38syl 17 . . . . . . . . . . . . 13  |-  ( ph  ->  sup ( ran  F ,  RR ,  <  )  e.  RR )
4039recnd 9687 . . . . . . . . . . . 12  |-  ( ph  ->  sup ( ran  F ,  RR ,  <  )  e.  CC )
4140adantr 472 . . . . . . . . . . 11  |-  ( (
ph  /\  F : X
--> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  ->  sup ( ran  F ,  RR ,  <  )  e.  CC )
428, 11, 41cnmptc 20754 . . . . . . . . . 10  |-  ( (
ph  /\  F : X
--> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  ->  (
z  e.  X  |->  sup ( ran  F ,  RR ,  <  ) )  e.  ( J  Cn  ( TopOpen ` fld ) ) )
4319feqmptd 5932 . . . . . . . . . . . 12  |-  ( ph  ->  F  =  ( z  e.  X  |->  ( F `
 z ) ) )
449cnfldtop 21882 . . . . . . . . . . . . . 14  |-  ( TopOpen ` fld )  e.  Top
45 cnrest2r 20380 . . . . . . . . . . . . . 14  |-  ( (
TopOpen ` fld )  e.  Top  ->  ( J  Cn  ( (
TopOpen ` fld )t  RR ) )  C_  ( J  Cn  ( TopOpen
` fld
) ) )
4644, 45ax-mp 5 . . . . . . . . . . . . 13  |-  ( J  Cn  ( ( TopOpen ` fld )t  RR ) )  C_  ( J  Cn  ( TopOpen ` fld ) )
479tgioo2 21899 . . . . . . . . . . . . . . . 16  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
482, 47eqtri 2493 . . . . . . . . . . . . . . 15  |-  K  =  ( ( TopOpen ` fld )t  RR )
4948oveq2i 6319 . . . . . . . . . . . . . 14  |-  ( J  Cn  K )  =  ( J  Cn  (
( TopOpen ` fld )t  RR ) )
5014, 49syl6eleq 2559 . . . . . . . . . . . . 13  |-  ( ph  ->  F  e.  ( J  Cn  ( ( TopOpen ` fld )t  RR ) ) )
5146, 50sseldi 3416 . . . . . . . . . . . 12  |-  ( ph  ->  F  e.  ( J  Cn  ( TopOpen ` fld ) ) )
5243, 51eqeltrrd 2550 . . . . . . . . . . 11  |-  ( ph  ->  ( z  e.  X  |->  ( F `  z
) )  e.  ( J  Cn  ( TopOpen ` fld )
) )
5352adantr 472 . . . . . . . . . 10  |-  ( (
ph  /\  F : X
--> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  ->  (
z  e.  X  |->  ( F `  z ) )  e.  ( J  Cn  ( TopOpen ` fld ) ) )
549subcn 21976 . . . . . . . . . . 11  |-  -  e.  ( ( ( TopOpen ` fld )  tX  ( TopOpen ` fld ) )  Cn  ( TopOpen
` fld
) )
5554a1i 11 . . . . . . . . . 10  |-  ( (
ph  /\  F : X
--> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  ->  -  e.  ( ( ( TopOpen ` fld )  tX  ( TopOpen ` fld ) )  Cn  ( TopOpen
` fld
) ) )
568, 42, 53, 55cnmpt12f 20758 . . . . . . . . 9  |-  ( (
ph  /\  F : X
--> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  ->  (
z  e.  X  |->  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  z ) ) )  e.  ( J  Cn  ( TopOpen ` fld ) ) )
5739ad2antrr 740 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  z  e.  X )  ->  sup ( ran  F ,  RR ,  <  )  e.  RR )
58 ffvelrn 6035 . . . . . . . . . . . . . . . . . 18  |-  ( ( F : X --> ( RR 
\  { sup ( ran  F ,  RR ,  <  ) } )  /\  z  e.  X )  ->  ( F `  z
)  e.  ( RR 
\  { sup ( ran  F ,  RR ,  <  ) } ) )
5958adantll 728 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  z  e.  X )  ->  ( F `  z
)  e.  ( RR 
\  { sup ( ran  F ,  RR ,  <  ) } ) )
60 eldifsn 4088 . . . . . . . . . . . . . . . . 17  |-  ( ( F `  z )  e.  ( RR  \  { sup ( ran  F ,  RR ,  <  ) } )  <->  ( ( F `  z )  e.  RR  /\  ( F `
 z )  =/= 
sup ( ran  F ,  RR ,  <  )
) )
6159, 60sylib 201 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  z  e.  X )  ->  ( ( F `  z )  e.  RR  /\  ( F `  z
)  =/=  sup ( ran  F ,  RR ,  <  ) ) )
6261simpld 466 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  z  e.  X )  ->  ( F `  z
)  e.  RR )
6357, 62resubcld 10068 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  z  e.  X )  ->  ( sup ( ran 
F ,  RR ,  <  )  -  ( F `
 z ) )  e.  RR )
6463recnd 9687 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  z  e.  X )  ->  ( sup ( ran 
F ,  RR ,  <  )  -  ( F `
 z ) )  e.  CC )
6557recnd 9687 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  z  e.  X )  ->  sup ( ran  F ,  RR ,  <  )  e.  CC )
6662recnd 9687 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  z  e.  X )  ->  ( F `  z
)  e.  CC )
6761simprd 470 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  z  e.  X )  ->  ( F `  z
)  =/=  sup ( ran  F ,  RR ,  <  ) )
6867necomd 2698 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  z  e.  X )  ->  sup ( ran  F ,  RR ,  <  )  =/=  ( F `  z
) )
6965, 66, 68subne0d 10014 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  z  e.  X )  ->  ( sup ( ran 
F ,  RR ,  <  )  -  ( F `
 z ) )  =/=  0 )
70 eldifsn 4088 . . . . . . . . . . . . 13  |-  ( ( sup ( ran  F ,  RR ,  <  )  -  ( F `  z ) )  e.  ( CC  \  {
0 } )  <->  ( ( sup ( ran  F ,  RR ,  <  )  -  ( F `  z ) )  e.  CC  /\  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  z ) )  =/=  0 ) )
7164, 69, 70sylanbrc 677 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  z  e.  X )  ->  ( sup ( ran 
F ,  RR ,  <  )  -  ( F `
 z ) )  e.  ( CC  \  { 0 } ) )
72 eqid 2471 . . . . . . . . . . . 12  |-  ( z  e.  X  |->  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  z ) ) )  =  ( z  e.  X  |->  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  z ) ) )
7371, 72fmptd 6061 . . . . . . . . . . 11  |-  ( (
ph  /\  F : X
--> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  ->  (
z  e.  X  |->  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  z ) ) ) : X --> ( CC 
\  { 0 } ) )
74 frn 5747 . . . . . . . . . . 11  |-  ( ( z  e.  X  |->  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  z ) ) ) : X --> ( CC 
\  { 0 } )  ->  ran  ( z  e.  X  |->  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  z ) ) )  C_  ( CC  \  { 0 } ) )
7573, 74syl 17 . . . . . . . . . 10  |-  ( (
ph  /\  F : X
--> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  ->  ran  ( z  e.  X  |->  ( sup ( ran 
F ,  RR ,  <  )  -  ( F `
 z ) ) )  C_  ( CC  \  { 0 } ) )
76 difssd 3550 . . . . . . . . . 10  |-  ( (
ph  /\  F : X
--> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  ->  ( CC  \  { 0 } )  C_  CC )
77 cnrest2 20379 . . . . . . . . . 10  |-  ( ( ( TopOpen ` fld )  e.  (TopOn `  CC )  /\  ran  ( z  e.  X  |->  ( sup ( ran 
F ,  RR ,  <  )  -  ( F `
 z ) ) )  C_  ( CC  \  { 0 } )  /\  ( CC  \  { 0 } ) 
C_  CC )  -> 
( ( z  e.  X  |->  ( sup ( ran  F ,  RR ,  <  )  -  ( F `
 z ) ) )  e.  ( J  Cn  ( TopOpen ` fld ) )  <->  ( z  e.  X  |->  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  z ) ) )  e.  ( J  Cn  ( (
TopOpen ` fld )t  ( CC  \  {
0 } ) ) ) ) )
7811, 75, 76, 77syl3anc 1292 . . . . . . . . 9  |-  ( (
ph  /\  F : X
--> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  ->  (
( z  e.  X  |->  ( sup ( ran 
F ,  RR ,  <  )  -  ( F `
 z ) ) )  e.  ( J  Cn  ( TopOpen ` fld ) )  <->  ( z  e.  X  |->  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  z ) ) )  e.  ( J  Cn  ( (
TopOpen ` fld )t  ( CC  \  {
0 } ) ) ) ) )
7956, 78mpbid 215 . . . . . . . 8  |-  ( (
ph  /\  F : X
--> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  ->  (
z  e.  X  |->  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  z ) ) )  e.  ( J  Cn  ( ( TopOpen ` fld )t  ( CC  \  { 0 } ) ) ) )
80 eqid 2471 . . . . . . . . . 10  |-  ( (
TopOpen ` fld )t  ( CC  \  {
0 } ) )  =  ( ( TopOpen ` fld )t  ( CC  \  { 0 } ) )
819, 80divcn 21978 . . . . . . . . 9  |-  /  e.  ( ( ( TopOpen ` fld )  tX  ( ( TopOpen ` fld )t  ( CC  \  { 0 } ) ) )  Cn  ( TopOpen
` fld
) )
8281a1i 11 . . . . . . . 8  |-  ( (
ph  /\  F : X
--> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  ->  /  e.  ( ( ( TopOpen ` fld )  tX  ( ( TopOpen ` fld )t  ( CC  \  { 0 } ) ) )  Cn  ( TopOpen
` fld
) ) )
838, 13, 79, 82cnmpt12f 20758 . . . . . . 7  |-  ( (
ph  /\  F : X
--> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  ->  (
z  e.  X  |->  ( 1  /  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  z ) ) ) )  e.  ( J  Cn  ( TopOpen
` fld
) ) )
8463, 69rereccld 10456 . . . . . . . . . 10  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  z  e.  X )  ->  ( 1  /  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  z ) ) )  e.  RR )
85 eqid 2471 . . . . . . . . . 10  |-  ( z  e.  X  |->  ( 1  /  ( sup ( ran  F ,  RR ,  <  )  -  ( F `
 z ) ) ) )  =  ( z  e.  X  |->  ( 1  /  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  z ) ) ) )
8684, 85fmptd 6061 . . . . . . . . 9  |-  ( (
ph  /\  F : X
--> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  ->  (
z  e.  X  |->  ( 1  /  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  z ) ) ) ) : X --> RR )
87 frn 5747 . . . . . . . . 9  |-  ( ( z  e.  X  |->  ( 1  /  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  z ) ) ) ) : X --> RR  ->  ran  ( z  e.  X  |->  ( 1  /  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  z ) ) ) )  C_  RR )
8886, 87syl 17 . . . . . . . 8  |-  ( (
ph  /\  F : X
--> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  ->  ran  ( z  e.  X  |->  ( 1  /  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  z ) ) ) )  C_  RR )
89 ax-resscn 9614 . . . . . . . . 9  |-  RR  C_  CC
9089a1i 11 . . . . . . . 8  |-  ( (
ph  /\  F : X
--> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  ->  RR  C_  CC )
91 cnrest2 20379 . . . . . . . 8  |-  ( ( ( TopOpen ` fld )  e.  (TopOn `  CC )  /\  ran  ( z  e.  X  |->  ( 1  /  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  z ) ) ) )  C_  RR  /\  RR  C_  CC )  ->  ( ( z  e.  X  |->  ( 1  /  ( sup ( ran  F ,  RR ,  <  )  -  ( F `
 z ) ) ) )  e.  ( J  Cn  ( TopOpen ` fld )
)  <->  ( z  e.  X  |->  ( 1  / 
( sup ( ran 
F ,  RR ,  <  )  -  ( F `
 z ) ) ) )  e.  ( J  Cn  ( (
TopOpen ` fld )t  RR ) ) ) )
9211, 88, 90, 91syl3anc 1292 . . . . . . 7  |-  ( (
ph  /\  F : X
--> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  ->  (
( z  e.  X  |->  ( 1  /  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  z ) ) ) )  e.  ( J  Cn  ( TopOpen
` fld
) )  <->  ( z  e.  X  |->  ( 1  /  ( sup ( ran  F ,  RR ,  <  )  -  ( F `
 z ) ) ) )  e.  ( J  Cn  ( (
TopOpen ` fld )t  RR ) ) ) )
9383, 92mpbid 215 . . . . . 6  |-  ( (
ph  /\  F : X
--> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  ->  (
z  e.  X  |->  ( 1  /  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  z ) ) ) )  e.  ( J  Cn  (
( TopOpen ` fld )t  RR ) ) )
9493, 49syl6eleqr 2560 . . . . 5  |-  ( (
ph  /\  F : X
--> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  ->  (
z  e.  X  |->  ( 1  /  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  z ) ) ) )  e.  ( J  Cn  K
) )
951, 2, 4, 94bndth 22064 . . . 4  |-  ( (
ph  /\  F : X
--> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  ->  E. x  e.  RR  A. y  e.  X  ( ( z  e.  X  |->  ( 1  /  ( sup ( ran  F ,  RR ,  <  )  -  ( F `
 z ) ) ) ) `  y
)  <_  x )
9639ad2antrr 740 . . . . . . . 8  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  x  e.  RR )  ->  sup ( ran  F ,  RR ,  <  )  e.  RR )
97 simpr 468 . . . . . . . . . . 11  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  x  e.  RR )  ->  x  e.  RR )
98 1re 9660 . . . . . . . . . . 11  |-  1  e.  RR
99 ifcl 3914 . . . . . . . . . . 11  |-  ( ( x  e.  RR  /\  1  e.  RR )  ->  if ( 1  <_  x ,  x , 
1 )  e.  RR )
10097, 98, 99sylancl 675 . . . . . . . . . 10  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  x  e.  RR )  ->  if ( 1  <_  x ,  x , 
1 )  e.  RR )
101 0red 9662 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  x  e.  RR )  ->  0  e.  RR )
10298a1i 11 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  x  e.  RR )  ->  1  e.  RR )
103 0lt1 10157 . . . . . . . . . . . . 13  |-  0  <  1
104103a1i 11 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  x  e.  RR )  ->  0  <  1 )
105 max1 11503 . . . . . . . . . . . . 13  |-  ( ( 1  e.  RR  /\  x  e.  RR )  ->  1  <_  if (
1  <_  x ,  x ,  1 ) )
10698, 97, 105sylancr 676 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  x  e.  RR )  ->  1  <_  if (
1  <_  x ,  x ,  1 ) )
107101, 102, 100, 104, 106ltletrd 9812 . . . . . . . . . . 11  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  x  e.  RR )  ->  0  <  if ( 1  <_  x ,  x ,  1 ) )
108107gt0ne0d 10199 . . . . . . . . . 10  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  x  e.  RR )  ->  if ( 1  <_  x ,  x , 
1 )  =/=  0
)
109100, 108rereccld 10456 . . . . . . . . 9  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  x  e.  RR )  ->  ( 1  /  if ( 1  <_  x ,  x ,  1 ) )  e.  RR )
110100, 107recgt0d 10563 . . . . . . . . 9  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  x  e.  RR )  ->  0  <  ( 1  /  if ( 1  <_  x ,  x ,  1 ) ) )
111109, 110elrpd 11361 . . . . . . . 8  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  x  e.  RR )  ->  ( 1  /  if ( 1  <_  x ,  x ,  1 ) )  e.  RR+ )
11296, 111ltsubrpd 11393 . . . . . . 7  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  x  e.  RR )  ->  ( sup ( ran 
F ,  RR ,  <  )  -  ( 1  /  if ( 1  <_  x ,  x ,  1 ) ) )  <  sup ( ran  F ,  RR ,  <  ) )
11396, 109resubcld 10068 . . . . . . . 8  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  x  e.  RR )  ->  ( sup ( ran 
F ,  RR ,  <  )  -  ( 1  /  if ( 1  <_  x ,  x ,  1 ) ) )  e.  RR )
114113, 96ltnled 9799 . . . . . . 7  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  x  e.  RR )  ->  ( ( sup ( ran  F ,  RR ,  <  )  -  ( 1  /  if ( 1  <_  x ,  x ,  1 ) ) )  <  sup ( ran  F ,  RR ,  <  )  <->  -.  sup ( ran  F ,  RR ,  <  )  <_  ( sup ( ran  F ,  RR ,  <  )  -  (
1  /  if ( 1  <_  x ,  x ,  1 ) ) ) ) )
115112, 114mpbid 215 . . . . . 6  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  x  e.  RR )  ->  -.  sup ( ran 
F ,  RR ,  <  )  <_  ( sup ( ran  F ,  RR ,  <  )  -  (
1  /  if ( 1  <_  x ,  x ,  1 ) ) ) )
116 simprl 772 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  x  e.  RR )
117 max2 11505 . . . . . . . . . . . 12  |-  ( ( 1  e.  RR  /\  x  e.  RR )  ->  x  <_  if (
1  <_  x ,  x ,  1 ) )
11898, 116, 117sylancr 676 . . . . . . . . . . 11  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  x  <_  if ( 1  <_  x ,  x ,  1 ) )
11939ad2antrr 740 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  sup ( ran  F ,  RR ,  <  )  e.  RR )
120 ffvelrn 6035 . . . . . . . . . . . . . . . . 17  |-  ( ( F : X --> ( RR 
\  { sup ( ran  F ,  RR ,  <  ) } )  /\  y  e.  X )  ->  ( F `  y
)  e.  ( RR 
\  { sup ( ran  F ,  RR ,  <  ) } ) )
121120ad2ant2l 760 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  ( F `  y )  e.  ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )
122 eldifsn 4088 . . . . . . . . . . . . . . . 16  |-  ( ( F `  y )  e.  ( RR  \  { sup ( ran  F ,  RR ,  <  ) } )  <->  ( ( F `  y )  e.  RR  /\  ( F `
 y )  =/= 
sup ( ran  F ,  RR ,  <  )
) )
123121, 122sylib 201 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  ( ( F `  y )  e.  RR  /\  ( F `
 y )  =/= 
sup ( ran  F ,  RR ,  <  )
) )
124123simpld 466 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  ( F `  y )  e.  RR )
125119, 124resubcld 10068 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  y )
)  e.  RR )
12637adantr 472 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  y  e.  X )  ->  ( ran  F  C_  RR  /\  ran  F  =/=  (/)  /\  E. x  e.  RR  A. z  e. 
ran  F  z  <_  x ) )
127 fnfvelrn 6034 . . . . . . . . . . . . . . . . . . 19  |-  ( ( F  Fn  X  /\  y  e.  X )  ->  ( F `  y
)  e.  ran  F
)
12831, 127sylan 479 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  y  e.  X )  ->  ( F `  y )  e.  ran  F )
129 suprub 10592 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ran  F  C_  RR  /\  ran  F  =/=  (/)  /\  E. x  e.  RR  A. z  e. 
ran  F  z  <_  x )  /\  ( F `
 y )  e. 
ran  F )  -> 
( F `  y
)  <_  sup ( ran  F ,  RR ,  <  ) )
130126, 128, 129syl2anc 673 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  y  e.  X )  ->  ( F `  y )  <_  sup ( ran  F ,  RR ,  <  )
)
131130ad2ant2rl 763 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  ( F `  y )  <_  sup ( ran  F ,  RR ,  <  ) )
132123simprd 470 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  ( F `  y )  =/=  sup ( ran  F ,  RR ,  <  ) )
133132necomd 2698 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  sup ( ran  F ,  RR ,  <  )  =/=  ( F `
 y ) )
134124, 119ltlend 9797 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  ( ( F `  y )  <  sup ( ran  F ,  RR ,  <  )  <->  ( ( F `  y
)  <_  sup ( ran  F ,  RR ,  <  )  /\  sup ( ran  F ,  RR ,  <  )  =/=  ( F `
 y ) ) ) )
135131, 133, 134mpbir2and 936 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  ( F `  y )  <  sup ( ran  F ,  RR ,  <  ) )
136124, 119posdifd 10221 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  ( ( F `  y )  <  sup ( ran  F ,  RR ,  <  )  <->  0  <  ( sup ( ran  F ,  RR ,  <  )  -  ( F `
 y ) ) ) )
137135, 136mpbid 215 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  0  <  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  y ) ) )
138137gt0ne0d 10199 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  y )
)  =/=  0 )
139125, 138rereccld 10456 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  ( 1  /  ( sup ( ran  F ,  RR ,  <  )  -  ( F `
 y ) ) )  e.  RR )
140116, 98, 99sylancl 675 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  if (
1  <_  x ,  x ,  1 )  e.  RR )
141 letr 9745 . . . . . . . . . . . 12  |-  ( ( ( 1  /  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  y ) ) )  e.  RR  /\  x  e.  RR  /\  if ( 1  <_  x ,  x ,  1 )  e.  RR )  -> 
( ( ( 1  /  ( sup ( ran  F ,  RR ,  <  )  -  ( F `
 y ) ) )  <_  x  /\  x  <_  if ( 1  <_  x ,  x ,  1 ) )  ->  ( 1  / 
( sup ( ran 
F ,  RR ,  <  )  -  ( F `
 y ) ) )  <_  if (
1  <_  x ,  x ,  1 ) ) )
142139, 116, 140, 141syl3anc 1292 . . . . . . . . . . 11  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  ( (
( 1  /  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  y ) ) )  <_  x  /\  x  <_  if ( 1  <_  x ,  x ,  1 ) )  ->  ( 1  /  ( sup ( ran  F ,  RR ,  <  )  -  ( F `
 y ) ) )  <_  if (
1  <_  x ,  x ,  1 ) ) )
143118, 142mpan2d 688 . . . . . . . . . 10  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  ( (
1  /  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  y ) ) )  <_  x  ->  ( 1  /  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  y ) ) )  <_  if ( 1  <_  x ,  x ,  1 ) ) )
144 fveq2 5879 . . . . . . . . . . . . . . 15  |-  ( z  =  y  ->  ( F `  z )  =  ( F `  y ) )
145144oveq2d 6324 . . . . . . . . . . . . . 14  |-  ( z  =  y  ->  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  z ) )  =  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  y ) ) )
146145oveq2d 6324 . . . . . . . . . . . . 13  |-  ( z  =  y  ->  (
1  /  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  z ) ) )  =  ( 1  /  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  y ) ) ) )
147 ovex 6336 . . . . . . . . . . . . 13  |-  ( 1  /  ( sup ( ran  F ,  RR ,  <  )  -  ( F `
 y ) ) )  e.  _V
148146, 85, 147fvmpt 5963 . . . . . . . . . . . 12  |-  ( y  e.  X  ->  (
( z  e.  X  |->  ( 1  /  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  z ) ) ) ) `  y )  =  ( 1  /  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  y ) ) ) )
149148breq1d 4405 . . . . . . . . . . 11  |-  ( y  e.  X  ->  (
( ( z  e.  X  |->  ( 1  / 
( sup ( ran 
F ,  RR ,  <  )  -  ( F `
 z ) ) ) ) `  y
)  <_  x  <->  ( 1  /  ( sup ( ran  F ,  RR ,  <  )  -  ( F `
 y ) ) )  <_  x )
)
150149ad2antll 743 . . . . . . . . . 10  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  ( (
( z  e.  X  |->  ( 1  /  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  z ) ) ) ) `  y )  <_  x  <->  ( 1  /  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  y ) ) )  <_  x
) )
151109adantrr 731 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  ( 1  /  if ( 1  <_  x ,  x ,  1 ) )  e.  RR )
152107adantrr 731 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  0  <  if ( 1  <_  x ,  x ,  1 ) )
153140, 152recgt0d 10563 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  0  <  ( 1  /  if ( 1  <_  x ,  x ,  1 ) ) )
154 lerec 10511 . . . . . . . . . . . 12  |-  ( ( ( ( 1  /  if ( 1  <_  x ,  x ,  1 ) )  e.  RR  /\  0  <  ( 1  /  if ( 1  <_  x ,  x ,  1 ) ) )  /\  (
( sup ( ran 
F ,  RR ,  <  )  -  ( F `
 y ) )  e.  RR  /\  0  <  ( sup ( ran 
F ,  RR ,  <  )  -  ( F `
 y ) ) ) )  ->  (
( 1  /  if ( 1  <_  x ,  x ,  1 ) )  <_  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  y )
)  <->  ( 1  / 
( sup ( ran 
F ,  RR ,  <  )  -  ( F `
 y ) ) )  <_  ( 1  /  ( 1  /  if ( 1  <_  x ,  x ,  1 ) ) ) ) )
155151, 153, 125, 137, 154syl22anc 1293 . . . . . . . . . . 11  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  ( (
1  /  if ( 1  <_  x ,  x ,  1 ) )  <_  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  y )
)  <->  ( 1  / 
( sup ( ran 
F ,  RR ,  <  )  -  ( F `
 y ) ) )  <_  ( 1  /  ( 1  /  if ( 1  <_  x ,  x ,  1 ) ) ) ) )
156 lesub 10114 . . . . . . . . . . . 12  |-  ( ( ( 1  /  if ( 1  <_  x ,  x ,  1 ) )  e.  RR  /\  sup ( ran  F ,  RR ,  <  )  e.  RR  /\  ( F `
 y )  e.  RR )  ->  (
( 1  /  if ( 1  <_  x ,  x ,  1 ) )  <_  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  y )
)  <->  ( F `  y )  <_  ( sup ( ran  F ,  RR ,  <  )  -  ( 1  /  if ( 1  <_  x ,  x ,  1 ) ) ) ) )
157151, 119, 124, 156syl3anc 1292 . . . . . . . . . . 11  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  ( (
1  /  if ( 1  <_  x ,  x ,  1 ) )  <_  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  y )
)  <->  ( F `  y )  <_  ( sup ( ran  F ,  RR ,  <  )  -  ( 1  /  if ( 1  <_  x ,  x ,  1 ) ) ) ) )
158140recnd 9687 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  if (
1  <_  x ,  x ,  1 )  e.  CC )
159108adantrr 731 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  if (
1  <_  x ,  x ,  1 )  =/=  0 )
160158, 159recrecd 10402 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  ( 1  /  ( 1  /  if ( 1  <_  x ,  x ,  1 ) ) )  =  if ( 1  <_  x ,  x ,  1 ) )
161160breq2d 4407 . . . . . . . . . . 11  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  ( (
1  /  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  y ) ) )  <_  (
1  /  ( 1  /  if ( 1  <_  x ,  x ,  1 ) ) )  <->  ( 1  / 
( sup ( ran 
F ,  RR ,  <  )  -  ( F `
 y ) ) )  <_  if (
1  <_  x ,  x ,  1 ) ) )
162155, 157, 1613bitr3d 291 . . . . . . . . . 10  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  ( ( F `  y )  <_  ( sup ( ran 
F ,  RR ,  <  )  -  ( 1  /  if ( 1  <_  x ,  x ,  1 ) ) )  <->  ( 1  / 
( sup ( ran 
F ,  RR ,  <  )  -  ( F `
 y ) ) )  <_  if (
1  <_  x ,  x ,  1 ) ) )
163143, 150, 1623imtr4d 276 . . . . . . . . 9  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  ( (
( z  e.  X  |->  ( 1  /  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  z ) ) ) ) `  y )  <_  x  ->  ( F `  y
)  <_  ( sup ( ran  F ,  RR ,  <  )  -  (
1  /  if ( 1  <_  x ,  x ,  1 ) ) ) ) )
164163anassrs 660 . . . . . . . 8  |-  ( ( ( ( ph  /\  F : X --> ( RR 
\  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  x  e.  RR )  /\  y  e.  X
)  ->  ( (
( z  e.  X  |->  ( 1  /  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  z ) ) ) ) `  y )  <_  x  ->  ( F `  y
)  <_  ( sup ( ran  F ,  RR ,  <  )  -  (
1  /  if ( 1  <_  x ,  x ,  1 ) ) ) ) )
165164ralimdva 2805 . . . . . . 7  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  x  e.  RR )  ->  ( A. y  e.  X  ( ( z  e.  X  |->  ( 1  /  ( sup ( ran  F ,  RR ,  <  )  -  ( F `
 z ) ) ) ) `  y
)  <_  x  ->  A. y  e.  X  ( F `  y )  <_  ( sup ( ran  F ,  RR ,  <  )  -  ( 1  /  if ( 1  <_  x ,  x ,  1 ) ) ) ) )
16637ad2antrr 740 . . . . . . . . 9  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  x  e.  RR )  ->  ( ran  F  C_  RR  /\  ran  F  =/=  (/)  /\  E. x  e.  RR  A. z  e. 
ran  F  z  <_  x ) )
167 suprleub 10595 . . . . . . . . 9  |-  ( ( ( ran  F  C_  RR  /\  ran  F  =/=  (/)  /\  E. x  e.  RR  A. z  e. 
ran  F  z  <_  x )  /\  ( sup ( ran  F ,  RR ,  <  )  -  ( 1  /  if ( 1  <_  x ,  x ,  1 ) ) )  e.  RR )  ->  ( sup ( ran  F ,  RR ,  <  )  <_  ( sup ( ran  F ,  RR ,  <  )  -  (
1  /  if ( 1  <_  x ,  x ,  1 ) ) )  <->  A. z  e.  ran  F  z  <_ 
( sup ( ran 
F ,  RR ,  <  )  -  ( 1  /  if ( 1  <_  x ,  x ,  1 ) ) ) ) )
168166, 113, 167syl2anc 673 . . . . . . . 8  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  x  e.  RR )  ->  ( sup ( ran 
F ,  RR ,  <  )  <_  ( sup ( ran  F ,  RR ,  <  )  -  (
1  /  if ( 1  <_  x ,  x ,  1 ) ) )  <->  A. z  e.  ran  F  z  <_ 
( sup ( ran 
F ,  RR ,  <  )  -  ( 1  /  if ( 1  <_  x ,  x ,  1 ) ) ) ) )
16931ad2antrr 740 . . . . . . . . 9  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  x  e.  RR )  ->  F  Fn  X )
170 breq1 4398 . . . . . . . . . 10  |-  ( z  =  ( F `  y )  ->  (
z  <_  ( sup ( ran  F ,  RR ,  <  )  -  (
1  /  if ( 1  <_  x ,  x ,  1 ) ) )  <->  ( F `  y )  <_  ( sup ( ran  F ,  RR ,  <  )  -  ( 1  /  if ( 1  <_  x ,  x ,  1 ) ) ) ) )
171170ralrn 6040 . . . . . . . . 9  |-  ( F  Fn  X  ->  ( A. z  e.  ran  F  z  <_  ( sup ( ran  F ,  RR ,  <  )  -  (
1  /  if ( 1  <_  x ,  x ,  1 ) ) )  <->  A. y  e.  X  ( F `  y )  <_  ( sup ( ran  F ,  RR ,  <  )  -  ( 1  /  if ( 1  <_  x ,  x ,  1 ) ) ) ) )
172169, 171syl 17 . . . . . . . 8  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  x  e.  RR )  ->  ( A. z  e. 
ran  F  z  <_  ( sup ( ran  F ,  RR ,  <  )  -  ( 1  /  if ( 1  <_  x ,  x ,  1 ) ) )  <->  A. y  e.  X  ( F `  y )  <_  ( sup ( ran  F ,  RR ,  <  )  -  ( 1  /  if ( 1  <_  x ,  x ,  1 ) ) ) ) )
173168, 172bitrd 261 . . . . . . 7  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  x  e.  RR )  ->  ( sup ( ran 
F ,  RR ,  <  )  <_  ( sup ( ran  F ,  RR ,  <  )  -  (
1  /  if ( 1  <_  x ,  x ,  1 ) ) )  <->  A. y  e.  X  ( F `  y )  <_  ( sup ( ran  F ,  RR ,  <  )  -  ( 1  /  if ( 1  <_  x ,  x ,  1 ) ) ) ) )
174165, 173sylibrd 242 . . . . . 6  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  x  e.  RR )  ->  ( A. y  e.  X  ( ( z  e.  X  |->  ( 1  /  ( sup ( ran  F ,  RR ,  <  )  -  ( F `
 z ) ) ) ) `  y
)  <_  x  ->  sup ( ran  F ,  RR ,  <  )  <_ 
( sup ( ran 
F ,  RR ,  <  )  -  ( 1  /  if ( 1  <_  x ,  x ,  1 ) ) ) ) )
175115, 174mtod 182 . . . . 5  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  x  e.  RR )  ->  -.  A. y  e.  X  ( ( z  e.  X  |->  ( 1  /  ( sup ( ran  F ,  RR ,  <  )  -  ( F `
 z ) ) ) ) `  y
)  <_  x )
176175nrexdv 2842 . . . 4  |-  ( (
ph  /\  F : X
--> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  ->  -.  E. x  e.  RR  A. y  e.  X  (
( z  e.  X  |->  ( 1  /  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  z ) ) ) ) `  y )  <_  x
)
17795, 176pm2.65da 586 . . 3  |-  ( ph  ->  -.  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )
178130ralrimiva 2809 . . . . . . . . 9  |-  ( ph  ->  A. y  e.  X  ( F `  y )  <_  sup ( ran  F ,  RR ,  <  )
)
179 breq2 4399 . . . . . . . . . 10  |-  ( ( F `  x )  =  sup ( ran 
F ,  RR ,  <  )  ->  ( ( F `  y )  <_  ( F `  x
)  <->  ( F `  y )  <_  sup ( ran  F ,  RR ,  <  ) ) )
180179ralbidv 2829 . . . . . . . . 9  |-  ( ( F `  x )  =  sup ( ran 
F ,  RR ,  <  )  ->  ( A. y  e.  X  ( F `  y )  <_  ( F `  x
)  <->  A. y  e.  X  ( F `  y )  <_  sup ( ran  F ,  RR ,  <  )
) )
181178, 180syl5ibrcom 230 . . . . . . . 8  |-  ( ph  ->  ( ( F `  x )  =  sup ( ran  F ,  RR ,  <  )  ->  A. y  e.  X  ( F `  y )  <_  ( F `  x )
) )
182181necon3bd 2657 . . . . . . 7  |-  ( ph  ->  ( -.  A. y  e.  X  ( F `  y )  <_  ( F `  x )  ->  ( F `  x
)  =/=  sup ( ran  F ,  RR ,  <  ) ) )
183182adantr 472 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  ( -.  A. y  e.  X  ( F `  y )  <_  ( F `  x )  ->  ( F `  x )  =/=  sup ( ran  F ,  RR ,  <  )
) )
18419ffvelrnda 6037 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  ( F `  x )  e.  RR )
185 eldifsn 4088 . . . . . . . 8  |-  ( ( F `  x )  e.  ( RR  \  { sup ( ran  F ,  RR ,  <  ) } )  <->  ( ( F `  x )  e.  RR  /\  ( F `
 x )  =/= 
sup ( ran  F ,  RR ,  <  )
) )
186185baib 919 . . . . . . 7  |-  ( ( F `  x )  e.  RR  ->  (
( F `  x
)  e.  ( RR 
\  { sup ( ran  F ,  RR ,  <  ) } )  <->  ( F `  x )  =/=  sup ( ran  F ,  RR ,  <  ) ) )
187184, 186syl 17 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  (
( F `  x
)  e.  ( RR 
\  { sup ( ran  F ,  RR ,  <  ) } )  <->  ( F `  x )  =/=  sup ( ran  F ,  RR ,  <  ) ) )
188183, 187sylibrd 242 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  ( -.  A. y  e.  X  ( F `  y )  <_  ( F `  x )  ->  ( F `  x )  e.  ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) ) )
189188ralimdva 2805 . . . 4  |-  ( ph  ->  ( A. x  e.  X  -.  A. y  e.  X  ( F `  y )  <_  ( F `  x )  ->  A. x  e.  X  ( F `  x )  e.  ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) ) )
190 ffnfv 6064 . . . . . 6  |-  ( F : X --> ( RR 
\  { sup ( ran  F ,  RR ,  <  ) } )  <->  ( F  Fn  X  /\  A. x  e.  X  ( F `  x )  e.  ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) ) )
191190baib 919 . . . . 5  |-  ( F  Fn  X  ->  ( F : X --> ( RR 
\  { sup ( ran  F ,  RR ,  <  ) } )  <->  A. x  e.  X  ( F `  x )  e.  ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) ) )
19231, 191syl 17 . . . 4  |-  ( ph  ->  ( F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } )  <->  A. x  e.  X  ( F `  x )  e.  ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) ) )
193189, 192sylibrd 242 . . 3  |-  ( ph  ->  ( A. x  e.  X  -.  A. y  e.  X  ( F `  y )  <_  ( F `  x )  ->  F : X --> ( RR 
\  { sup ( ran  F ,  RR ,  <  ) } ) ) )
194177, 193mtod 182 . 2  |-  ( ph  ->  -.  A. x  e.  X  -.  A. y  e.  X  ( F `  y )  <_  ( F `  x )
)
195 dfrex2 2837 . 2  |-  ( E. x  e.  X  A. y  e.  X  ( F `  y )  <_  ( F `  x
)  <->  -.  A. x  e.  X  -.  A. y  e.  X  ( F `  y )  <_  ( F `  x )
)
196194, 195sylibr 217 1  |-  ( ph  ->  E. x  e.  X  A. y  e.  X  ( F `  y )  <_  ( F `  x ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904    =/= wne 2641   A.wral 2756   E.wrex 2757    \ cdif 3387    C_ wss 3390   (/)c0 3722   ifcif 3872   {csn 3959   U.cuni 4190   class class class wbr 4395    |-> cmpt 4454   dom cdm 4839   ran crn 4840    Fn wfn 5584   -->wf 5585   ` cfv 5589  (class class class)co 6308   supcsup 7972   CCcc 9555   RRcr 9556   0cc0 9557   1c1 9558    < clt 9693    <_ cle 9694    - cmin 9880    / cdiv 10291   (,)cioo 11660   ↾t crest 15397   TopOpenctopn 15398   topGenctg 15414  ℂfldccnfld 19047   Topctop 19994  TopOnctopon 19995    Cn ccn 20317   Compccmp 20478    tX ctx 20652
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635  ax-mulf 9637
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-er 7381  df-map 7492  df-ixp 7541  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-fi 7943  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-q 11288  df-rp 11326  df-xneg 11432  df-xadd 11433  df-xmul 11434  df-ioo 11664  df-icc 11667  df-fz 11811  df-fzo 11943  df-seq 12252  df-exp 12311  df-hash 12554  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-starv 15283  df-sca 15284  df-vsca 15285  df-ip 15286  df-tset 15287  df-ple 15288  df-ds 15290  df-unif 15291  df-hom 15292  df-cco 15293  df-rest 15399  df-topn 15400  df-0g 15418  df-gsum 15419  df-topgen 15420  df-pt 15421  df-prds 15424  df-xrs 15478  df-qtop 15484  df-imas 15485  df-xps 15488  df-mre 15570  df-mrc 15571  df-acs 15573  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-submnd 16661  df-mulg 16754  df-cntz 17049  df-cmn 17510  df-psmet 19039  df-xmet 19040  df-met 19041  df-bl 19042  df-mopn 19043  df-cnfld 19048  df-top 19998  df-bases 19999  df-topon 20000  df-topsp 20001  df-cn 20320  df-cnp 20321  df-cmp 20479  df-tx 20654  df-hmeo 20847  df-xms 21413  df-ms 21414  df-tms 21415
This theorem is referenced by:  evth2  22066  evthicc  22488  evthf  37411  cncmpmax  37416
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