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Theorem evth 21924
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 466 . . . . 5  |-  ( (
ph  /\  F : X
--> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  ->  J  e.  Comp )
5 cmptop 20347 . . . . . . . . . 10  |-  ( J  e.  Comp  ->  J  e. 
Top )
64, 5syl 17 . . . . . . . . 9  |-  ( (
ph  /\  F : X
--> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  ->  J  e.  Top )
71toptopon 19885 . . . . . . . . 9  |-  ( J  e.  Top  <->  J  e.  (TopOn `  X ) )
86, 7sylib 199 . . . . . . . 8  |-  ( (
ph  /\  F : X
--> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  ->  J  e.  (TopOn `  X )
)
9 eqid 2423 . . . . . . . . . . 11  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
109cnfldtopon 21740 . . . . . . . . . 10  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
1110a1i 11 . . . . . . . . 9  |-  ( (
ph  /\  F : X
--> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  ->  ( TopOpen
` fld
)  e.  (TopOn `  CC ) )
12 1cnd 9605 . . . . . . . . 9  |-  ( (
ph  /\  F : X
--> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  ->  1  e.  CC )
138, 11, 12cnmptc 20614 . . . . . . . 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 21720 . . . . . . . . . . . . . . . . . . 19  |-  RR  =  U. ( topGen `  ran  (,) )
162unieqi 4166 . . . . . . . . . . . . . . . . . . 19  |-  U. K  =  U. ( topGen `  ran  (,) )
1715, 16eqtr4i 2448 . . . . . . . . . . . . . . . . . 18  |-  RR  =  U. K
181, 17cnf 20199 . . . . . . . . . . . . . . . . 17  |-  ( F  e.  ( J  Cn  K )  ->  F : X --> RR )
1914, 18syl 17 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  F : X --> RR )
20 frn 5690 . . . . . . . . . . . . . . . 16  |-  ( F : X --> RR  ->  ran 
F  C_  RR )
2119, 20syl 17 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ran  F  C_  RR )
22 fdm 5688 . . . . . . . . . . . . . . . . . 18  |-  ( F : X --> RR  ->  dom 
F  =  X )
2319, 22syl 17 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  dom  F  =  X )
24 evth.5 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  X  =/=  (/) )
2523, 24eqnetrd 2663 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  dom  F  =/=  (/) )
26 dm0rn0 5008 . . . . . . . . . . . . . . . . 17  |-  ( dom 
F  =  (/)  <->  ran  F  =  (/) )
2726necon3bii 2648 . . . . . . . . . . . . . . . 16  |-  ( dom 
F  =/=  (/)  <->  ran  F  =/=  (/) )
2825, 27sylib 199 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ran  F  =/=  (/) )
291, 2, 3, 14bndth 21923 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  E. x  e.  RR  A. y  e.  X  ( F `  y )  <_  x )
30 ffn 5684 . . . . . . . . . . . . . . . . . . 19  |-  ( F : X --> RR  ->  F  Fn  X )
3119, 30syl 17 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  F  Fn  X )
32 breq1 4364 . . . . . . . . . . . . . . . . . . 19  |-  ( z  =  ( F `  y )  ->  (
z  <_  x  <->  ( F `  y )  <_  x
) )
3332ralrn 5979 . . . . . . . . . . . . . . . . . 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 2873 . . . . . . . . . . . . . . . 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 235 . . . . . . . . . . . . . . 15  |-  ( ph  ->  E. x  e.  RR  A. z  e.  ran  F  z  <_  x )
3721, 28, 363jca 1185 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ran  F  C_  RR  /\  ran  F  =/=  (/)  /\  E. x  e.  RR  A. z  e. 
ran  F  z  <_  x ) )
38 suprcl 10515 . . . . . . . . . . . . . 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 9615 . . . . . . . . . . . 12  |-  ( ph  ->  sup ( ran  F ,  RR ,  <  )  e.  CC )
4140adantr 466 . . . . . . . . . . 11  |-  ( (
ph  /\  F : X
--> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  ->  sup ( ran  F ,  RR ,  <  )  e.  CC )
428, 11, 41cnmptc 20614 . . . . . . . . . 10  |-  ( (
ph  /\  F : X
--> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  ->  (
z  e.  X  |->  sup ( ran  F ,  RR ,  <  ) )  e.  ( J  Cn  ( TopOpen ` fld ) ) )
4319feqmptd 5873 . . . . . . . . . . . 12  |-  ( ph  ->  F  =  ( z  e.  X  |->  ( F `
 z ) ) )
449cnfldtop 21741 . . . . . . . . . . . . . 14  |-  ( TopOpen ` fld )  e.  Top
45 cnrest2r 20240 . . . . . . . . . . . . . 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 21758 . . . . . . . . . . . . . . . 16  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
482, 47eqtri 2445 . . . . . . . . . . . . . . 15  |-  K  =  ( ( TopOpen ` fld )t  RR )
4948oveq2i 6255 . . . . . . . . . . . . . 14  |-  ( J  Cn  K )  =  ( J  Cn  (
( TopOpen ` fld )t  RR ) )
5014, 49syl6eleq 2511 . . . . . . . . . . . . 13  |-  ( ph  ->  F  e.  ( J  Cn  ( ( TopOpen ` fld )t  RR ) ) )
5146, 50sseldi 3400 . . . . . . . . . . . 12  |-  ( ph  ->  F  e.  ( J  Cn  ( TopOpen ` fld ) ) )
5243, 51eqeltrrd 2502 . . . . . . . . . . 11  |-  ( ph  ->  ( z  e.  X  |->  ( F `  z
) )  e.  ( J  Cn  ( TopOpen ` fld )
) )
5352adantr 466 . . . . . . . . . 10  |-  ( (
ph  /\  F : X
--> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  ->  (
z  e.  X  |->  ( F `  z ) )  e.  ( J  Cn  ( TopOpen ` fld ) ) )
549subcn 21835 . . . . . . . . . . 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 20618 . . . . . . . . 9  |-  ( (
ph  /\  F : X
--> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  ->  (
z  e.  X  |->  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  z ) ) )  e.  ( J  Cn  ( TopOpen ` fld ) ) )
5739ad2antrr 730 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  z  e.  X )  ->  sup ( ran  F ,  RR ,  <  )  e.  RR )
58 ffvelrn 5974 . . . . . . . . . . . . . . . . . 18  |-  ( ( F : X --> ( RR 
\  { sup ( ran  F ,  RR ,  <  ) } )  /\  z  e.  X )  ->  ( F `  z
)  e.  ( RR 
\  { sup ( ran  F ,  RR ,  <  ) } ) )
5958adantll 718 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  z  e.  X )  ->  ( F `  z
)  e.  ( RR 
\  { sup ( ran  F ,  RR ,  <  ) } ) )
60 eldifsn 4063 . . . . . . . . . . . . . . . . 17  |-  ( ( F `  z )  e.  ( RR  \  { sup ( ran  F ,  RR ,  <  ) } )  <->  ( ( F `  z )  e.  RR  /\  ( F `
 z )  =/= 
sup ( ran  F ,  RR ,  <  )
) )
6159, 60sylib 199 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  z  e.  X )  ->  ( ( F `  z )  e.  RR  /\  ( F `  z
)  =/=  sup ( ran  F ,  RR ,  <  ) ) )
6261simpld 460 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  z  e.  X )  ->  ( F `  z
)  e.  RR )
6357, 62resubcld 9993 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  z  e.  X )  ->  ( sup ( ran 
F ,  RR ,  <  )  -  ( F `
 z ) )  e.  RR )
6463recnd 9615 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  z  e.  X )  ->  ( sup ( ran 
F ,  RR ,  <  )  -  ( F `
 z ) )  e.  CC )
6557recnd 9615 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  z  e.  X )  ->  sup ( ran  F ,  RR ,  <  )  e.  CC )
6662recnd 9615 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  z  e.  X )  ->  ( F `  z
)  e.  CC )
6761simprd 464 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  z  e.  X )  ->  ( F `  z
)  =/=  sup ( ran  F ,  RR ,  <  ) )
6867necomd 2651 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  z  e.  X )  ->  sup ( ran  F ,  RR ,  <  )  =/=  ( F `  z
) )
6965, 66, 68subne0d 9941 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  z  e.  X )  ->  ( sup ( ran 
F ,  RR ,  <  )  -  ( F `
 z ) )  =/=  0 )
70 eldifsn 4063 . . . . . . . . . . . . 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 668 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  z  e.  X )  ->  ( sup ( ran 
F ,  RR ,  <  )  -  ( F `
 z ) )  e.  ( CC  \  { 0 } ) )
72 eqid 2423 . . . . . . . . . . . 12  |-  ( z  e.  X  |->  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  z ) ) )  =  ( z  e.  X  |->  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  z ) ) )
7371, 72fmptd 6000 . . . . . . . . . . 11  |-  ( (
ph  /\  F : X
--> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  ->  (
z  e.  X  |->  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  z ) ) ) : X --> ( CC 
\  { 0 } ) )
74 frn 5690 . . . . . . . . . . 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 3531 . . . . . . . . . 10  |-  ( (
ph  /\  F : X
--> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  ->  ( CC  \  { 0 } )  C_  CC )
77 cnrest2 20239 . . . . . . . . . 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 1264 . . . . . . . . 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 213 . . . . . . . 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 2423 . . . . . . . . . 10  |-  ( (
TopOpen ` fld )t  ( CC  \  {
0 } ) )  =  ( ( TopOpen ` fld )t  ( CC  \  { 0 } ) )
819, 80divcn 21837 . . . . . . . . 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 20618 . . . . . . 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 10380 . . . . . . . . . 10  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  z  e.  X )  ->  ( 1  /  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  z ) ) )  e.  RR )
85 eqid 2423 . . . . . . . . . 10  |-  ( z  e.  X  |->  ( 1  /  ( sup ( ran  F ,  RR ,  <  )  -  ( F `
 z ) ) ) )  =  ( z  e.  X  |->  ( 1  /  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  z ) ) ) )
8684, 85fmptd 6000 . . . . . . . . 9  |-  ( (
ph  /\  F : X
--> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  ->  (
z  e.  X  |->  ( 1  /  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  z ) ) ) ) : X --> RR )
87 frn 5690 . . . . . . . . 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 9542 . . . . . . . . 9  |-  RR  C_  CC
9089a1i 11 . . . . . . . 8  |-  ( (
ph  /\  F : X
--> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  ->  RR  C_  CC )
91 cnrest2 20239 . . . . . . . 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 1264 . . . . . . 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 213 . . . . . 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 2512 . . . . 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 21923 . . . 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 730 . . . . . . . 8  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  x  e.  RR )  ->  sup ( ran  F ,  RR ,  <  )  e.  RR )
97 simpr 462 . . . . . . . . . . 11  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  x  e.  RR )  ->  x  e.  RR )
98 1re 9588 . . . . . . . . . . 11  |-  1  e.  RR
99 ifcl 3891 . . . . . . . . . . 11  |-  ( ( x  e.  RR  /\  1  e.  RR )  ->  if ( 1  <_  x ,  x , 
1 )  e.  RR )
10097, 98, 99sylancl 666 . . . . . . . . . 10  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  x  e.  RR )  ->  if ( 1  <_  x ,  x , 
1 )  e.  RR )
101 0red 9590 . . . . . . . . . . . 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 10082 . . . . . . . . . . . . 13  |-  0  <  1
104103a1i 11 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  x  e.  RR )  ->  0  <  1 )
105 max1 11426 . . . . . . . . . . . . 13  |-  ( ( 1  e.  RR  /\  x  e.  RR )  ->  1  <_  if (
1  <_  x ,  x ,  1 ) )
10698, 97, 105sylancr 667 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  x  e.  RR )  ->  1  <_  if (
1  <_  x ,  x ,  1 ) )
107101, 102, 100, 104, 106ltletrd 9741 . . . . . . . . . . 11  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  x  e.  RR )  ->  0  <  if ( 1  <_  x ,  x ,  1 ) )
108107gt0ne0d 10124 . . . . . . . . . 10  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  x  e.  RR )  ->  if ( 1  <_  x ,  x , 
1 )  =/=  0
)
109100, 108rereccld 10380 . . . . . . . . 9  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  x  e.  RR )  ->  ( 1  /  if ( 1  <_  x ,  x ,  1 ) )  e.  RR )
110100, 107recgt0d 10487 . . . . . . . . 9  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  x  e.  RR )  ->  0  <  ( 1  /  if ( 1  <_  x ,  x ,  1 ) ) )
111109, 110elrpd 11284 . . . . . . . 8  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  x  e.  RR )  ->  ( 1  /  if ( 1  <_  x ,  x ,  1 ) )  e.  RR+ )
11296, 111ltsubrpd 11316 . . . . . . 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 9993 . . . . . . . 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 9728 . . . . . . 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 213 . . . . . 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 762 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  x  e.  RR )
117 max2 11428 . . . . . . . . . . . 12  |-  ( ( 1  e.  RR  /\  x  e.  RR )  ->  x  <_  if (
1  <_  x ,  x ,  1 ) )
11898, 116, 117sylancr 667 . . . . . . . . . . 11  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  x  <_  if ( 1  <_  x ,  x ,  1 ) )
11939ad2antrr 730 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  sup ( ran  F ,  RR ,  <  )  e.  RR )
120 ffvelrn 5974 . . . . . . . . . . . . . . . . 17  |-  ( ( F : X --> ( RR 
\  { sup ( ran  F ,  RR ,  <  ) } )  /\  y  e.  X )  ->  ( F `  y
)  e.  ( RR 
\  { sup ( ran  F ,  RR ,  <  ) } ) )
121120ad2ant2l 750 . . . . . . . . . . . . . . . 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 4063 . . . . . . . . . . . . . . . 16  |-  ( ( F `  y )  e.  ( RR  \  { sup ( ran  F ,  RR ,  <  ) } )  <->  ( ( F `  y )  e.  RR  /\  ( F `
 y )  =/= 
sup ( ran  F ,  RR ,  <  )
) )
123121, 122sylib 199 . . . . . . . . . . . . . . 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 460 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  ( F `  y )  e.  RR )
125119, 124resubcld 9993 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  y )
)  e.  RR )
12637adantr 466 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  y  e.  X )  ->  ( ran  F  C_  RR  /\  ran  F  =/=  (/)  /\  E. x  e.  RR  A. z  e. 
ran  F  z  <_  x ) )
127 fnfvelrn 5973 . . . . . . . . . . . . . . . . . . 19  |-  ( ( F  Fn  X  /\  y  e.  X )  ->  ( F `  y
)  e.  ran  F
)
12831, 127sylan 473 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  y  e.  X )  ->  ( F `  y )  e.  ran  F )
129 suprub 10516 . . . . . . . . . . . . . . . . . 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 665 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  y  e.  X )  ->  ( F `  y )  <_  sup ( ran  F ,  RR ,  <  )
)
131130ad2ant2rl 753 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  ( F `  y )  <_  sup ( ran  F ,  RR ,  <  ) )
132123simprd 464 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  ( F `  y )  =/=  sup ( ran  F ,  RR ,  <  ) )
133132necomd 2651 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  sup ( ran  F ,  RR ,  <  )  =/=  ( F `
 y ) )
134124, 119ltlend 9726 . . . . . . . . . . . . . . . 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 930 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  ( F `  y )  <  sup ( ran  F ,  RR ,  <  ) )
136124, 119posdifd 10146 . . . . . . . . . . . . . . 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 213 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  0  <  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  y ) ) )
138137gt0ne0d 10124 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  y )
)  =/=  0 )
139125, 138rereccld 10380 . . . . . . . . . . . 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 666 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  if (
1  <_  x ,  x ,  1 )  e.  RR )
141 letr 9673 . . . . . . . . . . . 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 1264 . . . . . . . . . . 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 678 . . . . . . . . . 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 5820 . . . . . . . . . . . . . . 15  |-  ( z  =  y  ->  ( F `  z )  =  ( F `  y ) )
145144oveq2d 6260 . . . . . . . . . . . . . 14  |-  ( z  =  y  ->  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  z ) )  =  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  y ) ) )
146145oveq2d 6260 . . . . . . . . . . . . 13  |-  ( z  =  y  ->  (
1  /  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  z ) ) )  =  ( 1  /  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  y ) ) ) )
147 ovex 6272 . . . . . . . . . . . . 13  |-  ( 1  /  ( sup ( ran  F ,  RR ,  <  )  -  ( F `
 y ) ) )  e.  _V
148146, 85, 147fvmpt 5903 . . . . . . . . . . . 12  |-  ( y  e.  X  ->  (
( z  e.  X  |->  ( 1  /  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  z ) ) ) ) `  y )  =  ( 1  /  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  y ) ) ) )
149148breq1d 4371 . . . . . . . . . . 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 733 . . . . . . . . . 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 721 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  ( 1  /  if ( 1  <_  x ,  x ,  1 ) )  e.  RR )
152107adantrr 721 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  0  <  if ( 1  <_  x ,  x ,  1 ) )
153140, 152recgt0d 10487 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  0  <  ( 1  /  if ( 1  <_  x ,  x ,  1 ) ) )
154 lerec 10435 . . . . . . . . . . . 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 1265 . . . . . . . . . . 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 10039 . . . . . . . . . . . 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 1264 . . . . . . . . . . 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 9615 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  if (
1  <_  x ,  x ,  1 )  e.  CC )
159108adantrr 721 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  if (
1  <_  x ,  x ,  1 )  =/=  0 )
160158, 159recrecd 10326 . . . . . . . . . . . 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 4373 . . . . . . . . . . 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 286 . . . . . . . . . 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 271 . . . . . . . . 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 652 . . . . . . . 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 2768 . . . . . . 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 730 . . . . . . . . 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 10519 . . . . . . . . 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 665 . . . . . . . 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 730 . . . . . . . . 9  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  x  e.  RR )  ->  F  Fn  X )
170 breq1 4364 . . . . . . . . . 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 5979 . . . . . . . . 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 256 . . . . . . 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 237 . . . . . 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 180 . . . . 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 2815 . . . 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 578 . . 3  |-  ( ph  ->  -.  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )
178130ralrimiva 2774 . . . . . . . . 9  |-  ( ph  ->  A. y  e.  X  ( F `  y )  <_  sup ( ran  F ,  RR ,  <  )
)
179 breq2 4365 . . . . . . . . . 10  |-  ( ( F `  x )  =  sup ( ran 
F ,  RR ,  <  )  ->  ( ( F `  y )  <_  ( F `  x
)  <->  ( F `  y )  <_  sup ( ran  F ,  RR ,  <  ) ) )
180179ralbidv 2799 . . . . . . . . 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 225 . . . . . . . 8  |-  ( ph  ->  ( ( F `  x )  =  sup ( ran  F ,  RR ,  <  )  ->  A. y  e.  X  ( F `  y )  <_  ( F `  x )
) )
182181necon3bd 2610 . . . . . . 7  |-  ( ph  ->  ( -.  A. y  e.  X  ( F `  y )  <_  ( F `  x )  ->  ( F `  x
)  =/=  sup ( ran  F ,  RR ,  <  ) ) )
183182adantr 466 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  ( -.  A. y  e.  X  ( F `  y )  <_  ( F `  x )  ->  ( F `  x )  =/=  sup ( ran  F ,  RR ,  <  )
) )
18419ffvelrnda 5976 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  ( F `  x )  e.  RR )
185 eldifsn 4063 . . . . . . . 8  |-  ( ( F `  x )  e.  ( RR  \  { sup ( ran  F ,  RR ,  <  ) } )  <->  ( ( F `  x )  e.  RR  /\  ( F `
 x )  =/= 
sup ( ran  F ,  RR ,  <  )
) )
186185baib 911 . . . . . . 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 237 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  ( -.  A. y  e.  X  ( F `  y )  <_  ( F `  x )  ->  ( F `  x )  e.  ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) ) )
189188ralimdva 2768 . . . 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 6003 . . . . . 6  |-  ( F : X --> ( RR 
\  { sup ( ran  F ,  RR ,  <  ) } )  <->  ( F  Fn  X  /\  A. x  e.  X  ( F `  x )  e.  ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) ) )
191190baib 911 . . . . 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 237 . . 3  |-  ( ph  ->  ( A. x  e.  X  -.  A. y  e.  X  ( F `  y )  <_  ( F `  x )  ->  F : X --> ( RR 
\  { sup ( ran  F ,  RR ,  <  ) } ) ) )
194177, 193mtod 180 . 2  |-  ( ph  ->  -.  A. x  e.  X  -.  A. y  e.  X  ( F `  y )  <_  ( F `  x )
)
195 dfrex2 2810 . 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 215 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 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872    =/= wne 2594   A.wral 2709   E.wrex 2710    \ cdif 3371    C_ wss 3374   (/)c0 3699   ifcif 3849   {csn 3936   U.cuni 4157   class class class wbr 4361    |-> cmpt 4420   dom cdm 4791   ran crn 4792    Fn wfn 5534   -->wf 5535   ` cfv 5539  (class class class)co 6244   supcsup 7902   CCcc 9483   RRcr 9484   0cc0 9485   1c1 9486    < clt 9621    <_ cle 9622    - cmin 9806    / cdiv 10215   (,)cioo 11581   ↾t crest 15257   TopOpenctopn 15258   topGenctg 15274  ℂfldccnfld 18908   Topctop 19854  TopOnctopon 19855    Cn ccn 20177   Compccmp 20338    tX ctx 20512
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2058  ax-ext 2403  ax-rep 4474  ax-sep 4484  ax-nul 4493  ax-pow 4540  ax-pr 4598  ax-un 6536  ax-inf2 8094  ax-cnex 9541  ax-resscn 9542  ax-1cn 9543  ax-icn 9544  ax-addcl 9545  ax-addrcl 9546  ax-mulcl 9547  ax-mulrcl 9548  ax-mulcom 9549  ax-addass 9550  ax-mulass 9551  ax-distr 9552  ax-i2m1 9553  ax-1ne0 9554  ax-1rid 9555  ax-rnegex 9556  ax-rrecex 9557  ax-cnre 9558  ax-pre-lttri 9559  ax-pre-lttrn 9560  ax-pre-ltadd 9561  ax-pre-mulgt0 9562  ax-pre-sup 9563  ax-mulf 9565
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2275  df-mo 2276  df-clab 2410  df-cleq 2416  df-clel 2419  df-nfc 2553  df-ne 2596  df-nel 2597  df-ral 2714  df-rex 2715  df-reu 2716  df-rmo 2717  df-rab 2718  df-v 3019  df-sbc 3238  df-csb 3334  df-dif 3377  df-un 3379  df-in 3381  df-ss 3388  df-pss 3390  df-nul 3700  df-if 3850  df-pw 3921  df-sn 3937  df-pr 3939  df-tp 3941  df-op 3943  df-uni 4158  df-int 4194  df-iun 4239  df-iin 4240  df-br 4362  df-opab 4421  df-mpt 4422  df-tr 4457  df-eprel 4702  df-id 4706  df-po 4712  df-so 4713  df-fr 4750  df-se 4751  df-we 4752  df-xp 4797  df-rel 4798  df-cnv 4799  df-co 4800  df-dm 4801  df-rn 4802  df-res 4803  df-ima 4804  df-pred 5337  df-ord 5383  df-on 5384  df-lim 5385  df-suc 5386  df-iota 5503  df-fun 5541  df-fn 5542  df-f 5543  df-f1 5544  df-fo 5545  df-f1o 5546  df-fv 5547  df-isom 5548  df-riota 6206  df-ov 6247  df-oprab 6248  df-mpt2 6249  df-of 6484  df-om 6646  df-1st 6746  df-2nd 6747  df-supp 6865  df-wrecs 6978  df-recs 7040  df-rdg 7078  df-1o 7132  df-2o 7133  df-oadd 7136  df-er 7313  df-map 7424  df-ixp 7473  df-en 7520  df-dom 7521  df-sdom 7522  df-fin 7523  df-fsupp 7832  df-fi 7873  df-sup 7904  df-inf 7905  df-oi 7973  df-card 8320  df-cda 8544  df-pnf 9623  df-mnf 9624  df-xr 9625  df-ltxr 9626  df-le 9627  df-sub 9808  df-neg 9809  df-div 10216  df-nn 10556  df-2 10614  df-3 10615  df-4 10616  df-5 10617  df-6 10618  df-7 10619  df-8 10620  df-9 10621  df-10 10622  df-n0 10816  df-z 10884  df-dec 10998  df-uz 11106  df-q 11211  df-rp 11249  df-xneg 11355  df-xadd 11356  df-xmul 11357  df-ioo 11585  df-icc 11588  df-fz 11731  df-fzo 11862  df-seq 12159  df-exp 12218  df-hash 12461  df-cj 13101  df-re 13102  df-im 13103  df-sqrt 13237  df-abs 13238  df-struct 15061  df-ndx 15062  df-slot 15063  df-base 15064  df-sets 15065  df-ress 15066  df-plusg 15141  df-mulr 15142  df-starv 15143  df-sca 15144  df-vsca 15145  df-ip 15146  df-tset 15147  df-ple 15148  df-ds 15150  df-unif 15151  df-hom 15152  df-cco 15153  df-rest 15259  df-topn 15260  df-0g 15278  df-gsum 15279  df-topgen 15280  df-pt 15281  df-prds 15284  df-xrs 15338  df-qtop 15344  df-imas 15345  df-xps 15348  df-mre 15430  df-mrc 15431  df-acs 15433  df-mgm 16426  df-sgrp 16465  df-mnd 16475  df-submnd 16521  df-mulg 16614  df-cntz 16909  df-cmn 17370  df-psmet 18900  df-xmet 18901  df-met 18902  df-bl 18903  df-mopn 18904  df-cnfld 18909  df-top 19858  df-bases 19859  df-topon 19860  df-topsp 19861  df-cn 20180  df-cnp 20181  df-cmp 20339  df-tx 20514  df-hmeo 20707  df-xms 21272  df-ms 21273  df-tms 21274
This theorem is referenced by:  evth2  21925  evthicc  22347  evthf  37264  cncmpmax  37269
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