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Theorem evth 21194
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 465 . . . . 5  |-  ( (
ph  /\  F : X
--> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  ->  J  e.  Comp )
5 cmptop 19661 . . . . . . . . . 10  |-  ( J  e.  Comp  ->  J  e. 
Top )
64, 5syl 16 . . . . . . . . 9  |-  ( (
ph  /\  F : X
--> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  ->  J  e.  Top )
71toptopon 19201 . . . . . . . . 9  |-  ( J  e.  Top  <->  J  e.  (TopOn `  X ) )
86, 7sylib 196 . . . . . . . 8  |-  ( (
ph  /\  F : X
--> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  ->  J  e.  (TopOn `  X )
)
9 eqid 2467 . . . . . . . . . . 11  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
109cnfldtopon 21025 . . . . . . . . . 10  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
1110a1i 11 . . . . . . . . 9  |-  ( (
ph  /\  F : X
--> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  ->  ( TopOpen
` fld
)  e.  (TopOn `  CC ) )
12 1cnd 9608 . . . . . . . . 9  |-  ( (
ph  /\  F : X
--> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  ->  1  e.  CC )
138, 11, 12cnmptc 19898 . . . . . . . 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 21004 . . . . . . . . . . . . . . . . . . 19  |-  RR  =  U. ( topGen `  ran  (,) )
162unieqi 4254 . . . . . . . . . . . . . . . . . . 19  |-  U. K  =  U. ( topGen `  ran  (,) )
1715, 16eqtr4i 2499 . . . . . . . . . . . . . . . . . 18  |-  RR  =  U. K
181, 17cnf 19513 . . . . . . . . . . . . . . . . 17  |-  ( F  e.  ( J  Cn  K )  ->  F : X --> RR )
1914, 18syl 16 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  F : X --> RR )
20 frn 5735 . . . . . . . . . . . . . . . 16  |-  ( F : X --> RR  ->  ran 
F  C_  RR )
2119, 20syl 16 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ran  F  C_  RR )
22 fdm 5733 . . . . . . . . . . . . . . . . . 18  |-  ( F : X --> RR  ->  dom 
F  =  X )
2319, 22syl 16 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  dom  F  =  X )
24 evth.5 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  X  =/=  (/) )
2523, 24eqnetrd 2760 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  dom  F  =/=  (/) )
26 dm0rn0 5217 . . . . . . . . . . . . . . . . 17  |-  ( dom 
F  =  (/)  <->  ran  F  =  (/) )
2726necon3bii 2735 . . . . . . . . . . . . . . . 16  |-  ( dom 
F  =/=  (/)  <->  ran  F  =/=  (/) )
2825, 27sylib 196 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ran  F  =/=  (/) )
291, 2, 3, 14bndth 21193 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  E. x  e.  RR  A. y  e.  X  ( F `  y )  <_  x )
30 ffn 5729 . . . . . . . . . . . . . . . . . . 19  |-  ( F : X --> RR  ->  F  Fn  X )
3119, 30syl 16 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  F  Fn  X )
32 breq1 4450 . . . . . . . . . . . . . . . . . . 19  |-  ( z  =  ( F `  y )  ->  (
z  <_  x  <->  ( F `  y )  <_  x
) )
3332ralrn 6022 . . . . . . . . . . . . . . . . . 18  |-  ( F  Fn  X  ->  ( A. z  e.  ran  F  z  <_  x  <->  A. y  e.  X  ( F `  y )  <_  x
) )
3431, 33syl 16 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( A. z  e. 
ran  F  z  <_  x  <->  A. y  e.  X  ( F `  y )  <_  x ) )
3534rexbidv 2973 . . . . . . . . . . . . . . . 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 232 . . . . . . . . . . . . . . 15  |-  ( ph  ->  E. x  e.  RR  A. z  e.  ran  F  z  <_  x )
3721, 28, 363jca 1176 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ran  F  C_  RR  /\  ran  F  =/=  (/)  /\  E. x  e.  RR  A. z  e. 
ran  F  z  <_  x ) )
38 suprcl 10499 . . . . . . . . . . . . . 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 16 . . . . . . . . . . . . 13  |-  ( ph  ->  sup ( ran  F ,  RR ,  <  )  e.  RR )
4039recnd 9618 . . . . . . . . . . . 12  |-  ( ph  ->  sup ( ran  F ,  RR ,  <  )  e.  CC )
4140adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  F : X
--> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  ->  sup ( ran  F ,  RR ,  <  )  e.  CC )
428, 11, 41cnmptc 19898 . . . . . . . . . 10  |-  ( (
ph  /\  F : X
--> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  ->  (
z  e.  X  |->  sup ( ran  F ,  RR ,  <  ) )  e.  ( J  Cn  ( TopOpen ` fld ) ) )
4319feqmptd 5918 . . . . . . . . . . . 12  |-  ( ph  ->  F  =  ( z  e.  X  |->  ( F `
 z ) ) )
449cnfldtop 21026 . . . . . . . . . . . . . 14  |-  ( TopOpen ` fld )  e.  Top
45 cnrest2r 19554 . . . . . . . . . . . . . 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 21043 . . . . . . . . . . . . . . . 16  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
482, 47eqtri 2496 . . . . . . . . . . . . . . 15  |-  K  =  ( ( TopOpen ` fld )t  RR )
4948oveq2i 6293 . . . . . . . . . . . . . 14  |-  ( J  Cn  K )  =  ( J  Cn  (
( TopOpen ` fld )t  RR ) )
5014, 49syl6eleq 2565 . . . . . . . . . . . . 13  |-  ( ph  ->  F  e.  ( J  Cn  ( ( TopOpen ` fld )t  RR ) ) )
5146, 50sseldi 3502 . . . . . . . . . . . 12  |-  ( ph  ->  F  e.  ( J  Cn  ( TopOpen ` fld ) ) )
5243, 51eqeltrrd 2556 . . . . . . . . . . 11  |-  ( ph  ->  ( z  e.  X  |->  ( F `  z
) )  e.  ( J  Cn  ( TopOpen ` fld )
) )
5352adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  F : X
--> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  ->  (
z  e.  X  |->  ( F `  z ) )  e.  ( J  Cn  ( TopOpen ` fld ) ) )
549subcn 21105 . . . . . . . . . . 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 19902 . . . . . . . . 9  |-  ( (
ph  /\  F : X
--> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  ->  (
z  e.  X  |->  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  z ) ) )  e.  ( J  Cn  ( TopOpen ` fld ) ) )
5739ad2antrr 725 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  z  e.  X )  ->  sup ( ran  F ,  RR ,  <  )  e.  RR )
58 ffvelrn 6017 . . . . . . . . . . . . . . . . . 18  |-  ( ( F : X --> ( RR 
\  { sup ( ran  F ,  RR ,  <  ) } )  /\  z  e.  X )  ->  ( F `  z
)  e.  ( RR 
\  { sup ( ran  F ,  RR ,  <  ) } ) )
5958adantll 713 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  z  e.  X )  ->  ( F `  z
)  e.  ( RR 
\  { sup ( ran  F ,  RR ,  <  ) } ) )
60 eldifsn 4152 . . . . . . . . . . . . . . . . 17  |-  ( ( F `  z )  e.  ( RR  \  { sup ( ran  F ,  RR ,  <  ) } )  <->  ( ( F `  z )  e.  RR  /\  ( F `
 z )  =/= 
sup ( ran  F ,  RR ,  <  )
) )
6159, 60sylib 196 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  z  e.  X )  ->  ( ( F `  z )  e.  RR  /\  ( F `  z
)  =/=  sup ( ran  F ,  RR ,  <  ) ) )
6261simpld 459 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  z  e.  X )  ->  ( F `  z
)  e.  RR )
6357, 62resubcld 9983 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  z  e.  X )  ->  ( sup ( ran 
F ,  RR ,  <  )  -  ( F `
 z ) )  e.  RR )
6463recnd 9618 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  z  e.  X )  ->  ( sup ( ran 
F ,  RR ,  <  )  -  ( F `
 z ) )  e.  CC )
6557recnd 9618 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  z  e.  X )  ->  sup ( ran  F ,  RR ,  <  )  e.  CC )
6662recnd 9618 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  z  e.  X )  ->  ( F `  z
)  e.  CC )
6761simprd 463 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  z  e.  X )  ->  ( F `  z
)  =/=  sup ( ran  F ,  RR ,  <  ) )
6867necomd 2738 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  z  e.  X )  ->  sup ( ran  F ,  RR ,  <  )  =/=  ( F `  z
) )
6965, 66, 68subne0d 9935 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  z  e.  X )  ->  ( sup ( ran 
F ,  RR ,  <  )  -  ( F `
 z ) )  =/=  0 )
70 eldifsn 4152 . . . . . . . . . . . . 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 664 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  z  e.  X )  ->  ( sup ( ran 
F ,  RR ,  <  )  -  ( F `
 z ) )  e.  ( CC  \  { 0 } ) )
72 eqid 2467 . . . . . . . . . . . 12  |-  ( z  e.  X  |->  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  z ) ) )  =  ( z  e.  X  |->  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  z ) ) )
7371, 72fmptd 6043 . . . . . . . . . . 11  |-  ( (
ph  /\  F : X
--> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  ->  (
z  e.  X  |->  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  z ) ) ) : X --> ( CC 
\  { 0 } ) )
74 frn 5735 . . . . . . . . . . 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 16 . . . . . . . . . 10  |-  ( (
ph  /\  F : X
--> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  ->  ran  ( z  e.  X  |->  ( sup ( ran 
F ,  RR ,  <  )  -  ( F `
 z ) ) )  C_  ( CC  \  { 0 } ) )
76 difssd 3632 . . . . . . . . . 10  |-  ( (
ph  /\  F : X
--> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  ->  ( CC  \  { 0 } )  C_  CC )
77 cnrest2 19553 . . . . . . . . . 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 1228 . . . . . . . . 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 210 . . . . . . . 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 2467 . . . . . . . . . 10  |-  ( (
TopOpen ` fld )t  ( CC  \  {
0 } ) )  =  ( ( TopOpen ` fld )t  ( CC  \  { 0 } ) )
819, 80divcn 21107 . . . . . . . . 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 19902 . . . . . . 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 10367 . . . . . . . . . 10  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  z  e.  X )  ->  ( 1  /  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  z ) ) )  e.  RR )
85 eqid 2467 . . . . . . . . . 10  |-  ( z  e.  X  |->  ( 1  /  ( sup ( ran  F ,  RR ,  <  )  -  ( F `
 z ) ) ) )  =  ( z  e.  X  |->  ( 1  /  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  z ) ) ) )
8684, 85fmptd 6043 . . . . . . . . 9  |-  ( (
ph  /\  F : X
--> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  ->  (
z  e.  X  |->  ( 1  /  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  z ) ) ) ) : X --> RR )
87 frn 5735 . . . . . . . . 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 16 . . . . . . . 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 9545 . . . . . . . . 9  |-  RR  C_  CC
9089a1i 11 . . . . . . . 8  |-  ( (
ph  /\  F : X
--> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  ->  RR  C_  CC )
91 cnrest2 19553 . . . . . . . 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 1228 . . . . . . 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 210 . . . . . 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 2566 . . . . 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 21193 . . . 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 725 . . . . . . . 8  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  x  e.  RR )  ->  sup ( ran  F ,  RR ,  <  )  e.  RR )
97 simpr 461 . . . . . . . . . . 11  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  x  e.  RR )  ->  x  e.  RR )
98 1re 9591 . . . . . . . . . . 11  |-  1  e.  RR
99 ifcl 3981 . . . . . . . . . . 11  |-  ( ( x  e.  RR  /\  1  e.  RR )  ->  if ( 1  <_  x ,  x , 
1 )  e.  RR )
10097, 98, 99sylancl 662 . . . . . . . . . 10  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  x  e.  RR )  ->  if ( 1  <_  x ,  x , 
1 )  e.  RR )
101 0red 9593 . . . . . . . . . . . 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 10071 . . . . . . . . . . . . 13  |-  0  <  1
104103a1i 11 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  x  e.  RR )  ->  0  <  1 )
105 max1 11382 . . . . . . . . . . . . 13  |-  ( ( 1  e.  RR  /\  x  e.  RR )  ->  1  <_  if (
1  <_  x ,  x ,  1 ) )
10698, 97, 105sylancr 663 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  x  e.  RR )  ->  1  <_  if (
1  <_  x ,  x ,  1 ) )
107101, 102, 100, 104, 106ltletrd 9737 . . . . . . . . . . 11  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  x  e.  RR )  ->  0  <  if ( 1  <_  x ,  x ,  1 ) )
108107gt0ne0d 10113 . . . . . . . . . 10  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  x  e.  RR )  ->  if ( 1  <_  x ,  x , 
1 )  =/=  0
)
109100, 108rereccld 10367 . . . . . . . . 9  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  x  e.  RR )  ->  ( 1  /  if ( 1  <_  x ,  x ,  1 ) )  e.  RR )
110100, 107recgt0d 10476 . . . . . . . . 9  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  x  e.  RR )  ->  0  <  ( 1  /  if ( 1  <_  x ,  x ,  1 ) ) )
111109, 110elrpd 11250 . . . . . . . 8  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  x  e.  RR )  ->  ( 1  /  if ( 1  <_  x ,  x ,  1 ) )  e.  RR+ )
11296, 111ltsubrpd 11280 . . . . . . 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 9983 . . . . . . . 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 9727 . . . . . . 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 210 . . . . . 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 755 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  x  e.  RR )
117 max2 11384 . . . . . . . . . . . 12  |-  ( ( 1  e.  RR  /\  x  e.  RR )  ->  x  <_  if (
1  <_  x ,  x ,  1 ) )
11898, 116, 117sylancr 663 . . . . . . . . . . 11  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  x  <_  if ( 1  <_  x ,  x ,  1 ) )
11939ad2antrr 725 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  sup ( ran  F ,  RR ,  <  )  e.  RR )
120 ffvelrn 6017 . . . . . . . . . . . . . . . . 17  |-  ( ( F : X --> ( RR 
\  { sup ( ran  F ,  RR ,  <  ) } )  /\  y  e.  X )  ->  ( F `  y
)  e.  ( RR 
\  { sup ( ran  F ,  RR ,  <  ) } ) )
121120ad2ant2l 745 . . . . . . . . . . . . . . . 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 4152 . . . . . . . . . . . . . . . 16  |-  ( ( F `  y )  e.  ( RR  \  { sup ( ran  F ,  RR ,  <  ) } )  <->  ( ( F `  y )  e.  RR  /\  ( F `
 y )  =/= 
sup ( ran  F ,  RR ,  <  )
) )
123121, 122sylib 196 . . . . . . . . . . . . . . 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 459 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  ( F `  y )  e.  RR )
125119, 124resubcld 9983 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  y )
)  e.  RR )
12637adantr 465 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  y  e.  X )  ->  ( ran  F  C_  RR  /\  ran  F  =/=  (/)  /\  E. x  e.  RR  A. z  e. 
ran  F  z  <_  x ) )
127 fnfvelrn 6016 . . . . . . . . . . . . . . . . . . 19  |-  ( ( F  Fn  X  /\  y  e.  X )  ->  ( F `  y
)  e.  ran  F
)
12831, 127sylan 471 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  y  e.  X )  ->  ( F `  y )  e.  ran  F )
129 suprub 10500 . . . . . . . . . . . . . . . . . 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 661 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  y  e.  X )  ->  ( F `  y )  <_  sup ( ran  F ,  RR ,  <  )
)
131130ad2ant2rl 748 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  ( F `  y )  <_  sup ( ran  F ,  RR ,  <  ) )
132123simprd 463 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  ( F `  y )  =/=  sup ( ran  F ,  RR ,  <  ) )
133132necomd 2738 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  sup ( ran  F ,  RR ,  <  )  =/=  ( F `
 y ) )
134124, 119ltlend 9725 . . . . . . . . . . . . . . . 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 920 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  ( F `  y )  <  sup ( ran  F ,  RR ,  <  ) )
136124, 119posdifd 10135 . . . . . . . . . . . . . . 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 210 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  0  <  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  y ) ) )
138137gt0ne0d 10113 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  y )
)  =/=  0 )
139125, 138rereccld 10367 . . . . . . . . . . . 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 662 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  if (
1  <_  x ,  x ,  1 )  e.  RR )
141 letr 9674 . . . . . . . . . . . 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 1228 . . . . . . . . . . 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 674 . . . . . . . . . 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 5864 . . . . . . . . . . . . . . 15  |-  ( z  =  y  ->  ( F `  z )  =  ( F `  y ) )
145144oveq2d 6298 . . . . . . . . . . . . . 14  |-  ( z  =  y  ->  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  z ) )  =  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  y ) ) )
146145oveq2d 6298 . . . . . . . . . . . . 13  |-  ( z  =  y  ->  (
1  /  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  z ) ) )  =  ( 1  /  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  y ) ) ) )
147 ovex 6307 . . . . . . . . . . . . 13  |-  ( 1  /  ( sup ( ran  F ,  RR ,  <  )  -  ( F `
 y ) ) )  e.  _V
148146, 85, 147fvmpt 5948 . . . . . . . . . . . 12  |-  ( y  e.  X  ->  (
( z  e.  X  |->  ( 1  /  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  z ) ) ) ) `  y )  =  ( 1  /  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  y ) ) ) )
149148breq1d 4457 . . . . . . . . . . 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 728 . . . . . . . . . 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 716 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  ( 1  /  if ( 1  <_  x ,  x ,  1 ) )  e.  RR )
152107adantrr 716 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  0  <  if ( 1  <_  x ,  x ,  1 ) )
153140, 152recgt0d 10476 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  0  <  ( 1  /  if ( 1  <_  x ,  x ,  1 ) ) )
154 lerec 10423 . . . . . . . . . . . 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 1229 . . . . . . . . . . 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 10027 . . . . . . . . . . . 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 1228 . . . . . . . . . . 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 9618 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  if (
1  <_  x ,  x ,  1 )  e.  CC )
159108adantrr 716 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  if (
1  <_  x ,  x ,  1 )  =/=  0 )
160158, 159recrecd 10313 . . . . . . . . . . . 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 4459 . . . . . . . . . . 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 283 . . . . . . . . . 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 268 . . . . . . . . 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 648 . . . . . . . 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 2872 . . . . . . 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 725 . . . . . . . . 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 10503 . . . . . . . . 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 661 . . . . . . . 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 725 . . . . . . . . 9  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  x  e.  RR )  ->  F  Fn  X )
170 breq1 4450 . . . . . . . . . 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 6022 . . . . . . . . 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 16 . . . . . . . 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 253 . . . . . . 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 234 . . . . . 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 177 . . . . 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 2920 . . . 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 576 . . 3  |-  ( ph  ->  -.  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )
178130ralrimiva 2878 . . . . . . . . 9  |-  ( ph  ->  A. y  e.  X  ( F `  y )  <_  sup ( ran  F ,  RR ,  <  )
)
179 breq2 4451 . . . . . . . . . 10  |-  ( ( F `  x )  =  sup ( ran 
F ,  RR ,  <  )  ->  ( ( F `  y )  <_  ( F `  x
)  <->  ( F `  y )  <_  sup ( ran  F ,  RR ,  <  ) ) )
180179ralbidv 2903 . . . . . . . . 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 222 . . . . . . . 8  |-  ( ph  ->  ( ( F `  x )  =  sup ( ran  F ,  RR ,  <  )  ->  A. y  e.  X  ( F `  y )  <_  ( F `  x )
) )
182181necon3bd 2679 . . . . . . 7  |-  ( ph  ->  ( -.  A. y  e.  X  ( F `  y )  <_  ( F `  x )  ->  ( F `  x
)  =/=  sup ( ran  F ,  RR ,  <  ) ) )
183182adantr 465 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  ( -.  A. y  e.  X  ( F `  y )  <_  ( F `  x )  ->  ( F `  x )  =/=  sup ( ran  F ,  RR ,  <  )
) )
18419ffvelrnda 6019 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  ( F `  x )  e.  RR )
185 eldifsn 4152 . . . . . . . 8  |-  ( ( F `  x )  e.  ( RR  \  { sup ( ran  F ,  RR ,  <  ) } )  <->  ( ( F `  x )  e.  RR  /\  ( F `
 x )  =/= 
sup ( ran  F ,  RR ,  <  )
) )
186185baib 901 . . . . . . 7  |-  ( ( F `  x )  e.  RR  ->  (
( F `  x
)  e.  ( RR 
\  { sup ( ran  F ,  RR ,  <  ) } )  <->  ( F `  x )  =/=  sup ( ran  F ,  RR ,  <  ) ) )
187184, 186syl 16 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  (
( F `  x
)  e.  ( RR 
\  { sup ( ran  F ,  RR ,  <  ) } )  <->  ( F `  x )  =/=  sup ( ran  F ,  RR ,  <  ) ) )
188183, 187sylibrd 234 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  ( -.  A. y  e.  X  ( F `  y )  <_  ( F `  x )  ->  ( F `  x )  e.  ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) ) )
189188ralimdva 2872 . . . 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 6045 . . . . . 6  |-  ( F : X --> ( RR 
\  { sup ( ran  F ,  RR ,  <  ) } )  <->  ( F  Fn  X  /\  A. x  e.  X  ( F `  x )  e.  ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) ) )
191190baib 901 . . . . 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 16 . . . 4  |-  ( ph  ->  ( F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } )  <->  A. x  e.  X  ( F `  x )  e.  ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) ) )
193189, 192sylibrd 234 . . 3  |-  ( ph  ->  ( A. x  e.  X  -.  A. y  e.  X  ( F `  y )  <_  ( F `  x )  ->  F : X --> ( RR 
\  { sup ( ran  F ,  RR ,  <  ) } ) ) )
194177, 193mtod 177 . 2  |-  ( ph  ->  -.  A. x  e.  X  -.  A. y  e.  X  ( F `  y )  <_  ( F `  x )
)
195 dfrex2 2915 . 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 212 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 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   E.wrex 2815    \ cdif 3473    C_ wss 3476   (/)c0 3785   ifcif 3939   {csn 4027   U.cuni 4245   class class class wbr 4447    |-> cmpt 4505   dom cdm 4999   ran crn 5000    Fn wfn 5581   -->wf 5582   ` cfv 5586  (class class class)co 6282   supcsup 7896   CCcc 9486   RRcr 9487   0cc0 9488   1c1 9489    < clt 9624    <_ cle 9625    - cmin 9801    / cdiv 10202   (,)cioo 11525   ↾t crest 14672   TopOpenctopn 14673   topGenctg 14689  ℂfldccnfld 18191   Topctop 19161  TopOnctopon 19162    Cn ccn 19491   Compccmp 19652    tX ctx 19796
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566  ax-mulf 9568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-of 6522  df-om 6679  df-1st 6781  df-2nd 6782  df-supp 6899  df-recs 7039  df-rdg 7073  df-1o 7127  df-2o 7128  df-oadd 7131  df-er 7308  df-map 7419  df-ixp 7467  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-fsupp 7826  df-fi 7867  df-sup 7897  df-oi 7931  df-card 8316  df-cda 8544  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10973  df-uz 11079  df-q 11179  df-rp 11217  df-xneg 11314  df-xadd 11315  df-xmul 11316  df-ioo 11529  df-icc 11532  df-fz 11669  df-fzo 11789  df-seq 12072  df-exp 12131  df-hash 12370  df-cj 12891  df-re 12892  df-im 12893  df-sqrt 13027  df-abs 13028  df-struct 14488  df-ndx 14489  df-slot 14490  df-base 14491  df-sets 14492  df-ress 14493  df-plusg 14564  df-mulr 14565  df-starv 14566  df-sca 14567  df-vsca 14568  df-ip 14569  df-tset 14570  df-ple 14571  df-ds 14573  df-unif 14574  df-hom 14575  df-cco 14576  df-rest 14674  df-topn 14675  df-0g 14693  df-gsum 14694  df-topgen 14695  df-pt 14696  df-prds 14699  df-xrs 14753  df-qtop 14758  df-imas 14759  df-xps 14761  df-mre 14837  df-mrc 14838  df-acs 14840  df-mnd 15728  df-submnd 15778  df-mulg 15861  df-cntz 16150  df-cmn 16596  df-psmet 18182  df-xmet 18183  df-met 18184  df-bl 18185  df-mopn 18186  df-cnfld 18192  df-top 19166  df-bases 19168  df-topon 19169  df-topsp 19170  df-cn 19494  df-cnp 19495  df-cmp 19653  df-tx 19798  df-hmeo 19991  df-xms 20558  df-ms 20559  df-tms 20560
This theorem is referenced by:  evth2  21195  evthicc  21606  evthf  30980  cncmpmax  30985
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