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Theorem evth 18473
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 451 . . . . 5  |-  ( (
ph  /\  F : X
--> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  ->  J  e.  Comp )
5 cmptop 17138 . . . . . . . . . 10  |-  ( J  e.  Comp  ->  J  e. 
Top )
64, 5syl 15 . . . . . . . . 9  |-  ( (
ph  /\  F : X
--> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  ->  J  e.  Top )
71toptopon 16687 . . . . . . . . 9  |-  ( J  e.  Top  <->  J  e.  (TopOn `  X ) )
86, 7sylib 188 . . . . . . . 8  |-  ( (
ph  /\  F : X
--> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  ->  J  e.  (TopOn `  X )
)
9 eqid 2296 . . . . . . . . . . 11  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
109cnfldtopon 18308 . . . . . . . . . 10  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
1110a1i 10 . . . . . . . . 9  |-  ( (
ph  /\  F : X
--> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  ->  ( TopOpen
` fld
)  e.  (TopOn `  CC ) )
12 ax-1cn 8811 . . . . . . . . . 10  |-  1  e.  CC
1312a1i 10 . . . . . . . . 9  |-  ( (
ph  /\  F : X
--> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  ->  1  e.  CC )
148, 11, 13cnmptc 17372 . . . . . . . 8  |-  ( (
ph  /\  F : X
--> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  ->  (
z  e.  X  |->  1 )  e.  ( J  Cn  ( TopOpen ` fld ) ) )
15 bndth.4 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  F  e.  ( J  Cn  K ) )
16 uniretop 18287 . . . . . . . . . . . . . . . . . . 19  |-  RR  =  U. ( topGen `  ran  (,) )
172unieqi 3853 . . . . . . . . . . . . . . . . . . 19  |-  U. K  =  U. ( topGen `  ran  (,) )
1816, 17eqtr4i 2319 . . . . . . . . . . . . . . . . . 18  |-  RR  =  U. K
191, 18cnf 16992 . . . . . . . . . . . . . . . . 17  |-  ( F  e.  ( J  Cn  K )  ->  F : X --> RR )
2015, 19syl 15 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  F : X --> RR )
21 frn 5411 . . . . . . . . . . . . . . . 16  |-  ( F : X --> RR  ->  ran 
F  C_  RR )
2220, 21syl 15 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ran  F  C_  RR )
23 fdm 5409 . . . . . . . . . . . . . . . . . 18  |-  ( F : X --> RR  ->  dom 
F  =  X )
2420, 23syl 15 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  dom  F  =  X )
25 evth.5 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  X  =/=  (/) )
2624, 25eqnetrd 2477 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  dom  F  =/=  (/) )
27 dm0rn0 4911 . . . . . . . . . . . . . . . . 17  |-  ( dom 
F  =  (/)  <->  ran  F  =  (/) )
2827necon3bii 2491 . . . . . . . . . . . . . . . 16  |-  ( dom 
F  =/=  (/)  <->  ran  F  =/=  (/) )
2926, 28sylib 188 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ran  F  =/=  (/) )
301, 2, 3, 15bndth 18472 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  E. x  e.  RR  A. y  e.  X  ( F `  y )  <_  x )
31 ffn 5405 . . . . . . . . . . . . . . . . . . 19  |-  ( F : X --> RR  ->  F  Fn  X )
3220, 31syl 15 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  F  Fn  X )
33 breq1 4042 . . . . . . . . . . . . . . . . . . 19  |-  ( z  =  ( F `  y )  ->  (
z  <_  x  <->  ( F `  y )  <_  x
) )
3433ralrn 5684 . . . . . . . . . . . . . . . . . 18  |-  ( F  Fn  X  ->  ( A. z  e.  ran  F  z  <_  x  <->  A. y  e.  X  ( F `  y )  <_  x
) )
3532, 34syl 15 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( A. z  e. 
ran  F  z  <_  x  <->  A. y  e.  X  ( F `  y )  <_  x ) )
3635rexbidv 2577 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( E. x  e.  RR  A. z  e. 
ran  F  z  <_  x  <->  E. x  e.  RR  A. y  e.  X  ( F `  y )  <_  x ) )
3730, 36mpbird 223 . . . . . . . . . . . . . . 15  |-  ( ph  ->  E. x  e.  RR  A. z  e.  ran  F  z  <_  x )
3822, 29, 373jca 1132 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ran  F  C_  RR  /\  ran  F  =/=  (/)  /\  E. x  e.  RR  A. z  e. 
ran  F  z  <_  x ) )
39 suprcl 9730 . . . . . . . . . . . . . 14  |-  ( ( ran  F  C_  RR  /\ 
ran  F  =/=  (/)  /\  E. x  e.  RR  A. z  e.  ran  F  z  <_  x )  ->  sup ( ran  F ,  RR ,  <  )  e.  RR )
4038, 39syl 15 . . . . . . . . . . . . 13  |-  ( ph  ->  sup ( ran  F ,  RR ,  <  )  e.  RR )
4140recnd 8877 . . . . . . . . . . . 12  |-  ( ph  ->  sup ( ran  F ,  RR ,  <  )  e.  CC )
4241adantr 451 . . . . . . . . . . 11  |-  ( (
ph  /\  F : X
--> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  ->  sup ( ran  F ,  RR ,  <  )  e.  CC )
438, 11, 42cnmptc 17372 . . . . . . . . . 10  |-  ( (
ph  /\  F : X
--> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  ->  (
z  e.  X  |->  sup ( ran  F ,  RR ,  <  ) )  e.  ( J  Cn  ( TopOpen ` fld ) ) )
4420feqmptd 5591 . . . . . . . . . . . 12  |-  ( ph  ->  F  =  ( z  e.  X  |->  ( F `
 z ) ) )
459cnfldtop 18309 . . . . . . . . . . . . . 14  |-  ( TopOpen ` fld )  e.  Top
46 cnrest2r 17031 . . . . . . . . . . . . . 14  |-  ( (
TopOpen ` fld )  e.  Top  ->  ( J  Cn  ( (
TopOpen ` fld )t  RR ) )  C_  ( J  Cn  ( TopOpen
` fld
) ) )
4745, 46ax-mp 8 . . . . . . . . . . . . 13  |-  ( J  Cn  ( ( TopOpen ` fld )t  RR ) )  C_  ( J  Cn  ( TopOpen ` fld ) )
489tgioo2 18325 . . . . . . . . . . . . . . . 16  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
492, 48eqtri 2316 . . . . . . . . . . . . . . 15  |-  K  =  ( ( TopOpen ` fld )t  RR )
5049oveq2i 5885 . . . . . . . . . . . . . 14  |-  ( J  Cn  K )  =  ( J  Cn  (
( TopOpen ` fld )t  RR ) )
5115, 50syl6eleq 2386 . . . . . . . . . . . . 13  |-  ( ph  ->  F  e.  ( J  Cn  ( ( TopOpen ` fld )t  RR ) ) )
5247, 51sseldi 3191 . . . . . . . . . . . 12  |-  ( ph  ->  F  e.  ( J  Cn  ( TopOpen ` fld ) ) )
5344, 52eqeltrrd 2371 . . . . . . . . . . 11  |-  ( ph  ->  ( z  e.  X  |->  ( F `  z
) )  e.  ( J  Cn  ( TopOpen ` fld )
) )
5453adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  F : X
--> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  ->  (
z  e.  X  |->  ( F `  z ) )  e.  ( J  Cn  ( TopOpen ` fld ) ) )
559subcn 18386 . . . . . . . . . . 11  |-  -  e.  ( ( ( TopOpen ` fld )  tX  ( TopOpen ` fld ) )  Cn  ( TopOpen
` fld
) )
5655a1i 10 . . . . . . . . . 10  |-  ( (
ph  /\  F : X
--> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  ->  -  e.  ( ( ( TopOpen ` fld )  tX  ( TopOpen ` fld ) )  Cn  ( TopOpen
` fld
) ) )
578, 43, 54, 56cnmpt12f 17376 . . . . . . . . 9  |-  ( (
ph  /\  F : X
--> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  ->  (
z  e.  X  |->  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  z ) ) )  e.  ( J  Cn  ( TopOpen ` fld ) ) )
5840ad2antrr 706 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  z  e.  X )  ->  sup ( ran  F ,  RR ,  <  )  e.  RR )
59 ffvelrn 5679 . . . . . . . . . . . . . . . . . 18  |-  ( ( F : X --> ( RR 
\  { sup ( ran  F ,  RR ,  <  ) } )  /\  z  e.  X )  ->  ( F `  z
)  e.  ( RR 
\  { sup ( ran  F ,  RR ,  <  ) } ) )
6059adantll 694 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  z  e.  X )  ->  ( F `  z
)  e.  ( RR 
\  { sup ( ran  F ,  RR ,  <  ) } ) )
61 eldifsn 3762 . . . . . . . . . . . . . . . . 17  |-  ( ( F `  z )  e.  ( RR  \  { sup ( ran  F ,  RR ,  <  ) } )  <->  ( ( F `  z )  e.  RR  /\  ( F `
 z )  =/= 
sup ( ran  F ,  RR ,  <  )
) )
6260, 61sylib 188 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  z  e.  X )  ->  ( ( F `  z )  e.  RR  /\  ( F `  z
)  =/=  sup ( ran  F ,  RR ,  <  ) ) )
6362simpld 445 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  z  e.  X )  ->  ( F `  z
)  e.  RR )
6458, 63resubcld 9227 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  z  e.  X )  ->  ( sup ( ran 
F ,  RR ,  <  )  -  ( F `
 z ) )  e.  RR )
6564recnd 8877 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  z  e.  X )  ->  ( sup ( ran 
F ,  RR ,  <  )  -  ( F `
 z ) )  e.  CC )
6662simprd 449 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  z  e.  X )  ->  ( F `  z
)  =/=  sup ( ran  F ,  RR ,  <  ) )
6766necomd 2542 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  z  e.  X )  ->  sup ( ran  F ,  RR ,  <  )  =/=  ( F `  z
) )
6858recnd 8877 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  z  e.  X )  ->  sup ( ran  F ,  RR ,  <  )  e.  CC )
6963recnd 8877 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  z  e.  X )  ->  ( F `  z
)  e.  CC )
70 subeq0 9089 . . . . . . . . . . . . . . . 16  |-  ( ( sup ( ran  F ,  RR ,  <  )  e.  CC  /\  ( F `
 z )  e.  CC )  ->  (
( sup ( ran 
F ,  RR ,  <  )  -  ( F `
 z ) )  =  0  <->  sup ( ran  F ,  RR ,  <  )  =  ( F `
 z ) ) )
7168, 69, 70syl2anc 642 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  z  e.  X )  ->  ( ( sup ( ran  F ,  RR ,  <  )  -  ( F `
 z ) )  =  0  <->  sup ( ran  F ,  RR ,  <  )  =  ( F `
 z ) ) )
7271necon3bid 2494 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  z  e.  X )  ->  ( ( sup ( ran  F ,  RR ,  <  )  -  ( F `
 z ) )  =/=  0  <->  sup ( ran  F ,  RR ,  <  )  =/=  ( F `
 z ) ) )
7367, 72mpbird 223 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  z  e.  X )  ->  ( sup ( ran 
F ,  RR ,  <  )  -  ( F `
 z ) )  =/=  0 )
74 eldifsn 3762 . . . . . . . . . . . . 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 ) )
7565, 73, 74sylanbrc 645 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  z  e.  X )  ->  ( sup ( ran 
F ,  RR ,  <  )  -  ( F `
 z ) )  e.  ( CC  \  { 0 } ) )
76 eqid 2296 . . . . . . . . . . . 12  |-  ( z  e.  X  |->  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  z ) ) )  =  ( z  e.  X  |->  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  z ) ) )
7775, 76fmptd 5700 . . . . . . . . . . 11  |-  ( (
ph  /\  F : X
--> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  ->  (
z  e.  X  |->  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  z ) ) ) : X --> ( CC 
\  { 0 } ) )
78 frn 5411 . . . . . . . . . . 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 } ) )
7977, 78syl 15 . . . . . . . . . 10  |-  ( (
ph  /\  F : X
--> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  ->  ran  ( z  e.  X  |->  ( sup ( ran 
F ,  RR ,  <  )  -  ( F `
 z ) ) )  C_  ( CC  \  { 0 } ) )
80 difss 3316 . . . . . . . . . . 11  |-  ( CC 
\  { 0 } )  C_  CC
8180a1i 10 . . . . . . . . . 10  |-  ( (
ph  /\  F : X
--> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  ->  ( CC  \  { 0 } )  C_  CC )
82 cnrest2 17030 . . . . . . . . . 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 } ) ) ) ) )
8311, 79, 81, 82syl3anc 1182 . . . . . . . . 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 } ) ) ) ) )
8457, 83mpbid 201 . . . . . . . 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 } ) ) ) )
85 eqid 2296 . . . . . . . . . 10  |-  ( (
TopOpen ` fld )t  ( CC  \  {
0 } ) )  =  ( ( TopOpen ` fld )t  ( CC  \  { 0 } ) )
869, 85divcn 18388 . . . . . . . . 9  |-  /  e.  ( ( ( TopOpen ` fld )  tX  ( ( TopOpen ` fld )t  ( CC  \  { 0 } ) ) )  Cn  ( TopOpen
` fld
) )
8786a1i 10 . . . . . . . 8  |-  ( (
ph  /\  F : X
--> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  ->  /  e.  ( ( ( TopOpen ` fld )  tX  ( ( TopOpen ` fld )t  ( CC  \  { 0 } ) ) )  Cn  ( TopOpen
` fld
) ) )
888, 14, 84, 87cnmpt12f 17376 . . . . . . 7  |-  ( (
ph  /\  F : X
--> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  ->  (
z  e.  X  |->  ( 1  /  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  z ) ) ) )  e.  ( J  Cn  ( TopOpen
` fld
) ) )
8964, 73rereccld 9603 . . . . . . . . . 10  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  z  e.  X )  ->  ( 1  /  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  z ) ) )  e.  RR )
90 eqid 2296 . . . . . . . . . 10  |-  ( z  e.  X  |->  ( 1  /  ( sup ( ran  F ,  RR ,  <  )  -  ( F `
 z ) ) ) )  =  ( z  e.  X  |->  ( 1  /  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  z ) ) ) )
9189, 90fmptd 5700 . . . . . . . . 9  |-  ( (
ph  /\  F : X
--> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  ->  (
z  e.  X  |->  ( 1  /  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  z ) ) ) ) : X --> RR )
92 frn 5411 . . . . . . . . 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 )
9391, 92syl 15 . . . . . . . 8  |-  ( (
ph  /\  F : X
--> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  ->  ran  ( z  e.  X  |->  ( 1  /  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  z ) ) ) )  C_  RR )
94 ax-resscn 8810 . . . . . . . . 9  |-  RR  C_  CC
9594a1i 10 . . . . . . . 8  |-  ( (
ph  /\  F : X
--> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  ->  RR  C_  CC )
96 cnrest2 17030 . . . . . . . 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 ) ) ) )
9711, 93, 95, 96syl3anc 1182 . . . . . . 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 ) ) ) )
9888, 97mpbid 201 . . . . . 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 ) ) )
9998, 50syl6eleqr 2387 . . . . 5  |-  ( (
ph  /\  F : X
--> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  ->  (
z  e.  X  |->  ( 1  /  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  z ) ) ) )  e.  ( J  Cn  K
) )
1001, 2, 4, 99bndth 18472 . . . 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 )
10140ad2antrr 706 . . . . . . . 8  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  x  e.  RR )  ->  sup ( ran  F ,  RR ,  <  )  e.  RR )
102 simpr 447 . . . . . . . . . . 11  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  x  e.  RR )  ->  x  e.  RR )
103 1re 8853 . . . . . . . . . . 11  |-  1  e.  RR
104 ifcl 3614 . . . . . . . . . . 11  |-  ( ( x  e.  RR  /\  1  e.  RR )  ->  if ( 1  <_  x ,  x , 
1 )  e.  RR )
105102, 103, 104sylancl 643 . . . . . . . . . 10  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  x  e.  RR )  ->  if ( 1  <_  x ,  x , 
1 )  e.  RR )
106 0re 8854 . . . . . . . . . . . . 13  |-  0  e.  RR
107106a1i 10 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  x  e.  RR )  ->  0  e.  RR )
108103a1i 10 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  x  e.  RR )  ->  1  e.  RR )
109 0lt1 9312 . . . . . . . . . . . . 13  |-  0  <  1
110109a1i 10 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  x  e.  RR )  ->  0  <  1 )
111 max1 10530 . . . . . . . . . . . . 13  |-  ( ( 1  e.  RR  /\  x  e.  RR )  ->  1  <_  if (
1  <_  x ,  x ,  1 ) )
112103, 102, 111sylancr 644 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  x  e.  RR )  ->  1  <_  if (
1  <_  x ,  x ,  1 ) )
113107, 108, 105, 110, 112ltletrd 8992 . . . . . . . . . . 11  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  x  e.  RR )  ->  0  <  if ( 1  <_  x ,  x ,  1 ) )
114113gt0ne0d 9353 . . . . . . . . . 10  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  x  e.  RR )  ->  if ( 1  <_  x ,  x , 
1 )  =/=  0
)
115105, 114rereccld 9603 . . . . . . . . 9  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  x  e.  RR )  ->  ( 1  /  if ( 1  <_  x ,  x ,  1 ) )  e.  RR )
116105, 113recgt0d 9707 . . . . . . . . 9  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  x  e.  RR )  ->  0  <  ( 1  /  if ( 1  <_  x ,  x ,  1 ) ) )
117115, 116elrpd 10404 . . . . . . . 8  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  x  e.  RR )  ->  ( 1  /  if ( 1  <_  x ,  x ,  1 ) )  e.  RR+ )
118101, 117ltsubrpd 10434 . . . . . . 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 ,  <  ) )
119101, 115resubcld 9227 . . . . . . . 8  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  x  e.  RR )  ->  ( sup ( ran 
F ,  RR ,  <  )  -  ( 1  /  if ( 1  <_  x ,  x ,  1 ) ) )  e.  RR )
120119, 101ltnled 8982 . . . . . . 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 ) ) ) ) )
121118, 120mpbid 201 . . . . . 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 ) ) ) )
122 simprl 732 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  x  e.  RR )
123 max2 10532 . . . . . . . . . . . 12  |-  ( ( 1  e.  RR  /\  x  e.  RR )  ->  x  <_  if (
1  <_  x ,  x ,  1 ) )
124103, 122, 123sylancr 644 . . . . . . . . . . 11  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  x  <_  if ( 1  <_  x ,  x ,  1 ) )
12540ad2antrr 706 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  sup ( ran  F ,  RR ,  <  )  e.  RR )
126 ffvelrn 5679 . . . . . . . . . . . . . . . . 17  |-  ( ( F : X --> ( RR 
\  { sup ( ran  F ,  RR ,  <  ) } )  /\  y  e.  X )  ->  ( F `  y
)  e.  ( RR 
\  { sup ( ran  F ,  RR ,  <  ) } ) )
127126ad2ant2l 726 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  ( F `  y )  e.  ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )
128 eldifsn 3762 . . . . . . . . . . . . . . . 16  |-  ( ( F `  y )  e.  ( RR  \  { sup ( ran  F ,  RR ,  <  ) } )  <->  ( ( F `  y )  e.  RR  /\  ( F `
 y )  =/= 
sup ( ran  F ,  RR ,  <  )
) )
129127, 128sylib 188 . . . . . . . . . . . . . . 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 ,  <  )
) )
130129simpld 445 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  ( F `  y )  e.  RR )
131125, 130resubcld 9227 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  y )
)  e.  RR )
13238adantr 451 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  y  e.  X )  ->  ( ran  F  C_  RR  /\  ran  F  =/=  (/)  /\  E. x  e.  RR  A. z  e. 
ran  F  z  <_  x ) )
133 fnfvelrn 5678 . . . . . . . . . . . . . . . . . . 19  |-  ( ( F  Fn  X  /\  y  e.  X )  ->  ( F `  y
)  e.  ran  F
)
13432, 133sylan 457 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  y  e.  X )  ->  ( F `  y )  e.  ran  F )
135 suprub 9731 . . . . . . . . . . . . . . . . . 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 ,  <  ) )
136132, 134, 135syl2anc 642 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  y  e.  X )  ->  ( F `  y )  <_  sup ( ran  F ,  RR ,  <  )
)
137136ad2ant2rl 729 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  ( F `  y )  <_  sup ( ran  F ,  RR ,  <  ) )
138129simprd 449 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  ( F `  y )  =/=  sup ( ran  F ,  RR ,  <  ) )
139138necomd 2542 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  sup ( ran  F ,  RR ,  <  )  =/=  ( F `
 y ) )
140130, 125ltlend 8980 . . . . . . . . . . . . . . . 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 ) ) ) )
141137, 139, 140mpbir2and 888 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  ( F `  y )  <  sup ( ran  F ,  RR ,  <  ) )
142130, 125posdifd 9375 . . . . . . . . . . . . . . 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 ) ) ) )
143141, 142mpbid 201 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  0  <  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  y ) ) )
144143gt0ne0d 9353 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  y )
)  =/=  0 )
145131, 144rereccld 9603 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  ( 1  /  ( sup ( ran  F ,  RR ,  <  )  -  ( F `
 y ) ) )  e.  RR )
146122, 103, 104sylancl 643 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  if (
1  <_  x ,  x ,  1 )  e.  RR )
147 letr 8930 . . . . . . . . . . . 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 ) ) )
148145, 122, 146, 147syl3anc 1182 . . . . . . . . . . 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 ) ) )
149124, 148mpan2d 655 . . . . . . . . . 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 ) ) )
150 fveq2 5541 . . . . . . . . . . . . . . 15  |-  ( z  =  y  ->  ( F `  z )  =  ( F `  y ) )
151150oveq2d 5890 . . . . . . . . . . . . . 14  |-  ( z  =  y  ->  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  z ) )  =  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  y ) ) )
152151oveq2d 5890 . . . . . . . . . . . . 13  |-  ( z  =  y  ->  (
1  /  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  z ) ) )  =  ( 1  /  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  y ) ) ) )
153 ovex 5899 . . . . . . . . . . . . 13  |-  ( 1  /  ( sup ( ran  F ,  RR ,  <  )  -  ( F `
 y ) ) )  e.  _V
154152, 90, 153fvmpt 5618 . . . . . . . . . . . 12  |-  ( y  e.  X  ->  (
( z  e.  X  |->  ( 1  /  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  z ) ) ) ) `  y )  =  ( 1  /  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  y ) ) ) )
155154breq1d 4049 . . . . . . . . . . 11  |-  ( y  e.  X  ->  (
( ( z  e.  X  |->  ( 1  / 
( sup ( ran 
F ,  RR ,  <  )  -  ( F `
 z ) ) ) ) `  y
)  <_  x  <->  ( 1  /  ( sup ( ran  F ,  RR ,  <  )  -  ( F `
 y ) ) )  <_  x )
)
156155ad2antll 709 . . . . . . . . . 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
) )
157115adantrr 697 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  ( 1  /  if ( 1  <_  x ,  x ,  1 ) )  e.  RR )
158113adantrr 697 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  0  <  if ( 1  <_  x ,  x ,  1 ) )
159146, 158recgt0d 9707 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  0  <  ( 1  /  if ( 1  <_  x ,  x ,  1 ) ) )
160 lerec 9654 . . . . . . . . . . . 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 ) ) ) ) )
161157, 159, 131, 143, 160syl22anc 1183 . . . . . . . . . . 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 ) ) ) ) )
162 lesub 9269 . . . . . . . . . . . 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 ) ) ) ) )
163157, 125, 130, 162syl3anc 1182 . . . . . . . . . . 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 ) ) ) ) )
164146recnd 8877 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  if (
1  <_  x ,  x ,  1 )  e.  CC )
165114adantrr 697 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  if (
1  <_  x ,  x ,  1 )  =/=  0 )
166164, 165recrecd 9549 . . . . . . . . . . . 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 ) )
167166breq2d 4051 . . . . . . . . . . 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 ) ) )
168161, 163, 1673bitr3d 274 . . . . . . . . . 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 ) ) )
169149, 156, 1683imtr4d 259 . . . . . . . . 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 ) ) ) ) )
170169anassrs 629 . . . . . . . 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 ) ) ) ) )
171170ralimdva 2634 . . . . . . 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 ) ) ) ) )
17238ad2antrr 706 . . . . . . . . 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 ) )
173 suprleub 9734 . . . . . . . . 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 ) ) ) ) )
174172, 119, 173syl2anc 642 . . . . . . . 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 ) ) ) ) )
17532ad2antrr 706 . . . . . . . . 9  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  x  e.  RR )  ->  F  Fn  X )
176 breq1 4042 . . . . . . . . . 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 ) ) ) ) )
177176ralrn 5684 . . . . . . . . 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 ) ) ) ) )
178175, 177syl 15 . . . . . . . 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 ) ) ) ) )
179174, 178bitrd 244 . . . . . . 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 ) ) ) ) )
180171, 179sylibrd 225 . . . . . 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 ) ) ) ) )
181121, 180mtod 168 . . . . 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 )
182181nrexdv 2659 . . . 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
)
183100, 182pm2.65da 559 . . 3  |-  ( ph  ->  -.  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )
184136ralrimiva 2639 . . . . . . . . 9  |-  ( ph  ->  A. y  e.  X  ( F `  y )  <_  sup ( ran  F ,  RR ,  <  )
)
185 breq2 4043 . . . . . . . . . 10  |-  ( ( F `  x )  =  sup ( ran 
F ,  RR ,  <  )  ->  ( ( F `  y )  <_  ( F `  x
)  <->  ( F `  y )  <_  sup ( ran  F ,  RR ,  <  ) ) )
186185ralbidv 2576 . . . . . . . . 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 ,  <  )
) )
187184, 186syl5ibrcom 213 . . . . . . . 8  |-  ( ph  ->  ( ( F `  x )  =  sup ( ran  F ,  RR ,  <  )  ->  A. y  e.  X  ( F `  y )  <_  ( F `  x )
) )
188187necon3bd 2496 . . . . . . 7  |-  ( ph  ->  ( -.  A. y  e.  X  ( F `  y )  <_  ( F `  x )  ->  ( F `  x
)  =/=  sup ( ran  F ,  RR ,  <  ) ) )
189188adantr 451 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  ( -.  A. y  e.  X  ( F `  y )  <_  ( F `  x )  ->  ( F `  x )  =/=  sup ( ran  F ,  RR ,  <  )
) )
190 ffvelrn 5679 . . . . . . . 8  |-  ( ( F : X --> RR  /\  x  e.  X )  ->  ( F `  x
)  e.  RR )
19120, 190sylan 457 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  ( F `  x )  e.  RR )
192 eldifsn 3762 . . . . . . . 8  |-  ( ( F `  x )  e.  ( RR  \  { sup ( ran  F ,  RR ,  <  ) } )  <->  ( ( F `  x )  e.  RR  /\  ( F `
 x )  =/= 
sup ( ran  F ,  RR ,  <  )
) )
193192baib 871 . . . . . . 7  |-  ( ( F `  x )  e.  RR  ->  (
( F `  x
)  e.  ( RR 
\  { sup ( ran  F ,  RR ,  <  ) } )  <->  ( F `  x )  =/=  sup ( ran  F ,  RR ,  <  ) ) )
194191, 193syl 15 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  (
( F `  x
)  e.  ( RR 
\  { sup ( ran  F ,  RR ,  <  ) } )  <->  ( F `  x )  =/=  sup ( ran  F ,  RR ,  <  ) ) )
195189, 194sylibrd 225 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  ( -.  A. y  e.  X  ( F `  y )  <_  ( F `  x )  ->  ( F `  x )  e.  ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) ) )
196195ralimdva 2634 . . . 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 ,  <  ) } ) ) )
197 ffnfv 5701 . . . . . 6  |-  ( F : X --> ( RR 
\  { sup ( ran  F ,  RR ,  <  ) } )  <->  ( F  Fn  X  /\  A. x  e.  X  ( F `  x )  e.  ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) ) )
198197baib 871 . . . . 5  |-  ( F  Fn  X  ->  ( F : X --> ( RR 
\  { sup ( ran  F ,  RR ,  <  ) } )  <->  A. x  e.  X  ( F `  x )  e.  ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) ) )
19932, 198syl 15 . . . 4  |-  ( ph  ->  ( F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } )  <->  A. x  e.  X  ( F `  x )  e.  ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) ) )
200196, 199sylibrd 225 . . 3  |-  ( ph  ->  ( A. x  e.  X  -.  A. y  e.  X  ( F `  y )  <_  ( F `  x )  ->  F : X --> ( RR 
\  { sup ( ran  F ,  RR ,  <  ) } ) ) )
201183, 200mtod 168 . 2  |-  ( ph  ->  -.  A. x  e.  X  -.  A. y  e.  X  ( F `  y )  <_  ( F `  x )
)
202 dfrex2 2569 . 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 )
)
203201, 202sylibr 203 1  |-  ( ph  ->  E. x  e.  X  A. y  e.  X  ( F `  y )  <_  ( F `  x ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   E.wrex 2557    \ cdif 3162    C_ wss 3165   (/)c0 3468   ifcif 3578   {csn 3653   U.cuni 3843   class class class wbr 4039    e. cmpt 4093   dom cdm 4705   ran crn 4706    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874   supcsup 7209   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754    < clt 8883    <_ cle 8884    - cmin 9053    / cdiv 9439   (,)cioo 10672   ↾t crest 13341   TopOpenctopn 13342   topGenctg 13358  ℂfldccnfld 16393   Topctop 16647  TopOnctopon 16648    Cn ccn 16970   Compccmp 17129    tX ctx 17271
This theorem is referenced by:  evth2  18474  evthicc  18835  evthf  27801  cncmpmax  27806
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-mulf 8833
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-fi 7181  df-sup 7210  df-oi 7241  df-card 7588  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-ioo 10676  df-icc 10679  df-fz 10799  df-fzo 10887  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-starv 13239  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-hom 13248  df-cco 13249  df-rest 13343  df-topn 13344  df-topgen 13360  df-pt 13361  df-prds 13364  df-xrs 13419  df-0g 13420  df-gsum 13421  df-qtop 13426  df-imas 13427  df-xps 13429  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-submnd 14432  df-mulg 14508  df-cntz 14809  df-cmn 15107  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-cnfld 16394  df-top 16652  df-bases 16654  df-topon 16655  df-topsp 16656  df-cn 16973  df-cnp 16974  df-cmp 17130  df-tx 17273  df-hmeo 17462  df-xms 17901  df-ms 17902  df-tms 17903
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