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Theorem evth 18937
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 452 . . . . 5  |-  ( (
ph  /\  F : X
--> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  ->  J  e.  Comp )
5 cmptop 17412 . . . . . . . . . 10  |-  ( J  e.  Comp  ->  J  e. 
Top )
64, 5syl 16 . . . . . . . . 9  |-  ( (
ph  /\  F : X
--> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  ->  J  e.  Top )
71toptopon 16953 . . . . . . . . 9  |-  ( J  e.  Top  <->  J  e.  (TopOn `  X ) )
86, 7sylib 189 . . . . . . . 8  |-  ( (
ph  /\  F : X
--> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  ->  J  e.  (TopOn `  X )
)
9 eqid 2404 . . . . . . . . . . 11  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
109cnfldtopon 18770 . . . . . . . . . 10  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
1110a1i 11 . . . . . . . . 9  |-  ( (
ph  /\  F : X
--> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  ->  ( TopOpen
` fld
)  e.  (TopOn `  CC ) )
12 ax-1cn 9004 . . . . . . . . . 10  |-  1  e.  CC
1312a1i 11 . . . . . . . . 9  |-  ( (
ph  /\  F : X
--> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  ->  1  e.  CC )
148, 11, 13cnmptc 17647 . . . . . . . 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 18749 . . . . . . . . . . . . . . . . . . 19  |-  RR  =  U. ( topGen `  ran  (,) )
172unieqi 3985 . . . . . . . . . . . . . . . . . . 19  |-  U. K  =  U. ( topGen `  ran  (,) )
1816, 17eqtr4i 2427 . . . . . . . . . . . . . . . . . 18  |-  RR  =  U. K
191, 18cnf 17264 . . . . . . . . . . . . . . . . 17  |-  ( F  e.  ( J  Cn  K )  ->  F : X --> RR )
2015, 19syl 16 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  F : X --> RR )
21 frn 5556 . . . . . . . . . . . . . . . 16  |-  ( F : X --> RR  ->  ran 
F  C_  RR )
2220, 21syl 16 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ran  F  C_  RR )
23 fdm 5554 . . . . . . . . . . . . . . . . . 18  |-  ( F : X --> RR  ->  dom 
F  =  X )
2420, 23syl 16 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  dom  F  =  X )
25 evth.5 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  X  =/=  (/) )
2624, 25eqnetrd 2585 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  dom  F  =/=  (/) )
27 dm0rn0 5045 . . . . . . . . . . . . . . . . 17  |-  ( dom 
F  =  (/)  <->  ran  F  =  (/) )
2827necon3bii 2599 . . . . . . . . . . . . . . . 16  |-  ( dom 
F  =/=  (/)  <->  ran  F  =/=  (/) )
2926, 28sylib 189 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ran  F  =/=  (/) )
301, 2, 3, 15bndth 18936 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  E. x  e.  RR  A. y  e.  X  ( F `  y )  <_  x )
31 ffn 5550 . . . . . . . . . . . . . . . . . . 19  |-  ( F : X --> RR  ->  F  Fn  X )
3220, 31syl 16 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  F  Fn  X )
33 breq1 4175 . . . . . . . . . . . . . . . . . . 19  |-  ( z  =  ( F `  y )  ->  (
z  <_  x  <->  ( F `  y )  <_  x
) )
3433ralrn 5832 . . . . . . . . . . . . . . . . . 18  |-  ( F  Fn  X  ->  ( A. z  e.  ran  F  z  <_  x  <->  A. y  e.  X  ( F `  y )  <_  x
) )
3532, 34syl 16 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( A. z  e. 
ran  F  z  <_  x  <->  A. y  e.  X  ( F `  y )  <_  x ) )
3635rexbidv 2687 . . . . . . . . . . . . . . . 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 224 . . . . . . . . . . . . . . 15  |-  ( ph  ->  E. x  e.  RR  A. z  e.  ran  F  z  <_  x )
3822, 29, 373jca 1134 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ran  F  C_  RR  /\  ran  F  =/=  (/)  /\  E. x  e.  RR  A. z  e. 
ran  F  z  <_  x ) )
39 suprcl 9924 . . . . . . . . . . . . . 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 16 . . . . . . . . . . . . 13  |-  ( ph  ->  sup ( ran  F ,  RR ,  <  )  e.  RR )
4140recnd 9070 . . . . . . . . . . . 12  |-  ( ph  ->  sup ( ran  F ,  RR ,  <  )  e.  CC )
4241adantr 452 . . . . . . . . . . 11  |-  ( (
ph  /\  F : X
--> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  ->  sup ( ran  F ,  RR ,  <  )  e.  CC )
438, 11, 42cnmptc 17647 . . . . . . . . . 10  |-  ( (
ph  /\  F : X
--> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  ->  (
z  e.  X  |->  sup ( ran  F ,  RR ,  <  ) )  e.  ( J  Cn  ( TopOpen ` fld ) ) )
4420feqmptd 5738 . . . . . . . . . . . 12  |-  ( ph  ->  F  =  ( z  e.  X  |->  ( F `
 z ) ) )
459cnfldtop 18771 . . . . . . . . . . . . . 14  |-  ( TopOpen ` fld )  e.  Top
46 cnrest2r 17305 . . . . . . . . . . . . . 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 18787 . . . . . . . . . . . . . . . 16  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
492, 48eqtri 2424 . . . . . . . . . . . . . . 15  |-  K  =  ( ( TopOpen ` fld )t  RR )
5049oveq2i 6051 . . . . . . . . . . . . . 14  |-  ( J  Cn  K )  =  ( J  Cn  (
( TopOpen ` fld )t  RR ) )
5115, 50syl6eleq 2494 . . . . . . . . . . . . 13  |-  ( ph  ->  F  e.  ( J  Cn  ( ( TopOpen ` fld )t  RR ) ) )
5247, 51sseldi 3306 . . . . . . . . . . . 12  |-  ( ph  ->  F  e.  ( J  Cn  ( TopOpen ` fld ) ) )
5344, 52eqeltrrd 2479 . . . . . . . . . . 11  |-  ( ph  ->  ( z  e.  X  |->  ( F `  z
) )  e.  ( J  Cn  ( TopOpen ` fld )
) )
5453adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  F : X
--> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  ->  (
z  e.  X  |->  ( F `  z ) )  e.  ( J  Cn  ( TopOpen ` fld ) ) )
559subcn 18849 . . . . . . . . . . 11  |-  -  e.  ( ( ( TopOpen ` fld )  tX  ( TopOpen ` fld ) )  Cn  ( TopOpen
` fld
) )
5655a1i 11 . . . . . . . . . 10  |-  ( (
ph  /\  F : X
--> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  ->  -  e.  ( ( ( TopOpen ` fld )  tX  ( TopOpen ` fld ) )  Cn  ( TopOpen
` fld
) ) )
578, 43, 54, 56cnmpt12f 17651 . . . . . . . . 9  |-  ( (
ph  /\  F : X
--> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  ->  (
z  e.  X  |->  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  z ) ) )  e.  ( J  Cn  ( TopOpen ` fld ) ) )
5840ad2antrr 707 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  z  e.  X )  ->  sup ( ran  F ,  RR ,  <  )  e.  RR )
59 ffvelrn 5827 . . . . . . . . . . . . . . . . . 18  |-  ( ( F : X --> ( RR 
\  { sup ( ran  F ,  RR ,  <  ) } )  /\  z  e.  X )  ->  ( F `  z
)  e.  ( RR 
\  { sup ( ran  F ,  RR ,  <  ) } ) )
6059adantll 695 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  z  e.  X )  ->  ( F `  z
)  e.  ( RR 
\  { sup ( ran  F ,  RR ,  <  ) } ) )
61 eldifsn 3887 . . . . . . . . . . . . . . . . 17  |-  ( ( F `  z )  e.  ( RR  \  { sup ( ran  F ,  RR ,  <  ) } )  <->  ( ( F `  z )  e.  RR  /\  ( F `
 z )  =/= 
sup ( ran  F ,  RR ,  <  )
) )
6260, 61sylib 189 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  z  e.  X )  ->  ( ( F `  z )  e.  RR  /\  ( F `  z
)  =/=  sup ( ran  F ,  RR ,  <  ) ) )
6362simpld 446 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  z  e.  X )  ->  ( F `  z
)  e.  RR )
6458, 63resubcld 9421 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  z  e.  X )  ->  ( sup ( ran 
F ,  RR ,  <  )  -  ( F `
 z ) )  e.  RR )
6564recnd 9070 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  z  e.  X )  ->  ( sup ( ran 
F ,  RR ,  <  )  -  ( F `
 z ) )  e.  CC )
6658recnd 9070 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  z  e.  X )  ->  sup ( ran  F ,  RR ,  <  )  e.  CC )
6763recnd 9070 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  z  e.  X )  ->  ( F `  z
)  e.  CC )
6862simprd 450 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  z  e.  X )  ->  ( F `  z
)  =/=  sup ( ran  F ,  RR ,  <  ) )
6968necomd 2650 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  z  e.  X )  ->  sup ( ran  F ,  RR ,  <  )  =/=  ( F `  z
) )
7066, 67, 69subne0d 9376 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  z  e.  X )  ->  ( sup ( ran 
F ,  RR ,  <  )  -  ( F `
 z ) )  =/=  0 )
71 eldifsn 3887 . . . . . . . . . . . . 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 ) )
7265, 70, 71sylanbrc 646 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  z  e.  X )  ->  ( sup ( ran 
F ,  RR ,  <  )  -  ( F `
 z ) )  e.  ( CC  \  { 0 } ) )
73 eqid 2404 . . . . . . . . . . . 12  |-  ( z  e.  X  |->  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  z ) ) )  =  ( z  e.  X  |->  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  z ) ) )
7472, 73fmptd 5852 . . . . . . . . . . 11  |-  ( (
ph  /\  F : X
--> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  ->  (
z  e.  X  |->  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  z ) ) ) : X --> ( CC 
\  { 0 } ) )
75 frn 5556 . . . . . . . . . . 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 } ) )
7674, 75syl 16 . . . . . . . . . 10  |-  ( (
ph  /\  F : X
--> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  ->  ran  ( z  e.  X  |->  ( sup ( ran 
F ,  RR ,  <  )  -  ( F `
 z ) ) )  C_  ( CC  \  { 0 } ) )
77 difssd 3435 . . . . . . . . . 10  |-  ( (
ph  /\  F : X
--> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  ->  ( CC  \  { 0 } )  C_  CC )
78 cnrest2 17304 . . . . . . . . . 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 } ) ) ) ) )
7911, 76, 77, 78syl3anc 1184 . . . . . . . . 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 } ) ) ) ) )
8057, 79mpbid 202 . . . . . . . 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 } ) ) ) )
81 eqid 2404 . . . . . . . . . 10  |-  ( (
TopOpen ` fld )t  ( CC  \  {
0 } ) )  =  ( ( TopOpen ` fld )t  ( CC  \  { 0 } ) )
829, 81divcn 18851 . . . . . . . . 9  |-  /  e.  ( ( ( TopOpen ` fld )  tX  ( ( TopOpen ` fld )t  ( CC  \  { 0 } ) ) )  Cn  ( TopOpen
` fld
) )
8382a1i 11 . . . . . . . 8  |-  ( (
ph  /\  F : X
--> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  ->  /  e.  ( ( ( TopOpen ` fld )  tX  ( ( TopOpen ` fld )t  ( CC  \  { 0 } ) ) )  Cn  ( TopOpen
` fld
) ) )
848, 14, 80, 83cnmpt12f 17651 . . . . . . 7  |-  ( (
ph  /\  F : X
--> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  ->  (
z  e.  X  |->  ( 1  /  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  z ) ) ) )  e.  ( J  Cn  ( TopOpen
` fld
) ) )
8564, 70rereccld 9797 . . . . . . . . . 10  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  z  e.  X )  ->  ( 1  /  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  z ) ) )  e.  RR )
86 eqid 2404 . . . . . . . . . 10  |-  ( z  e.  X  |->  ( 1  /  ( sup ( ran  F ,  RR ,  <  )  -  ( F `
 z ) ) ) )  =  ( z  e.  X  |->  ( 1  /  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  z ) ) ) )
8785, 86fmptd 5852 . . . . . . . . 9  |-  ( (
ph  /\  F : X
--> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  ->  (
z  e.  X  |->  ( 1  /  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  z ) ) ) ) : X --> RR )
88 frn 5556 . . . . . . . . 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 )
8987, 88syl 16 . . . . . . . 8  |-  ( (
ph  /\  F : X
--> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  ->  ran  ( z  e.  X  |->  ( 1  /  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  z ) ) ) )  C_  RR )
90 ax-resscn 9003 . . . . . . . . 9  |-  RR  C_  CC
9190a1i 11 . . . . . . . 8  |-  ( (
ph  /\  F : X
--> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  ->  RR  C_  CC )
92 cnrest2 17304 . . . . . . . 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 ) ) ) )
9311, 89, 91, 92syl3anc 1184 . . . . . . 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 ) ) ) )
9484, 93mpbid 202 . . . . . 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 ) ) )
9594, 50syl6eleqr 2495 . . . . 5  |-  ( (
ph  /\  F : X
--> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  ->  (
z  e.  X  |->  ( 1  /  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  z ) ) ) )  e.  ( J  Cn  K
) )
961, 2, 4, 95bndth 18936 . . . 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 )
9740ad2antrr 707 . . . . . . . 8  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  x  e.  RR )  ->  sup ( ran  F ,  RR ,  <  )  e.  RR )
98 simpr 448 . . . . . . . . . . 11  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  x  e.  RR )  ->  x  e.  RR )
99 1re 9046 . . . . . . . . . . 11  |-  1  e.  RR
100 ifcl 3735 . . . . . . . . . . 11  |-  ( ( x  e.  RR  /\  1  e.  RR )  ->  if ( 1  <_  x ,  x , 
1 )  e.  RR )
10198, 99, 100sylancl 644 . . . . . . . . . 10  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  x  e.  RR )  ->  if ( 1  <_  x ,  x , 
1 )  e.  RR )
102 0re 9047 . . . . . . . . . . . . 13  |-  0  e.  RR
103102a1i 11 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  x  e.  RR )  ->  0  e.  RR )
10499a1i 11 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  x  e.  RR )  ->  1  e.  RR )
105 0lt1 9506 . . . . . . . . . . . . 13  |-  0  <  1
106105a1i 11 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  x  e.  RR )  ->  0  <  1 )
107 max1 10729 . . . . . . . . . . . . 13  |-  ( ( 1  e.  RR  /\  x  e.  RR )  ->  1  <_  if (
1  <_  x ,  x ,  1 ) )
10899, 98, 107sylancr 645 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  x  e.  RR )  ->  1  <_  if (
1  <_  x ,  x ,  1 ) )
109103, 104, 101, 106, 108ltletrd 9186 . . . . . . . . . . 11  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  x  e.  RR )  ->  0  <  if ( 1  <_  x ,  x ,  1 ) )
110109gt0ne0d 9547 . . . . . . . . . 10  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  x  e.  RR )  ->  if ( 1  <_  x ,  x , 
1 )  =/=  0
)
111101, 110rereccld 9797 . . . . . . . . 9  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  x  e.  RR )  ->  ( 1  /  if ( 1  <_  x ,  x ,  1 ) )  e.  RR )
112101, 109recgt0d 9901 . . . . . . . . 9  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  x  e.  RR )  ->  0  <  ( 1  /  if ( 1  <_  x ,  x ,  1 ) ) )
113111, 112elrpd 10602 . . . . . . . 8  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  x  e.  RR )  ->  ( 1  /  if ( 1  <_  x ,  x ,  1 ) )  e.  RR+ )
11497, 113ltsubrpd 10632 . . . . . . 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 ,  <  ) )
11597, 111resubcld 9421 . . . . . . . 8  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  x  e.  RR )  ->  ( sup ( ran 
F ,  RR ,  <  )  -  ( 1  /  if ( 1  <_  x ,  x ,  1 ) ) )  e.  RR )
116115, 97ltnled 9176 . . . . . . 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 ) ) ) ) )
117114, 116mpbid 202 . . . . . 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 ) ) ) )
118 simprl 733 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  x  e.  RR )
119 max2 10731 . . . . . . . . . . . 12  |-  ( ( 1  e.  RR  /\  x  e.  RR )  ->  x  <_  if (
1  <_  x ,  x ,  1 ) )
12099, 118, 119sylancr 645 . . . . . . . . . . 11  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  x  <_  if ( 1  <_  x ,  x ,  1 ) )
12140ad2antrr 707 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  sup ( ran  F ,  RR ,  <  )  e.  RR )
122 ffvelrn 5827 . . . . . . . . . . . . . . . . 17  |-  ( ( F : X --> ( RR 
\  { sup ( ran  F ,  RR ,  <  ) } )  /\  y  e.  X )  ->  ( F `  y
)  e.  ( RR 
\  { sup ( ran  F ,  RR ,  <  ) } ) )
123122ad2ant2l 727 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  ( F `  y )  e.  ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )
124 eldifsn 3887 . . . . . . . . . . . . . . . 16  |-  ( ( F `  y )  e.  ( RR  \  { sup ( ran  F ,  RR ,  <  ) } )  <->  ( ( F `  y )  e.  RR  /\  ( F `
 y )  =/= 
sup ( ran  F ,  RR ,  <  )
) )
125123, 124sylib 189 . . . . . . . . . . . . . . 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 ,  <  )
) )
126125simpld 446 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  ( F `  y )  e.  RR )
127121, 126resubcld 9421 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  y )
)  e.  RR )
12838adantr 452 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  y  e.  X )  ->  ( ran  F  C_  RR  /\  ran  F  =/=  (/)  /\  E. x  e.  RR  A. z  e. 
ran  F  z  <_  x ) )
129 fnfvelrn 5826 . . . . . . . . . . . . . . . . . . 19  |-  ( ( F  Fn  X  /\  y  e.  X )  ->  ( F `  y
)  e.  ran  F
)
13032, 129sylan 458 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  y  e.  X )  ->  ( F `  y )  e.  ran  F )
131 suprub 9925 . . . . . . . . . . . . . . . . . 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 ,  <  ) )
132128, 130, 131syl2anc 643 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  y  e.  X )  ->  ( F `  y )  <_  sup ( ran  F ,  RR ,  <  )
)
133132ad2ant2rl 730 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  ( F `  y )  <_  sup ( ran  F ,  RR ,  <  ) )
134125simprd 450 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  ( F `  y )  =/=  sup ( ran  F ,  RR ,  <  ) )
135134necomd 2650 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  sup ( ran  F ,  RR ,  <  )  =/=  ( F `
 y ) )
136126, 121ltlend 9174 . . . . . . . . . . . . . . . 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 ) ) ) )
137133, 135, 136mpbir2and 889 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  ( F `  y )  <  sup ( ran  F ,  RR ,  <  ) )
138126, 121posdifd 9569 . . . . . . . . . . . . . . 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 ) ) ) )
139137, 138mpbid 202 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  0  <  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  y ) ) )
140139gt0ne0d 9547 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  y )
)  =/=  0 )
141127, 140rereccld 9797 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  ( 1  /  ( sup ( ran  F ,  RR ,  <  )  -  ( F `
 y ) ) )  e.  RR )
142118, 99, 100sylancl 644 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  if (
1  <_  x ,  x ,  1 )  e.  RR )
143 letr 9123 . . . . . . . . . . . 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 ) ) )
144141, 118, 142, 143syl3anc 1184 . . . . . . . . . . 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 ) ) )
145120, 144mpan2d 656 . . . . . . . . . 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 ) ) )
146 fveq2 5687 . . . . . . . . . . . . . . 15  |-  ( z  =  y  ->  ( F `  z )  =  ( F `  y ) )
147146oveq2d 6056 . . . . . . . . . . . . . 14  |-  ( z  =  y  ->  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  z ) )  =  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  y ) ) )
148147oveq2d 6056 . . . . . . . . . . . . 13  |-  ( z  =  y  ->  (
1  /  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  z ) ) )  =  ( 1  /  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  y ) ) ) )
149 ovex 6065 . . . . . . . . . . . . 13  |-  ( 1  /  ( sup ( ran  F ,  RR ,  <  )  -  ( F `
 y ) ) )  e.  _V
150148, 86, 149fvmpt 5765 . . . . . . . . . . . 12  |-  ( y  e.  X  ->  (
( z  e.  X  |->  ( 1  /  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  z ) ) ) ) `  y )  =  ( 1  /  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  y ) ) ) )
151150breq1d 4182 . . . . . . . . . . 11  |-  ( y  e.  X  ->  (
( ( z  e.  X  |->  ( 1  / 
( sup ( ran 
F ,  RR ,  <  )  -  ( F `
 z ) ) ) ) `  y
)  <_  x  <->  ( 1  /  ( sup ( ran  F ,  RR ,  <  )  -  ( F `
 y ) ) )  <_  x )
)
152151ad2antll 710 . . . . . . . . . 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
) )
153111adantrr 698 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  ( 1  /  if ( 1  <_  x ,  x ,  1 ) )  e.  RR )
154109adantrr 698 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  0  <  if ( 1  <_  x ,  x ,  1 ) )
155142, 154recgt0d 9901 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  0  <  ( 1  /  if ( 1  <_  x ,  x ,  1 ) ) )
156 lerec 9848 . . . . . . . . . . . 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 ) ) ) ) )
157153, 155, 127, 139, 156syl22anc 1185 . . . . . . . . . . 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 ) ) ) ) )
158 lesub 9463 . . . . . . . . . . . 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 ) ) ) ) )
159153, 121, 126, 158syl3anc 1184 . . . . . . . . . . 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 ) ) ) ) )
160142recnd 9070 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  if (
1  <_  x ,  x ,  1 )  e.  CC )
161110adantrr 698 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  ( x  e.  RR  /\  y  e.  X ) )  ->  if (
1  <_  x ,  x ,  1 )  =/=  0 )
162160, 161recrecd 9743 . . . . . . . . . . . 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 ) )
163162breq2d 4184 . . . . . . . . . . 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 ) ) )
164157, 159, 1633bitr3d 275 . . . . . . . . . 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 ) ) )
165145, 152, 1643imtr4d 260 . . . . . . . . 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 ) ) ) ) )
166165anassrs 630 . . . . . . . 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 ) ) ) ) )
167166ralimdva 2744 . . . . . . 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 ) ) ) ) )
16838ad2antrr 707 . . . . . . . . 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 ) )
169 suprleub 9928 . . . . . . . . 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 ) ) ) ) )
170168, 115, 169syl2anc 643 . . . . . . . 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 ) ) ) ) )
17132ad2antrr 707 . . . . . . . . 9  |-  ( ( ( ph  /\  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )  /\  x  e.  RR )  ->  F  Fn  X )
172 breq1 4175 . . . . . . . . . 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 ) ) ) ) )
173172ralrn 5832 . . . . . . . . 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 ) ) ) ) )
174171, 173syl 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 ) ) ) ) )
175170, 174bitrd 245 . . . . . . 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 ) ) ) ) )
176167, 175sylibrd 226 . . . . . 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 ) ) ) ) )
177117, 176mtod 170 . . . . 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 )
178177nrexdv 2769 . . . 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
)
17996, 178pm2.65da 560 . . 3  |-  ( ph  ->  -.  F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) )
180132ralrimiva 2749 . . . . . . . . 9  |-  ( ph  ->  A. y  e.  X  ( F `  y )  <_  sup ( ran  F ,  RR ,  <  )
)
181 breq2 4176 . . . . . . . . . 10  |-  ( ( F `  x )  =  sup ( ran 
F ,  RR ,  <  )  ->  ( ( F `  y )  <_  ( F `  x
)  <->  ( F `  y )  <_  sup ( ran  F ,  RR ,  <  ) ) )
182181ralbidv 2686 . . . . . . . . 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 ,  <  )
) )
183180, 182syl5ibrcom 214 . . . . . . . 8  |-  ( ph  ->  ( ( F `  x )  =  sup ( ran  F ,  RR ,  <  )  ->  A. y  e.  X  ( F `  y )  <_  ( F `  x )
) )
184183necon3bd 2604 . . . . . . 7  |-  ( ph  ->  ( -.  A. y  e.  X  ( F `  y )  <_  ( F `  x )  ->  ( F `  x
)  =/=  sup ( ran  F ,  RR ,  <  ) ) )
185184adantr 452 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  ( -.  A. y  e.  X  ( F `  y )  <_  ( F `  x )  ->  ( F `  x )  =/=  sup ( ran  F ,  RR ,  <  )
) )
18620ffvelrnda 5829 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  ( F `  x )  e.  RR )
187 eldifsn 3887 . . . . . . . 8  |-  ( ( F `  x )  e.  ( RR  \  { sup ( ran  F ,  RR ,  <  ) } )  <->  ( ( F `  x )  e.  RR  /\  ( F `
 x )  =/= 
sup ( ran  F ,  RR ,  <  )
) )
188187baib 872 . . . . . . 7  |-  ( ( F `  x )  e.  RR  ->  (
( F `  x
)  e.  ( RR 
\  { sup ( ran  F ,  RR ,  <  ) } )  <->  ( F `  x )  =/=  sup ( ran  F ,  RR ,  <  ) ) )
189186, 188syl 16 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  (
( F `  x
)  e.  ( RR 
\  { sup ( ran  F ,  RR ,  <  ) } )  <->  ( F `  x )  =/=  sup ( ran  F ,  RR ,  <  ) ) )
190185, 189sylibrd 226 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  ( -.  A. y  e.  X  ( F `  y )  <_  ( F `  x )  ->  ( F `  x )  e.  ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) ) )
191190ralimdva 2744 . . . 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 ,  <  ) } ) ) )
192 ffnfv 5853 . . . . . 6  |-  ( F : X --> ( RR 
\  { sup ( ran  F ,  RR ,  <  ) } )  <->  ( F  Fn  X  /\  A. x  e.  X  ( F `  x )  e.  ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) ) )
193192baib 872 . . . . 5  |-  ( F  Fn  X  ->  ( F : X --> ( RR 
\  { sup ( ran  F ,  RR ,  <  ) } )  <->  A. x  e.  X  ( F `  x )  e.  ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) ) )
19432, 193syl 16 . . . 4  |-  ( ph  ->  ( F : X --> ( RR  \  { sup ( ran  F ,  RR ,  <  ) } )  <->  A. x  e.  X  ( F `  x )  e.  ( RR  \  { sup ( ran  F ,  RR ,  <  ) } ) ) )
195191, 194sylibrd 226 . . 3  |-  ( ph  ->  ( A. x  e.  X  -.  A. y  e.  X  ( F `  y )  <_  ( F `  x )  ->  F : X --> ( RR 
\  { sup ( ran  F ,  RR ,  <  ) } ) ) )
196179, 195mtod 170 . 2  |-  ( ph  ->  -.  A. x  e.  X  -.  A. y  e.  X  ( F `  y )  <_  ( F `  x )
)
197 dfrex2 2679 . 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 )
)
198196, 197sylibr 204 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 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567   A.wral 2666   E.wrex 2667    \ cdif 3277    C_ wss 3280   (/)c0 3588   ifcif 3699   {csn 3774   U.cuni 3975   class class class wbr 4172    e. cmpt 4226   dom cdm 4837   ran crn 4838    Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6040   supcsup 7403   CCcc 8944   RRcr 8945   0cc0 8946   1c1 8947    < clt 9076    <_ cle 9077    - cmin 9247    / cdiv 9633   (,)cioo 10872   ↾t crest 13603   TopOpenctopn 13604   topGenctg 13620  ℂfldccnfld 16658   Topctop 16913  TopOnctopon 16914    Cn ccn 17242   Compccmp 17403    tX ctx 17545
This theorem is referenced by:  evth2  18938  evthicc  19309  evthf  27565  cncmpmax  27570
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-mulf 9026
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-sup 7404  df-oi 7435  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-ioo 10876  df-icc 10879  df-fz 11000  df-fzo 11091  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-hom 13508  df-cco 13509  df-rest 13605  df-topn 13606  df-topgen 13622  df-pt 13623  df-prds 13626  df-xrs 13681  df-0g 13682  df-gsum 13683  df-qtop 13688  df-imas 13689  df-xps 13691  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-submnd 14694  df-mulg 14770  df-cntz 15071  df-cmn 15369  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-cnfld 16659  df-top 16918  df-bases 16920  df-topon 16921  df-topsp 16922  df-cn 17245  df-cnp 17246  df-cmp 17404  df-tx 17547  df-hmeo 17740  df-xms 18303  df-ms 18304  df-tms 18305
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