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Theorem climinf 27708
Description: A bounded monotonic non increasing sequence converges to the infimum of its range. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
Hypotheses
Ref Expression
climinf.3  |-  Z  =  ( ZZ>= `  M )
climinf.4  |-  ( ph  ->  M  e.  ZZ )
climinf.5  |-  ( ph  ->  F : Z --> RR )
climinf.6  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  ( k  +  1 ) )  <_  ( F `  k ) )
climinf.7  |-  ( ph  ->  E. x  e.  RR  A. k  e.  Z  x  <_  ( F `  k ) )
Assertion
Ref Expression
climinf  |-  ( ph  ->  F  ~~>  sup ( ran  F ,  RR ,  `'  <  ) )
Distinct variable groups:    ph, k    x, k, F    k, Z, x
Allowed substitution hints:    ph( x)    M( x, k)

Proof of Theorem climinf
Dummy variables  j  n  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 climinf.5 . . . . . . . . . . . 12  |-  ( ph  ->  F : Z --> RR )
2 frn 5597 . . . . . . . . . . . 12  |-  ( F : Z --> RR  ->  ran 
F  C_  RR )
31, 2syl 16 . . . . . . . . . . 11  |-  ( ph  ->  ran  F  C_  RR )
4 ffn 5591 . . . . . . . . . . . . . 14  |-  ( F : Z --> RR  ->  F  Fn  Z )
51, 4syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  F  Fn  Z )
6 climinf.4 . . . . . . . . . . . . . . 15  |-  ( ph  ->  M  e.  ZZ )
7 uzid 10500 . . . . . . . . . . . . . . 15  |-  ( M  e.  ZZ  ->  M  e.  ( ZZ>= `  M )
)
86, 7syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  M  e.  ( ZZ>= `  M ) )
9 climinf.3 . . . . . . . . . . . . . 14  |-  Z  =  ( ZZ>= `  M )
108, 9syl6eleqr 2527 . . . . . . . . . . . . 13  |-  ( ph  ->  M  e.  Z )
11 fnfvelrn 5867 . . . . . . . . . . . . 13  |-  ( ( F  Fn  Z  /\  M  e.  Z )  ->  ( F `  M
)  e.  ran  F
)
125, 10, 11syl2anc 643 . . . . . . . . . . . 12  |-  ( ph  ->  ( F `  M
)  e.  ran  F
)
13 ne0i 3634 . . . . . . . . . . . 12  |-  ( ( F `  M )  e.  ran  F  ->  ran  F  =/=  (/) )
1412, 13syl 16 . . . . . . . . . . 11  |-  ( ph  ->  ran  F  =/=  (/) )
15 climinf.7 . . . . . . . . . . . 12  |-  ( ph  ->  E. x  e.  RR  A. k  e.  Z  x  <_  ( F `  k ) )
16 breq2 4216 . . . . . . . . . . . . . . 15  |-  ( y  =  ( F `  k )  ->  (
x  <_  y  <->  x  <_  ( F `  k ) ) )
1716ralrn 5873 . . . . . . . . . . . . . 14  |-  ( F  Fn  Z  ->  ( A. y  e.  ran  F  x  <_  y  <->  A. k  e.  Z  x  <_  ( F `  k ) ) )
1817rexbidv 2726 . . . . . . . . . . . . 13  |-  ( F  Fn  Z  ->  ( E. x  e.  RR  A. y  e.  ran  F  x  <_  y  <->  E. x  e.  RR  A. k  e.  Z  x  <_  ( F `  k )
) )
195, 18syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  ( E. x  e.  RR  A. y  e. 
ran  F  x  <_  y  <->  E. x  e.  RR  A. k  e.  Z  x  <_  ( F `  k ) ) )
2015, 19mpbird 224 . . . . . . . . . . 11  |-  ( ph  ->  E. x  e.  RR  A. y  e.  ran  F  x  <_  y )
213, 14, 203jca 1134 . . . . . . . . . 10  |-  ( ph  ->  ( ran  F  C_  RR  /\  ran  F  =/=  (/)  /\  E. x  e.  RR  A. y  e. 
ran  F  x  <_  y ) )
2221adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  RR+ )  ->  ( ran  F 
C_  RR  /\  ran  F  =/=  (/)  /\  E. x  e.  RR  A. y  e. 
ran  F  x  <_  y ) )
23 infmrcl 9987 . . . . . . . . 9  |-  ( ( ran  F  C_  RR  /\ 
ran  F  =/=  (/)  /\  E. x  e.  RR  A. y  e.  ran  F  x  <_ 
y )  ->  sup ( ran  F ,  RR ,  `'  <  )  e.  RR )
2422, 23syl 16 . . . . . . . 8  |-  ( (
ph  /\  y  e.  RR+ )  ->  sup ( ran  F ,  RR ,  `'  <  )  e.  RR )
25 simpr 448 . . . . . . . 8  |-  ( (
ph  /\  y  e.  RR+ )  ->  y  e.  RR+ )
2624, 25ltaddrpd 10677 . . . . . . 7  |-  ( (
ph  /\  y  e.  RR+ )  ->  sup ( ran  F ,  RR ,  `'  <  )  <  ( sup ( ran  F ,  RR ,  `'  <  )  +  y ) )
27 rpre 10618 . . . . . . . . . 10  |-  ( y  e.  RR+  ->  y  e.  RR )
2827adantl 453 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  RR+ )  ->  y  e.  RR )
2924, 28readdcld 9115 . . . . . . . 8  |-  ( (
ph  /\  y  e.  RR+ )  ->  ( sup ( ran  F ,  RR ,  `'  <  )  +  y )  e.  RR )
30 infrglb 27698 . . . . . . . 8  |-  ( ( ( ran  F  C_  RR  /\  ran  F  =/=  (/)  /\  E. x  e.  RR  A. y  e. 
ran  F  x  <_  y )  /\  ( sup ( ran  F ,  RR ,  `'  <  )  +  y )  e.  RR )  ->  ( sup ( ran  F ,  RR ,  `'  <  )  <  ( sup ( ran  F ,  RR ,  `'  <  )  +  y )  <->  E. k  e.  ran  F  k  <  ( sup ( ran  F ,  RR ,  `'  <  )  +  y ) ) )
3122, 29, 30syl2anc 643 . . . . . . 7  |-  ( (
ph  /\  y  e.  RR+ )  ->  ( sup ( ran  F ,  RR ,  `'  <  )  < 
( sup ( ran 
F ,  RR ,  `'  <  )  +  y )  <->  E. k  e.  ran  F  k  <  ( sup ( ran  F ,  RR ,  `'  <  )  +  y ) ) )
3226, 31mpbid 202 . . . . . 6  |-  ( (
ph  /\  y  e.  RR+ )  ->  E. k  e.  ran  F  k  < 
( sup ( ran 
F ,  RR ,  `'  <  )  +  y ) )
333sselda 3348 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  ran  F )  ->  k  e.  RR )
3433adantlr 696 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  k  e.  ran  F )  -> 
k  e.  RR )
3524adantr 452 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  k  e.  ran  F )  ->  sup ( ran  F ,  RR ,  `'  <  )  e.  RR )
3627ad2antlr 708 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  k  e.  ran  F )  -> 
y  e.  RR )
3735, 36readdcld 9115 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  k  e.  ran  F )  -> 
( sup ( ran 
F ,  RR ,  `'  <  )  +  y )  e.  RR )
3834, 37, 36ltsub1d 9635 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  k  e.  ran  F )  -> 
( k  <  ( sup ( ran  F ,  RR ,  `'  <  )  +  y )  <->  ( k  -  y )  < 
( ( sup ( ran  F ,  RR ,  `'  <  )  +  y )  -  y ) ) )
393, 14, 20, 23syl3anc 1184 . . . . . . . . . . . . 13  |-  ( ph  ->  sup ( ran  F ,  RR ,  `'  <  )  e.  RR )
4039recnd 9114 . . . . . . . . . . . 12  |-  ( ph  ->  sup ( ran  F ,  RR ,  `'  <  )  e.  CC )
4140ad2antrr 707 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  k  e.  ran  F )  ->  sup ( ran  F ,  RR ,  `'  <  )  e.  CC )
4236recnd 9114 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  k  e.  ran  F )  -> 
y  e.  CC )
4341, 42pncand 9412 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  k  e.  ran  F )  -> 
( ( sup ( ran  F ,  RR ,  `'  <  )  +  y )  -  y )  =  sup ( ran 
F ,  RR ,  `'  <  ) )
4443breq2d 4224 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  k  e.  ran  F )  -> 
( ( k  -  y )  <  (
( sup ( ran 
F ,  RR ,  `'  <  )  +  y )  -  y )  <-> 
( k  -  y
)  <  sup ( ran  F ,  RR ,  `'  <  ) ) )
4538, 44bitrd 245 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  k  e.  ran  F )  -> 
( k  <  ( sup ( ran  F ,  RR ,  `'  <  )  +  y )  <->  ( k  -  y )  <  sup ( ran  F ,  RR ,  `'  <  ) ) )
4645biimpd 199 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  k  e.  ran  F )  -> 
( k  <  ( sup ( ran  F ,  RR ,  `'  <  )  +  y )  -> 
( k  -  y
)  <  sup ( ran  F ,  RR ,  `'  <  ) ) )
4746reximdva 2818 . . . . . 6  |-  ( (
ph  /\  y  e.  RR+ )  ->  ( E. k  e.  ran  F  k  <  ( sup ( ran  F ,  RR ,  `'  <  )  +  y )  ->  E. k  e.  ran  F ( k  -  y )  <  sup ( ran  F ,  RR ,  `'  <  ) ) )
4832, 47mpd 15 . . . . 5  |-  ( (
ph  /\  y  e.  RR+ )  ->  E. k  e.  ran  F ( k  -  y )  <  sup ( ran  F ,  RR ,  `'  <  ) )
49 oveq1 6088 . . . . . . . . 9  |-  ( k  =  ( F `  j )  ->  (
k  -  y )  =  ( ( F `
 j )  -  y ) )
5049breq1d 4222 . . . . . . . 8  |-  ( k  =  ( F `  j )  ->  (
( k  -  y
)  <  sup ( ran  F ,  RR ,  `'  <  )  <->  ( ( F `  j )  -  y )  <  sup ( ran  F ,  RR ,  `'  <  ) ) )
5150rexrn 5872 . . . . . . 7  |-  ( F  Fn  Z  ->  ( E. k  e.  ran  F ( k  -  y
)  <  sup ( ran  F ,  RR ,  `'  <  )  <->  E. j  e.  Z  ( ( F `  j )  -  y )  <  sup ( ran  F ,  RR ,  `'  <  ) ) )
525, 51syl 16 . . . . . 6  |-  ( ph  ->  ( E. k  e. 
ran  F ( k  -  y )  <  sup ( ran  F ,  RR ,  `'  <  )  <->  E. j  e.  Z  ( ( F `  j )  -  y
)  <  sup ( ran  F ,  RR ,  `'  <  ) ) )
5352biimpa 471 . . . . 5  |-  ( (
ph  /\  E. k  e.  ran  F ( k  -  y )  <  sup ( ran  F ,  RR ,  `'  <  ) )  ->  E. j  e.  Z  ( ( F `  j )  -  y )  <  sup ( ran  F ,  RR ,  `'  <  ) )
5448, 53syldan 457 . . . 4  |-  ( (
ph  /\  y  e.  RR+ )  ->  E. j  e.  Z  ( ( F `  j )  -  y )  <  sup ( ran  F ,  RR ,  `'  <  ) )
551adantr 452 . . . . . . . . . . 11  |-  ( (
ph  /\  y  e.  RR+ )  ->  F : Z
--> RR )
569uztrn2 10503 . . . . . . . . . . 11  |-  ( ( j  e.  Z  /\  k  e.  ( ZZ>= `  j ) )  -> 
k  e.  Z )
57 ffvelrn 5868 . . . . . . . . . . 11  |-  ( ( F : Z --> RR  /\  k  e.  Z )  ->  ( F `  k
)  e.  RR )
5855, 56, 57syl2an 464 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( F `  k )  e.  RR )
59 simpl 444 . . . . . . . . . . 11  |-  ( ( j  e.  Z  /\  k  e.  ( ZZ>= `  j ) )  -> 
j  e.  Z )
60 ffvelrn 5868 . . . . . . . . . . 11  |-  ( ( F : Z --> RR  /\  j  e.  Z )  ->  ( F `  j
)  e.  RR )
6155, 59, 60syl2an 464 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( F `  j )  e.  RR )
6239ad2antrr 707 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  sup ( ran  F ,  RR ,  `'  <  )  e.  RR )
63 simprr 734 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  k  e.  (
ZZ>= `  j ) )
64 fzssuz 11093 . . . . . . . . . . . . . 14  |-  ( j ... k )  C_  ( ZZ>= `  j )
65 uzss 10506 . . . . . . . . . . . . . . . . 17  |-  ( j  e.  ( ZZ>= `  M
)  ->  ( ZZ>= `  j )  C_  ( ZZ>=
`  M ) )
6665, 9syl6sseqr 3395 . . . . . . . . . . . . . . . 16  |-  ( j  e.  ( ZZ>= `  M
)  ->  ( ZZ>= `  j )  C_  Z
)
6766, 9eleq2s 2528 . . . . . . . . . . . . . . 15  |-  ( j  e.  Z  ->  ( ZZ>=
`  j )  C_  Z )
6867ad2antrl 709 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( ZZ>= `  j
)  C_  Z )
6964, 68syl5ss 3359 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( j ... k )  C_  Z
)
70 ffvelrn 5868 . . . . . . . . . . . . . . . 16  |-  ( ( F : Z --> RR  /\  n  e.  Z )  ->  ( F `  n
)  e.  RR )
7170ralrimiva 2789 . . . . . . . . . . . . . . 15  |-  ( F : Z --> RR  ->  A. n  e.  Z  ( F `  n )  e.  RR )
721, 71syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  A. n  e.  Z  ( F `  n )  e.  RR )
7372ad2antrr 707 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  A. n  e.  Z  ( F `  n )  e.  RR )
74 ssralv 3407 . . . . . . . . . . . . 13  |-  ( ( j ... k ) 
C_  Z  ->  ( A. n  e.  Z  ( F `  n )  e.  RR  ->  A. n  e.  ( j ... k
) ( F `  n )  e.  RR ) )
7569, 73, 74sylc 58 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  A. n  e.  ( j ... k ) ( F `  n
)  e.  RR )
7675r19.21bi 2804 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  y  e.  RR+ )  /\  ( j  e.  Z  /\  k  e.  ( ZZ>=
`  j ) ) )  /\  n  e.  ( j ... k
) )  ->  ( F `  n )  e.  RR )
77 fzssuz 11093 . . . . . . . . . . . . . 14  |-  ( j ... ( k  - 
1 ) )  C_  ( ZZ>= `  j )
7877, 68syl5ss 3359 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( j ... ( k  -  1 ) )  C_  Z
)
7978sselda 3348 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  y  e.  RR+ )  /\  ( j  e.  Z  /\  k  e.  ( ZZ>=
`  j ) ) )  /\  n  e.  ( j ... (
k  -  1 ) ) )  ->  n  e.  Z )
80 climinf.6 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  ( k  +  1 ) )  <_  ( F `  k ) )
8180ralrimiva 2789 . . . . . . . . . . . . . 14  |-  ( ph  ->  A. k  e.  Z  ( F `  ( k  +  1 ) )  <_  ( F `  k ) )
8281ad2antrr 707 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  A. k  e.  Z  ( F `  ( k  +  1 ) )  <_  ( F `  k ) )
83 oveq1 6088 . . . . . . . . . . . . . . . 16  |-  ( k  =  n  ->  (
k  +  1 )  =  ( n  + 
1 ) )
8483fveq2d 5732 . . . . . . . . . . . . . . 15  |-  ( k  =  n  ->  ( F `  ( k  +  1 ) )  =  ( F `  ( n  +  1
) ) )
85 fveq2 5728 . . . . . . . . . . . . . . 15  |-  ( k  =  n  ->  ( F `  k )  =  ( F `  n ) )
8684, 85breq12d 4225 . . . . . . . . . . . . . 14  |-  ( k  =  n  ->  (
( F `  (
k  +  1 ) )  <_  ( F `  k )  <->  ( F `  ( n  +  1 ) )  <_  ( F `  n )
) )
8786rspccva 3051 . . . . . . . . . . . . 13  |-  ( ( A. k  e.  Z  ( F `  ( k  +  1 ) )  <_  ( F `  k )  /\  n  e.  Z )  ->  ( F `  ( n  +  1 ) )  <_  ( F `  n ) )
8882, 87sylan 458 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  y  e.  RR+ )  /\  ( j  e.  Z  /\  k  e.  ( ZZ>=
`  j ) ) )  /\  n  e.  Z )  ->  ( F `  ( n  +  1 ) )  <_  ( F `  n ) )
8979, 88syldan 457 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  y  e.  RR+ )  /\  ( j  e.  Z  /\  k  e.  ( ZZ>=
`  j ) ) )  /\  n  e.  ( j ... (
k  -  1 ) ) )  ->  ( F `  ( n  +  1 ) )  <_  ( F `  n ) )
9063, 76, 89monoord2 11354 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( F `  k )  <_  ( F `  j )
)
9158, 61, 62, 90lesub1dd 9642 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( ( F `
 k )  -  sup ( ran  F ,  RR ,  `'  <  ) )  <_  ( ( F `  j )  -  sup ( ran  F ,  RR ,  `'  <  ) ) )
9258, 62resubcld 9465 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( ( F `
 k )  -  sup ( ran  F ,  RR ,  `'  <  ) )  e.  RR )
9361, 62resubcld 9465 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( ( F `
 j )  -  sup ( ran  F ,  RR ,  `'  <  ) )  e.  RR )
9427ad2antlr 708 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  y  e.  RR )
95 lelttr 9165 . . . . . . . . . 10  |-  ( ( ( ( F `  k )  -  sup ( ran  F ,  RR ,  `'  <  ) )  e.  RR  /\  (
( F `  j
)  -  sup ( ran  F ,  RR ,  `'  <  ) )  e.  RR  /\  y  e.  RR )  ->  (
( ( ( F `
 k )  -  sup ( ran  F ,  RR ,  `'  <  ) )  <_  ( ( F `  j )  -  sup ( ran  F ,  RR ,  `'  <  ) )  /\  ( ( F `  j )  -  sup ( ran 
F ,  RR ,  `'  <  ) )  < 
y )  ->  (
( F `  k
)  -  sup ( ran  F ,  RR ,  `'  <  ) )  < 
y ) )
9692, 93, 94, 95syl3anc 1184 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( ( ( ( F `  k
)  -  sup ( ran  F ,  RR ,  `'  <  ) )  <_ 
( ( F `  j )  -  sup ( ran  F ,  RR ,  `'  <  ) )  /\  ( ( F `
 j )  -  sup ( ran  F ,  RR ,  `'  <  ) )  <  y )  ->  ( ( F `
 k )  -  sup ( ran  F ,  RR ,  `'  <  ) )  <  y ) )
9791, 96mpand 657 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( ( ( F `  j )  -  sup ( ran 
F ,  RR ,  `'  <  ) )  < 
y  ->  ( ( F `  k )  -  sup ( ran  F ,  RR ,  `'  <  ) )  <  y ) )
98 ltsub23 9508 . . . . . . . . 9  |-  ( ( ( F `  j
)  e.  RR  /\  y  e.  RR  /\  sup ( ran  F ,  RR ,  `'  <  )  e.  RR )  ->  (
( ( F `  j )  -  y
)  <  sup ( ran  F ,  RR ,  `'  <  )  <->  ( ( F `  j )  -  sup ( ran  F ,  RR ,  `'  <  ) )  <  y ) )
9961, 94, 62, 98syl3anc 1184 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( ( ( F `  j )  -  y )  <  sup ( ran  F ,  RR ,  `'  <  )  <-> 
( ( F `  j )  -  sup ( ran  F ,  RR ,  `'  <  ) )  <  y ) )
1003ad2antrr 707 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ran  F  C_  RR )
1015adantr 452 . . . . . . . . . . . 12  |-  ( (
ph  /\  y  e.  RR+ )  ->  F  Fn  Z )
102 fnfvelrn 5867 . . . . . . . . . . . 12  |-  ( ( F  Fn  Z  /\  k  e.  Z )  ->  ( F `  k
)  e.  ran  F
)
103101, 56, 102syl2an 464 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( F `  k )  e.  ran  F )
104100, 103sseldd 3349 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( F `  k )  e.  RR )
10520ad2antrr 707 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  E. x  e.  RR  A. y  e.  ran  F  x  <_  y )
106 infmrlb 9989 . . . . . . . . . . 11  |-  ( ( ran  F  C_  RR  /\ 
E. x  e.  RR  A. y  e.  ran  F  x  <_  y  /\  ( F `  k )  e.  ran  F )  ->  sup ( ran  F ,  RR ,  `'  <  )  <_  ( F `  k ) )
107100, 105, 103, 106syl3anc 1184 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  sup ( ran  F ,  RR ,  `'  <  )  <_  ( F `  k ) )
10862, 104, 107abssubge0d 12234 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( abs `  (
( F `  k
)  -  sup ( ran  F ,  RR ,  `'  <  ) ) )  =  ( ( F `
 k )  -  sup ( ran  F ,  RR ,  `'  <  ) ) )
109108breq1d 4222 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( ( abs `  ( ( F `  k )  -  sup ( ran  F ,  RR ,  `'  <  ) ) )  <  y  <->  ( ( F `  k )  -  sup ( ran  F ,  RR ,  `'  <  ) )  <  y ) )
11097, 99, 1093imtr4d 260 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( ( ( F `  j )  -  y )  <  sup ( ran  F ,  RR ,  `'  <  )  ->  ( abs `  (
( F `  k
)  -  sup ( ran  F ,  RR ,  `'  <  ) ) )  <  y ) )
111110anassrs 630 . . . . . 6  |-  ( ( ( ( ph  /\  y  e.  RR+ )  /\  j  e.  Z )  /\  k  e.  ( ZZ>=
`  j ) )  ->  ( ( ( F `  j )  -  y )  <  sup ( ran  F ,  RR ,  `'  <  )  ->  ( abs `  (
( F `  k
)  -  sup ( ran  F ,  RR ,  `'  <  ) ) )  <  y ) )
112111ralrimdva 2796 . . . . 5  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  j  e.  Z )  ->  (
( ( F `  j )  -  y
)  <  sup ( ran  F ,  RR ,  `'  <  )  ->  A. k  e.  ( ZZ>= `  j )
( abs `  (
( F `  k
)  -  sup ( ran  F ,  RR ,  `'  <  ) ) )  <  y ) )
113112reximdva 2818 . . . 4  |-  ( (
ph  /\  y  e.  RR+ )  ->  ( E. j  e.  Z  (
( F `  j
)  -  y )  <  sup ( ran  F ,  RR ,  `'  <  )  ->  E. j  e.  Z  A. k  e.  ( ZZ>=
`  j ) ( abs `  ( ( F `  k )  -  sup ( ran 
F ,  RR ,  `'  <  ) ) )  <  y ) )
11454, 113mpd 15 . . 3  |-  ( (
ph  /\  y  e.  RR+ )  ->  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( abs `  (
( F `  k
)  -  sup ( ran  F ,  RR ,  `'  <  ) ) )  <  y )
115114ralrimiva 2789 . 2  |-  ( ph  ->  A. y  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j ) ( abs `  ( ( F `  k )  -  sup ( ran  F ,  RR ,  `'  <  ) ) )  <  y )
116 fvex 5742 . . . . 5  |-  ( ZZ>= `  M )  e.  _V
1179, 116eqeltri 2506 . . . 4  |-  Z  e. 
_V
118 fex 5969 . . . 4  |-  ( ( F : Z --> RR  /\  Z  e.  _V )  ->  F  e.  _V )
1191, 117, 118sylancl 644 . . 3  |-  ( ph  ->  F  e.  _V )
120 eqidd 2437 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  =  ( F `  k ) )
1211ffvelrnda 5870 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  RR )
122121recnd 9114 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  CC )
1239, 6, 119, 120, 40, 122clim2c 12299 . 2  |-  ( ph  ->  ( F  ~~>  sup ( ran  F ,  RR ,  `'  <  )  <->  A. y  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( abs `  (
( F `  k
)  -  sup ( ran  F ,  RR ,  `'  <  ) ) )  <  y ) )
124115, 123mpbird 224 1  |-  ( ph  ->  F  ~~>  sup ( ran  F ,  RR ,  `'  <  ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2599   A.wral 2705   E.wrex 2706   _Vcvv 2956    C_ wss 3320   (/)c0 3628   class class class wbr 4212   `'ccnv 4877   ran crn 4879    Fn wfn 5449   -->wf 5450   ` cfv 5454  (class class class)co 6081   supcsup 7445   CCcc 8988   RRcr 8989   1c1 8991    + caddc 8993    < clt 9120    <_ cle 9121    - cmin 9291   ZZcz 10282   ZZ>=cuz 10488   RR+crp 10612   ...cfz 11043   abscabs 12039    ~~> cli 12278
This theorem is referenced by:  climinff  27713  stirlinglem13  27811
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-sup 7446  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-n0 10222  df-z 10283  df-uz 10489  df-rp 10613  df-fz 11044  df-seq 11324  df-exp 11383  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-clim 12282
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