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Theorem climinf 29947
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 5676 . . . . . . . . . . . 12  |-  ( F : Z --> RR  ->  ran 
F  C_  RR )
31, 2syl 16 . . . . . . . . . . 11  |-  ( ph  ->  ran  F  C_  RR )
4 ffn 5670 . . . . . . . . . . . . . 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 10989 . . . . . . . . . . . . . . 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 2553 . . . . . . . . . . . . 13  |-  ( ph  ->  M  e.  Z )
11 fnfvelrn 5952 . . . . . . . . . . . . 13  |-  ( ( F  Fn  Z  /\  M  e.  Z )  ->  ( F `  M
)  e.  ran  F
)
125, 10, 11syl2anc 661 . . . . . . . . . . . 12  |-  ( ph  ->  ( F `  M
)  e.  ran  F
)
13 ne0i 3754 . . . . . . . . . . . 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 4407 . . . . . . . . . . . . . . 15  |-  ( y  =  ( F `  k )  ->  (
x  <_  y  <->  x  <_  ( F `  k ) ) )
1716ralrn 5958 . . . . . . . . . . . . . 14  |-  ( F  Fn  Z  ->  ( A. y  e.  ran  F  x  <_  y  <->  A. k  e.  Z  x  <_  ( F `  k ) ) )
1817rexbidv 2868 . . . . . . . . . . . . 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 232 . . . . . . . . . . 11  |-  ( ph  ->  E. x  e.  RR  A. y  e.  ran  F  x  <_  y )
213, 14, 203jca 1168 . . . . . . . . . 10  |-  ( ph  ->  ( ran  F  C_  RR  /\  ran  F  =/=  (/)  /\  E. x  e.  RR  A. y  e. 
ran  F  x  <_  y ) )
2221adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  RR+ )  ->  ( ran  F 
C_  RR  /\  ran  F  =/=  (/)  /\  E. x  e.  RR  A. y  e. 
ran  F  x  <_  y ) )
23 infmrcl 10423 . . . . . . . . 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 461 . . . . . . . 8  |-  ( (
ph  /\  y  e.  RR+ )  ->  y  e.  RR+ )
2624, 25ltaddrpd 11170 . . . . . . 7  |-  ( (
ph  /\  y  e.  RR+ )  ->  sup ( ran  F ,  RR ,  `'  <  )  <  ( sup ( ran  F ,  RR ,  `'  <  )  +  y ) )
27 rpre 11111 . . . . . . . . . 10  |-  ( y  e.  RR+  ->  y  e.  RR )
2827adantl 466 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  RR+ )  ->  y  e.  RR )
2924, 28readdcld 9527 . . . . . . . 8  |-  ( (
ph  /\  y  e.  RR+ )  ->  ( sup ( ran  F ,  RR ,  `'  <  )  +  y )  e.  RR )
30 infrglb 29939 . . . . . . . 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 661 . . . . . . 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 210 . . . . . 6  |-  ( (
ph  /\  y  e.  RR+ )  ->  E. k  e.  ran  F  k  < 
( sup ( ran 
F ,  RR ,  `'  <  )  +  y ) )
333sselda 3467 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  ran  F )  ->  k  e.  RR )
3433adantlr 714 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  k  e.  ran  F )  -> 
k  e.  RR )
3524adantr 465 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  k  e.  ran  F )  ->  sup ( ran  F ,  RR ,  `'  <  )  e.  RR )
3627ad2antlr 726 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  k  e.  ran  F )  -> 
y  e.  RR )
3735, 36readdcld 9527 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  k  e.  ran  F )  -> 
( sup ( ran 
F ,  RR ,  `'  <  )  +  y )  e.  RR )
3834, 37, 36ltsub1d 10062 . . . . . . . . 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 1219 . . . . . . . . . . . . 13  |-  ( ph  ->  sup ( ran  F ,  RR ,  `'  <  )  e.  RR )
4039recnd 9526 . . . . . . . . . . . 12  |-  ( ph  ->  sup ( ran  F ,  RR ,  `'  <  )  e.  CC )
4140ad2antrr 725 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  k  e.  ran  F )  ->  sup ( ran  F ,  RR ,  `'  <  )  e.  CC )
4236recnd 9526 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  k  e.  ran  F )  -> 
y  e.  CC )
4341, 42pncand 9834 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  k  e.  ran  F )  -> 
( ( sup ( ran  F ,  RR ,  `'  <  )  +  y )  -  y )  =  sup ( ran 
F ,  RR ,  `'  <  ) )
4443breq2d 4415 . . . . . . . . 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 253 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  k  e.  ran  F )  -> 
( k  <  ( sup ( ran  F ,  RR ,  `'  <  )  +  y )  <->  ( k  -  y )  <  sup ( ran  F ,  RR ,  `'  <  ) ) )
4645biimpd 207 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  k  e.  ran  F )  -> 
( k  <  ( sup ( ran  F ,  RR ,  `'  <  )  +  y )  -> 
( k  -  y
)  <  sup ( ran  F ,  RR ,  `'  <  ) ) )
4746reximdva 2934 . . . . . 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 6210 . . . . . . . . 9  |-  ( k  =  ( F `  j )  ->  (
k  -  y )  =  ( ( F `
 j )  -  y ) )
5049breq1d 4413 . . . . . . . 8  |-  ( k  =  ( F `  j )  ->  (
( k  -  y
)  <  sup ( ran  F ,  RR ,  `'  <  )  <->  ( ( F `  j )  -  y )  <  sup ( ran  F ,  RR ,  `'  <  ) ) )
5150rexrn 5957 . . . . . . 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 484 . . . . 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 470 . . . 4  |-  ( (
ph  /\  y  e.  RR+ )  ->  E. j  e.  Z  ( ( F `  j )  -  y )  <  sup ( ran  F ,  RR ,  `'  <  ) )
551adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  y  e.  RR+ )  ->  F : Z
--> RR )
569uztrn2 10992 . . . . . . . . . . 11  |-  ( ( j  e.  Z  /\  k  e.  ( ZZ>= `  j ) )  -> 
k  e.  Z )
57 ffvelrn 5953 . . . . . . . . . . 11  |-  ( ( F : Z --> RR  /\  k  e.  Z )  ->  ( F `  k
)  e.  RR )
5855, 56, 57syl2an 477 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( F `  k )  e.  RR )
59 simpl 457 . . . . . . . . . . 11  |-  ( ( j  e.  Z  /\  k  e.  ( ZZ>= `  j ) )  -> 
j  e.  Z )
60 ffvelrn 5953 . . . . . . . . . . 11  |-  ( ( F : Z --> RR  /\  j  e.  Z )  ->  ( F `  j
)  e.  RR )
6155, 59, 60syl2an 477 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( F `  j )  e.  RR )
6239ad2antrr 725 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  sup ( ran  F ,  RR ,  `'  <  )  e.  RR )
63 simprr 756 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  k  e.  (
ZZ>= `  j ) )
64 fzssuz 11619 . . . . . . . . . . . . . 14  |-  ( j ... k )  C_  ( ZZ>= `  j )
65 uzss 10995 . . . . . . . . . . . . . . . . 17  |-  ( j  e.  ( ZZ>= `  M
)  ->  ( ZZ>= `  j )  C_  ( ZZ>=
`  M ) )
6665, 9syl6sseqr 3514 . . . . . . . . . . . . . . . 16  |-  ( j  e.  ( ZZ>= `  M
)  ->  ( ZZ>= `  j )  C_  Z
)
6766, 9eleq2s 2562 . . . . . . . . . . . . . . 15  |-  ( j  e.  Z  ->  ( ZZ>=
`  j )  C_  Z )
6867ad2antrl 727 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( ZZ>= `  j
)  C_  Z )
6964, 68syl5ss 3478 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( j ... k )  C_  Z
)
70 ffvelrn 5953 . . . . . . . . . . . . . . . 16  |-  ( ( F : Z --> RR  /\  n  e.  Z )  ->  ( F `  n
)  e.  RR )
7170ralrimiva 2830 . . . . . . . . . . . . . . 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 725 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  A. n  e.  Z  ( F `  n )  e.  RR )
74 ssralv 3527 . . . . . . . . . . . . 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 60 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  A. n  e.  ( j ... k ) ( F `  n
)  e.  RR )
7675r19.21bi 2920 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  y  e.  RR+ )  /\  ( j  e.  Z  /\  k  e.  ( ZZ>=
`  j ) ) )  /\  n  e.  ( j ... k
) )  ->  ( F `  n )  e.  RR )
77 fzssuz 11619 . . . . . . . . . . . . . 14  |-  ( j ... ( k  - 
1 ) )  C_  ( ZZ>= `  j )
7877, 68syl5ss 3478 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( j ... ( k  -  1 ) )  C_  Z
)
7978sselda 3467 . . . . . . . . . . . 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 2830 . . . . . . . . . . . . . 14  |-  ( ph  ->  A. k  e.  Z  ( F `  ( k  +  1 ) )  <_  ( F `  k ) )
8281ad2antrr 725 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  A. k  e.  Z  ( F `  ( k  +  1 ) )  <_  ( F `  k ) )
83 oveq1 6210 . . . . . . . . . . . . . . . 16  |-  ( k  =  n  ->  (
k  +  1 )  =  ( n  + 
1 ) )
8483fveq2d 5806 . . . . . . . . . . . . . . 15  |-  ( k  =  n  ->  ( F `  ( k  +  1 ) )  =  ( F `  ( n  +  1
) ) )
85 fveq2 5802 . . . . . . . . . . . . . . 15  |-  ( k  =  n  ->  ( F `  k )  =  ( F `  n ) )
8684, 85breq12d 4416 . . . . . . . . . . . . . 14  |-  ( k  =  n  ->  (
( F `  (
k  +  1 ) )  <_  ( F `  k )  <->  ( F `  ( n  +  1 ) )  <_  ( F `  n )
) )
8786rspccva 3178 . . . . . . . . . . . . 13  |-  ( ( A. k  e.  Z  ( F `  ( k  +  1 ) )  <_  ( F `  k )  /\  n  e.  Z )  ->  ( F `  ( n  +  1 ) )  <_  ( F `  n ) )
8882, 87sylan 471 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  y  e.  RR+ )  /\  ( j  e.  Z  /\  k  e.  ( ZZ>=
`  j ) ) )  /\  n  e.  Z )  ->  ( F `  ( n  +  1 ) )  <_  ( F `  n ) )
8979, 88syldan 470 . . . . . . . . . . 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 11957 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( F `  k )  <_  ( F `  j )
)
9158, 61, 62, 90lesub1dd 10069 . . . . . . . . 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 9890 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( ( F `
 k )  -  sup ( ran  F ,  RR ,  `'  <  ) )  e.  RR )
9361, 62resubcld 9890 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( ( F `
 j )  -  sup ( ran  F ,  RR ,  `'  <  ) )  e.  RR )
9427ad2antlr 726 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  y  e.  RR )
95 lelttr 9579 . . . . . . . . . 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 1219 . . . . . . . . 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 675 . . . . . . . 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 9933 . . . . . . . . 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 1219 . . . . . . . 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 725 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ran  F  C_  RR )
1015adantr 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  y  e.  RR+ )  ->  F  Fn  Z )
102 fnfvelrn 5952 . . . . . . . . . . . 12  |-  ( ( F  Fn  Z  /\  k  e.  Z )  ->  ( F `  k
)  e.  ran  F
)
103101, 56, 102syl2an 477 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( F `  k )  e.  ran  F )
104100, 103sseldd 3468 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( F `  k )  e.  RR )
10520ad2antrr 725 . . . . . . . . . . 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 10425 . . . . . . . . . . 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 1219 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  sup ( ran  F ,  RR ,  `'  <  )  <_  ( F `  k ) )
10862, 104, 107abssubge0d 13039 . . . . . . . . 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 4413 . . . . . . . 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 268 . . . . . . 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 648 . . . . . 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 2912 . . . . 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 2934 . . . 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 2830 . 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 5812 . . . . 5  |-  ( ZZ>= `  M )  e.  _V
1179, 116eqeltri 2538 . . . 4  |-  Z  e. 
_V
118 fex 6062 . . . 4  |-  ( ( F : Z --> RR  /\  Z  e.  _V )  ->  F  e.  _V )
1191, 117, 118sylancl 662 . . 3  |-  ( ph  ->  F  e.  _V )
120 eqidd 2455 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  =  ( F `  k ) )
1211ffvelrnda 5955 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  RR )
122121recnd 9526 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  CC )
1239, 6, 119, 120, 40, 122clim2c 13104 . 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 232 1  |-  ( ph  ->  F  ~~>  sup ( ran  F ,  RR ,  `'  <  ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2648   A.wral 2799   E.wrex 2800   _Vcvv 3078    C_ wss 3439   (/)c0 3748   class class class wbr 4403   `'ccnv 4950   ran crn 4952    Fn wfn 5524   -->wf 5525   ` cfv 5529  (class class class)co 6203   supcsup 7804   CCcc 9394   RRcr 9395   1c1 9397    + caddc 9399    < clt 9532    <_ cle 9533    - cmin 9709   ZZcz 10760   ZZ>=cuz 10975   RR+crp 11105   ...cfz 11557   abscabs 12844    ~~> cli 13083
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9452  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473  ax-pre-sup 9474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-isom 5538  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-sup 7805  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-div 10108  df-nn 10437  df-2 10494  df-3 10495  df-n0 10694  df-z 10761  df-uz 10976  df-rp 11106  df-fz 11558  df-seq 11927  df-exp 11986  df-cj 12709  df-re 12710  df-im 12711  df-sqr 12845  df-abs 12846  df-clim 13087
This theorem is referenced by:  climinff  29952  stirlinglem13  30049
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