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Theorem climsup 13721
Description: A bounded monotonic sequence converges to the supremum of its range. Theorem 12-5.1 of [Gleason] p. 180. (Contributed by NM, 13-Mar-2005.) (Revised by Mario Carneiro, 10-Feb-2014.)
Hypotheses
Ref Expression
climsup.1  |-  Z  =  ( ZZ>= `  M )
climsup.2  |-  ( ph  ->  M  e.  ZZ )
climsup.3  |-  ( ph  ->  F : Z --> RR )
climsup.4  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  <_  ( F `  (
k  +  1 ) ) )
climsup.5  |-  ( ph  ->  E. x  e.  RR  A. k  e.  Z  ( F `  k )  <_  x )
Assertion
Ref Expression
climsup  |-  ( ph  ->  F  ~~>  sup ( ran  F ,  RR ,  <  )
)
Distinct variable groups:    x, k, F    ph, k    k, Z, x
Allowed substitution hints:    ph( x)    M( x, k)

Proof of Theorem climsup
Dummy variables  j  n  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 climsup.3 . . . . . . . . . 10  |-  ( ph  ->  F : Z --> RR )
2 frn 5749 . . . . . . . . . 10  |-  ( F : Z --> RR  ->  ran 
F  C_  RR )
31, 2syl 17 . . . . . . . . 9  |-  ( ph  ->  ran  F  C_  RR )
4 ffn 5743 . . . . . . . . . . . 12  |-  ( F : Z --> RR  ->  F  Fn  Z )
51, 4syl 17 . . . . . . . . . . 11  |-  ( ph  ->  F  Fn  Z )
6 climsup.2 . . . . . . . . . . . . 13  |-  ( ph  ->  M  e.  ZZ )
7 uzid 11174 . . . . . . . . . . . . 13  |-  ( M  e.  ZZ  ->  M  e.  ( ZZ>= `  M )
)
86, 7syl 17 . . . . . . . . . . . 12  |-  ( ph  ->  M  e.  ( ZZ>= `  M ) )
9 climsup.1 . . . . . . . . . . . 12  |-  Z  =  ( ZZ>= `  M )
108, 9syl6eleqr 2521 . . . . . . . . . . 11  |-  ( ph  ->  M  e.  Z )
11 fnfvelrn 6031 . . . . . . . . . . 11  |-  ( ( F  Fn  Z  /\  M  e.  Z )  ->  ( F `  M
)  e.  ran  F
)
125, 10, 11syl2anc 665 . . . . . . . . . 10  |-  ( ph  ->  ( F `  M
)  e.  ran  F
)
13 ne0i 3767 . . . . . . . . . 10  |-  ( ( F `  M )  e.  ran  F  ->  ran  F  =/=  (/) )
1412, 13syl 17 . . . . . . . . 9  |-  ( ph  ->  ran  F  =/=  (/) )
15 climsup.5 . . . . . . . . . 10  |-  ( ph  ->  E. x  e.  RR  A. k  e.  Z  ( F `  k )  <_  x )
16 breq1 4423 . . . . . . . . . . . . 13  |-  ( y  =  ( F `  k )  ->  (
y  <_  x  <->  ( F `  k )  <_  x
) )
1716ralrn 6037 . . . . . . . . . . . 12  |-  ( F  Fn  Z  ->  ( A. y  e.  ran  F  y  <_  x  <->  A. k  e.  Z  ( F `  k )  <_  x
) )
1817rexbidv 2939 . . . . . . . . . . 11  |-  ( F  Fn  Z  ->  ( E. x  e.  RR  A. y  e.  ran  F  y  <_  x  <->  E. x  e.  RR  A. k  e.  Z  ( F `  k )  <_  x
) )
195, 18syl 17 . . . . . . . . . 10  |-  ( ph  ->  ( E. x  e.  RR  A. y  e. 
ran  F  y  <_  x  <->  E. x  e.  RR  A. k  e.  Z  ( F `  k )  <_  x ) )
2015, 19mpbird 235 . . . . . . . . 9  |-  ( ph  ->  E. x  e.  RR  A. y  e.  ran  F  y  <_  x )
213, 14, 203jca 1185 . . . . . . . 8  |-  ( ph  ->  ( ran  F  C_  RR  /\  ran  F  =/=  (/)  /\  E. x  e.  RR  A. y  e. 
ran  F  y  <_  x ) )
22 suprcl 10570 . . . . . . . 8  |-  ( ( ran  F  C_  RR  /\ 
ran  F  =/=  (/)  /\  E. x  e.  RR  A. y  e.  ran  F  y  <_  x )  ->  sup ( ran  F ,  RR ,  <  )  e.  RR )
2321, 22syl 17 . . . . . . 7  |-  ( ph  ->  sup ( ran  F ,  RR ,  <  )  e.  RR )
24 ltsubrp 11336 . . . . . . 7  |-  ( ( sup ( ran  F ,  RR ,  <  )  e.  RR  /\  y  e.  RR+ )  ->  ( sup ( ran  F ,  RR ,  <  )  -  y )  <  sup ( ran  F ,  RR ,  <  ) )
2523, 24sylan 473 . . . . . 6  |-  ( (
ph  /\  y  e.  RR+ )  ->  ( sup ( ran  F ,  RR ,  <  )  -  y
)  <  sup ( ran  F ,  RR ,  <  ) )
2621adantr 466 . . . . . . 7  |-  ( (
ph  /\  y  e.  RR+ )  ->  ( ran  F 
C_  RR  /\  ran  F  =/=  (/)  /\  E. x  e.  RR  A. y  e. 
ran  F  y  <_  x ) )
27 rpre 11309 . . . . . . . 8  |-  ( y  e.  RR+  ->  y  e.  RR )
28 resubcl 9939 . . . . . . . 8  |-  ( ( sup ( ran  F ,  RR ,  <  )  e.  RR  /\  y  e.  RR )  ->  ( sup ( ran  F ,  RR ,  <  )  -  y )  e.  RR )
2923, 27, 28syl2an 479 . . . . . . 7  |-  ( (
ph  /\  y  e.  RR+ )  ->  ( sup ( ran  F ,  RR ,  <  )  -  y
)  e.  RR )
30 suprlub 10572 . . . . . . 7  |-  ( ( ( ran  F  C_  RR  /\  ran  F  =/=  (/)  /\  E. x  e.  RR  A. y  e. 
ran  F  y  <_  x )  /\  ( sup ( ran  F ,  RR ,  <  )  -  y )  e.  RR )  ->  ( ( sup ( ran  F ,  RR ,  <  )  -  y )  <  sup ( ran  F ,  RR ,  <  )  <->  E. k  e.  ran  F ( sup ( ran  F ,  RR ,  <  )  -  y )  <  k
) )
3126, 29, 30syl2anc 665 . . . . . 6  |-  ( (
ph  /\  y  e.  RR+ )  ->  ( ( sup ( ran  F ,  RR ,  <  )  -  y )  <  sup ( ran  F ,  RR ,  <  )  <->  E. k  e.  ran  F ( sup ( ran  F ,  RR ,  <  )  -  y )  <  k
) )
3225, 31mpbid 213 . . . . 5  |-  ( (
ph  /\  y  e.  RR+ )  ->  E. k  e.  ran  F ( sup ( ran  F ,  RR ,  <  )  -  y )  <  k
)
33 breq2 4424 . . . . . . . 8  |-  ( k  =  ( F `  j )  ->  (
( sup ( ran 
F ,  RR ,  <  )  -  y )  <  k  <->  ( sup ( ran  F ,  RR ,  <  )  -  y
)  <  ( F `  j ) ) )
3433rexrn 6036 . . . . . . 7  |-  ( F  Fn  Z  ->  ( E. k  e.  ran  F ( sup ( ran 
F ,  RR ,  <  )  -  y )  <  k  <->  E. j  e.  Z  ( sup ( ran  F ,  RR ,  <  )  -  y
)  <  ( F `  j ) ) )
355, 34syl 17 . . . . . 6  |-  ( ph  ->  ( E. k  e. 
ran  F ( sup ( ran  F ,  RR ,  <  )  -  y )  <  k  <->  E. j  e.  Z  ( sup ( ran  F ,  RR ,  <  )  -  y )  < 
( F `  j
) ) )
3635biimpa 486 . . . . 5  |-  ( (
ph  /\  E. k  e.  ran  F ( sup ( ran  F ,  RR ,  <  )  -  y )  <  k
)  ->  E. j  e.  Z  ( sup ( ran  F ,  RR ,  <  )  -  y
)  <  ( F `  j ) )
3732, 36syldan 472 . . . 4  |-  ( (
ph  /\  y  e.  RR+ )  ->  E. j  e.  Z  ( sup ( ran  F ,  RR ,  <  )  -  y
)  <  ( F `  j ) )
38 ffvelrn 6032 . . . . . . . . . . . 12  |-  ( ( F : Z --> RR  /\  j  e.  Z )  ->  ( F `  j
)  e.  RR )
391, 38sylan 473 . . . . . . . . . . 11  |-  ( (
ph  /\  j  e.  Z )  ->  ( F `  j )  e.  RR )
4039ad2ant2r 751 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( F `  j )  e.  RR )
411adantr 466 . . . . . . . . . . 11  |-  ( (
ph  /\  y  e.  RR+ )  ->  F : Z
--> RR )
429uztrn2 11177 . . . . . . . . . . 11  |-  ( ( j  e.  Z  /\  k  e.  ( ZZ>= `  j ) )  -> 
k  e.  Z )
43 ffvelrn 6032 . . . . . . . . . . 11  |-  ( ( F : Z --> RR  /\  k  e.  Z )  ->  ( F `  k
)  e.  RR )
4441, 42, 43syl2an 479 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( F `  k )  e.  RR )
4523ad2antrr 730 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  sup ( ran  F ,  RR ,  <  )  e.  RR )
46 simprr 764 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  k  e.  (
ZZ>= `  j ) )
47 fzssuz 11840 . . . . . . . . . . . . . 14  |-  ( j ... k )  C_  ( ZZ>= `  j )
48 uzss 11180 . . . . . . . . . . . . . . . . 17  |-  ( j  e.  ( ZZ>= `  M
)  ->  ( ZZ>= `  j )  C_  ( ZZ>=
`  M ) )
4948, 9syl6sseqr 3511 . . . . . . . . . . . . . . . 16  |-  ( j  e.  ( ZZ>= `  M
)  ->  ( ZZ>= `  j )  C_  Z
)
5049, 9eleq2s 2530 . . . . . . . . . . . . . . 15  |-  ( j  e.  Z  ->  ( ZZ>=
`  j )  C_  Z )
5150ad2antrl 732 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( ZZ>= `  j
)  C_  Z )
5247, 51syl5ss 3475 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( j ... k )  C_  Z
)
53 ffvelrn 6032 . . . . . . . . . . . . . . . 16  |-  ( ( F : Z --> RR  /\  n  e.  Z )  ->  ( F `  n
)  e.  RR )
5453ralrimiva 2839 . . . . . . . . . . . . . . 15  |-  ( F : Z --> RR  ->  A. n  e.  Z  ( F `  n )  e.  RR )
551, 54syl 17 . . . . . . . . . . . . . 14  |-  ( ph  ->  A. n  e.  Z  ( F `  n )  e.  RR )
5655ad2antrr 730 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  A. n  e.  Z  ( F `  n )  e.  RR )
57 ssralv 3525 . . . . . . . . . . . . 13  |-  ( ( j ... k ) 
C_  Z  ->  ( A. n  e.  Z  ( F `  n )  e.  RR  ->  A. n  e.  ( j ... k
) ( F `  n )  e.  RR ) )
5852, 56, 57sylc 62 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  A. n  e.  ( j ... k ) ( F `  n
)  e.  RR )
5958r19.21bi 2794 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  y  e.  RR+ )  /\  ( j  e.  Z  /\  k  e.  ( ZZ>=
`  j ) ) )  /\  n  e.  ( j ... k
) )  ->  ( F `  n )  e.  RR )
60 fzssuz 11840 . . . . . . . . . . . . . 14  |-  ( j ... ( k  - 
1 ) )  C_  ( ZZ>= `  j )
6160, 51syl5ss 3475 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( j ... ( k  -  1 ) )  C_  Z
)
6261sselda 3464 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  y  e.  RR+ )  /\  ( j  e.  Z  /\  k  e.  ( ZZ>=
`  j ) ) )  /\  n  e.  ( j ... (
k  -  1 ) ) )  ->  n  e.  Z )
63 climsup.4 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  <_  ( F `  (
k  +  1 ) ) )
6463ralrimiva 2839 . . . . . . . . . . . . . 14  |-  ( ph  ->  A. k  e.  Z  ( F `  k )  <_  ( F `  ( k  +  1 ) ) )
6564ad2antrr 730 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  A. k  e.  Z  ( F `  k )  <_  ( F `  ( k  +  1 ) ) )
66 fveq2 5878 . . . . . . . . . . . . . . 15  |-  ( k  =  n  ->  ( F `  k )  =  ( F `  n ) )
67 oveq1 6309 . . . . . . . . . . . . . . . 16  |-  ( k  =  n  ->  (
k  +  1 )  =  ( n  + 
1 ) )
6867fveq2d 5882 . . . . . . . . . . . . . . 15  |-  ( k  =  n  ->  ( F `  ( k  +  1 ) )  =  ( F `  ( n  +  1
) ) )
6966, 68breq12d 4433 . . . . . . . . . . . . . 14  |-  ( k  =  n  ->  (
( F `  k
)  <_  ( F `  ( k  +  1 ) )  <->  ( F `  n )  <_  ( F `  ( n  +  1 ) ) ) )
7069rspccva 3181 . . . . . . . . . . . . 13  |-  ( ( A. k  e.  Z  ( F `  k )  <_  ( F `  ( k  +  1 ) )  /\  n  e.  Z )  ->  ( F `  n )  <_  ( F `  (
n  +  1 ) ) )
7165, 70sylan 473 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  y  e.  RR+ )  /\  ( j  e.  Z  /\  k  e.  ( ZZ>=
`  j ) ) )  /\  n  e.  Z )  ->  ( F `  n )  <_  ( F `  (
n  +  1 ) ) )
7262, 71syldan 472 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  y  e.  RR+ )  /\  ( j  e.  Z  /\  k  e.  ( ZZ>=
`  j ) ) )  /\  n  e.  ( j ... (
k  -  1 ) ) )  ->  ( F `  n )  <_  ( F `  (
n  +  1 ) ) )
7346, 59, 72monoord 12243 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( F `  j )  <_  ( F `  k )
)
7440, 44, 45, 73lesub2dd 10231 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( sup ( ran  F ,  RR ,  <  )  -  ( F `
 k ) )  <_  ( sup ( ran  F ,  RR ,  <  )  -  ( F `
 j ) ) )
7545, 44resubcld 10048 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( sup ( ran  F ,  RR ,  <  )  -  ( F `
 k ) )  e.  RR )
7645, 40resubcld 10048 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( sup ( ran  F ,  RR ,  <  )  -  ( F `
 j ) )  e.  RR )
7727ad2antlr 731 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  y  e.  RR )
78 lelttr 9725 . . . . . . . . . 10  |-  ( ( ( sup ( ran 
F ,  RR ,  <  )  -  ( F `
 k ) )  e.  RR  /\  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  j ) )  e.  RR  /\  y  e.  RR )  ->  ( ( ( sup ( ran  F ,  RR ,  <  )  -  ( F `  k ) )  <_  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  j )
)  /\  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  j )
)  <  y )  ->  ( sup ( ran 
F ,  RR ,  <  )  -  ( F `
 k ) )  <  y ) )
7975, 76, 77, 78syl3anc 1264 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( ( ( sup ( ran  F ,  RR ,  <  )  -  ( F `  k ) )  <_ 
( sup ( ran 
F ,  RR ,  <  )  -  ( F `
 j ) )  /\  ( sup ( ran  F ,  RR ,  <  )  -  ( F `
 j ) )  <  y )  -> 
( sup ( ran 
F ,  RR ,  <  )  -  ( F `
 k ) )  <  y ) )
8074, 79mpand 679 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( ( sup ( ran  F ,  RR ,  <  )  -  ( F `  j ) )  <  y  -> 
( sup ( ran 
F ,  RR ,  <  )  -  ( F `
 k ) )  <  y ) )
81 ltsub23 10095 . . . . . . . . 9  |-  ( ( sup ( ran  F ,  RR ,  <  )  e.  RR  /\  y  e.  RR  /\  ( F `
 j )  e.  RR )  ->  (
( sup ( ran 
F ,  RR ,  <  )  -  y )  <  ( F `  j )  <->  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  j )
)  <  y )
)
8245, 77, 40, 81syl3anc 1264 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( ( sup ( ran  F ,  RR ,  <  )  -  y )  <  ( F `  j )  <->  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  j ) )  < 
y ) )
8321ad2antrr 730 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( ran  F  C_  RR  /\  ran  F  =/=  (/)  /\  E. x  e.  RR  A. y  e. 
ran  F  y  <_  x ) )
845adantr 466 . . . . . . . . . . . 12  |-  ( (
ph  /\  y  e.  RR+ )  ->  F  Fn  Z )
85 fnfvelrn 6031 . . . . . . . . . . . 12  |-  ( ( F  Fn  Z  /\  k  e.  Z )  ->  ( F `  k
)  e.  ran  F
)
8684, 42, 85syl2an 479 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( F `  k )  e.  ran  F )
87 suprub 10571 . . . . . . . . . . 11  |-  ( ( ( ran  F  C_  RR  /\  ran  F  =/=  (/)  /\  E. x  e.  RR  A. y  e. 
ran  F  y  <_  x )  /\  ( F `
 k )  e. 
ran  F )  -> 
( F `  k
)  <_  sup ( ran  F ,  RR ,  <  ) )
8883, 86, 87syl2anc 665 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( F `  k )  <_  sup ( ran  F ,  RR ,  <  ) )
8944, 45, 88abssuble0d 13483 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( abs `  (
( F `  k
)  -  sup ( ran  F ,  RR ,  <  ) ) )  =  ( sup ( ran 
F ,  RR ,  <  )  -  ( F `
 k ) ) )
9089breq1d 4430 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( ( abs `  ( ( F `  k )  -  sup ( ran  F ,  RR ,  <  ) ) )  <  y  <->  ( sup ( ran  F ,  RR ,  <  )  -  ( F `  k )
)  <  y )
)
9180, 82, 903imtr4d 271 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( ( sup ( ran  F ,  RR ,  <  )  -  y )  <  ( F `  j )  ->  ( abs `  (
( F `  k
)  -  sup ( ran  F ,  RR ,  <  ) ) )  < 
y ) )
9291anassrs 652 . . . . . 6  |-  ( ( ( ( ph  /\  y  e.  RR+ )  /\  j  e.  Z )  /\  k  e.  ( ZZ>=
`  j ) )  ->  ( ( sup ( ran  F ,  RR ,  <  )  -  y )  <  ( F `  j )  ->  ( abs `  (
( F `  k
)  -  sup ( ran  F ,  RR ,  <  ) ) )  < 
y ) )
9392ralrimdva 2843 . . . . 5  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  j  e.  Z )  ->  (
( sup ( ran 
F ,  RR ,  <  )  -  y )  <  ( F `  j )  ->  A. k  e.  ( ZZ>= `  j )
( abs `  (
( F `  k
)  -  sup ( ran  F ,  RR ,  <  ) ) )  < 
y ) )
9493reximdva 2900 . . . 4  |-  ( (
ph  /\  y  e.  RR+ )  ->  ( E. j  e.  Z  ( sup ( ran  F ,  RR ,  <  )  -  y )  <  ( F `  j )  ->  E. j  e.  Z  A. k  e.  ( ZZ>=
`  j ) ( abs `  ( ( F `  k )  -  sup ( ran 
F ,  RR ,  <  ) ) )  < 
y ) )
9537, 94mpd 15 . . 3  |-  ( (
ph  /\  y  e.  RR+ )  ->  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( abs `  (
( F `  k
)  -  sup ( ran  F ,  RR ,  <  ) ) )  < 
y )
9695ralrimiva 2839 . 2  |-  ( ph  ->  A. y  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j ) ( abs `  ( ( F `  k )  -  sup ( ran  F ,  RR ,  <  ) ) )  <  y )
97 fvex 5888 . . . . 5  |-  ( ZZ>= `  M )  e.  _V
989, 97eqeltri 2506 . . . 4  |-  Z  e. 
_V
99 fex 6150 . . . 4  |-  ( ( F : Z --> RR  /\  Z  e.  _V )  ->  F  e.  _V )
1001, 98, 99sylancl 666 . . 3  |-  ( ph  ->  F  e.  _V )
101 eqidd 2423 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  =  ( F `  k ) )
10223recnd 9670 . . 3  |-  ( ph  ->  sup ( ran  F ,  RR ,  <  )  e.  CC )
1031, 43sylan 473 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  RR )
104103recnd 9670 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  CC )
1059, 6, 100, 101, 102, 104clim2c 13557 . 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 ) )
10696, 105mpbird 235 1  |-  ( ph  ->  F  ~~>  sup ( ran  F ,  RR ,  <  )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1868    =/= wne 2618   A.wral 2775   E.wrex 2776   _Vcvv 3081    C_ wss 3436   (/)c0 3761   class class class wbr 4420   ran crn 4851    Fn wfn 5593   -->wf 5594   ` cfv 5598  (class class class)co 6302   supcsup 7957   RRcr 9539   1c1 9541    + caddc 9543    < clt 9676    <_ cle 9677    - cmin 9861   ZZcz 10938   ZZ>=cuz 11160   RR+crp 11303   ...cfz 11785   abscabs 13286    ~~> cli 13536
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4552  ax-pow 4599  ax-pr 4657  ax-un 6594  ax-cnex 9596  ax-resscn 9597  ax-1cn 9598  ax-icn 9599  ax-addcl 9600  ax-addrcl 9601  ax-mulcl 9602  ax-mulrcl 9603  ax-mulcom 9604  ax-addass 9605  ax-mulass 9606  ax-distr 9607  ax-i2m1 9608  ax-1ne0 9609  ax-1rid 9610  ax-rnegex 9611  ax-rrecex 9612  ax-cnre 9613  ax-pre-lttri 9614  ax-pre-lttrn 9615  ax-pre-ltadd 9616  ax-pre-mulgt0 9617  ax-pre-sup 9618
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-iun 4298  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4761  df-id 4765  df-po 4771  df-so 4772  df-fr 4809  df-we 4811  df-xp 4856  df-rel 4857  df-cnv 4858  df-co 4859  df-dm 4860  df-rn 4861  df-res 4862  df-ima 4863  df-pred 5396  df-ord 5442  df-on 5443  df-lim 5444  df-suc 5445  df-iota 5562  df-fun 5600  df-fn 5601  df-f 5602  df-f1 5603  df-fo 5604  df-f1o 5605  df-fv 5606  df-riota 6264  df-ov 6305  df-oprab 6306  df-mpt2 6307  df-om 6704  df-1st 6804  df-2nd 6805  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-er 7368  df-en 7575  df-dom 7576  df-sdom 7577  df-sup 7959  df-pnf 9678  df-mnf 9679  df-xr 9680  df-ltxr 9681  df-le 9682  df-sub 9863  df-neg 9864  df-div 10271  df-nn 10611  df-2 10669  df-3 10670  df-n0 10871  df-z 10939  df-uz 11161  df-rp 11304  df-fz 11786  df-seq 12214  df-exp 12273  df-cj 13151  df-re 13152  df-im 13153  df-sqrt 13287  df-abs 13288  df-clim 13540
This theorem is referenced by:  isumsup2  13892  climcnds  13897  itg1climres  22659  itg2monolem1  22695  itg2i1fseq  22700  itg2i1fseq2  22701  emcllem6  23913  lmdvg  28755  esumpcvgval  28895
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