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Theorem climsup 12143
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 5395 . . . . . . . . . 10  |-  ( F : Z --> RR  ->  ran 
F  C_  RR )
31, 2syl 15 . . . . . . . . 9  |-  ( ph  ->  ran  F  C_  RR )
4 ffn 5389 . . . . . . . . . . . 12  |-  ( F : Z --> RR  ->  F  Fn  Z )
51, 4syl 15 . . . . . . . . . . 11  |-  ( ph  ->  F  Fn  Z )
6 climsup.2 . . . . . . . . . . . . 13  |-  ( ph  ->  M  e.  ZZ )
7 uzid 10242 . . . . . . . . . . . . 13  |-  ( M  e.  ZZ  ->  M  e.  ( ZZ>= `  M )
)
86, 7syl 15 . . . . . . . . . . . 12  |-  ( ph  ->  M  e.  ( ZZ>= `  M ) )
9 climsup.1 . . . . . . . . . . . 12  |-  Z  =  ( ZZ>= `  M )
108, 9syl6eleqr 2374 . . . . . . . . . . 11  |-  ( ph  ->  M  e.  Z )
11 fnfvelrn 5662 . . . . . . . . . . 11  |-  ( ( F  Fn  Z  /\  M  e.  Z )  ->  ( F `  M
)  e.  ran  F
)
125, 10, 11syl2anc 642 . . . . . . . . . 10  |-  ( ph  ->  ( F `  M
)  e.  ran  F
)
13 ne0i 3461 . . . . . . . . . 10  |-  ( ( F `  M )  e.  ran  F  ->  ran  F  =/=  (/) )
1412, 13syl 15 . . . . . . . . 9  |-  ( ph  ->  ran  F  =/=  (/) )
15 climsup.5 . . . . . . . . . 10  |-  ( ph  ->  E. x  e.  RR  A. k  e.  Z  ( F `  k )  <_  x )
16 breq1 4026 . . . . . . . . . . . . 13  |-  ( y  =  ( F `  k )  ->  (
y  <_  x  <->  ( F `  k )  <_  x
) )
1716ralrn 5668 . . . . . . . . . . . 12  |-  ( F  Fn  Z  ->  ( A. y  e.  ran  F  y  <_  x  <->  A. k  e.  Z  ( F `  k )  <_  x
) )
1817rexbidv 2564 . . . . . . . . . . 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 15 . . . . . . . . . 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 223 . . . . . . . . 9  |-  ( ph  ->  E. x  e.  RR  A. y  e.  ran  F  y  <_  x )
213, 14, 203jca 1132 . . . . . . . 8  |-  ( ph  ->  ( ran  F  C_  RR  /\  ran  F  =/=  (/)  /\  E. x  e.  RR  A. y  e. 
ran  F  y  <_  x ) )
22 suprcl 9714 . . . . . . . 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 15 . . . . . . 7  |-  ( ph  ->  sup ( ran  F ,  RR ,  <  )  e.  RR )
24 ltsubrp 10385 . . . . . . 7  |-  ( ( sup ( ran  F ,  RR ,  <  )  e.  RR  /\  y  e.  RR+ )  ->  ( sup ( ran  F ,  RR ,  <  )  -  y )  <  sup ( ran  F ,  RR ,  <  ) )
2523, 24sylan 457 . . . . . 6  |-  ( (
ph  /\  y  e.  RR+ )  ->  ( sup ( ran  F ,  RR ,  <  )  -  y
)  <  sup ( ran  F ,  RR ,  <  ) )
2621adantr 451 . . . . . . 7  |-  ( (
ph  /\  y  e.  RR+ )  ->  ( ran  F 
C_  RR  /\  ran  F  =/=  (/)  /\  E. x  e.  RR  A. y  e. 
ran  F  y  <_  x ) )
27 rpre 10360 . . . . . . . 8  |-  ( y  e.  RR+  ->  y  e.  RR )
28 resubcl 9111 . . . . . . . 8  |-  ( ( sup ( ran  F ,  RR ,  <  )  e.  RR  /\  y  e.  RR )  ->  ( sup ( ran  F ,  RR ,  <  )  -  y )  e.  RR )
2923, 27, 28syl2an 463 . . . . . . 7  |-  ( (
ph  /\  y  e.  RR+ )  ->  ( sup ( ran  F ,  RR ,  <  )  -  y
)  e.  RR )
30 suprlub 9716 . . . . . . 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 642 . . . . . 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 201 . . . . 5  |-  ( (
ph  /\  y  e.  RR+ )  ->  E. k  e.  ran  F ( sup ( ran  F ,  RR ,  <  )  -  y )  <  k
)
33 breq2 4027 . . . . . . . 8  |-  ( k  =  ( F `  j )  ->  (
( sup ( ran 
F ,  RR ,  <  )  -  y )  <  k  <->  ( sup ( ran  F ,  RR ,  <  )  -  y
)  <  ( F `  j ) ) )
3433rexrn 5667 . . . . . . 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 15 . . . . . 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 470 . . . . 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 456 . . . 4  |-  ( (
ph  /\  y  e.  RR+ )  ->  E. j  e.  Z  ( sup ( ran  F ,  RR ,  <  )  -  y
)  <  ( F `  j ) )
38 ffvelrn 5663 . . . . . . . . . . . 12  |-  ( ( F : Z --> RR  /\  j  e.  Z )  ->  ( F `  j
)  e.  RR )
391, 38sylan 457 . . . . . . . . . . 11  |-  ( (
ph  /\  j  e.  Z )  ->  ( F `  j )  e.  RR )
4039ad2ant2r 727 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( F `  j )  e.  RR )
411adantr 451 . . . . . . . . . . 11  |-  ( (
ph  /\  y  e.  RR+ )  ->  F : Z
--> RR )
429uztrn2 10245 . . . . . . . . . . 11  |-  ( ( j  e.  Z  /\  k  e.  ( ZZ>= `  j ) )  -> 
k  e.  Z )
43 ffvelrn 5663 . . . . . . . . . . 11  |-  ( ( F : Z --> RR  /\  k  e.  Z )  ->  ( F `  k
)  e.  RR )
4441, 42, 43syl2an 463 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( F `  k )  e.  RR )
4523ad2antrr 706 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  sup ( ran  F ,  RR ,  <  )  e.  RR )
46 simprr 733 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  k  e.  (
ZZ>= `  j ) )
47 fzssuz 10832 . . . . . . . . . . . . . 14  |-  ( j ... k )  C_  ( ZZ>= `  j )
48 uzss 10248 . . . . . . . . . . . . . . . . 17  |-  ( j  e.  ( ZZ>= `  M
)  ->  ( ZZ>= `  j )  C_  ( ZZ>=
`  M ) )
4948, 9syl6sseqr 3225 . . . . . . . . . . . . . . . 16  |-  ( j  e.  ( ZZ>= `  M
)  ->  ( ZZ>= `  j )  C_  Z
)
5049, 9eleq2s 2375 . . . . . . . . . . . . . . 15  |-  ( j  e.  Z  ->  ( ZZ>=
`  j )  C_  Z )
5150ad2antrl 708 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( ZZ>= `  j
)  C_  Z )
5247, 51syl5ss 3190 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( j ... k )  C_  Z
)
53 ffvelrn 5663 . . . . . . . . . . . . . . . 16  |-  ( ( F : Z --> RR  /\  n  e.  Z )  ->  ( F `  n
)  e.  RR )
5453ralrimiva 2626 . . . . . . . . . . . . . . 15  |-  ( F : Z --> RR  ->  A. n  e.  Z  ( F `  n )  e.  RR )
551, 54syl 15 . . . . . . . . . . . . . 14  |-  ( ph  ->  A. n  e.  Z  ( F `  n )  e.  RR )
5655ad2antrr 706 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  A. n  e.  Z  ( F `  n )  e.  RR )
57 ssralv 3237 . . . . . . . . . . . . 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 56 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  A. n  e.  ( j ... k ) ( F `  n
)  e.  RR )
5958r19.21bi 2641 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  y  e.  RR+ )  /\  ( j  e.  Z  /\  k  e.  ( ZZ>=
`  j ) ) )  /\  n  e.  ( j ... k
) )  ->  ( F `  n )  e.  RR )
60 fzssuz 10832 . . . . . . . . . . . . . 14  |-  ( j ... ( k  - 
1 ) )  C_  ( ZZ>= `  j )
6160, 51syl5ss 3190 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( j ... ( k  -  1 ) )  C_  Z
)
6261sselda 3180 . . . . . . . . . . . 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 2626 . . . . . . . . . . . . . 14  |-  ( ph  ->  A. k  e.  Z  ( F `  k )  <_  ( F `  ( k  +  1 ) ) )
6564ad2antrr 706 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  A. k  e.  Z  ( F `  k )  <_  ( F `  ( k  +  1 ) ) )
66 fveq2 5525 . . . . . . . . . . . . . . 15  |-  ( k  =  n  ->  ( F `  k )  =  ( F `  n ) )
67 oveq1 5865 . . . . . . . . . . . . . . . 16  |-  ( k  =  n  ->  (
k  +  1 )  =  ( n  + 
1 ) )
6867fveq2d 5529 . . . . . . . . . . . . . . 15  |-  ( k  =  n  ->  ( F `  ( k  +  1 ) )  =  ( F `  ( n  +  1
) ) )
6966, 68breq12d 4036 . . . . . . . . . . . . . 14  |-  ( k  =  n  ->  (
( F `  k
)  <_  ( F `  ( k  +  1 ) )  <->  ( F `  n )  <_  ( F `  ( n  +  1 ) ) ) )
7069rspccva 2883 . . . . . . . . . . . . 13  |-  ( ( A. k  e.  Z  ( F `  k )  <_  ( F `  ( k  +  1 ) )  /\  n  e.  Z )  ->  ( F `  n )  <_  ( F `  (
n  +  1 ) ) )
7165, 70sylan 457 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  y  e.  RR+ )  /\  ( j  e.  Z  /\  k  e.  ( ZZ>=
`  j ) ) )  /\  n  e.  Z )  ->  ( F `  n )  <_  ( F `  (
n  +  1 ) ) )
7262, 71syldan 456 . . . . . . . . . . 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 11076 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( F `  j )  <_  ( F `  k )
)
7440, 44, 45, 73lesub2dd 9389 . . . . . . . . 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 9211 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( sup ( ran  F ,  RR ,  <  )  -  ( F `
 k ) )  e.  RR )
7645, 40resubcld 9211 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( sup ( ran  F ,  RR ,  <  )  -  ( F `
 j ) )  e.  RR )
7727ad2antlr 707 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  y  e.  RR )
78 lelttr 8912 . . . . . . . . . 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 1182 . . . . . . . . 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 656 . . . . . . . 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 9254 . . . . . . . . 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 1182 . . . . . . . 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 706 . . . . . . . . . . 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 451 . . . . . . . . . . . 12  |-  ( (
ph  /\  y  e.  RR+ )  ->  F  Fn  Z )
85 fnfvelrn 5662 . . . . . . . . . . . 12  |-  ( ( F  Fn  Z  /\  k  e.  Z )  ->  ( F `  k
)  e.  ran  F
)
8684, 42, 85syl2an 463 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( F `  k )  e.  ran  F )
87 suprub 9715 . . . . . . . . . . 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 642 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( F `  k )  <_  sup ( ran  F ,  RR ,  <  ) )
8944, 45, 88abssuble0d 11915 . . . . . . . . 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 4033 . . . . . . . 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 259 . . . . . . 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 629 . . . . . 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 2633 . . . . 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 2655 . . . 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 14 . . 3  |-  ( (
ph  /\  y  e.  RR+ )  ->  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( abs `  (
( F `  k
)  -  sup ( ran  F ,  RR ,  <  ) ) )  < 
y )
9695ralrimiva 2626 . 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 5539 . . . . 5  |-  ( ZZ>= `  M )  e.  _V
989, 97eqeltri 2353 . . . 4  |-  Z  e. 
_V
99 fex 5749 . . . 4  |-  ( ( F : Z --> RR  /\  Z  e.  _V )  ->  F  e.  _V )
1001, 98, 99sylancl 643 . . 3  |-  ( ph  ->  F  e.  _V )
101 eqidd 2284 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  =  ( F `  k ) )
10223recnd 8861 . . 3  |-  ( ph  ->  sup ( ran  F ,  RR ,  <  )  e.  CC )
1031, 43sylan 457 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  RR )
104103recnd 8861 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  CC )
1059, 6, 100, 101, 102, 104clim2c 11979 . 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 223 1  |-  ( ph  ->  F  ~~>  sup ( ran  F ,  RR ,  <  )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544   _Vcvv 2788    C_ wss 3152   (/)c0 3455   class class class wbr 4023   ran crn 4690    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   supcsup 7193   RRcr 8736   1c1 8738    + caddc 8740    < clt 8867    <_ cle 8868    - cmin 9037   ZZcz 10024   ZZ>=cuz 10230   RR+crp 10354   ...cfz 10782   abscabs 11719    ~~> cli 11958
This theorem is referenced by:  isumsup2  12305  climcnds  12310  itg1climres  19069  itg2monolem1  19105  itg2i1fseq  19110  itg2i1fseq2  19111  emcllem6  20294  lmdvg  23376  esumpcvgval  23446
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-fz 10783  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962
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