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Theorem climsup 13810
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 5747 . . . . . . . . . 10  |-  ( F : Z --> RR  ->  ran 
F  C_  RR )
31, 2syl 17 . . . . . . . . 9  |-  ( ph  ->  ran  F  C_  RR )
4 ffn 5739 . . . . . . . . . . . 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 11197 . . . . . . . . . . . . 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 2560 . . . . . . . . . . 11  |-  ( ph  ->  M  e.  Z )
11 fnfvelrn 6034 . . . . . . . . . . 11  |-  ( ( F  Fn  Z  /\  M  e.  Z )  ->  ( F `  M
)  e.  ran  F
)
125, 10, 11syl2anc 673 . . . . . . . . . 10  |-  ( ph  ->  ( F `  M
)  e.  ran  F
)
13 ne0i 3728 . . . . . . . . . 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 4398 . . . . . . . . . . . . 13  |-  ( y  =  ( F `  k )  ->  (
y  <_  x  <->  ( F `  k )  <_  x
) )
1716ralrn 6040 . . . . . . . . . . . 12  |-  ( F  Fn  Z  ->  ( A. y  e.  ran  F  y  <_  x  <->  A. k  e.  Z  ( F `  k )  <_  x
) )
1817rexbidv 2892 . . . . . . . . . . 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 240 . . . . . . . . 9  |-  ( ph  ->  E. x  e.  RR  A. y  e.  ran  F  y  <_  x )
213, 14, 203jca 1210 . . . . . . . 8  |-  ( ph  ->  ( ran  F  C_  RR  /\  ran  F  =/=  (/)  /\  E. x  e.  RR  A. y  e. 
ran  F  y  <_  x ) )
22 suprcl 10591 . . . . . . . 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 11358 . . . . . . 7  |-  ( ( sup ( ran  F ,  RR ,  <  )  e.  RR  /\  y  e.  RR+ )  ->  ( sup ( ran  F ,  RR ,  <  )  -  y )  <  sup ( ran  F ,  RR ,  <  ) )
2523, 24sylan 479 . . . . . 6  |-  ( (
ph  /\  y  e.  RR+ )  ->  ( sup ( ran  F ,  RR ,  <  )  -  y
)  <  sup ( ran  F ,  RR ,  <  ) )
2621adantr 472 . . . . . . 7  |-  ( (
ph  /\  y  e.  RR+ )  ->  ( ran  F 
C_  RR  /\  ran  F  =/=  (/)  /\  E. x  e.  RR  A. y  e. 
ran  F  y  <_  x ) )
27 rpre 11331 . . . . . . . 8  |-  ( y  e.  RR+  ->  y  e.  RR )
28 resubcl 9958 . . . . . . . 8  |-  ( ( sup ( ran  F ,  RR ,  <  )  e.  RR  /\  y  e.  RR )  ->  ( sup ( ran  F ,  RR ,  <  )  -  y )  e.  RR )
2923, 27, 28syl2an 485 . . . . . . 7  |-  ( (
ph  /\  y  e.  RR+ )  ->  ( sup ( ran  F ,  RR ,  <  )  -  y
)  e.  RR )
30 suprlub 10593 . . . . . . 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 673 . . . . . 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 215 . . . . 5  |-  ( (
ph  /\  y  e.  RR+ )  ->  E. k  e.  ran  F ( sup ( ran  F ,  RR ,  <  )  -  y )  <  k
)
33 breq2 4399 . . . . . . . 8  |-  ( k  =  ( F `  j )  ->  (
( sup ( ran 
F ,  RR ,  <  )  -  y )  <  k  <->  ( sup ( ran  F ,  RR ,  <  )  -  y
)  <  ( F `  j ) ) )
3433rexrn 6039 . . . . . . 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 492 . . . . 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 478 . . . 4  |-  ( (
ph  /\  y  e.  RR+ )  ->  E. j  e.  Z  ( sup ( ran  F ,  RR ,  <  )  -  y
)  <  ( F `  j ) )
38 ffvelrn 6035 . . . . . . . . . . . 12  |-  ( ( F : Z --> RR  /\  j  e.  Z )  ->  ( F `  j
)  e.  RR )
391, 38sylan 479 . . . . . . . . . . 11  |-  ( (
ph  /\  j  e.  Z )  ->  ( F `  j )  e.  RR )
4039ad2ant2r 761 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( F `  j )  e.  RR )
411adantr 472 . . . . . . . . . . 11  |-  ( (
ph  /\  y  e.  RR+ )  ->  F : Z
--> RR )
429uztrn2 11200 . . . . . . . . . . 11  |-  ( ( j  e.  Z  /\  k  e.  ( ZZ>= `  j ) )  -> 
k  e.  Z )
43 ffvelrn 6035 . . . . . . . . . . 11  |-  ( ( F : Z --> RR  /\  k  e.  Z )  ->  ( F `  k
)  e.  RR )
4441, 42, 43syl2an 485 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( F `  k )  e.  RR )
4523ad2antrr 740 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  sup ( ran  F ,  RR ,  <  )  e.  RR )
46 simprr 774 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  k  e.  (
ZZ>= `  j ) )
47 fzssuz 11865 . . . . . . . . . . . . . 14  |-  ( j ... k )  C_  ( ZZ>= `  j )
48 uzss 11203 . . . . . . . . . . . . . . . . 17  |-  ( j  e.  ( ZZ>= `  M
)  ->  ( ZZ>= `  j )  C_  ( ZZ>=
`  M ) )
4948, 9syl6sseqr 3465 . . . . . . . . . . . . . . . 16  |-  ( j  e.  ( ZZ>= `  M
)  ->  ( ZZ>= `  j )  C_  Z
)
5049, 9eleq2s 2567 . . . . . . . . . . . . . . 15  |-  ( j  e.  Z  ->  ( ZZ>=
`  j )  C_  Z )
5150ad2antrl 742 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( ZZ>= `  j
)  C_  Z )
5247, 51syl5ss 3429 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( j ... k )  C_  Z
)
53 ffvelrn 6035 . . . . . . . . . . . . . . . 16  |-  ( ( F : Z --> RR  /\  n  e.  Z )  ->  ( F `  n
)  e.  RR )
5453ralrimiva 2809 . . . . . . . . . . . . . . 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 740 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  A. n  e.  Z  ( F `  n )  e.  RR )
57 ssralv 3479 . . . . . . . . . . . . 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 61 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  A. n  e.  ( j ... k ) ( F `  n
)  e.  RR )
5958r19.21bi 2776 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  y  e.  RR+ )  /\  ( j  e.  Z  /\  k  e.  ( ZZ>=
`  j ) ) )  /\  n  e.  ( j ... k
) )  ->  ( F `  n )  e.  RR )
60 fzssuz 11865 . . . . . . . . . . . . . 14  |-  ( j ... ( k  - 
1 ) )  C_  ( ZZ>= `  j )
6160, 51syl5ss 3429 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( j ... ( k  -  1 ) )  C_  Z
)
6261sselda 3418 . . . . . . . . . . . 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 2809 . . . . . . . . . . . . . 14  |-  ( ph  ->  A. k  e.  Z  ( F `  k )  <_  ( F `  ( k  +  1 ) ) )
6564ad2antrr 740 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  A. k  e.  Z  ( F `  k )  <_  ( F `  ( k  +  1 ) ) )
66 fveq2 5879 . . . . . . . . . . . . . . 15  |-  ( k  =  n  ->  ( F `  k )  =  ( F `  n ) )
67 oveq1 6315 . . . . . . . . . . . . . . . 16  |-  ( k  =  n  ->  (
k  +  1 )  =  ( n  + 
1 ) )
6867fveq2d 5883 . . . . . . . . . . . . . . 15  |-  ( k  =  n  ->  ( F `  ( k  +  1 ) )  =  ( F `  ( n  +  1
) ) )
6966, 68breq12d 4408 . . . . . . . . . . . . . 14  |-  ( k  =  n  ->  (
( F `  k
)  <_  ( F `  ( k  +  1 ) )  <->  ( F `  n )  <_  ( F `  ( n  +  1 ) ) ) )
7069rspccva 3135 . . . . . . . . . . . . 13  |-  ( ( A. k  e.  Z  ( F `  k )  <_  ( F `  ( k  +  1 ) )  /\  n  e.  Z )  ->  ( F `  n )  <_  ( F `  (
n  +  1 ) ) )
7165, 70sylan 479 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  y  e.  RR+ )  /\  ( j  e.  Z  /\  k  e.  ( ZZ>=
`  j ) ) )  /\  n  e.  Z )  ->  ( F `  n )  <_  ( F `  (
n  +  1 ) ) )
7262, 71syldan 478 . . . . . . . . . . 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 12281 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( F `  j )  <_  ( F `  k )
)
7440, 44, 45, 73lesub2dd 10251 . . . . . . . . 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 10068 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( sup ( ran  F ,  RR ,  <  )  -  ( F `
 k ) )  e.  RR )
7645, 40resubcld 10068 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( sup ( ran  F ,  RR ,  <  )  -  ( F `
 j ) )  e.  RR )
7727ad2antlr 741 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  y  e.  RR )
78 lelttr 9742 . . . . . . . . . 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 1292 . . . . . . . . 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 689 . . . . . . . 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 10115 . . . . . . . . 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 1292 . . . . . . . 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 740 . . . . . . . . . . 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 472 . . . . . . . . . . . 12  |-  ( (
ph  /\  y  e.  RR+ )  ->  F  Fn  Z )
85 fnfvelrn 6034 . . . . . . . . . . . 12  |-  ( ( F  Fn  Z  /\  k  e.  Z )  ->  ( F `  k
)  e.  ran  F
)
8684, 42, 85syl2an 485 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( F `  k )  e.  ran  F )
87 suprub 10592 . . . . . . . . . . 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 673 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( F `  k )  <_  sup ( ran  F ,  RR ,  <  ) )
8944, 45, 88abssuble0d 13571 . . . . . . . . 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 4405 . . . . . . . 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 276 . . . . . . 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 660 . . . . . 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 2812 . . . . 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 2858 . . . 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 2809 . 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 5889 . . . . 5  |-  ( ZZ>= `  M )  e.  _V
989, 97eqeltri 2545 . . . 4  |-  Z  e. 
_V
99 fex 6155 . . . 4  |-  ( ( F : Z --> RR  /\  Z  e.  _V )  ->  F  e.  _V )
1001, 98, 99sylancl 675 . . 3  |-  ( ph  ->  F  e.  _V )
101 eqidd 2472 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  =  ( F `  k ) )
10223recnd 9687 . . 3  |-  ( ph  ->  sup ( ran  F ,  RR ,  <  )  e.  CC )
1031, 43sylan 479 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  RR )
104103recnd 9687 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  CC )
1059, 6, 100, 101, 102, 104clim2c 13646 . 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 240 1  |-  ( ph  ->  F  ~~>  sup ( ran  F ,  RR ,  <  )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904    =/= wne 2641   A.wral 2756   E.wrex 2757   _Vcvv 3031    C_ wss 3390   (/)c0 3722   class class class wbr 4395   ran crn 4840    Fn wfn 5584   -->wf 5585   ` cfv 5589  (class class class)co 6308   supcsup 7972   RRcr 9556   1c1 9558    + caddc 9560    < clt 9693    <_ cle 9694    - cmin 9880   ZZcz 10961   ZZ>=cuz 11182   RR+crp 11325   ...cfz 11810   abscabs 13374    ~~> cli 13625
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-sup 7974  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-n0 10894  df-z 10962  df-uz 11183  df-rp 11326  df-fz 11811  df-seq 12252  df-exp 12311  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-clim 13629
This theorem is referenced by:  isumsup2  13981  climcnds  13986  itg1climres  22751  itg2monolem1  22787  itg2i1fseq  22792  itg2i1fseq2  22793  emcllem6  24005  lmdvg  28833  esumpcvgval  28973
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