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Theorem climsup 12137
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
Dummy variables  j  n  y are mutually distinct and distinct from all other variables.
Allowed substitution hints:    ph( x)    M( x, k)

Proof of Theorem climsup
StepHypRef Expression
1 climsup.3 . . . . . . . . . 10  |-  ( ph  ->  F : Z --> RR )
2 frn 5360 . . . . . . . . . 10  |-  ( F : Z --> RR  ->  ran 
F  C_  RR )
31, 2syl 17 . . . . . . . . 9  |-  ( ph  ->  ran  F  C_  RR )
4 ffn 5354 . . . . . . . . . . . 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 10237 . . . . . . . . . . . . 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 2375 . . . . . . . . . . 11  |-  ( ph  ->  M  e.  Z )
11 fnfvelrn 5623 . . . . . . . . . . 11  |-  ( ( F  Fn  Z  /\  M  e.  Z )  ->  ( F `  M
)  e.  ran  F
)
125, 10, 11syl2anc 644 . . . . . . . . . 10  |-  ( ph  ->  ( F `  M
)  e.  ran  F
)
13 ne0i 3462 . . . . . . . . . 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 4027 . . . . . . . . . . . . 13  |-  ( y  =  ( F `  k )  ->  (
y  <_  x  <->  ( F `  k )  <_  x
) )
1716ralrn 5629 . . . . . . . . . . . 12  |-  ( F  Fn  Z  ->  ( A. y  e.  ran  F  y  <_  x  <->  A. k  e.  Z  ( F `  k )  <_  x
) )
1817rexbidv 2565 . . . . . . . . . . 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 225 . . . . . . . . 9  |-  ( ph  ->  E. x  e.  RR  A. y  e.  ran  F  y  <_  x )
213, 14, 203jca 1134 . . . . . . . 8  |-  ( ph  ->  ( ran  F  C_  RR  /\  ran  F  =/=  (/)  /\  E. x  e.  RR  A. y  e. 
ran  F  y  <_  x ) )
22 suprcl 9709 . . . . . . . 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 10380 . . . . . . 7  |-  ( ( sup ( ran  F ,  RR ,  <  )  e.  RR  /\  y  e.  RR+ )  ->  ( sup ( ran  F ,  RR ,  <  )  -  y )  <  sup ( ran  F ,  RR ,  <  ) )
2523, 24sylan 459 . . . . . 6  |-  ( (
ph  /\  y  e.  RR+ )  ->  ( sup ( ran  F ,  RR ,  <  )  -  y
)  <  sup ( ran  F ,  RR ,  <  ) )
2621adantr 453 . . . . . . 7  |-  ( (
ph  /\  y  e.  RR+ )  ->  ( ran  F 
C_  RR  /\  ran  F  =/=  (/)  /\  E. x  e.  RR  A. y  e. 
ran  F  y  <_  x ) )
27 rpre 10355 . . . . . . . 8  |-  ( y  e.  RR+  ->  y  e.  RR )
28 resubcl 9106 . . . . . . . 8  |-  ( ( sup ( ran  F ,  RR ,  <  )  e.  RR  /\  y  e.  RR )  ->  ( sup ( ran  F ,  RR ,  <  )  -  y )  e.  RR )
2923, 27, 28syl2an 465 . . . . . . 7  |-  ( (
ph  /\  y  e.  RR+ )  ->  ( sup ( ran  F ,  RR ,  <  )  -  y
)  e.  RR )
30 suprlub 9711 . . . . . . 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 644 . . . . . 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 203 . . . . 5  |-  ( (
ph  /\  y  e.  RR+ )  ->  E. k  e.  ran  F ( sup ( ran  F ,  RR ,  <  )  -  y )  <  k
)
33 breq2 4028 . . . . . . . 8  |-  ( k  =  ( F `  j )  ->  (
( sup ( ran 
F ,  RR ,  <  )  -  y )  <  k  <->  ( sup ( ran  F ,  RR ,  <  )  -  y
)  <  ( F `  j ) ) )
3433rexrn 5628 . . . . . . 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 472 . . . . 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 458 . . . 4  |-  ( (
ph  /\  y  e.  RR+ )  ->  E. j  e.  Z  ( sup ( ran  F ,  RR ,  <  )  -  y
)  <  ( F `  j ) )
38 ffvelrn 5624 . . . . . . . . . . . 12  |-  ( ( F : Z --> RR  /\  j  e.  Z )  ->  ( F `  j
)  e.  RR )
391, 38sylan 459 . . . . . . . . . . 11  |-  ( (
ph  /\  j  e.  Z )  ->  ( F `  j )  e.  RR )
4039ad2ant2r 729 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( F `  j )  e.  RR )
411adantr 453 . . . . . . . . . . 11  |-  ( (
ph  /\  y  e.  RR+ )  ->  F : Z
--> RR )
429uztrn2 10240 . . . . . . . . . . 11  |-  ( ( j  e.  Z  /\  k  e.  ( ZZ>= `  j ) )  -> 
k  e.  Z )
43 ffvelrn 5624 . . . . . . . . . . 11  |-  ( ( F : Z --> RR  /\  k  e.  Z )  ->  ( F `  k
)  e.  RR )
4441, 42, 43syl2an 465 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( F `  k )  e.  RR )
4523ad2antrr 708 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  sup ( ran  F ,  RR ,  <  )  e.  RR )
46 simprr 735 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  k  e.  (
ZZ>= `  j ) )
47 fzssuz 10826 . . . . . . . . . . . . . 14  |-  ( j ... k )  C_  ( ZZ>= `  j )
48 uzss 10243 . . . . . . . . . . . . . . . . 17  |-  ( j  e.  ( ZZ>= `  M
)  ->  ( ZZ>= `  j )  C_  ( ZZ>=
`  M ) )
4948, 9syl6sseqr 3226 . . . . . . . . . . . . . . . 16  |-  ( j  e.  ( ZZ>= `  M
)  ->  ( ZZ>= `  j )  C_  Z
)
5049, 9eleq2s 2376 . . . . . . . . . . . . . . 15  |-  ( j  e.  Z  ->  ( ZZ>=
`  j )  C_  Z )
5150ad2antrl 710 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( ZZ>= `  j
)  C_  Z )
5247, 51syl5ss 3191 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( j ... k )  C_  Z
)
53 ffvelrn 5624 . . . . . . . . . . . . . . . 16  |-  ( ( F : Z --> RR  /\  n  e.  Z )  ->  ( F `  n
)  e.  RR )
5453ralrimiva 2627 . . . . . . . . . . . . . . 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 708 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  A. n  e.  Z  ( F `  n )  e.  RR )
57 ssralv 3238 . . . . . . . . . . . . 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 58 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  A. n  e.  ( j ... k ) ( F `  n
)  e.  RR )
5958r19.21bi 2642 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  y  e.  RR+ )  /\  ( j  e.  Z  /\  k  e.  ( ZZ>=
`  j ) ) )  /\  n  e.  ( j ... k
) )  ->  ( F `  n )  e.  RR )
60 fzssuz 10826 . . . . . . . . . . . . . 14  |-  ( j ... ( k  - 
1 ) )  C_  ( ZZ>= `  j )
6160, 51syl5ss 3191 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( j ... ( k  -  1 ) )  C_  Z
)
6261sselda 3181 . . . . . . . . . . . 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 2627 . . . . . . . . . . . . . 14  |-  ( ph  ->  A. k  e.  Z  ( F `  k )  <_  ( F `  ( k  +  1 ) ) )
6564ad2antrr 708 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  A. k  e.  Z  ( F `  k )  <_  ( F `  ( k  +  1 ) ) )
66 fveq2 5485 . . . . . . . . . . . . . . 15  |-  ( k  =  n  ->  ( F `  k )  =  ( F `  n ) )
67 oveq1 5826 . . . . . . . . . . . . . . . 16  |-  ( k  =  n  ->  (
k  +  1 )  =  ( n  + 
1 ) )
6867fveq2d 5489 . . . . . . . . . . . . . . 15  |-  ( k  =  n  ->  ( F `  ( k  +  1 ) )  =  ( F `  ( n  +  1
) ) )
6966, 68breq12d 4037 . . . . . . . . . . . . . 14  |-  ( k  =  n  ->  (
( F `  k
)  <_  ( F `  ( k  +  1 ) )  <->  ( F `  n )  <_  ( F `  ( n  +  1 ) ) ) )
7069rspccva 2884 . . . . . . . . . . . . 13  |-  ( ( A. k  e.  Z  ( F `  k )  <_  ( F `  ( k  +  1 ) )  /\  n  e.  Z )  ->  ( F `  n )  <_  ( F `  (
n  +  1 ) ) )
7165, 70sylan 459 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  y  e.  RR+ )  /\  ( j  e.  Z  /\  k  e.  ( ZZ>=
`  j ) ) )  /\  n  e.  Z )  ->  ( F `  n )  <_  ( F `  (
n  +  1 ) ) )
7262, 71syldan 458 . . . . . . . . . . 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 11070 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( F `  j )  <_  ( F `  k )
)
7440, 44, 45, 73lesub2dd 9384 . . . . . . . . 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 9206 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( sup ( ran  F ,  RR ,  <  )  -  ( F `
 k ) )  e.  RR )
7645, 40resubcld 9206 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( sup ( ran  F ,  RR ,  <  )  -  ( F `
 j ) )  e.  RR )
7727ad2antlr 709 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  y  e.  RR )
78 lelttr 8907 . . . . . . . . . 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 1184 . . . . . . . . 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 658 . . . . . . . 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 9249 . . . . . . . . 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 1184 . . . . . . . 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 708 . . . . . . . . . . 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 453 . . . . . . . . . . . 12  |-  ( (
ph  /\  y  e.  RR+ )  ->  F  Fn  Z )
85 fnfvelrn 5623 . . . . . . . . . . . 12  |-  ( ( F  Fn  Z  /\  k  e.  Z )  ->  ( F `  k
)  e.  ran  F
)
8684, 42, 85syl2an 465 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( F `  k )  e.  ran  F )
87 suprub 9710 . . . . . . . . . . 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 644 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( F `  k )  <_  sup ( ran  F ,  RR ,  <  ) )
8944, 45, 88abssuble0d 11909 . . . . . . . . 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 4034 . . . . . . . 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 261 . . . . . . 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 631 . . . . . 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 2634 . . . . 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 2656 . . . 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 16 . . 3  |-  ( (
ph  /\  y  e.  RR+ )  ->  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( abs `  (
( F `  k
)  -  sup ( ran  F ,  RR ,  <  ) ) )  < 
y )
9695ralrimiva 2627 . 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 5499 . . . . 5  |-  ( ZZ>= `  M )  e.  _V
989, 97eqeltri 2354 . . . 4  |-  Z  e. 
_V
99 fex 5710 . . . 4  |-  ( ( F : Z --> RR  /\  Z  e.  _V )  ->  F  e.  _V )
1001, 98, 99sylancl 645 . . 3  |-  ( ph  ->  F  e.  _V )
101 eqidd 2285 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  =  ( F `  k ) )
10223recnd 8856 . . 3  |-  ( ph  ->  sup ( ran  F ,  RR ,  <  )  e.  CC )
1031, 43sylan 459 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  RR )
104103recnd 8856 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  CC )
1059, 6, 100, 101, 102, 104clim2c 11973 . 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 225 1  |-  ( ph  ->  F  ~~>  sup ( ran  F ,  RR ,  <  )
)
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    /\ w3a 936    = wceq 1624    e. wcel 1685    =/= wne 2447   A.wral 2544   E.wrex 2545   _Vcvv 2789    C_ wss 3153   (/)c0 3456   class class class wbr 4024   ran crn 4689    Fn wfn 5216   -->wf 5217   ` cfv 5221  (class class class)co 5819   supcsup 7188   RRcr 8731   1c1 8733    + caddc 8735    < clt 8862    <_ cle 8863    - cmin 9032   ZZcz 10019   ZZ>=cuz 10225   RR+crp 10349   ...cfz 10776   abscabs 11713    ~~> cli 11952
This theorem is referenced by:  isumsup2  12299  climcnds  12304  itg1climres  19063  itg2monolem1  19099  itg2i1fseq  19104  itg2i1fseq2  19105  emcllem6  20288
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-cnex 8788  ax-resscn 8789  ax-1cn 8790  ax-icn 8791  ax-addcl 8792  ax-addrcl 8793  ax-mulcl 8794  ax-mulrcl 8795  ax-mulcom 8796  ax-addass 8797  ax-mulass 8798  ax-distr 8799  ax-i2m1 8800  ax-1ne0 8801  ax-1rid 8802  ax-rnegex 8803  ax-rrecex 8804  ax-cnre 8805  ax-pre-lttri 8806  ax-pre-lttrn 8807  ax-pre-ltadd 8808  ax-pre-mulgt0 8809  ax-pre-sup 8810
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-1st 6083  df-2nd 6084  df-iota 6252  df-riota 6299  df-recs 6383  df-rdg 6418  df-er 6655  df-en 6859  df-dom 6860  df-sdom 6861  df-sup 7189  df-pnf 8864  df-mnf 8865  df-xr 8866  df-ltxr 8867  df-le 8868  df-sub 9034  df-neg 9035  df-div 9419  df-nn 9742  df-2 9799  df-3 9800  df-n0 9961  df-z 10020  df-uz 10226  df-rp 10350  df-fz 10777  df-seq 11041  df-exp 11099  df-cj 11578  df-re 11579  df-im 11580  df-sqr 11714  df-abs 11715  df-clim 11956
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