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Theorem climinf 27732
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 5395 . . . . . . . . . . . 12  |-  ( F : Z --> RR  ->  ran 
F  C_  RR )
31, 2syl 15 . . . . . . . . . . 11  |-  ( ph  ->  ran  F  C_  RR )
4 ffn 5389 . . . . . . . . . . . . . 14  |-  ( F : Z --> RR  ->  F  Fn  Z )
51, 4syl 15 . . . . . . . . . . . . 13  |-  ( ph  ->  F  Fn  Z )
6 climinf.4 . . . . . . . . . . . . . . 15  |-  ( ph  ->  M  e.  ZZ )
7 uzid 10242 . . . . . . . . . . . . . . 15  |-  ( M  e.  ZZ  ->  M  e.  ( ZZ>= `  M )
)
86, 7syl 15 . . . . . . . . . . . . . 14  |-  ( ph  ->  M  e.  ( ZZ>= `  M ) )
9 climinf.3 . . . . . . . . . . . . . 14  |-  Z  =  ( ZZ>= `  M )
108, 9syl6eleqr 2374 . . . . . . . . . . . . 13  |-  ( ph  ->  M  e.  Z )
11 fnfvelrn 5662 . . . . . . . . . . . . 13  |-  ( ( F  Fn  Z  /\  M  e.  Z )  ->  ( F `  M
)  e.  ran  F
)
125, 10, 11syl2anc 642 . . . . . . . . . . . 12  |-  ( ph  ->  ( F `  M
)  e.  ran  F
)
13 ne0i 3461 . . . . . . . . . . . 12  |-  ( ( F `  M )  e.  ran  F  ->  ran  F  =/=  (/) )
1412, 13syl 15 . . . . . . . . . . 11  |-  ( ph  ->  ran  F  =/=  (/) )
15 climinf.7 . . . . . . . . . . . 12  |-  ( ph  ->  E. x  e.  RR  A. k  e.  Z  x  <_  ( F `  k ) )
16 breq2 4027 . . . . . . . . . . . . . . 15  |-  ( y  =  ( F `  k )  ->  (
x  <_  y  <->  x  <_  ( F `  k ) ) )
1716ralrn 5668 . . . . . . . . . . . . . 14  |-  ( F  Fn  Z  ->  ( A. y  e.  ran  F  x  <_  y  <->  A. k  e.  Z  x  <_  ( F `  k ) ) )
1817rexbidv 2564 . . . . . . . . . . . . 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 15 . . . . . . . . . . . 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 223 . . . . . . . . . . 11  |-  ( ph  ->  E. x  e.  RR  A. y  e.  ran  F  x  <_  y )
213, 14, 203jca 1132 . . . . . . . . . 10  |-  ( ph  ->  ( ran  F  C_  RR  /\  ran  F  =/=  (/)  /\  E. x  e.  RR  A. y  e. 
ran  F  x  <_  y ) )
2221adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  RR+ )  ->  ( ran  F 
C_  RR  /\  ran  F  =/=  (/)  /\  E. x  e.  RR  A. y  e. 
ran  F  x  <_  y ) )
23 infmrcl 9733 . . . . . . . . 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 15 . . . . . . . 8  |-  ( (
ph  /\  y  e.  RR+ )  ->  sup ( ran  F ,  RR ,  `'  <  )  e.  RR )
25 simpr 447 . . . . . . . 8  |-  ( (
ph  /\  y  e.  RR+ )  ->  y  e.  RR+ )
2624, 25ltaddrpd 10419 . . . . . . 7  |-  ( (
ph  /\  y  e.  RR+ )  ->  sup ( ran  F ,  RR ,  `'  <  )  <  ( sup ( ran  F ,  RR ,  `'  <  )  +  y ) )
27 rpre 10360 . . . . . . . . . . 11  |-  ( y  e.  RR+  ->  y  e.  RR )
2827adantl 452 . . . . . . . . . 10  |-  ( (
ph  /\  y  e.  RR+ )  ->  y  e.  RR )
2924, 28readdcld 8862 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  RR+ )  ->  ( sup ( ran  F ,  RR ,  `'  <  )  +  y )  e.  RR )
3022, 29jca 518 . . . . . . . 8  |-  ( (
ph  /\  y  e.  RR+ )  ->  ( ( ran  F  C_  RR  /\  ran  F  =/=  (/)  /\  E. x  e.  RR  A. y  e. 
ran  F  x  <_  y )  /\  ( sup ( ran  F ,  RR ,  `'  <  )  +  y )  e.  RR ) )
31 infrglb 27722 . . . . . . . 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 ) ) )
3230, 31syl 15 . . . . . . 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 ) ) )
3326, 32mpbid 201 . . . . . 6  |-  ( (
ph  /\  y  e.  RR+ )  ->  E. k  e.  ran  F  k  < 
( sup ( ran 
F ,  RR ,  `'  <  )  +  y ) )
34 nfv 1605 . . . . . . 7  |-  F/ k ( ph  /\  y  e.  RR+ )
353sselda 3180 . . . . . . . . . . . . 13  |-  ( (
ph  /\  k  e.  ran  F )  ->  k  e.  RR )
3635adantlr 695 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  k  e.  ran  F )  -> 
k  e.  RR )
3724adantr 451 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  k  e.  ran  F )  ->  sup ( ran  F ,  RR ,  `'  <  )  e.  RR )
3828adantr 451 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  k  e.  ran  F )  -> 
y  e.  RR )
3937, 38readdcld 8862 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  k  e.  ran  F )  -> 
( sup ( ran 
F ,  RR ,  `'  <  )  +  y )  e.  RR )
4036, 39, 383jca 1132 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  k  e.  ran  F )  -> 
( k  e.  RR  /\  ( sup ( ran 
F ,  RR ,  `'  <  )  +  y )  e.  RR  /\  y  e.  RR )
)
41 ltsub1 9270 . . . . . . . . . . 11  |-  ( ( k  e.  RR  /\  ( sup ( ran  F ,  RR ,  `'  <  )  +  y )  e.  RR  /\  y  e.  RR )  ->  (
k  <  ( sup ( ran  F ,  RR ,  `'  <  )  +  y )  <->  ( k  -  y )  < 
( ( sup ( ran  F ,  RR ,  `'  <  )  +  y )  -  y ) ) )
4240, 41syl 15 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  k  e.  ran  F )  -> 
( k  <  ( sup ( ran  F ,  RR ,  `'  <  )  +  y )  <->  ( k  -  y )  < 
( ( sup ( ran  F ,  RR ,  `'  <  )  +  y )  -  y ) ) )
4321, 23syl 15 . . . . . . . . . . . . . 14  |-  ( ph  ->  sup ( ran  F ,  RR ,  `'  <  )  e.  RR )
4443recnd 8861 . . . . . . . . . . . . 13  |-  ( ph  ->  sup ( ran  F ,  RR ,  `'  <  )  e.  CC )
4544ad2antrr 706 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  k  e.  ran  F )  ->  sup ( ran  F ,  RR ,  `'  <  )  e.  CC )
4638recnd 8861 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  k  e.  ran  F )  -> 
y  e.  CC )
4745, 46pncand 9158 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  k  e.  ran  F )  -> 
( ( sup ( ran  F ,  RR ,  `'  <  )  +  y )  -  y )  =  sup ( ran 
F ,  RR ,  `'  <  ) )
4847breq2d 4035 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  k  e.  ran  F )  -> 
( ( k  -  y )  <  (
( sup ( ran 
F ,  RR ,  `'  <  )  +  y )  -  y )  <-> 
( k  -  y
)  <  sup ( ran  F ,  RR ,  `'  <  ) ) )
4942, 48bitrd 244 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  k  e.  ran  F )  -> 
( k  <  ( sup ( ran  F ,  RR ,  `'  <  )  +  y )  <->  ( k  -  y )  <  sup ( ran  F ,  RR ,  `'  <  ) ) )
5049biimpd 198 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  k  e.  ran  F )  -> 
( k  <  ( sup ( ran  F ,  RR ,  `'  <  )  +  y )  -> 
( k  -  y
)  <  sup ( ran  F ,  RR ,  `'  <  ) ) )
5150ex 423 . . . . . . 7  |-  ( (
ph  /\  y  e.  RR+ )  ->  ( k  e.  ran  F  ->  (
k  <  ( sup ( ran  F ,  RR ,  `'  <  )  +  y )  ->  (
k  -  y )  <  sup ( ran  F ,  RR ,  `'  <  ) ) ) )
5234, 51reximdai 2651 . . . . . 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 ,  `'  <  ) ) )
5333, 52mpd 14 . . . . 5  |-  ( (
ph  /\  y  e.  RR+ )  ->  E. k  e.  ran  F ( k  -  y )  <  sup ( ran  F ,  RR ,  `'  <  ) )
54 oveq1 5865 . . . . . . . . 9  |-  ( k  =  ( F `  j )  ->  (
k  -  y )  =  ( ( F `
 j )  -  y ) )
5554breq1d 4033 . . . . . . . 8  |-  ( k  =  ( F `  j )  ->  (
( k  -  y
)  <  sup ( ran  F ,  RR ,  `'  <  )  <->  ( ( F `  j )  -  y )  <  sup ( ran  F ,  RR ,  `'  <  ) ) )
5655rexrn 5667 . . . . . . 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 ,  `'  <  ) ) )
575, 56syl 15 . . . . . 6  |-  ( ph  ->  ( E. k  e. 
ran  F ( k  -  y )  <  sup ( ran  F ,  RR ,  `'  <  )  <->  E. j  e.  Z  ( ( F `  j )  -  y
)  <  sup ( ran  F ,  RR ,  `'  <  ) ) )
5857biimpa 470 . . . . 5  |-  ( (
ph  /\  E. k  e.  ran  F ( k  -  y )  <  sup ( ran  F ,  RR ,  `'  <  ) )  ->  E. j  e.  Z  ( ( F `  j )  -  y )  <  sup ( ran  F ,  RR ,  `'  <  ) )
5953, 58syldan 456 . . . 4  |-  ( (
ph  /\  y  e.  RR+ )  ->  E. j  e.  Z  ( ( F `  j )  -  y )  <  sup ( ran  F ,  RR ,  `'  <  ) )
601adantr 451 . . . . . . . . . . 11  |-  ( (
ph  /\  y  e.  RR+ )  ->  F : Z
--> RR )
619uztrn2 10245 . . . . . . . . . . 11  |-  ( ( j  e.  Z  /\  k  e.  ( ZZ>= `  j ) )  -> 
k  e.  Z )
62 ffvelrn 5663 . . . . . . . . . . 11  |-  ( ( F : Z --> RR  /\  k  e.  Z )  ->  ( F `  k
)  e.  RR )
6360, 61, 62syl2an 463 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( F `  k )  e.  RR )
64 simpl 443 . . . . . . . . . . 11  |-  ( ( j  e.  Z  /\  k  e.  ( ZZ>= `  j ) )  -> 
j  e.  Z )
65 ffvelrn 5663 . . . . . . . . . . 11  |-  ( ( F : Z --> RR  /\  j  e.  Z )  ->  ( F `  j
)  e.  RR )
6660, 64, 65syl2an 463 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( F `  j )  e.  RR )
6743ad2antrr 706 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  sup ( ran  F ,  RR ,  `'  <  )  e.  RR )
68 simprr 733 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  k  e.  (
ZZ>= `  j ) )
69 fzssuz 10832 . . . . . . . . . . . . . 14  |-  ( j ... k )  C_  ( ZZ>= `  j )
70 uzss 10248 . . . . . . . . . . . . . . . . 17  |-  ( j  e.  ( ZZ>= `  M
)  ->  ( ZZ>= `  j )  C_  ( ZZ>=
`  M ) )
7170, 9syl6sseqr 3225 . . . . . . . . . . . . . . . 16  |-  ( j  e.  ( ZZ>= `  M
)  ->  ( ZZ>= `  j )  C_  Z
)
7271, 9eleq2s 2375 . . . . . . . . . . . . . . 15  |-  ( j  e.  Z  ->  ( ZZ>=
`  j )  C_  Z )
7372ad2antrl 708 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( ZZ>= `  j
)  C_  Z )
7469, 73syl5ss 3190 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( j ... k )  C_  Z
)
75 ffvelrn 5663 . . . . . . . . . . . . . . . 16  |-  ( ( F : Z --> RR  /\  n  e.  Z )  ->  ( F `  n
)  e.  RR )
7675ralrimiva 2626 . . . . . . . . . . . . . . 15  |-  ( F : Z --> RR  ->  A. n  e.  Z  ( F `  n )  e.  RR )
771, 76syl 15 . . . . . . . . . . . . . 14  |-  ( ph  ->  A. n  e.  Z  ( F `  n )  e.  RR )
7877ad2antrr 706 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  A. n  e.  Z  ( F `  n )  e.  RR )
79 ssralv 3237 . . . . . . . . . . . . 13  |-  ( ( j ... k ) 
C_  Z  ->  ( A. n  e.  Z  ( F `  n )  e.  RR  ->  A. n  e.  ( j ... k
) ( F `  n )  e.  RR ) )
8074, 78, 79sylc 56 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  A. n  e.  ( j ... k ) ( F `  n
)  e.  RR )
8180r19.21bi 2641 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  y  e.  RR+ )  /\  ( j  e.  Z  /\  k  e.  ( ZZ>=
`  j ) ) )  /\  n  e.  ( j ... k
) )  ->  ( F `  n )  e.  RR )
82 fzssuz 10832 . . . . . . . . . . . . . 14  |-  ( j ... ( k  - 
1 ) )  C_  ( ZZ>= `  j )
8382, 73syl5ss 3190 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( j ... ( k  -  1 ) )  C_  Z
)
8483sselda 3180 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  y  e.  RR+ )  /\  ( j  e.  Z  /\  k  e.  ( ZZ>=
`  j ) ) )  /\  n  e.  ( j ... (
k  -  1 ) ) )  ->  n  e.  Z )
85 climinf.6 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  ( k  +  1 ) )  <_  ( F `  k ) )
8685ralrimiva 2626 . . . . . . . . . . . . . 14  |-  ( ph  ->  A. k  e.  Z  ( F `  ( k  +  1 ) )  <_  ( F `  k ) )
8786ad2antrr 706 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  A. k  e.  Z  ( F `  ( k  +  1 ) )  <_  ( F `  k ) )
88 oveq1 5865 . . . . . . . . . . . . . . . 16  |-  ( k  =  n  ->  (
k  +  1 )  =  ( n  + 
1 ) )
8988fveq2d 5529 . . . . . . . . . . . . . . 15  |-  ( k  =  n  ->  ( F `  ( k  +  1 ) )  =  ( F `  ( n  +  1
) ) )
90 fveq2 5525 . . . . . . . . . . . . . . 15  |-  ( k  =  n  ->  ( F `  k )  =  ( F `  n ) )
9189, 90breq12d 4036 . . . . . . . . . . . . . 14  |-  ( k  =  n  ->  (
( F `  (
k  +  1 ) )  <_  ( F `  k )  <->  ( F `  ( n  +  1 ) )  <_  ( F `  n )
) )
9291rspccva 2883 . . . . . . . . . . . . 13  |-  ( ( A. k  e.  Z  ( F `  ( k  +  1 ) )  <_  ( F `  k )  /\  n  e.  Z )  ->  ( F `  ( n  +  1 ) )  <_  ( F `  n ) )
9387, 92sylan 457 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  y  e.  RR+ )  /\  ( j  e.  Z  /\  k  e.  ( ZZ>=
`  j ) ) )  /\  n  e.  Z )  ->  ( F `  ( n  +  1 ) )  <_  ( F `  n ) )
9484, 93syldan 456 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  y  e.  RR+ )  /\  ( j  e.  Z  /\  k  e.  ( ZZ>=
`  j ) ) )  /\  n  e.  ( j ... (
k  -  1 ) ) )  ->  ( F `  ( n  +  1 ) )  <_  ( F `  n ) )
9568, 81, 94monoord2 11077 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( F `  k )  <_  ( F `  j )
)
9663, 66, 67, 95lesub1dd 9388 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( ( F `
 k )  -  sup ( ran  F ,  RR ,  `'  <  ) )  <_  ( ( F `  j )  -  sup ( ran  F ,  RR ,  `'  <  ) ) )
9763, 67resubcld 9211 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( ( F `
 k )  -  sup ( ran  F ,  RR ,  `'  <  ) )  e.  RR )
9866, 67resubcld 9211 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( ( F `
 j )  -  sup ( ran  F ,  RR ,  `'  <  ) )  e.  RR )
9927ad2antlr 707 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  y  e.  RR )
100 lelttr 8912 . . . . . . . . . 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 ) )
10197, 98, 99, 100syl3anc 1182 . . . . . . . . 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 ) )
10296, 101mpand 656 . . . . . . . 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 ) )
103 ltsub23 9254 . . . . . . . . 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 ) )
10466, 99, 67, 103syl3anc 1182 . . . . . . . 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 ) )
1053ad2antrr 706 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ran  F  C_  RR )
1065adantr 451 . . . . . . . . . . . 12  |-  ( (
ph  /\  y  e.  RR+ )  ->  F  Fn  Z )
107 fnfvelrn 5662 . . . . . . . . . . . 12  |-  ( ( F  Fn  Z  /\  k  e.  Z )  ->  ( F `  k
)  e.  ran  F
)
108106, 61, 107syl2an 463 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( F `  k )  e.  ran  F )
109105, 108sseldd 3181 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( F `  k )  e.  RR )
11020ad2antrr 706 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  E. x  e.  RR  A. y  e.  ran  F  x  <_  y )
111105, 110, 1083jca 1132 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( ran  F  C_  RR  /\  E. x  e.  RR  A. y  e. 
ran  F  x  <_  y  /\  ( F `  k )  e.  ran  F ) )
112 infmrlb 9735 . . . . . . . . . . 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 ) )
113111, 112syl 15 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  sup ( ran  F ,  RR ,  `'  <  )  <_  ( F `  k ) )
11467, 109, 113abssubge0d 11914 . . . . . . . . 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 ,  `'  <  ) ) )
115114breq1d 4033 . . . . . . . 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 ) )
116102, 104, 1153imtr4d 259 . . . . . . 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 ) )
117116anassrs 629 . . . . . 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 ) )
118117ralrimdva 2633 . . . . 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 ) )
119118reximdva 2655 . . . 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 ) )
12059, 119mpd 14 . . 3  |-  ( (
ph  /\  y  e.  RR+ )  ->  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( abs `  (
( F `  k
)  -  sup ( ran  F ,  RR ,  `'  <  ) ) )  <  y )
121120ralrimiva 2626 . 2  |-  ( ph  ->  A. y  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j ) ( abs `  ( ( F `  k )  -  sup ( ran  F ,  RR ,  `'  <  ) ) )  <  y )
122 fvex 5539 . . . . 5  |-  ( ZZ>= `  M )  e.  _V
1239, 122eqeltri 2353 . . . 4  |-  Z  e. 
_V
124 fex 5749 . . . 4  |-  ( ( F : Z --> RR  /\  Z  e.  _V )  ->  F  e.  _V )
1251, 123, 124sylancl 643 . . 3  |-  ( ph  ->  F  e.  _V )
126 eqidd 2284 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  =  ( F `  k ) )
1271ffvelrnda 5665 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  RR )
128127recnd 8861 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  CC )
1299, 6, 125, 126, 44, 128clim2c 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 ) )
130121, 129mpbird 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   `'ccnv 4688   ran crn 4690    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   supcsup 7193   CCcc 8735   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:  climinff  27737
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-isom 5264  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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