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Theorem climinf 26880
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 5433 . . . . . . . . . . . 12  |-  ( F : Z --> RR  ->  ran 
F  C_  RR )
31, 2syl 15 . . . . . . . . . . 11  |-  ( ph  ->  ran  F  C_  RR )
4 ffn 5427 . . . . . . . . . . . . . 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 10289 . . . . . . . . . . . . . . 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 2407 . . . . . . . . . . . . 13  |-  ( ph  ->  M  e.  Z )
11 fnfvelrn 5700 . . . . . . . . . . . . 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 3495 . . . . . . . . . . . 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 4064 . . . . . . . . . . . . . . 15  |-  ( y  =  ( F `  k )  ->  (
x  <_  y  <->  x  <_  ( F `  k ) ) )
1716ralrn 5706 . . . . . . . . . . . . . 14  |-  ( F  Fn  Z  ->  ( A. y  e.  ran  F  x  <_  y  <->  A. k  e.  Z  x  <_  ( F `  k ) ) )
1817rexbidv 2598 . . . . . . . . . . . . 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 9778 . . . . . . . . 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 10466 . . . . . . 7  |-  ( (
ph  /\  y  e.  RR+ )  ->  sup ( ran  F ,  RR ,  `'  <  )  <  ( sup ( ran  F ,  RR ,  `'  <  )  +  y ) )
27 rpre 10407 . . . . . . . . . . 11  |-  ( y  e.  RR+  ->  y  e.  RR )
2827adantl 452 . . . . . . . . . 10  |-  ( (
ph  /\  y  e.  RR+ )  ->  y  e.  RR )
2924, 28readdcld 8907 . . . . . . . . 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 26870 . . . . . . . 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 1610 . . . . . . 7  |-  F/ k ( ph  /\  y  e.  RR+ )
353sselda 3214 . . . . . . . . . . . . 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 8907 . . . . . . . . . . . 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 9315 . . . . . . . . . . 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 8906 . . . . . . . . . . . . 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 8906 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  k  e.  ran  F )  -> 
y  e.  CC )
4745, 46pncand 9203 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  k  e.  ran  F )  -> 
( ( sup ( ran  F ,  RR ,  `'  <  )  +  y )  -  y )  =  sup ( ran 
F ,  RR ,  `'  <  ) )
4847breq2d 4072 . . . . . . . . . 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 2685 . . . . . 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 5907 . . . . . . . . 9  |-  ( k  =  ( F `  j )  ->  (
k  -  y )  =  ( ( F `
 j )  -  y ) )
5554breq1d 4070 . . . . . . . 8  |-  ( k  =  ( F `  j )  ->  (
( k  -  y
)  <  sup ( ran  F ,  RR ,  `'  <  )  <->  ( ( F `  j )  -  y )  <  sup ( ran  F ,  RR ,  `'  <  ) ) )
5655rexrn 5705 . . . . . . 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 10292 . . . . . . . . . . 11  |-  ( ( j  e.  Z  /\  k  e.  ( ZZ>= `  j ) )  -> 
k  e.  Z )
62 ffvelrn 5701 . . . . . . . . . . 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 5701 . . . . . . . . . . 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 10879 . . . . . . . . . . . . . 14  |-  ( j ... k )  C_  ( ZZ>= `  j )
70 uzss 10295 . . . . . . . . . . . . . . . . 17  |-  ( j  e.  ( ZZ>= `  M
)  ->  ( ZZ>= `  j )  C_  ( ZZ>=
`  M ) )
7170, 9syl6sseqr 3259 . . . . . . . . . . . . . . . 16  |-  ( j  e.  ( ZZ>= `  M
)  ->  ( ZZ>= `  j )  C_  Z
)
7271, 9eleq2s 2408 . . . . . . . . . . . . . . 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 3224 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( j ... k )  C_  Z
)
75 ffvelrn 5701 . . . . . . . . . . . . . . . 16  |-  ( ( F : Z --> RR  /\  n  e.  Z )  ->  ( F `  n
)  e.  RR )
7675ralrimiva 2660 . . . . . . . . . . . . . . 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 3271 . . . . . . . . . . . . 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 2675 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  y  e.  RR+ )  /\  ( j  e.  Z  /\  k  e.  ( ZZ>=
`  j ) ) )  /\  n  e.  ( j ... k
) )  ->  ( F `  n )  e.  RR )
82 fzssuz 10879 . . . . . . . . . . . . . 14  |-  ( j ... ( k  - 
1 ) )  C_  ( ZZ>= `  j )
8382, 73syl5ss 3224 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( j ... ( k  -  1 ) )  C_  Z
)
8483sselda 3214 . . . . . . . . . . . 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 2660 . . . . . . . . . . . . . 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 5907 . . . . . . . . . . . . . . . 16  |-  ( k  =  n  ->  (
k  +  1 )  =  ( n  + 
1 ) )
8988fveq2d 5567 . . . . . . . . . . . . . . 15  |-  ( k  =  n  ->  ( F `  ( k  +  1 ) )  =  ( F `  ( n  +  1
) ) )
90 fveq2 5563 . . . . . . . . . . . . . . 15  |-  ( k  =  n  ->  ( F `  k )  =  ( F `  n ) )
9189, 90breq12d 4073 . . . . . . . . . . . . . 14  |-  ( k  =  n  ->  (
( F `  (
k  +  1 ) )  <_  ( F `  k )  <->  ( F `  ( n  +  1 ) )  <_  ( F `  n )
) )
9291rspccva 2917 . . . . . . . . . . . . 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 11124 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( F `  k )  <_  ( F `  j )
)
9663, 66, 67, 95lesub1dd 9433 . . . . . . . . 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 9256 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( ( F `
 k )  -  sup ( ran  F ,  RR ,  `'  <  ) )  e.  RR )
9866, 67resubcld 9256 . . . . . . . . . 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 8957 . . . . . . . . . 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 9299 . . . . . . . . 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 5700 . . . . . . . . . . . 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 3215 . . . . . . . . . 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 9780 . . . . . . . . . . 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 11961 . . . . . . . . 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 4070 . . . . . . . 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 2667 . . . . 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 2689 . . . 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 2660 . 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 5577 . . . . 5  |-  ( ZZ>= `  M )  e.  _V
1239, 122eqeltri 2386 . . . 4  |-  Z  e. 
_V
124 fex 5790 . . . 4  |-  ( ( F : Z --> RR  /\  Z  e.  _V )  ->  F  e.  _V )
1251, 123, 124sylancl 643 . . 3  |-  ( ph  ->  F  e.  _V )
126 eqidd 2317 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  =  ( F `  k ) )
1271ffvelrnda 5703 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  RR )
128127recnd 8906 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  CC )
1299, 6, 125, 126, 44, 128clim2c 12026 . 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 1633    e. wcel 1701    =/= wne 2479   A.wral 2577   E.wrex 2578   _Vcvv 2822    C_ wss 3186   (/)c0 3489   class class class wbr 4060   `'ccnv 4725   ran crn 4727    Fn wfn 5287   -->wf 5288   ` cfv 5292  (class class class)co 5900   supcsup 7238   CCcc 8780   RRcr 8781   1c1 8783    + caddc 8785    < clt 8912    <_ cle 8913    - cmin 9082   ZZcz 10071   ZZ>=cuz 10277   RR+crp 10401   ...cfz 10829   abscabs 11766    ~~> cli 12005
This theorem is referenced by:  climinff  26885
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-cnex 8838  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-mulcom 8846  ax-addass 8847  ax-mulass 8848  ax-distr 8849  ax-i2m1 8850  ax-1ne0 8851  ax-1rid 8852  ax-rnegex 8853  ax-rrecex 8854  ax-cnre 8855  ax-pre-lttri 8856  ax-pre-lttrn 8857  ax-pre-ltadd 8858  ax-pre-mulgt0 8859  ax-pre-sup 8860
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 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-isom 5301  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-riota 6346  df-recs 6430  df-rdg 6465  df-er 6702  df-en 6907  df-dom 6908  df-sdom 6909  df-sup 7239  df-pnf 8914  df-mnf 8915  df-xr 8916  df-ltxr 8917  df-le 8918  df-sub 9084  df-neg 9085  df-div 9469  df-nn 9792  df-2 9849  df-3 9850  df-n0 10013  df-z 10072  df-uz 10278  df-rp 10402  df-fz 10830  df-seq 11094  df-exp 11152  df-cj 11631  df-re 11632  df-im 11633  df-sqr 11767  df-abs 11768  df-clim 12009
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