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Theorem climinfOLD 37782
Description: A bounded monotonic non increasing sequence converges to the infimum of its range. (Contributed by Glauco Siliprandi, 29-Jun-2017.) Obsolete version of climinf 37781 as of 15-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.)
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
climinfOLD  |-  ( 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 climinfOLD
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 5747 . . . . . . . . . . . 12  |-  ( F : Z --> RR  ->  ran 
F  C_  RR )
31, 2syl 17 . . . . . . . . . . 11  |-  ( ph  ->  ran  F  C_  RR )
4 ffn 5739 . . . . . . . . . . . . . 14  |-  ( F : Z --> RR  ->  F  Fn  Z )
51, 4syl 17 . . . . . . . . . . . . 13  |-  ( ph  ->  F  Fn  Z )
6 climinf.4 . . . . . . . . . . . . . . 15  |-  ( ph  ->  M  e.  ZZ )
7 uzid 11197 . . . . . . . . . . . . . . 15  |-  ( M  e.  ZZ  ->  M  e.  ( ZZ>= `  M )
)
86, 7syl 17 . . . . . . . . . . . . . 14  |-  ( ph  ->  M  e.  ( ZZ>= `  M ) )
9 climinf.3 . . . . . . . . . . . . . 14  |-  Z  =  ( ZZ>= `  M )
108, 9syl6eleqr 2560 . . . . . . . . . . . . 13  |-  ( ph  ->  M  e.  Z )
11 fnfvelrn 6034 . . . . . . . . . . . . 13  |-  ( ( F  Fn  Z  /\  M  e.  Z )  ->  ( F `  M
)  e.  ran  F
)
125, 10, 11syl2anc 673 . . . . . . . . . . . 12  |-  ( ph  ->  ( F `  M
)  e.  ran  F
)
13 ne0i 3728 . . . . . . . . . . . 12  |-  ( ( F `  M )  e.  ran  F  ->  ran  F  =/=  (/) )
1412, 13syl 17 . . . . . . . . . . 11  |-  ( ph  ->  ran  F  =/=  (/) )
15 climinf.7 . . . . . . . . . . . 12  |-  ( ph  ->  E. x  e.  RR  A. k  e.  Z  x  <_  ( F `  k ) )
16 breq2 4399 . . . . . . . . . . . . . . 15  |-  ( y  =  ( F `  k )  ->  (
x  <_  y  <->  x  <_  ( F `  k ) ) )
1716ralrn 6040 . . . . . . . . . . . . . 14  |-  ( F  Fn  Z  ->  ( A. y  e.  ran  F  x  <_  y  <->  A. k  e.  Z  x  <_  ( F `  k ) ) )
1817rexbidv 2892 . . . . . . . . . . . . 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 17 . . . . . . . . . . . 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 240 . . . . . . . . . . 11  |-  ( ph  ->  E. x  e.  RR  A. y  e.  ran  F  x  <_  y )
213, 14, 203jca 1210 . . . . . . . . . 10  |-  ( ph  ->  ( ran  F  C_  RR  /\  ran  F  =/=  (/)  /\  E. x  e.  RR  A. y  e. 
ran  F  x  <_  y ) )
2221adantr 472 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  RR+ )  ->  ( ran  F 
C_  RR  /\  ran  F  =/=  (/)  /\  E. x  e.  RR  A. y  e. 
ran  F  x  <_  y ) )
23 infmrclOLD 10615 . . . . . . . . 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 17 . . . . . . . 8  |-  ( (
ph  /\  y  e.  RR+ )  ->  sup ( ran  F ,  RR ,  `'  <  )  e.  RR )
25 simpr 468 . . . . . . . 8  |-  ( (
ph  /\  y  e.  RR+ )  ->  y  e.  RR+ )
2624, 25ltaddrpd 11394 . . . . . . 7  |-  ( (
ph  /\  y  e.  RR+ )  ->  sup ( ran  F ,  RR ,  `'  <  )  <  ( sup ( ran  F ,  RR ,  `'  <  )  +  y ) )
27 rpre 11331 . . . . . . . . . 10  |-  ( y  e.  RR+  ->  y  e.  RR )
2827adantl 473 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  RR+ )  ->  y  e.  RR )
2924, 28readdcld 9688 . . . . . . . 8  |-  ( (
ph  /\  y  e.  RR+ )  ->  ( sup ( ran  F ,  RR ,  `'  <  )  +  y )  e.  RR )
30 infrglbOLD 37766 . . . . . . . 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 ) ) )
3122, 29, 30syl2anc 673 . . . . . . 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 ) ) )
3226, 31mpbid 215 . . . . . 6  |-  ( (
ph  /\  y  e.  RR+ )  ->  E. k  e.  ran  F  k  < 
( sup ( ran 
F ,  RR ,  `'  <  )  +  y ) )
333sselda 3418 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  ran  F )  ->  k  e.  RR )
3433adantlr 729 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  k  e.  ran  F )  -> 
k  e.  RR )
3524adantr 472 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  k  e.  ran  F )  ->  sup ( ran  F ,  RR ,  `'  <  )  e.  RR )
3627ad2antlr 741 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  k  e.  ran  F )  -> 
y  e.  RR )
3735, 36readdcld 9688 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  k  e.  ran  F )  -> 
( sup ( ran 
F ,  RR ,  `'  <  )  +  y )  e.  RR )
3834, 37, 36ltsub1d 10243 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  k  e.  ran  F )  -> 
( k  <  ( sup ( ran  F ,  RR ,  `'  <  )  +  y )  <->  ( k  -  y )  < 
( ( sup ( ran  F ,  RR ,  `'  <  )  +  y )  -  y ) ) )
393, 14, 20, 23syl3anc 1292 . . . . . . . . . . . . 13  |-  ( ph  ->  sup ( ran  F ,  RR ,  `'  <  )  e.  RR )
4039recnd 9687 . . . . . . . . . . . 12  |-  ( ph  ->  sup ( ran  F ,  RR ,  `'  <  )  e.  CC )
4140ad2antrr 740 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  k  e.  ran  F )  ->  sup ( ran  F ,  RR ,  `'  <  )  e.  CC )
4236recnd 9687 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  k  e.  ran  F )  -> 
y  e.  CC )
4341, 42pncand 10006 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  k  e.  ran  F )  -> 
( ( sup ( ran  F ,  RR ,  `'  <  )  +  y )  -  y )  =  sup ( ran 
F ,  RR ,  `'  <  ) )
4443breq2d 4407 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  k  e.  ran  F )  -> 
( ( k  -  y )  <  (
( sup ( ran 
F ,  RR ,  `'  <  )  +  y )  -  y )  <-> 
( k  -  y
)  <  sup ( ran  F ,  RR ,  `'  <  ) ) )
4538, 44bitrd 261 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  k  e.  ran  F )  -> 
( k  <  ( sup ( ran  F ,  RR ,  `'  <  )  +  y )  <->  ( k  -  y )  <  sup ( ran  F ,  RR ,  `'  <  ) ) )
4645biimpd 212 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  k  e.  ran  F )  -> 
( k  <  ( sup ( ran  F ,  RR ,  `'  <  )  +  y )  -> 
( k  -  y
)  <  sup ( ran  F ,  RR ,  `'  <  ) ) )
4746reximdva 2858 . . . . . 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 ,  `'  <  ) ) )
4832, 47mpd 15 . . . . 5  |-  ( (
ph  /\  y  e.  RR+ )  ->  E. k  e.  ran  F ( k  -  y )  <  sup ( ran  F ,  RR ,  `'  <  ) )
49 oveq1 6315 . . . . . . . . 9  |-  ( k  =  ( F `  j )  ->  (
k  -  y )  =  ( ( F `
 j )  -  y ) )
5049breq1d 4405 . . . . . . . 8  |-  ( k  =  ( F `  j )  ->  (
( k  -  y
)  <  sup ( ran  F ,  RR ,  `'  <  )  <->  ( ( F `  j )  -  y )  <  sup ( ran  F ,  RR ,  `'  <  ) ) )
5150rexrn 6039 . . . . . . 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 ,  `'  <  ) ) )
525, 51syl 17 . . . . . 6  |-  ( ph  ->  ( E. k  e. 
ran  F ( k  -  y )  <  sup ( ran  F ,  RR ,  `'  <  )  <->  E. j  e.  Z  ( ( F `  j )  -  y
)  <  sup ( ran  F ,  RR ,  `'  <  ) ) )
5352biimpa 492 . . . . 5  |-  ( (
ph  /\  E. k  e.  ran  F ( k  -  y )  <  sup ( ran  F ,  RR ,  `'  <  ) )  ->  E. j  e.  Z  ( ( F `  j )  -  y )  <  sup ( ran  F ,  RR ,  `'  <  ) )
5448, 53syldan 478 . . . 4  |-  ( (
ph  /\  y  e.  RR+ )  ->  E. j  e.  Z  ( ( F `  j )  -  y )  <  sup ( ran  F ,  RR ,  `'  <  ) )
551adantr 472 . . . . . . . . . . 11  |-  ( (
ph  /\  y  e.  RR+ )  ->  F : Z
--> RR )
569uztrn2 11200 . . . . . . . . . . 11  |-  ( ( j  e.  Z  /\  k  e.  ( ZZ>= `  j ) )  -> 
k  e.  Z )
57 ffvelrn 6035 . . . . . . . . . . 11  |-  ( ( F : Z --> RR  /\  k  e.  Z )  ->  ( F `  k
)  e.  RR )
5855, 56, 57syl2an 485 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( F `  k )  e.  RR )
59 simpl 464 . . . . . . . . . . 11  |-  ( ( j  e.  Z  /\  k  e.  ( ZZ>= `  j ) )  -> 
j  e.  Z )
60 ffvelrn 6035 . . . . . . . . . . 11  |-  ( ( F : Z --> RR  /\  j  e.  Z )  ->  ( F `  j
)  e.  RR )
6155, 59, 60syl2an 485 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( F `  j )  e.  RR )
6239ad2antrr 740 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  sup ( ran  F ,  RR ,  `'  <  )  e.  RR )
63 simprr 774 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  k  e.  (
ZZ>= `  j ) )
64 fzssuz 11865 . . . . . . . . . . . . . 14  |-  ( j ... k )  C_  ( ZZ>= `  j )
65 uzss 11203 . . . . . . . . . . . . . . . . 17  |-  ( j  e.  ( ZZ>= `  M
)  ->  ( ZZ>= `  j )  C_  ( ZZ>=
`  M ) )
6665, 9syl6sseqr 3465 . . . . . . . . . . . . . . . 16  |-  ( j  e.  ( ZZ>= `  M
)  ->  ( ZZ>= `  j )  C_  Z
)
6766, 9eleq2s 2567 . . . . . . . . . . . . . . 15  |-  ( j  e.  Z  ->  ( ZZ>=
`  j )  C_  Z )
6867ad2antrl 742 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( ZZ>= `  j
)  C_  Z )
6964, 68syl5ss 3429 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( j ... k )  C_  Z
)
70 ffvelrn 6035 . . . . . . . . . . . . . . . 16  |-  ( ( F : Z --> RR  /\  n  e.  Z )  ->  ( F `  n
)  e.  RR )
7170ralrimiva 2809 . . . . . . . . . . . . . . 15  |-  ( F : Z --> RR  ->  A. n  e.  Z  ( F `  n )  e.  RR )
721, 71syl 17 . . . . . . . . . . . . . 14  |-  ( ph  ->  A. n  e.  Z  ( F `  n )  e.  RR )
7372ad2antrr 740 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  A. n  e.  Z  ( F `  n )  e.  RR )
74 ssralv 3479 . . . . . . . . . . . . 13  |-  ( ( j ... k ) 
C_  Z  ->  ( A. n  e.  Z  ( F `  n )  e.  RR  ->  A. n  e.  ( j ... k
) ( F `  n )  e.  RR ) )
7569, 73, 74sylc 61 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  A. n  e.  ( j ... k ) ( F `  n
)  e.  RR )
7675r19.21bi 2776 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  y  e.  RR+ )  /\  ( j  e.  Z  /\  k  e.  ( ZZ>=
`  j ) ) )  /\  n  e.  ( j ... k
) )  ->  ( F `  n )  e.  RR )
77 fzssuz 11865 . . . . . . . . . . . . . 14  |-  ( j ... ( k  - 
1 ) )  C_  ( ZZ>= `  j )
7877, 68syl5ss 3429 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( j ... ( k  -  1 ) )  C_  Z
)
7978sselda 3418 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  y  e.  RR+ )  /\  ( j  e.  Z  /\  k  e.  ( ZZ>=
`  j ) ) )  /\  n  e.  ( j ... (
k  -  1 ) ) )  ->  n  e.  Z )
80 climinf.6 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  ( k  +  1 ) )  <_  ( F `  k ) )
8180ralrimiva 2809 . . . . . . . . . . . . . 14  |-  ( ph  ->  A. k  e.  Z  ( F `  ( k  +  1 ) )  <_  ( F `  k ) )
8281ad2antrr 740 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  A. k  e.  Z  ( F `  ( k  +  1 ) )  <_  ( F `  k ) )
83 oveq1 6315 . . . . . . . . . . . . . . . 16  |-  ( k  =  n  ->  (
k  +  1 )  =  ( n  + 
1 ) )
8483fveq2d 5883 . . . . . . . . . . . . . . 15  |-  ( k  =  n  ->  ( F `  ( k  +  1 ) )  =  ( F `  ( n  +  1
) ) )
85 fveq2 5879 . . . . . . . . . . . . . . 15  |-  ( k  =  n  ->  ( F `  k )  =  ( F `  n ) )
8684, 85breq12d 4408 . . . . . . . . . . . . . 14  |-  ( k  =  n  ->  (
( F `  (
k  +  1 ) )  <_  ( F `  k )  <->  ( F `  ( n  +  1 ) )  <_  ( F `  n )
) )
8786rspccva 3135 . . . . . . . . . . . . 13  |-  ( ( A. k  e.  Z  ( F `  ( k  +  1 ) )  <_  ( F `  k )  /\  n  e.  Z )  ->  ( F `  ( n  +  1 ) )  <_  ( F `  n ) )
8882, 87sylan 479 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  y  e.  RR+ )  /\  ( j  e.  Z  /\  k  e.  ( ZZ>=
`  j ) ) )  /\  n  e.  Z )  ->  ( F `  ( n  +  1 ) )  <_  ( F `  n ) )
8979, 88syldan 478 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  y  e.  RR+ )  /\  ( j  e.  Z  /\  k  e.  ( ZZ>=
`  j ) ) )  /\  n  e.  ( j ... (
k  -  1 ) ) )  ->  ( F `  ( n  +  1 ) )  <_  ( F `  n ) )
9063, 76, 89monoord2 12282 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( F `  k )  <_  ( F `  j )
)
9158, 61, 62, 90lesub1dd 10250 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( ( F `
 k )  -  sup ( ran  F ,  RR ,  `'  <  ) )  <_  ( ( F `  j )  -  sup ( ran  F ,  RR ,  `'  <  ) ) )
9258, 62resubcld 10068 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( ( F `
 k )  -  sup ( ran  F ,  RR ,  `'  <  ) )  e.  RR )
9361, 62resubcld 10068 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( ( F `
 j )  -  sup ( ran  F ,  RR ,  `'  <  ) )  e.  RR )
9427ad2antlr 741 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  y  e.  RR )
95 lelttr 9742 . . . . . . . . . 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 ) )
9692, 93, 94, 95syl3anc 1292 . . . . . . . . 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 ) )
9791, 96mpand 689 . . . . . . . 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 ) )
98 ltsub23 10115 . . . . . . . . 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 ) )
9961, 94, 62, 98syl3anc 1292 . . . . . . . 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 ) )
1003ad2antrr 740 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ran  F  C_  RR )
1015adantr 472 . . . . . . . . . . . 12  |-  ( (
ph  /\  y  e.  RR+ )  ->  F  Fn  Z )
102 fnfvelrn 6034 . . . . . . . . . . . 12  |-  ( ( F  Fn  Z  /\  k  e.  Z )  ->  ( F `  k
)  e.  ran  F
)
103101, 56, 102syl2an 485 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( F `  k )  e.  ran  F )
104100, 103sseldd 3419 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( F `  k )  e.  RR )
10520ad2antrr 740 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  E. x  e.  RR  A. y  e.  ran  F  x  <_  y )
106 infmrlbOLD 10619 . . . . . . . . . . 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 ) )
107100, 105, 103, 106syl3anc 1292 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  sup ( ran  F ,  RR ,  `'  <  )  <_  ( F `  k ) )
10862, 104, 107abssubge0d 13570 . . . . . . . . 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 ,  `'  <  ) ) )
109108breq1d 4405 . . . . . . . 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 ) )
11097, 99, 1093imtr4d 276 . . . . . . 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 ) )
111110anassrs 660 . . . . . 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 ) )
112111ralrimdva 2812 . . . . 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 ) )
113112reximdva 2858 . . . 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 ) )
11454, 113mpd 15 . . 3  |-  ( (
ph  /\  y  e.  RR+ )  ->  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( abs `  (
( F `  k
)  -  sup ( ran  F ,  RR ,  `'  <  ) ) )  <  y )
115114ralrimiva 2809 . 2  |-  ( ph  ->  A. y  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j ) ( abs `  ( ( F `  k )  -  sup ( ran  F ,  RR ,  `'  <  ) ) )  <  y )
116 fvex 5889 . . . . 5  |-  ( ZZ>= `  M )  e.  _V
1179, 116eqeltri 2545 . . . 4  |-  Z  e. 
_V
118 fex 6155 . . . 4  |-  ( ( F : Z --> RR  /\  Z  e.  _V )  ->  F  e.  _V )
1191, 117, 118sylancl 675 . . 3  |-  ( ph  ->  F  e.  _V )
120 eqidd 2472 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  =  ( F `  k ) )
1211ffvelrnda 6037 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  RR )
122121recnd 9687 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  CC )
1239, 6, 119, 120, 40, 122clim2c 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 ) )
124115, 123mpbird 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   `'ccnv 4838   ran crn 4840    Fn wfn 5584   -->wf 5585   ` cfv 5589  (class class class)co 6308   supcsup 7972   CCcc 9555   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-isom 5598  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:  climinffOLD  37788
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