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Theorem clim2c 13624
Description: Express the predicate  F converges to  A. (Contributed by NM, 24-Feb-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)
Hypotheses
Ref Expression
clim2.1  |-  Z  =  ( ZZ>= `  M )
clim2.2  |-  ( ph  ->  M  e.  ZZ )
clim2.3  |-  ( ph  ->  F  e.  V )
clim2.4  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  =  B )
clim2c.5  |-  ( ph  ->  A  e.  CC )
clim2c.6  |-  ( (
ph  /\  k  e.  Z )  ->  B  e.  CC )
Assertion
Ref Expression
clim2c  |-  ( ph  ->  ( F  ~~>  A  <->  A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( abs `  ( B  -  A )
)  <  x )
)
Distinct variable groups:    j, k, x, A    j, F, k, x    j, M    ph, j,
k, x    j, Z, k
Allowed substitution hints:    B( x, j, k)    M( x, k)    V( x, j, k)    Z( x)

Proof of Theorem clim2c
StepHypRef Expression
1 clim2c.5 . . 3  |-  ( ph  ->  A  e.  CC )
21biantrurd 515 . 2  |-  ( ph  ->  ( A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>=
`  j ) ( B  e.  CC  /\  ( abs `  ( B  -  A ) )  <  x )  <->  ( A  e.  CC  /\  A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( B  e.  CC  /\  ( abs `  ( B  -  A )
)  <  x )
) ) )
3 clim2.1 . . . . . . . 8  |-  Z  =  ( ZZ>= `  M )
43uztrn2 11210 . . . . . . 7  |-  ( ( j  e.  Z  /\  k  e.  ( ZZ>= `  j ) )  -> 
k  e.  Z )
5 clim2c.6 . . . . . . . 8  |-  ( (
ph  /\  k  e.  Z )  ->  B  e.  CC )
65biantrurd 515 . . . . . . 7  |-  ( (
ph  /\  k  e.  Z )  ->  (
( abs `  ( B  -  A )
)  <  x  <->  ( B  e.  CC  /\  ( abs `  ( B  -  A
) )  <  x
) ) )
74, 6sylan2 481 . . . . . 6  |-  ( (
ph  /\  ( j  e.  Z  /\  k  e.  ( ZZ>= `  j )
) )  ->  (
( abs `  ( B  -  A )
)  <  x  <->  ( B  e.  CC  /\  ( abs `  ( B  -  A
) )  <  x
) ) )
87anassrs 658 . . . . 5  |-  ( ( ( ph  /\  j  e.  Z )  /\  k  e.  ( ZZ>= `  j )
)  ->  ( ( abs `  ( B  -  A ) )  < 
x  <->  ( B  e.  CC  /\  ( abs `  ( B  -  A
) )  <  x
) ) )
98ralbidva 2836 . . . 4  |-  ( (
ph  /\  j  e.  Z )  ->  ( A. k  e.  ( ZZ>=
`  j ) ( abs `  ( B  -  A ) )  <  x  <->  A. k  e.  ( ZZ>= `  j )
( B  e.  CC  /\  ( abs `  ( B  -  A )
)  <  x )
) )
109rexbidva 2910 . . 3  |-  ( ph  ->  ( E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( abs `  ( B  -  A )
)  <  x  <->  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( B  e.  CC  /\  ( abs `  ( B  -  A )
)  <  x )
) )
1110ralbidv 2839 . 2  |-  ( ph  ->  ( A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>=
`  j ) ( abs `  ( B  -  A ) )  <  x  <->  A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( B  e.  CC  /\  ( abs `  ( B  -  A )
)  <  x )
) )
12 clim2.2 . . 3  |-  ( ph  ->  M  e.  ZZ )
13 clim2.3 . . 3  |-  ( ph  ->  F  e.  V )
14 clim2.4 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  =  B )
153, 12, 13, 14clim2 13623 . 2  |-  ( ph  ->  ( F  ~~>  A  <->  ( A  e.  CC  /\  A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( B  e.  CC  /\  ( abs `  ( B  -  A )
)  <  x )
) ) )
162, 11, 153bitr4rd 294 1  |-  ( ph  ->  ( F  ~~>  A  <->  A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( abs `  ( B  -  A )
)  <  x )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 375    = wceq 1455    e. wcel 1898   A.wral 2749   E.wrex 2750   class class class wbr 4418   ` cfv 5605  (class class class)co 6320   CCcc 9568    < clt 9706    - cmin 9891   ZZcz 10971   ZZ>=cuz 11193   RR+crp 11336   abscabs 13352    ~~> cli 13603
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4541  ax-nul 4550  ax-pow 4598  ax-pr 4656  ax-un 6615  ax-cnex 9626  ax-resscn 9627  ax-pre-lttri 9644  ax-pre-lttrn 9645
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-nel 2636  df-ral 2754  df-rex 2755  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-br 4419  df-opab 4478  df-mpt 4479  df-id 4771  df-po 4777  df-so 4778  df-xp 4862  df-rel 4863  df-cnv 4864  df-co 4865  df-dm 4866  df-rn 4867  df-res 4868  df-ima 4869  df-iota 5569  df-fun 5607  df-fn 5608  df-f 5609  df-f1 5610  df-fo 5611  df-f1o 5612  df-fv 5613  df-ov 6323  df-er 7394  df-en 7601  df-dom 7602  df-sdom 7603  df-pnf 9708  df-mnf 9709  df-xr 9710  df-ltxr 9711  df-le 9712  df-neg 9894  df-z 10972  df-uz 11194  df-clim 13607
This theorem is referenced by:  clim0c  13626  climconst  13662  rlimclim1  13664  2clim  13691  climcn1  13710  climcn2  13711  climsqz  13759  climsqz2  13760  climsup  13788  ulmclm  23398  itgulm  23419  climinf  37770  climinfOLD  37771
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