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

Step	Hyp	Ref	Expression
1		clim2c.5	. . 3 |- ( ph -> A e. CC )
2	1	biantrurd 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 )
4	3	uztrn2 11210	. . . . . . 7 |- ( ( j e. Z /\ k e. ( ZZ>= ` j ) ) -> k e. Z )
5		clim2c.6	. . . . . . . 8 |- ( ( ph /\ k e. Z ) -> B e. CC )
6	5	biantrurd 515	. . . . . . 7 |- ( ( ph /\ k e. Z ) -> ( ( abs ` ( B - A ) ) < x <-> ( B e. CC /\ ( abs ` ( B - A ) ) < x ) ) )
7	4, 6	sylan2 481	. . . . . 6 |- ( ( ph /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) -> ( ( abs ` ( B - A ) ) < x <-> ( B e. CC /\ ( abs ` ( B - A ) ) < x ) ) )
8	7	anassrs 658	. . . . 5 |- ( ( ( ph /\ j e. Z ) /\ k e. ( ZZ>= ` j ) ) -> ( ( abs ` ( B - A ) ) < x <-> ( B e. CC /\ ( abs ` ( B - A ) ) < x ) ) )
9	8	ralbidva 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 ) ) )
10	9	rexbidva 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 ) ) )
11	10	ralbidv 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 )
15	3, 12, 13, 14	clim2 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 ) ) ) )
16	2, 11, 15	3bitr4rd 294	1 |- ( ph -> ( F ~~> A <-> A. x e. RR+ E. j e. Z A. k e. ( ZZ>= ` j ) ( abs ` ( B - A ) ) < x ) )

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