Theorem clim2c 12028

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 305	. 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 9919	. . . . . . 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 305	. . . . . . 7 |- ( ( ph /\ k e. Z ) -> ( ( abs ` ( B - A ) ) < x <-> ( B e. CC /\ ( abs ` ( B - A ) ) < x ) ) )
7	4, 6	sylan2 286	. . . . . 6 |- ( ( ph /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) -> ( ( abs ` ( B - A ) ) < x <-> ( B e. CC /\ ( abs ` ( B - A ) ) < x ) ) )
8	7	anassrs 404	. . . . 5 |- ( ( ( ph /\ j e. Z ) /\ k e. ( ZZ>= ` j ) ) -> ( ( abs ` ( B - A ) ) < x <-> ( B e. CC /\ ( abs ` ( B - A ) ) < x ) ) )
9	8	ralbidva 2546	. . . 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 2547	. . 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 2550	. 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 12027	. 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 221	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   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   A.wral 2528   E.wrex 2529   class class class wbr 4125   ` cfv 5372  (class class class)co 6075   CCcc 8167   < clt 8350   - cmin 8487   ZZcz 9623   ZZ>=cuz 9900   RR+crp 10033   abscabs 11741   ~~> cli 12022

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-0id 8277  ax-rnegex 8278  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-inn 9284  df-n0 9543  df-z 9624  df-uz 9901  df-clim 12023

This theorem is referenced by:  clim0c  12030  climconst  12034  2clim  12045  climcn1  12052  climcn2  12053  climsqz  12079  climsqz2  12080  climrecvg1n  12092