Theorem clim0 11596

Description: Express the predicate F converges to 0. (Contributed by NM, 24-Feb-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)

Hypotheses

Ref	Expression
clim0.1	|- Z = ( ZZ>= ` M )
clim0.2	|- ( ph -> M e. ZZ )
clim0.3	|- ( ph -> F e. V )
clim0.4	|- ( ( ph /\ k e. Z ) -> ( F ` k ) = B )

Assertion

Ref	Expression
clim0	|- ( ph -> ( F ~~> 0 <-> A. x e. RR+ E. j e. Z A. k e. ( ZZ>= ` j ) ( B e. CC /\ ( abs ` B ) < x ) ) )

Distinct variable groups:   j, k, x, F   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 clim0

Step	Hyp	Ref	Expression
1		clim0.1	. . 3 |- Z = ( ZZ>= ` M )
2		clim0.2	. . 3 |- ( ph -> M e. ZZ )
3		clim0.3	. . 3 |- ( ph -> F e. V )
4		clim0.4	. . 3 |- ( ( ph /\ k e. Z ) -> ( F ` k ) = B )
5	1, 2, 3, 4	clim2 11594	. 2 |- ( ph -> ( F ~~> 0 <-> ( 0 e. CC /\ A. x e. RR+ E. j e. Z A. k e. ( ZZ>= ` j ) ( B e. CC /\ ( abs ` ( B - 0 ) ) < x ) ) ) )
6		0cn 8064	. . . 4 |- 0 e. CC
7	6	biantrur 303	. . 3 |- ( A. x e. RR+ E. j e. Z A. k e. ( ZZ>= ` j ) ( B e. CC /\ ( abs ` ( B - 0 ) ) < x ) <-> ( 0 e. CC /\ A. x e. RR+ E. j e. Z A. k e. ( ZZ>= ` j ) ( B e. CC /\ ( abs ` ( B - 0 ) ) < x ) ) )
8		subid1 8292	. . . . . . . . 9 |- ( B e. CC -> ( B - 0 ) = B )
9	8	fveq2d 5580	. . . . . . . 8 |- ( B e. CC -> ( abs ` ( B - 0 ) ) = ( abs ` B ) )
10	9	breq1d 4054	. . . . . . 7 |- ( B e. CC -> ( ( abs ` ( B - 0 ) ) < x <-> ( abs ` B ) < x ) )
11	10	pm5.32i 454	. . . . . 6 |- ( ( B e. CC /\ ( abs ` ( B - 0 ) ) < x ) <-> ( B e. CC /\ ( abs ` B ) < x ) )
12	11	ralbii 2512	. . . . 5 |- ( A. k e. ( ZZ>= ` j ) ( B e. CC /\ ( abs ` ( B - 0 ) ) < x ) <-> A. k e. ( ZZ>= ` j ) ( B e. CC /\ ( abs ` B ) < x ) )
13	12	rexbii 2513	. . . 4 |- ( E. j e. Z A. k e. ( ZZ>= ` j ) ( B e. CC /\ ( abs ` ( B - 0 ) ) < x ) <-> E. j e. Z A. k e. ( ZZ>= ` j ) ( B e. CC /\ ( abs ` B ) < x ) )
14	13	ralbii 2512	. . 3 |- ( A. x e. RR+ E. j e. Z A. k e. ( ZZ>= ` j ) ( B e. CC /\ ( abs ` ( B - 0 ) ) < x ) <-> A. x e. RR+ E. j e. Z A. k e. ( ZZ>= ` j ) ( B e. CC /\ ( abs ` B ) < x ) )
15	7, 14	bitr3i 186	. 2 |- ( ( 0 e. CC /\ A. x e. RR+ E. j e. Z A. k e. ( ZZ>= ` j ) ( B e. CC /\ ( abs ` ( B - 0 ) ) < x ) ) <-> A. x e. RR+ E. j e. Z A. k e. ( ZZ>= ` j ) ( B e. CC /\ ( abs ` B ) < x ) )
16	5, 15	bitrdi 196	1 |- ( ph -> ( F ~~> 0 <-> A. x e. RR+ E. j e. Z A. k e. ( ZZ>= ` j ) ( B e. CC /\ ( abs ` B ) < x ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   e. wcel 2176   A.wral 2484   E.wrex 2485   class class class wbr 4044   ` cfv 5271  (class class class)co 5944   CCcc 7923   0cc0 7925   < clt 8107   - cmin 8243   ZZcz 9372   ZZ>=cuz 9648   RR+crp 9775   abscabs 11308   ~~> cli 11589

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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-cnex 8016  ax-resscn 8017  ax-1cn 8018  ax-1re 8019  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-addcom 8025  ax-addass 8027  ax-distr 8029  ax-i2m1 8030  ax-0lt1 8031  ax-0id 8033  ax-rnegex 8034  ax-cnre 8036  ax-pre-ltirr 8037  ax-pre-ltwlin 8038  ax-pre-lttrn 8039  ax-pre-apti 8040  ax-pre-ltadd 8041

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-if 3572  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-pnf 8109  df-mnf 8110  df-xr 8111  df-ltxr 8112  df-le 8113  df-sub 8245  df-neg 8246  df-inn 9037  df-n0 9296  df-z 9373  df-uz 9649  df-clim 11590

This theorem is referenced by: (None)