Theorem climconst 13684

Description: An (eventually) constant sequence converges to its value. (Contributed by NM, 28-Aug-2005.) (Revised by Mario Carneiro, 31-Jan-2014.)

Hypotheses

Ref	Expression
climconst.1	|- Z = ( ZZ>= ` M )
climconst.2	|- ( ph -> M e. ZZ )
climconst.3	|- ( ph -> F e. V )
climconst.4	|- ( ph -> A e. CC )
climconst.5	|- ( ( ph /\ k e. Z ) -> ( F ` k ) = A )

Assertion

Ref	Expression
climconst	|- ( ph -> F ~~> A )

Distinct variable groups:   A, k   k, F   ph, k   k, Z

Allowed substitution hints:   M( k)   V( k)

Proof of Theorem climconst

Dummy variables j x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		climconst.2	. . . . . . 7 |- ( ph -> M e. ZZ )
2		uzid 11197	. . . . . . 7 |- ( M e. ZZ -> M e. ( ZZ>= ` M ) )
3	1, 2	syl 17	. . . . . 6 |- ( ph -> M e. ( ZZ>= ` M ) )
4		climconst.1	. . . . . 6 |- Z = ( ZZ>= ` M )
5	3, 4	syl6eleqr 2560	. . . . 5 |- ( ph -> M e. Z )
6	5	adantr 472	. . . 4 |- ( ( ph /\ x e. RR+ ) -> M e. Z )
7		climconst.4	. . . . . . . . . 10 |- ( ph -> A e. CC )
8	7	subidd 9993	. . . . . . . . 9 |- ( ph -> ( A - A ) = 0 )
9	8	fveq2d 5883	. . . . . . . 8 |- ( ph -> ( abs ` ( A - A ) ) = ( abs ` 0 ) )
10		abs0 13425	. . . . . . . 8 |- ( abs ` 0 ) = 0
11	9, 10	syl6eq 2521	. . . . . . 7 |- ( ph -> ( abs ` ( A - A ) ) = 0 )
12	11	adantr 472	. . . . . 6 |- ( ( ph /\ x e. RR+ ) -> ( abs ` ( A - A ) ) = 0 )
13		rpgt0 11336	. . . . . . 7 |- ( x e. RR+ -> 0 < x )
14	13	adantl 473	. . . . . 6 |- ( ( ph /\ x e. RR+ ) -> 0 < x )
15	12, 14	eqbrtrd 4416	. . . . 5 |- ( ( ph /\ x e. RR+ ) -> ( abs ` ( A - A ) ) < x )
16	15	ralrimivw 2810	. . . 4 |- ( ( ph /\ x e. RR+ ) -> A. k e. Z ( abs ` ( A - A ) ) < x )
17		fveq2 5879	. . . . . . 7 |- ( j = M -> ( ZZ>= ` j ) = ( ZZ>= ` M ) )
18	17, 4	syl6eqr 2523	. . . . . 6 |- ( j = M -> ( ZZ>= ` j ) = Z )
19	18	raleqdv 2979	. . . . 5 |- ( j = M -> ( A. k e. ( ZZ>= ` j ) ( abs ` ( A - A ) ) < x <-> A. k e. Z ( abs ` ( A - A ) ) < x ) )
20	19	rspcev 3136	. . . 4 |- ( ( M e. Z /\ A. k e. Z ( abs ` ( A - A ) ) < x ) -> E. j e. Z A. k e. ( ZZ>= ` j ) ( abs ` ( A - A ) ) < x )
21	6, 16, 20	syl2anc 673	. . 3 |- ( ( ph /\ x e. RR+ ) -> E. j e. Z A. k e. ( ZZ>= ` j ) ( abs ` ( A - A ) ) < x )
22	21	ralrimiva 2809	. 2 |- ( ph -> A. x e. RR+ E. j e. Z A. k e. ( ZZ>= ` j ) ( abs ` ( A - A ) ) < x )
23		climconst.3	. . 3 |- ( ph -> F e. V )
24		climconst.5	. . 3 |- ( ( ph /\ k e. Z ) -> ( F ` k ) = A )
25	7	adantr 472	. . 3 |- ( ( ph /\ k e. Z ) -> A e. CC )
26	4, 1, 23, 24, 7, 25	clim2c 13646	. 2 |- ( ph -> ( F ~~> A <-> A. x e. RR+ E. j e. Z A. k e. ( ZZ>= ` j ) ( abs ` ( A - A ) ) < x ) )
27	22, 26	mpbird 240	1 |- ( ph -> F ~~> A )

Syntax hints:   -> wi 4   /\ wa 376   = wceq 1452   e. wcel 1904   A.wral 2756   E.wrex 2757   class class class wbr 4395   ` cfv 5589  (class class class)co 6308   CCcc 9555   0cc0 9557   < clt 9693   - cmin 9880   ZZcz 10961   ZZ>=cuz 11182   RR+crp 11325   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-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

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-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  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-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-n0 10894  df-z 10962  df-uz 11183  df-rp 11326  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:  climconst2  13689  fsumcvg  13855  expcnv  13999  ntrivcvgfvn0  14032  fprodcvg  14061  fprodntriv  14073  faclim2  30455  clim1fr1  37776  climneg  37786  ioodvbdlimc1lem2  37901  ioodvbdlimc1lem2OLD  37903  ioodvbdlimc2lem  37905  ioodvbdlimc2lemOLD  37906  fourierdlem103  38185  fourierdlem104  38186  etransclem48OLD  38259  etransclem48  38260