Theorem rlimclim 14277

Description: A sequence on an upper integer set converges in the real sense iff it converges in the integer sense. (Contributed by Mario Carneiro, 16-Sep-2014.)

Hypotheses

Ref	Expression
rlimclim.1	|- Z = ( ZZ>= ` M )
rlimclim.2	|- ( ph -> M e. ZZ )
rlimclim.3	|- ( ph -> F : Z --> CC )

Assertion

Ref	Expression
rlimclim	|- ( ph -> ( F ~~> r A <-> F ~~> A ) )

Proof of Theorem rlimclim

Dummy variables w k y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rlimclim.1	. . 3 |- Z = ( ZZ>= ` M )
2		rlimclim.2	. . . 4 |- ( ph -> M e. ZZ )
3	2	adantr 481	. . 3 |- ( ( ph /\ F ~~> r A ) -> M e. ZZ )
4		simpr 477	. . 3 |- ( ( ph /\ F ~~> r A ) -> F ~~> r A )
5		rlimclim.3	. . . . 5 |- ( ph -> F : Z --> CC )
6		fdm 6051	. . . . 5 |- ( F : Z --> CC -> dom F = Z )
7		eqimss2 3658	. . . . 5 |- ( dom F = Z -> Z C_ dom F )
8	5, 6, 7	3syl 18	. . . 4 |- ( ph -> Z C_ dom F )
9	8	adantr 481	. . 3 |- ( ( ph /\ F ~~> r A ) -> Z C_ dom F )
10	1, 3, 4, 9	rlimclim1 14276	. 2 |- ( ( ph /\ F ~~> r A ) -> F ~~> A )
11		climcl 14230	. . . 4 |- ( F ~~> A -> A e. CC )
12	11	adantl 482	. . 3 |- ( ( ph /\ F ~~> A ) -> A e. CC )
13	2	ad2antrr 762	. . . . . 6 |- ( ( ( ph /\ F ~~> A ) /\ y e. RR+ ) -> M e. ZZ )
14		simpr 477	. . . . . 6 |- ( ( ( ph /\ F ~~> A ) /\ y e. RR+ ) -> y e. RR+ )
15		eqidd 2623	. . . . . 6 |- ( ( ( ( ph /\ F ~~> A ) /\ y e. RR+ ) /\ k e. Z ) -> ( F ` k ) = ( F ` k ) )
16		simplr 792	. . . . . 6 |- ( ( ( ph /\ F ~~> A ) /\ y e. RR+ ) -> F ~~> A )
17	1, 13, 14, 15, 16	climi2 14242	. . . . 5 |- ( ( ( ph /\ F ~~> A ) /\ y e. RR+ ) -> E. z e. Z A. k e. ( ZZ>= ` z ) ( abs ` ( ( F ` k ) - A ) ) < y )
18		uzssz 11707	. . . . . . . . . . . . . 14 |- ( ZZ>= ` M ) C_ ZZ
19	1, 18	eqsstri 3635	. . . . . . . . . . . . 13 |- Z C_ ZZ
20		simplrl 800	. . . . . . . . . . . . 13 |- ( ( ( ( ( ph /\ F ~~> A ) /\ y e. RR+ ) /\ ( z e. Z /\ A. k e. ( ZZ>= ` z ) ( abs ` ( ( F ` k ) - A ) ) < y ) ) /\ ( w e. Z /\ z <_ w ) ) -> z e. Z )
21	19, 20	sseldi 3601	. . . . . . . . . . . 12 |- ( ( ( ( ( ph /\ F ~~> A ) /\ y e. RR+ ) /\ ( z e. Z /\ A. k e. ( ZZ>= ` z ) ( abs ` ( ( F ` k ) - A ) ) < y ) ) /\ ( w e. Z /\ z <_ w ) ) -> z e. ZZ )
22		simprl 794	. . . . . . . . . . . . 13 |- ( ( ( ( ( ph /\ F ~~> A ) /\ y e. RR+ ) /\ ( z e. Z /\ A. k e. ( ZZ>= ` z ) ( abs ` ( ( F ` k ) - A ) ) < y ) ) /\ ( w e. Z /\ z <_ w ) ) -> w e. Z )
23	19, 22	sseldi 3601	. . . . . . . . . . . 12 |- ( ( ( ( ( ph /\ F ~~> A ) /\ y e. RR+ ) /\ ( z e. Z /\ A. k e. ( ZZ>= ` z ) ( abs ` ( ( F ` k ) - A ) ) < y ) ) /\ ( w e. Z /\ z <_ w ) ) -> w e. ZZ )
24		simprr 796	. . . . . . . . . . . 12 |- ( ( ( ( ( ph /\ F ~~> A ) /\ y e. RR+ ) /\ ( z e. Z /\ A. k e. ( ZZ>= ` z ) ( abs ` ( ( F ` k ) - A ) ) < y ) ) /\ ( w e. Z /\ z <_ w ) ) -> z <_ w )
25		eluz2 11693	. . . . . . . . . . . 12 |- ( w e. ( ZZ>= ` z ) <-> ( z e. ZZ /\ w e. ZZ /\ z <_ w ) )
26	21, 23, 24, 25	syl3anbrc 1246	. . . . . . . . . . 11 |- ( ( ( ( ( ph /\ F ~~> A ) /\ y e. RR+ ) /\ ( z e. Z /\ A. k e. ( ZZ>= ` z ) ( abs ` ( ( F ` k ) - A ) ) < y ) ) /\ ( w e. Z /\ z <_ w ) ) -> w e. ( ZZ>= ` z ) )
27		simplrr 801	. . . . . . . . . . 11 |- ( ( ( ( ( ph /\ F ~~> A ) /\ y e. RR+ ) /\ ( z e. Z /\ A. k e. ( ZZ>= ` z ) ( abs ` ( ( F ` k ) - A ) ) < y ) ) /\ ( w e. Z /\ z <_ w ) ) -> A. k e. ( ZZ>= ` z ) ( abs ` ( ( F ` k ) - A ) ) < y )
28		fveq2 6191	. . . . . . . . . . . . . . 15 |- ( k = w -> ( F ` k ) = ( F ` w ) )
29	28	oveq1d 6665	. . . . . . . . . . . . . 14 |- ( k = w -> ( ( F ` k ) - A ) = ( ( F ` w ) - A ) )
30	29	fveq2d 6195	. . . . . . . . . . . . 13 |- ( k = w -> ( abs ` ( ( F ` k ) - A ) ) = ( abs ` ( ( F ` w ) - A ) ) )
31	30	breq1d 4663	. . . . . . . . . . . 12 |- ( k = w -> ( ( abs ` ( ( F ` k ) - A ) ) < y <-> ( abs ` ( ( F ` w ) - A ) ) < y ) )
32	31	rspcv 3305	. . . . . . . . . . 11 |- ( w e. ( ZZ>= ` z ) -> ( A. k e. ( ZZ>= ` z ) ( abs ` ( ( F ` k ) - A ) ) < y -> ( abs ` ( ( F ` w ) - A ) ) < y ) )
33	26, 27, 32	sylc 65	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ F ~~> A ) /\ y e. RR+ ) /\ ( z e. Z /\ A. k e. ( ZZ>= ` z ) ( abs ` ( ( F ` k ) - A ) ) < y ) ) /\ ( w e. Z /\ z <_ w ) ) -> ( abs ` ( ( F ` w ) - A ) ) < y )
34	33	expr 643	. . . . . . . . 9 |- ( ( ( ( ( ph /\ F ~~> A ) /\ y e. RR+ ) /\ ( z e. Z /\ A. k e. ( ZZ>= ` z ) ( abs ` ( ( F ` k ) - A ) ) < y ) ) /\ w e. Z ) -> ( z <_ w -> ( abs ` ( ( F ` w ) - A ) ) < y ) )
35	34	ralrimiva 2966	. . . . . . . 8 |- ( ( ( ( ph /\ F ~~> A ) /\ y e. RR+ ) /\ ( z e. Z /\ A. k e. ( ZZ>= ` z ) ( abs ` ( ( F ` k ) - A ) ) < y ) ) -> A. w e. Z ( z <_ w -> ( abs ` ( ( F ` w ) - A ) ) < y ) )
36	35	expr 643	. . . . . . 7 |- ( ( ( ( ph /\ F ~~> A ) /\ y e. RR+ ) /\ z e. Z ) -> ( A. k e. ( ZZ>= ` z ) ( abs ` ( ( F ` k ) - A ) ) < y -> A. w e. Z ( z <_ w -> ( abs ` ( ( F ` w ) - A ) ) < y ) ) )
37	36	reximdva 3017	. . . . . 6 |- ( ( ( ph /\ F ~~> A ) /\ y e. RR+ ) -> ( E. z e. Z A. k e. ( ZZ>= ` z ) ( abs ` ( ( F ` k ) - A ) ) < y -> E. z e. Z A. w e. Z ( z <_ w -> ( abs ` ( ( F ` w ) - A ) ) < y ) ) )
38		zssre 11384	. . . . . . . 8 |- ZZ C_ RR
39	19, 38	sstri 3612	. . . . . . 7 |- Z C_ RR
40		ssrexv 3667	. . . . . . 7 |- ( Z C_ RR -> ( E. z e. Z A. w e. Z ( z <_ w -> ( abs ` ( ( F ` w ) - A ) ) < y ) -> E. z e. RR A. w e. Z ( z <_ w -> ( abs ` ( ( F ` w ) - A ) ) < y ) ) )
41	39, 40	ax-mp 5	. . . . . 6 |- ( E. z e. Z A. w e. Z ( z <_ w -> ( abs ` ( ( F ` w ) - A ) ) < y ) -> E. z e. RR A. w e. Z ( z <_ w -> ( abs ` ( ( F ` w ) - A ) ) < y ) )
42	37, 41	syl6 35	. . . . 5 |- ( ( ( ph /\ F ~~> A ) /\ y e. RR+ ) -> ( E. z e. Z A. k e. ( ZZ>= ` z ) ( abs ` ( ( F ` k ) - A ) ) < y -> E. z e. RR A. w e. Z ( z <_ w -> ( abs ` ( ( F ` w ) - A ) ) < y ) ) )
43	17, 42	mpd 15	. . . 4 |- ( ( ( ph /\ F ~~> A ) /\ y e. RR+ ) -> E. z e. RR A. w e. Z ( z <_ w -> ( abs ` ( ( F ` w ) - A ) ) < y ) )
44	43	ralrimiva 2966	. . 3 |- ( ( ph /\ F ~~> A ) -> A. y e. RR+ E. z e. RR A. w e. Z ( z <_ w -> ( abs ` ( ( F ` w ) - A ) ) < y ) )
45	5	adantr 481	. . . 4 |- ( ( ph /\ F ~~> A ) -> F : Z --> CC )
46	39	a1i 11	. . . 4 |- ( ( ph /\ F ~~> A ) -> Z C_ RR )
47		eqidd 2623	. . . 4 |- ( ( ( ph /\ F ~~> A ) /\ w e. Z ) -> ( F ` w ) = ( F ` w ) )
48	45, 46, 47	rlim 14226	. . 3 |- ( ( ph /\ F ~~> A ) -> ( F ~~> r A <-> ( A e. CC /\ A. y e. RR+ E. z e. RR A. w e. Z ( z <_ w -> ( abs ` ( ( F ` w ) - A ) ) < y ) ) ) )
49	12, 44, 48	mpbir2and 957	. 2 |- ( ( ph /\ F ~~> A ) -> F ~~> r A )
50	10, 49	impbida 877	1 |- ( ph -> ( F ~~> r A <-> F ~~> A ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   C_ wss 3574   class class class wbr 4653   dom cdm 5114   -->wf 5884   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   < clt 10074   <_ cle 10075   - cmin 10266   ZZcz 11377   ZZ>=cuz 11687   RR+crp 11832   abscabs 13974   ~~> cli 14215   ~~> r crli 14216

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fl 12593  df-clim 14219  df-rlim 14220

This theorem is referenced by:  climmpt2  14304  climrecl  14314  climge0  14315  caurcvg  14407  caucvg  14409  climfsum  14552  divcnv  14585  dfef2  24697