Theorem divcnv 11681

Description: The sequence of reciprocals of positive integers, multiplied by the factor A, converges to zero. (Contributed by NM, 6-Feb-2008.) (Revised by Jim Kingdon, 22-Oct-2022.)

Assertion

Ref	Expression
divcnv	|- ( A e. CC -> ( n e. NN |-> ( A / n ) ) ~~> 0 )

Distinct variable group:   A, n

Proof of Theorem divcnv

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

Step	Hyp	Ref	Expression
1		simpl 109	. . . . . . 7 |- ( ( A e. CC /\ x e. RR+ ) -> A e. CC )
2	1	abscld 11365	. . . . . 6 |- ( ( A e. CC /\ x e. RR+ ) -> ( abs ` A ) e. RR )
3		simpr 110	. . . . . 6 |- ( ( A e. CC /\ x e. RR+ ) -> x e. RR+ )
4	2, 3	rerpdivcld 9822	. . . . 5 |- ( ( A e. CC /\ x e. RR+ ) -> ( ( abs ` A ) / x ) e. RR )
5		arch 9265	. . . . 5 |- ( ( ( abs ` A ) / x ) e. RR -> E. j e. NN ( ( abs ` A ) / x ) < j )
6	4, 5	syl 14	. . . 4 |- ( ( A e. CC /\ x e. RR+ ) -> E. j e. NN ( ( abs ` A ) / x ) < j )
7	1	ad3antrrr 492	. . . . . . . . . 10 |- ( ( ( ( ( A e. CC /\ x e. RR+ ) /\ j e. NN ) /\ ( ( abs ` A ) / x ) < j ) /\ k e. ( ZZ>= ` j ) ) -> A e. CC )
8		eluzelz 9629	. . . . . . . . . . . 12 |- ( k e. ( ZZ>= ` j ) -> k e. ZZ )
9	8	adantl 277	. . . . . . . . . . 11 |- ( ( ( ( ( A e. CC /\ x e. RR+ ) /\ j e. NN ) /\ ( ( abs ` A ) / x ) < j ) /\ k e. ( ZZ>= ` j ) ) -> k e. ZZ )
10	9	zcnd 9468	. . . . . . . . . 10 |- ( ( ( ( ( A e. CC /\ x e. RR+ ) /\ j e. NN ) /\ ( ( abs ` A ) / x ) < j ) /\ k e. ( ZZ>= ` j ) ) -> k e. CC )
11	9	zred 9467	. . . . . . . . . . 11 |- ( ( ( ( ( A e. CC /\ x e. RR+ ) /\ j e. NN ) /\ ( ( abs ` A ) / x ) < j ) /\ k e. ( ZZ>= ` j ) ) -> k e. RR )
12		0red 8046	. . . . . . . . . . . 12 |- ( ( ( ( ( A e. CC /\ x e. RR+ ) /\ j e. NN ) /\ ( ( abs ` A ) / x ) < j ) /\ k e. ( ZZ>= ` j ) ) -> 0 e. RR )
13		simpllr 534	. . . . . . . . . . . . 13 |- ( ( ( ( ( A e. CC /\ x e. RR+ ) /\ j e. NN ) /\ ( ( abs ` A ) / x ) < j ) /\ k e. ( ZZ>= ` j ) ) -> j e. NN )
14	13	nnred 9022	. . . . . . . . . . . 12 |- ( ( ( ( ( A e. CC /\ x e. RR+ ) /\ j e. NN ) /\ ( ( abs ` A ) / x ) < j ) /\ k e. ( ZZ>= ` j ) ) -> j e. RR )
15	13	nngt0d 9053	. . . . . . . . . . . 12 |- ( ( ( ( ( A e. CC /\ x e. RR+ ) /\ j e. NN ) /\ ( ( abs ` A ) / x ) < j ) /\ k e. ( ZZ>= ` j ) ) -> 0 < j )
16		eluzle 9632	. . . . . . . . . . . . 13 |- ( k e. ( ZZ>= ` j ) -> j <_ k )
17	16	adantl 277	. . . . . . . . . . . 12 |- ( ( ( ( ( A e. CC /\ x e. RR+ ) /\ j e. NN ) /\ ( ( abs ` A ) / x ) < j ) /\ k e. ( ZZ>= ` j ) ) -> j <_ k )
18	12, 14, 11, 15, 17	ltletrd 8469	. . . . . . . . . . 11 |- ( ( ( ( ( A e. CC /\ x e. RR+ ) /\ j e. NN ) /\ ( ( abs ` A ) / x ) < j ) /\ k e. ( ZZ>= ` j ) ) -> 0 < k )
19	11, 18	gt0ap0d 8675	. . . . . . . . . 10 |- ( ( ( ( ( A e. CC /\ x e. RR+ ) /\ j e. NN ) /\ ( ( abs ` A ) / x ) < j ) /\ k e. ( ZZ>= ` j ) ) -> k # 0 )
20	7, 10, 19	absdivapd 11379	. . . . . . . . 9 |- ( ( ( ( ( A e. CC /\ x e. RR+ ) /\ j e. NN ) /\ ( ( abs ` A ) / x ) < j ) /\ k e. ( ZZ>= ` j ) ) -> ( abs ` ( A / k ) ) = ( ( abs ` A ) / ( abs ` k ) ) )
21	12, 11, 18	ltled 8164	. . . . . . . . . . 11 |- ( ( ( ( ( A e. CC /\ x e. RR+ ) /\ j e. NN ) /\ ( ( abs ` A ) / x ) < j ) /\ k e. ( ZZ>= ` j ) ) -> 0 <_ k )
22	11, 21	absidd 11351	. . . . . . . . . 10 |- ( ( ( ( ( A e. CC /\ x e. RR+ ) /\ j e. NN ) /\ ( ( abs ` A ) / x ) < j ) /\ k e. ( ZZ>= ` j ) ) -> ( abs ` k ) = k )
23	22	oveq2d 5941	. . . . . . . . 9 |- ( ( ( ( ( A e. CC /\ x e. RR+ ) /\ j e. NN ) /\ ( ( abs ` A ) / x ) < j ) /\ k e. ( ZZ>= ` j ) ) -> ( ( abs ` A ) / ( abs ` k ) ) = ( ( abs ` A ) / k ) )
24	20, 23	eqtrd 2229	. . . . . . . 8 |- ( ( ( ( ( A e. CC /\ x e. RR+ ) /\ j e. NN ) /\ ( ( abs ` A ) / x ) < j ) /\ k e. ( ZZ>= ` j ) ) -> ( abs ` ( A / k ) ) = ( ( abs ` A ) / k ) )
25	2	ad3antrrr 492	. . . . . . . . 9 |- ( ( ( ( ( A e. CC /\ x e. RR+ ) /\ j e. NN ) /\ ( ( abs ` A ) / x ) < j ) /\ k e. ( ZZ>= ` j ) ) -> ( abs ` A ) e. RR )
26	3	ad3antrrr 492	. . . . . . . . 9 |- ( ( ( ( ( A e. CC /\ x e. RR+ ) /\ j e. NN ) /\ ( ( abs ` A ) / x ) < j ) /\ k e. ( ZZ>= ` j ) ) -> x e. RR+ )
27	11, 18	elrpd 9787	. . . . . . . . 9 |- ( ( ( ( ( A e. CC /\ x e. RR+ ) /\ j e. NN ) /\ ( ( abs ` A ) / x ) < j ) /\ k e. ( ZZ>= ` j ) ) -> k e. RR+ )
28	4	ad3antrrr 492	. . . . . . . . . 10 |- ( ( ( ( ( A e. CC /\ x e. RR+ ) /\ j e. NN ) /\ ( ( abs ` A ) / x ) < j ) /\ k e. ( ZZ>= ` j ) ) -> ( ( abs ` A ) / x ) e. RR )
29		simplr 528	. . . . . . . . . 10 |- ( ( ( ( ( A e. CC /\ x e. RR+ ) /\ j e. NN ) /\ ( ( abs ` A ) / x ) < j ) /\ k e. ( ZZ>= ` j ) ) -> ( ( abs ` A ) / x ) < j )
30	28, 14, 11, 29, 17	ltletrd 8469	. . . . . . . . 9 |- ( ( ( ( ( A e. CC /\ x e. RR+ ) /\ j e. NN ) /\ ( ( abs ` A ) / x ) < j ) /\ k e. ( ZZ>= ` j ) ) -> ( ( abs ` A ) / x ) < k )
31	25, 26, 27, 30	ltdiv23d 9851	. . . . . . . 8 |- ( ( ( ( ( A e. CC /\ x e. RR+ ) /\ j e. NN ) /\ ( ( abs ` A ) / x ) < j ) /\ k e. ( ZZ>= ` j ) ) -> ( ( abs ` A ) / k ) < x )
32	24, 31	eqbrtrd 4056	. . . . . . 7 |- ( ( ( ( ( A e. CC /\ x e. RR+ ) /\ j e. NN ) /\ ( ( abs ` A ) / x ) < j ) /\ k e. ( ZZ>= ` j ) ) -> ( abs ` ( A / k ) ) < x )
33	32	ralrimiva 2570	. . . . . 6 |- ( ( ( ( A e. CC /\ x e. RR+ ) /\ j e. NN ) /\ ( ( abs ` A ) / x ) < j ) -> A. k e. ( ZZ>= ` j ) ( abs ` ( A / k ) ) < x )
34	33	ex 115	. . . . 5 |- ( ( ( A e. CC /\ x e. RR+ ) /\ j e. NN ) -> ( ( ( abs ` A ) / x ) < j -> A. k e. ( ZZ>= ` j ) ( abs ` ( A / k ) ) < x ) )
35	34	reximdva 2599	. . . 4 |- ( ( A e. CC /\ x e. RR+ ) -> ( E. j e. NN ( ( abs ` A ) / x ) < j -> E. j e. NN A. k e. ( ZZ>= ` j ) ( abs ` ( A / k ) ) < x ) )
36	6, 35	mpd 13	. . 3 |- ( ( A e. CC /\ x e. RR+ ) -> E. j e. NN A. k e. ( ZZ>= ` j ) ( abs ` ( A / k ) ) < x )
37	36	ralrimiva 2570	. 2 |- ( A e. CC -> A. x e. RR+ E. j e. NN A. k e. ( ZZ>= ` j ) ( abs ` ( A / k ) ) < x )
38		nnuz 9656	. . 3 |- NN = ( ZZ>= ` 1 )
39		1zzd 9372	. . 3 |- ( A e. CC -> 1 e. ZZ )
40		nnex 9015	. . . . 5 |- NN e. _V
41	40	mptex 5791	. . . 4 |- ( n e. NN |-> ( A / n ) ) e. _V
42	41	a1i 9	. . 3 |- ( A e. CC -> ( n e. NN |-> ( A / n ) ) e. _V )
43		simpr 110	. . . 4 |- ( ( A e. CC /\ k e. NN ) -> k e. NN )
44		simpl 109	. . . . 5 |- ( ( A e. CC /\ k e. NN ) -> A e. CC )
45	43	nncnd 9023	. . . . 5 |- ( ( A e. CC /\ k e. NN ) -> k e. CC )
46	43	nnap0d 9055	. . . . 5 |- ( ( A e. CC /\ k e. NN ) -> k # 0 )
47	44, 45, 46	divclapd 8836	. . . 4 |- ( ( A e. CC /\ k e. NN ) -> ( A / k ) e. CC )
48		oveq2 5933	. . . . 5 |- ( n = k -> ( A / n ) = ( A / k ) )
49		eqid 2196	. . . . 5 |- ( n e. NN |-> ( A / n ) ) = ( n e. NN |-> ( A / n ) )
50	48, 49	fvmptg 5640	. . . 4 |- ( ( k e. NN /\ ( A / k ) e. CC ) -> ( ( n e. NN |-> ( A / n ) ) ` k ) = ( A / k ) )
51	43, 47, 50	syl2anc 411	. . 3 |- ( ( A e. CC /\ k e. NN ) -> ( ( n e. NN |-> ( A / n ) ) ` k ) = ( A / k ) )
52	38, 39, 42, 51, 47	clim0c 11470	. 2 |- ( A e. CC -> ( ( n e. NN |-> ( A / n ) ) ~~> 0 <-> A. x e. RR+ E. j e. NN A. k e. ( ZZ>= ` j ) ( abs ` ( A / k ) ) < x ) )
53	37, 52	mpbird 167	1 |- ( A e. CC -> ( n e. NN |-> ( A / n ) ) ~~> 0 )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2167   A.wral 2475   E.wrex 2476   _Vcvv 2763   class class class wbr 4034   |-> cmpt 4095   ` cfv 5259  (class class class)co 5925   CCcc 7896   RRcr 7897   0cc0 7898   1c1 7899   < clt 8080   <_ cle 8081   / cdiv 8718   NNcn 9009   ZZcz 9345   ZZ>=cuz 9620   RR+crp 9747   abscabs 11181   ~~> cli 11462

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7989  ax-resscn 7990  ax-1cn 7991  ax-1re 7992  ax-icn 7993  ax-addcl 7994  ax-addrcl 7995  ax-mulcl 7996  ax-mulrcl 7997  ax-addcom 7998  ax-mulcom 7999  ax-addass 8000  ax-mulass 8001  ax-distr 8002  ax-i2m1 8003  ax-0lt1 8004  ax-1rid 8005  ax-0id 8006  ax-rnegex 8007  ax-precex 8008  ax-cnre 8009  ax-pre-ltirr 8010  ax-pre-ltwlin 8011  ax-pre-lttrn 8012  ax-pre-apti 8013  ax-pre-ltadd 8014  ax-pre-mulgt0 8015  ax-pre-mulext 8016  ax-arch 8017  ax-caucvg 8018

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-frec 6458  df-pnf 8082  df-mnf 8083  df-xr 8084  df-ltxr 8085  df-le 8086  df-sub 8218  df-neg 8219  df-reap 8621  df-ap 8628  df-div 8719  df-inn 9010  df-2 9068  df-3 9069  df-4 9070  df-n0 9269  df-z 9346  df-uz 9621  df-rp 9748  df-seqfrec 10559  df-exp 10650  df-cj 11026  df-re 11027  df-im 11028  df-rsqrt 11182  df-abs 11183  df-clim 11463

This theorem is referenced by:  trireciplem  11684  expcnvap0  11686