Theorem clim1fr1 39833

Description: A class of sequences of fractions that converge to 1. (Contributed by Glauco Siliprandi, 29-Jun-2017.)

Hypotheses

Ref	Expression
clim1fr1.1	|- F = ( n e. NN |-> ( ( ( A x. n ) + B ) / ( A x. n ) ) )
clim1fr1.2	|- ( ph -> A e. CC )
clim1fr1.3	|- ( ph -> A =/= 0 )
clim1fr1.4	|- ( ph -> B e. CC )

Assertion

Ref	Expression
clim1fr1	|- ( ph -> F ~~> 1 )

Distinct variable groups:   ph, n   A, n   B, n

Allowed substitution hint:   F( n)

Proof of Theorem clim1fr1

Dummy variable k is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nnuz 11723	. . 3 |- NN = ( ZZ>= ` 1 )
2		1zzd 11408	. . 3 |- ( ph -> 1 e. ZZ )
3		nnex 11026	. . . . . 6 |- NN e. _V
4	3	mptex 6486	. . . . 5 |- ( n e. NN |-> 1 ) e. _V
5	4	a1i 11	. . . 4 |- ( ph -> ( n e. NN |-> 1 ) e. _V )
6		1cnd 10056	. . . 4 |- ( ph -> 1 e. CC )
7		eqidd 2623	. . . . . 6 |- ( k e. NN -> ( n e. NN |-> 1 ) = ( n e. NN |-> 1 ) )
8		eqidd 2623	. . . . . 6 |- ( ( k e. NN /\ n = k ) -> 1 = 1 )
9		id 22	. . . . . 6 |- ( k e. NN -> k e. NN )
10		1cnd 10056	. . . . . 6 |- ( k e. NN -> 1 e. CC )
11	7, 8, 9, 10	fvmptd 6288	. . . . 5 |- ( k e. NN -> ( ( n e. NN |-> 1 ) ` k ) = 1 )
12	11	adantl 482	. . . 4 |- ( ( ph /\ k e. NN ) -> ( ( n e. NN |-> 1 ) ` k ) = 1 )
13	1, 2, 5, 6, 12	climconst 14274	. . 3 |- ( ph -> ( n e. NN |-> 1 ) ~~> 1 )
14		clim1fr1.1	. . . . 5 |- F = ( n e. NN |-> ( ( ( A x. n ) + B ) / ( A x. n ) ) )
15	3	mptex 6486	. . . . 5 |- ( n e. NN |-> ( ( ( A x. n ) + B ) / ( A x. n ) ) ) e. _V
16	14, 15	eqeltri 2697	. . . 4 |- F e. _V
17	16	a1i 11	. . 3 |- ( ph -> F e. _V )
18		clim1fr1.4	. . . . . . 7 |- ( ph -> B e. CC )
19	18	adantr 481	. . . . . 6 |- ( ( ph /\ n e. NN ) -> B e. CC )
20		clim1fr1.2	. . . . . . 7 |- ( ph -> A e. CC )
21	20	adantr 481	. . . . . 6 |- ( ( ph /\ n e. NN ) -> A e. CC )
22		nncn 11028	. . . . . . 7 |- ( n e. NN -> n e. CC )
23	22	adantl 482	. . . . . 6 |- ( ( ph /\ n e. NN ) -> n e. CC )
24		clim1fr1.3	. . . . . . 7 |- ( ph -> A =/= 0 )
25	24	adantr 481	. . . . . 6 |- ( ( ph /\ n e. NN ) -> A =/= 0 )
26		nnne0 11053	. . . . . . 7 |- ( n e. NN -> n =/= 0 )
27	26	adantl 482	. . . . . 6 |- ( ( ph /\ n e. NN ) -> n =/= 0 )
28	19, 21, 23, 25, 27	divdiv1d 10832	. . . . 5 |- ( ( ph /\ n e. NN ) -> ( ( B / A ) / n ) = ( B / ( A x. n ) ) )
29	28	mpteq2dva 4744	. . . 4 |- ( ph -> ( n e. NN |-> ( ( B / A ) / n ) ) = ( n e. NN |-> ( B / ( A x. n ) ) ) )
30	18, 20, 24	divcld 10801	. . . . 5 |- ( ph -> ( B / A ) e. CC )
31		divcnv 14585	. . . . 5 |- ( ( B / A ) e. CC -> ( n e. NN |-> ( ( B / A ) / n ) ) ~~> 0 )
32	30, 31	syl 17	. . . 4 |- ( ph -> ( n e. NN |-> ( ( B / A ) / n ) ) ~~> 0 )
33	29, 32	eqbrtrrd 4677	. . 3 |- ( ph -> ( n e. NN |-> ( B / ( A x. n ) ) ) ~~> 0 )
34		eqid 2622	. . . . . 6 |- ( n e. NN |-> 1 ) = ( n e. NN |-> 1 )
35		1cnd 10056	. . . . . 6 |- ( n e. NN -> 1 e. CC )
36	34, 35	fmpti 6383	. . . . 5 |- ( n e. NN |-> 1 ) : NN --> CC
37	36	a1i 11	. . . 4 |- ( ph -> ( n e. NN |-> 1 ) : NN --> CC )
38	37	ffvelrnda 6359	. . 3 |- ( ( ph /\ k e. NN ) -> ( ( n e. NN |-> 1 ) ` k ) e. CC )
39	21, 23	mulcld 10060	. . . . . 6 |- ( ( ph /\ n e. NN ) -> ( A x. n ) e. CC )
40	21, 23, 25, 27	mulne0d 10679	. . . . . 6 |- ( ( ph /\ n e. NN ) -> ( A x. n ) =/= 0 )
41	19, 39, 40	divcld 10801	. . . . 5 |- ( ( ph /\ n e. NN ) -> ( B / ( A x. n ) ) e. CC )
42		eqid 2622	. . . . 5 |- ( n e. NN |-> ( B / ( A x. n ) ) ) = ( n e. NN |-> ( B / ( A x. n ) ) )
43	41, 42	fmptd 6385	. . . 4 |- ( ph -> ( n e. NN |-> ( B / ( A x. n ) ) ) : NN --> CC )
44	43	ffvelrnda 6359	. . 3 |- ( ( ph /\ k e. NN ) -> ( ( n e. NN |-> ( B / ( A x. n ) ) ) ` k ) e. CC )
45	14	a1i 11	. . . . 5 |- ( ( ph /\ k e. NN ) -> F = ( n e. NN |-> ( ( ( A x. n ) + B ) / ( A x. n ) ) ) )
46		oveq2 6658	. . . . . . . 8 |- ( n = k -> ( A x. n ) = ( A x. k ) )
47	46	oveq1d 6665	. . . . . . 7 |- ( n = k -> ( ( A x. n ) + B ) = ( ( A x. k ) + B ) )
48	47, 46	oveq12d 6668	. . . . . 6 |- ( n = k -> ( ( ( A x. n ) + B ) / ( A x. n ) ) = ( ( ( A x. k ) + B ) / ( A x. k ) ) )
49	48	adantl 482	. . . . 5 |- ( ( ( ph /\ k e. NN ) /\ n = k ) -> ( ( ( A x. n ) + B ) / ( A x. n ) ) = ( ( ( A x. k ) + B ) / ( A x. k ) ) )
50		simpr 477	. . . . 5 |- ( ( ph /\ k e. NN ) -> k e. NN )
51	20	adantr 481	. . . . . . . 8 |- ( ( ph /\ k e. NN ) -> A e. CC )
52	50	nncnd 11036	. . . . . . . 8 |- ( ( ph /\ k e. NN ) -> k e. CC )
53	51, 52	mulcld 10060	. . . . . . 7 |- ( ( ph /\ k e. NN ) -> ( A x. k ) e. CC )
54	18	adantr 481	. . . . . . 7 |- ( ( ph /\ k e. NN ) -> B e. CC )
55	53, 54	addcld 10059	. . . . . 6 |- ( ( ph /\ k e. NN ) -> ( ( A x. k ) + B ) e. CC )
56	24	adantr 481	. . . . . . 7 |- ( ( ph /\ k e. NN ) -> A =/= 0 )
57	50	nnne0d 11065	. . . . . . 7 |- ( ( ph /\ k e. NN ) -> k =/= 0 )
58	51, 52, 56, 57	mulne0d 10679	. . . . . 6 |- ( ( ph /\ k e. NN ) -> ( A x. k ) =/= 0 )
59	55, 53, 58	divcld 10801	. . . . 5 |- ( ( ph /\ k e. NN ) -> ( ( ( A x. k ) + B ) / ( A x. k ) ) e. CC )
60	45, 49, 50, 59	fvmptd 6288	. . . 4 |- ( ( ph /\ k e. NN ) -> ( F ` k ) = ( ( ( A x. k ) + B ) / ( A x. k ) ) )
61	53, 54, 53, 58	divdird 10839	. . . . 5 |- ( ( ph /\ k e. NN ) -> ( ( ( A x. k ) + B ) / ( A x. k ) ) = ( ( ( A x. k ) / ( A x. k ) ) + ( B / ( A x. k ) ) ) )
62	53, 58	dividd 10799	. . . . . 6 |- ( ( ph /\ k e. NN ) -> ( ( A x. k ) / ( A x. k ) ) = 1 )
63	62	oveq1d 6665	. . . . 5 |- ( ( ph /\ k e. NN ) -> ( ( ( A x. k ) / ( A x. k ) ) + ( B / ( A x. k ) ) ) = ( 1 + ( B / ( A x. k ) ) ) )
64	61, 63	eqtrd 2656	. . . 4 |- ( ( ph /\ k e. NN ) -> ( ( ( A x. k ) + B ) / ( A x. k ) ) = ( 1 + ( B / ( A x. k ) ) ) )
65	12	eqcomd 2628	. . . . 5 |- ( ( ph /\ k e. NN ) -> 1 = ( ( n e. NN |-> 1 ) ` k ) )
66		eqidd 2623	. . . . . . 7 |- ( ( ph /\ k e. NN ) -> ( n e. NN |-> ( B / ( A x. n ) ) ) = ( n e. NN |-> ( B / ( A x. n ) ) ) )
67		simpr 477	. . . . . . . . 9 |- ( ( ( ph /\ k e. NN ) /\ n = k ) -> n = k )
68	67	oveq2d 6666	. . . . . . . 8 |- ( ( ( ph /\ k e. NN ) /\ n = k ) -> ( A x. n ) = ( A x. k ) )
69	68	oveq2d 6666	. . . . . . 7 |- ( ( ( ph /\ k e. NN ) /\ n = k ) -> ( B / ( A x. n ) ) = ( B / ( A x. k ) ) )
70	54, 53, 58	divcld 10801	. . . . . . 7 |- ( ( ph /\ k e. NN ) -> ( B / ( A x. k ) ) e. CC )
71	66, 69, 50, 70	fvmptd 6288	. . . . . 6 |- ( ( ph /\ k e. NN ) -> ( ( n e. NN |-> ( B / ( A x. n ) ) ) ` k ) = ( B / ( A x. k ) ) )
72	71	eqcomd 2628	. . . . 5 |- ( ( ph /\ k e. NN ) -> ( B / ( A x. k ) ) = ( ( n e. NN |-> ( B / ( A x. n ) ) ) ` k ) )
73	65, 72	oveq12d 6668	. . . 4 |- ( ( ph /\ k e. NN ) -> ( 1 + ( B / ( A x. k ) ) ) = ( ( ( n e. NN |-> 1 ) ` k ) + ( ( n e. NN |-> ( B / ( A x. n ) ) ) ` k ) ) )
74	60, 64, 73	3eqtrd 2660	. . 3 |- ( ( ph /\ k e. NN ) -> ( F ` k ) = ( ( ( n e. NN |-> 1 ) ` k ) + ( ( n e. NN |-> ( B / ( A x. n ) ) ) ` k ) ) )
75	1, 2, 13, 17, 33, 38, 44, 74	climadd 14362	. 2 |- ( ph -> F ~~> ( 1 + 0 ) )
76		1p0e1 11133	. 2 |- ( 1 + 0 ) = 1
77	75, 76	syl6breq 4694	1 |- ( ph -> F ~~> 1 )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   class class class wbr 4653   |-> cmpt 4729   -->wf 5884   ` cfv 5888  (class class class)co 6650   CCcc 9934   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   / cdiv 10684   NNcn 11020   ~~> cli 14215

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-rep 4771  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-2nd 7169  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-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fl 12593  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-rlim 14220

This theorem is referenced by:  wallispilem5  40286