Theorem logtayllem 24405

Description: Lemma for logtayl 24406. (Contributed by Mario Carneiro, 1-Apr-2015.)

Assertion

Ref	Expression
logtayllem	|- ( ( A e. CC /\ ( abs ` A ) < 1 ) -> seq 0 ( + , ( n e. NN0 |-> ( if ( n = 0 , 0 , ( 1 / n ) ) x. ( A ^ n ) ) ) ) e. dom ~~> )

Distinct variable group:   A, n

Proof of Theorem logtayllem

Dummy variable k is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nn0uz 11722	. 2 |- NN0 = ( ZZ>= ` 0 )
2		1nn0 11308	. . 3 |- 1 e. NN0
3	2	a1i 11	. 2 |- ( ( A e. CC /\ ( abs ` A ) < 1 ) -> 1 e. NN0 )
4		oveq2 6658	. . . . 5 |- ( n = k -> ( ( abs ` A ) ^ n ) = ( ( abs ` A ) ^ k ) )
5		eqid 2622	. . . . 5 |- ( n e. NN0 |-> ( ( abs ` A ) ^ n ) ) = ( n e. NN0 |-> ( ( abs ` A ) ^ n ) )
6		ovex 6678	. . . . 5 |- ( ( abs ` A ) ^ k ) e. _V
7	4, 5, 6	fvmpt 6282	. . . 4 |- ( k e. NN0 -> ( ( n e. NN0 |-> ( ( abs ` A ) ^ n ) ) ` k ) = ( ( abs ` A ) ^ k ) )
8	7	adantl 482	. . 3 |- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ k e. NN0 ) -> ( ( n e. NN0 |-> ( ( abs ` A ) ^ n ) ) ` k ) = ( ( abs ` A ) ^ k ) )
9		abscl 14018	. . . . 5 |- ( A e. CC -> ( abs ` A ) e. RR )
10	9	adantr 481	. . . 4 |- ( ( A e. CC /\ ( abs ` A ) < 1 ) -> ( abs ` A ) e. RR )
11		reexpcl 12877	. . . 4 |- ( ( ( abs ` A ) e. RR /\ k e. NN0 ) -> ( ( abs ` A ) ^ k ) e. RR )
12	10, 11	sylan 488	. . 3 |- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ k e. NN0 ) -> ( ( abs ` A ) ^ k ) e. RR )
13	8, 12	eqeltrd 2701	. 2 |- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ k e. NN0 ) -> ( ( n e. NN0 |-> ( ( abs ` A ) ^ n ) ) ` k ) e. RR )
14		eqeq1 2626	. . . . . . 7 |- ( n = k -> ( n = 0 <-> k = 0 ) )
15		oveq2 6658	. . . . . . 7 |- ( n = k -> ( 1 / n ) = ( 1 / k ) )
16	14, 15	ifbieq2d 4111	. . . . . 6 |- ( n = k -> if ( n = 0 , 0 , ( 1 / n ) ) = if ( k = 0 , 0 , ( 1 / k ) ) )
17		oveq2 6658	. . . . . 6 |- ( n = k -> ( A ^ n ) = ( A ^ k ) )
18	16, 17	oveq12d 6668	. . . . 5 |- ( n = k -> ( if ( n = 0 , 0 , ( 1 / n ) ) x. ( A ^ n ) ) = ( if ( k = 0 , 0 , ( 1 / k ) ) x. ( A ^ k ) ) )
19		eqid 2622	. . . . 5 |- ( n e. NN0 |-> ( if ( n = 0 , 0 , ( 1 / n ) ) x. ( A ^ n ) ) ) = ( n e. NN0 |-> ( if ( n = 0 , 0 , ( 1 / n ) ) x. ( A ^ n ) ) )
20		ovex 6678	. . . . 5 |- ( if ( k = 0 , 0 , ( 1 / k ) ) x. ( A ^ k ) ) e. _V
21	18, 19, 20	fvmpt 6282	. . . 4 |- ( k e. NN0 -> ( ( n e. NN0 |-> ( if ( n = 0 , 0 , ( 1 / n ) ) x. ( A ^ n ) ) ) ` k ) = ( if ( k = 0 , 0 , ( 1 / k ) ) x. ( A ^ k ) ) )
22	21	adantl 482	. . 3 |- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ k e. NN0 ) -> ( ( n e. NN0 |-> ( if ( n = 0 , 0 , ( 1 / n ) ) x. ( A ^ n ) ) ) ` k ) = ( if ( k = 0 , 0 , ( 1 / k ) ) x. ( A ^ k ) ) )
23		0cnd 10033	. . . . 5 |- ( ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ k e. NN0 ) /\ k = 0 ) -> 0 e. CC )
24		nn0cn 11302	. . . . . . 7 |- ( k e. NN0 -> k e. CC )
25	24	adantl 482	. . . . . 6 |- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ k e. NN0 ) -> k e. CC )
26		df-ne 2795	. . . . . . 7 |- ( k =/= 0 <-> -. k = 0 )
27	26	biimpri 218	. . . . . 6 |- ( -. k = 0 -> k =/= 0 )
28		reccl 10692	. . . . . 6 |- ( ( k e. CC /\ k =/= 0 ) -> ( 1 / k ) e. CC )
29	25, 27, 28	syl2an 494	. . . . 5 |- ( ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ k e. NN0 ) /\ -. k = 0 ) -> ( 1 / k ) e. CC )
30	23, 29	ifclda 4120	. . . 4 |- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ k e. NN0 ) -> if ( k = 0 , 0 , ( 1 / k ) ) e. CC )
31		expcl 12878	. . . . 5 |- ( ( A e. CC /\ k e. NN0 ) -> ( A ^ k ) e. CC )
32	31	adantlr 751	. . . 4 |- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ k e. NN0 ) -> ( A ^ k ) e. CC )
33	30, 32	mulcld 10060	. . 3 |- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ k e. NN0 ) -> ( if ( k = 0 , 0 , ( 1 / k ) ) x. ( A ^ k ) ) e. CC )
34	22, 33	eqeltrd 2701	. 2 |- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ k e. NN0 ) -> ( ( n e. NN0 |-> ( if ( n = 0 , 0 , ( 1 / n ) ) x. ( A ^ n ) ) ) ` k ) e. CC )
35	10	recnd 10068	. . . 4 |- ( ( A e. CC /\ ( abs ` A ) < 1 ) -> ( abs ` A ) e. CC )
36		absidm 14063	. . . . . 6 |- ( A e. CC -> ( abs ` ( abs ` A ) ) = ( abs ` A ) )
37	36	adantr 481	. . . . 5 |- ( ( A e. CC /\ ( abs ` A ) < 1 ) -> ( abs ` ( abs ` A ) ) = ( abs ` A ) )
38		simpr 477	. . . . 5 |- ( ( A e. CC /\ ( abs ` A ) < 1 ) -> ( abs ` A ) < 1 )
39	37, 38	eqbrtrd 4675	. . . 4 |- ( ( A e. CC /\ ( abs ` A ) < 1 ) -> ( abs ` ( abs ` A ) ) < 1 )
40	35, 39, 8	geolim 14601	. . 3 |- ( ( A e. CC /\ ( abs ` A ) < 1 ) -> seq 0 ( + , ( n e. NN0 |-> ( ( abs ` A ) ^ n ) ) ) ~~> ( 1 / ( 1 - ( abs ` A ) ) ) )
41		seqex 12803	. . . 4 |- seq 0 ( + , ( n e. NN0 |-> ( ( abs ` A ) ^ n ) ) ) e. _V
42		ovex 6678	. . . 4 |- ( 1 / ( 1 - ( abs ` A ) ) ) e. _V
43	41, 42	breldm 5329	. . 3 |- ( seq 0 ( + , ( n e. NN0 |-> ( ( abs ` A ) ^ n ) ) ) ~~> ( 1 / ( 1 - ( abs ` A ) ) ) -> seq 0 ( + , ( n e. NN0 |-> ( ( abs ` A ) ^ n ) ) ) e. dom ~~> )
44	40, 43	syl 17	. 2 |- ( ( A e. CC /\ ( abs ` A ) < 1 ) -> seq 0 ( + , ( n e. NN0 |-> ( ( abs ` A ) ^ n ) ) ) e. dom ~~> )
45		1red 10055	. 2 |- ( ( A e. CC /\ ( abs ` A ) < 1 ) -> 1 e. RR )
46		elnnuz 11724	. . 3 |- ( k e. NN <-> k e. ( ZZ>= ` 1 ) )
47		nnrecre 11057	. . . . . . . . 9 |- ( k e. NN -> ( 1 / k ) e. RR )
48	47	adantl 482	. . . . . . . 8 |- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ k e. NN ) -> ( 1 / k ) e. RR )
49	48	recnd 10068	. . . . . . 7 |- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ k e. NN ) -> ( 1 / k ) e. CC )
50		nnnn0 11299	. . . . . . . 8 |- ( k e. NN -> k e. NN0 )
51	50, 32	sylan2 491	. . . . . . 7 |- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ k e. NN ) -> ( A ^ k ) e. CC )
52	49, 51	absmuld 14193	. . . . . 6 |- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ k e. NN ) -> ( abs ` ( ( 1 / k ) x. ( A ^ k ) ) ) = ( ( abs ` ( 1 / k ) ) x. ( abs ` ( A ^ k ) ) ) )
53		nnrp 11842	. . . . . . . . . . 11 |- ( k e. NN -> k e. RR+ )
54	53	adantl 482	. . . . . . . . . 10 |- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ k e. NN ) -> k e. RR+ )
55	54	rpreccld 11882	. . . . . . . . 9 |- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ k e. NN ) -> ( 1 / k ) e. RR+ )
56	55	rpge0d 11876	. . . . . . . 8 |- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ k e. NN ) -> 0 <_ ( 1 / k ) )
57	48, 56	absidd 14161	. . . . . . 7 |- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ k e. NN ) -> ( abs ` ( 1 / k ) ) = ( 1 / k ) )
58		simpl 473	. . . . . . . 8 |- ( ( A e. CC /\ ( abs ` A ) < 1 ) -> A e. CC )
59		absexp 14044	. . . . . . . 8 |- ( ( A e. CC /\ k e. NN0 ) -> ( abs ` ( A ^ k ) ) = ( ( abs ` A ) ^ k ) )
60	58, 50, 59	syl2an 494	. . . . . . 7 |- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ k e. NN ) -> ( abs ` ( A ^ k ) ) = ( ( abs ` A ) ^ k ) )
61	57, 60	oveq12d 6668	. . . . . 6 |- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ k e. NN ) -> ( ( abs ` ( 1 / k ) ) x. ( abs ` ( A ^ k ) ) ) = ( ( 1 / k ) x. ( ( abs ` A ) ^ k ) ) )
62	52, 61	eqtrd 2656	. . . . 5 |- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ k e. NN ) -> ( abs ` ( ( 1 / k ) x. ( A ^ k ) ) ) = ( ( 1 / k ) x. ( ( abs ` A ) ^ k ) ) )
63		1red 10055	. . . . . 6 |- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ k e. NN ) -> 1 e. RR )
64	50, 12	sylan2 491	. . . . . 6 |- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ k e. NN ) -> ( ( abs ` A ) ^ k ) e. RR )
65	51	absge0d 14183	. . . . . . 7 |- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ k e. NN ) -> 0 <_ ( abs ` ( A ^ k ) ) )
66	65, 60	breqtrd 4679	. . . . . 6 |- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ k e. NN ) -> 0 <_ ( ( abs ` A ) ^ k ) )
67		nnge1 11046	. . . . . . . . 9 |- ( k e. NN -> 1 <_ k )
68	67	adantl 482	. . . . . . . 8 |- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ k e. NN ) -> 1 <_ k )
69		0lt1 10550	. . . . . . . . . 10 |- 0 < 1
70	69	a1i 11	. . . . . . . . 9 |- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ k e. NN ) -> 0 < 1 )
71		nnre 11027	. . . . . . . . . 10 |- ( k e. NN -> k e. RR )
72	71	adantl 482	. . . . . . . . 9 |- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ k e. NN ) -> k e. RR )
73		nngt0 11049	. . . . . . . . . 10 |- ( k e. NN -> 0 < k )
74	73	adantl 482	. . . . . . . . 9 |- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ k e. NN ) -> 0 < k )
75		lerec 10906	. . . . . . . . 9 |- ( ( ( 1 e. RR /\ 0 < 1 ) /\ ( k e. RR /\ 0 < k ) ) -> ( 1 <_ k <-> ( 1 / k ) <_ ( 1 / 1 ) ) )
76	63, 70, 72, 74, 75	syl22anc 1327	. . . . . . . 8 |- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ k e. NN ) -> ( 1 <_ k <-> ( 1 / k ) <_ ( 1 / 1 ) ) )
77	68, 76	mpbid 222	. . . . . . 7 |- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ k e. NN ) -> ( 1 / k ) <_ ( 1 / 1 ) )
78		1div1e1 10717	. . . . . . 7 |- ( 1 / 1 ) = 1
79	77, 78	syl6breq 4694	. . . . . 6 |- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ k e. NN ) -> ( 1 / k ) <_ 1 )
80	48, 63, 64, 66, 79	lemul1ad 10963	. . . . 5 |- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ k e. NN ) -> ( ( 1 / k ) x. ( ( abs ` A ) ^ k ) ) <_ ( 1 x. ( ( abs ` A ) ^ k ) ) )
81	62, 80	eqbrtrd 4675	. . . 4 |- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ k e. NN ) -> ( abs ` ( ( 1 / k ) x. ( A ^ k ) ) ) <_ ( 1 x. ( ( abs ` A ) ^ k ) ) )
82	50, 22	sylan2 491	. . . . . 6 |- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ k e. NN ) -> ( ( n e. NN0 |-> ( if ( n = 0 , 0 , ( 1 / n ) ) x. ( A ^ n ) ) ) ` k ) = ( if ( k = 0 , 0 , ( 1 / k ) ) x. ( A ^ k ) ) )
83		nnne0 11053	. . . . . . . . . 10 |- ( k e. NN -> k =/= 0 )
84	83	adantl 482	. . . . . . . . 9 |- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ k e. NN ) -> k =/= 0 )
85	84	neneqd 2799	. . . . . . . 8 |- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ k e. NN ) -> -. k = 0 )
86	85	iffalsed 4097	. . . . . . 7 |- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ k e. NN ) -> if ( k = 0 , 0 , ( 1 / k ) ) = ( 1 / k ) )
87	86	oveq1d 6665	. . . . . 6 |- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ k e. NN ) -> ( if ( k = 0 , 0 , ( 1 / k ) ) x. ( A ^ k ) ) = ( ( 1 / k ) x. ( A ^ k ) ) )
88	82, 87	eqtrd 2656	. . . . 5 |- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ k e. NN ) -> ( ( n e. NN0 |-> ( if ( n = 0 , 0 , ( 1 / n ) ) x. ( A ^ n ) ) ) ` k ) = ( ( 1 / k ) x. ( A ^ k ) ) )
89	88	fveq2d 6195	. . . 4 |- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ k e. NN ) -> ( abs ` ( ( n e. NN0 |-> ( if ( n = 0 , 0 , ( 1 / n ) ) x. ( A ^ n ) ) ) ` k ) ) = ( abs ` ( ( 1 / k ) x. ( A ^ k ) ) ) )
90	50, 8	sylan2 491	. . . . 5 |- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ k e. NN ) -> ( ( n e. NN0 |-> ( ( abs ` A ) ^ n ) ) ` k ) = ( ( abs ` A ) ^ k ) )
91	90	oveq2d 6666	. . . 4 |- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ k e. NN ) -> ( 1 x. ( ( n e. NN0 |-> ( ( abs ` A ) ^ n ) ) ` k ) ) = ( 1 x. ( ( abs ` A ) ^ k ) ) )
92	81, 89, 91	3brtr4d 4685	. . 3 |- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ k e. NN ) -> ( abs ` ( ( n e. NN0 |-> ( if ( n = 0 , 0 , ( 1 / n ) ) x. ( A ^ n ) ) ) ` k ) ) <_ ( 1 x. ( ( n e. NN0 |-> ( ( abs ` A ) ^ n ) ) ` k ) ) )
93	46, 92	sylan2br 493	. 2 |- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ k e. ( ZZ>= ` 1 ) ) -> ( abs ` ( ( n e. NN0 |-> ( if ( n = 0 , 0 , ( 1 / n ) ) x. ( A ^ n ) ) ) ` k ) ) <_ ( 1 x. ( ( n e. NN0 |-> ( ( abs ` A ) ^ n ) ) ` k ) ) )
94	1, 3, 13, 34, 44, 45, 93	cvgcmpce 14550	1 |- ( ( A e. CC /\ ( abs ` A ) < 1 ) -> seq 0 ( + , ( n e. NN0 |-> ( if ( n = 0 , 0 , ( 1 / n ) ) x. ( A ^ n ) ) ) ) e. dom ~~> )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   ifcif 4086   class class class wbr 4653   |-> cmpt 4729   dom cdm 5114   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   < clt 10074   <_ cle 10075   - cmin 10266   / cdiv 10684   NNcn 11020   NN0cn0 11292   ZZ>=cuz 11687   RR+crp 11832   seqcseq 12801   ^cexp 12860   abscabs 13974   ~~> 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-inf2 8538  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  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  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-int 4476  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-se 5074  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-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  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-ico 12181  df-fz 12327  df-fzo 12466  df-fl 12593  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-limsup 14202  df-clim 14219  df-rlim 14220  df-sum 14417

This theorem is referenced by:  logtayl  24406