Theorem clim2divap 12307

Description: The limit of an infinite product with an initial segment removed. (Contributed by Scott Fenton, 20-Dec-2017.)

Hypotheses

Ref	Expression
clim2div.1	|- Z = ( ZZ>= ` M )
clim2div.2	|- ( ph -> N e. Z )
clim2div.3	|- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )
clim2div.4	|- ( ph -> seq M ( x. , F ) ~~> A )
clim2divap.5	|- ( ph -> ( seq M ( x. , F ) ` N ) # 0 )

Assertion

Ref	Expression
clim2divap	|- ( ph -> seq ( N + 1 ) ( x. , F ) ~~> ( A / ( seq M ( x. , F ) ` N ) ) )

Distinct variable groups:   k, F   ph, k   k, M   k, N   k, Z

Allowed substitution hint:   A( k)

Proof of Theorem clim2divap

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

Step	Hyp	Ref	Expression
1		eqid 2238	. . 3 |- ( ZZ>= ` ( N + 1 ) ) = ( ZZ>= ` ( N + 1 ) )
2		clim2div.2	. . . . 5 |- ( ph -> N e. Z )
3		eluzelz 9931	. . . . . 6 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
4		clim2div.1	. . . . . 6 |- Z = ( ZZ>= ` M )
5	3, 4	eleq2s 2333	. . . . 5 |- ( N e. Z -> N e. ZZ )
6	2, 5	syl 14	. . . 4 |- ( ph -> N e. ZZ )
7	6	peano2zd 9771	. . 3 |- ( ph -> ( N + 1 ) e. ZZ )
8		clim2div.4	. . 3 |- ( ph -> seq M ( x. , F ) ~~> A )
9		eluzel2 9926	. . . . . . . 8 |- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
10	9, 4	eleq2s 2333	. . . . . . 7 |- ( N e. Z -> M e. ZZ )
11	2, 10	syl 14	. . . . . 6 |- ( ph -> M e. ZZ )
12		clim2div.3	. . . . . 6 |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )
13	4, 11, 12	prodf 12305	. . . . 5 |- ( ph -> seq M ( x. , F ) : Z --> CC )
14	13, 2	ffvelcdmd 5844	. . . 4 |- ( ph -> ( seq M ( x. , F ) ` N ) e. CC )
15		clim2divap.5	. . . 4 |- ( ph -> ( seq M ( x. , F ) ` N ) # 0 )
16	14, 15	recclapd 9111	. . 3 |- ( ph -> ( 1 / ( seq M ( x. , F ) ` N ) ) e. CC )
17		seqex 10886	. . . 4 |- seq ( N + 1 ) ( x. , F ) e. _V
18	17	a1i 9	. . 3 |- ( ph -> seq ( N + 1 ) ( x. , F ) e. _V )
19	2, 4	eleqtrdi 2331	. . . . . . 7 |- ( ph -> N e. ( ZZ>= ` M ) )
20		peano2uz 9983	. . . . . . 7 |- ( N e. ( ZZ>= ` M ) -> ( N + 1 ) e. ( ZZ>= ` M ) )
21	19, 20	syl 14	. . . . . 6 |- ( ph -> ( N + 1 ) e. ( ZZ>= ` M ) )
22	21, 4	eleqtrrdi 2332	. . . . 5 |- ( ph -> ( N + 1 ) e. Z )
23	4	uztrn2 9940	. . . . 5 |- ( ( ( N + 1 ) e. Z /\ j e. ( ZZ>= ` ( N + 1 ) ) ) -> j e. Z )
24	22, 23	sylan 283	. . . 4 |- ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) -> j e. Z )
25	13	ffvelcdmda 5843	. . . 4 |- ( ( ph /\ j e. Z ) -> ( seq M ( x. , F ) ` j ) e. CC )
26	24, 25	syldan 282	. . 3 |- ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) -> ( seq M ( x. , F ) ` j ) e. CC )
27		mulcl 8306	. . . . . . . 8 |- ( ( k e. CC /\ x e. CC ) -> ( k x. x ) e. CC )
28	27	adantl 277	. . . . . . 7 |- ( ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) /\ ( k e. CC /\ x e. CC ) ) -> ( k x. x ) e. CC )
29		mulass 8310	. . . . . . . 8 |- ( ( k e. CC /\ x e. CC /\ y e. CC ) -> ( ( k x. x ) x. y ) = ( k x. ( x x. y ) ) )
30	29	adantl 277	. . . . . . 7 |- ( ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) /\ ( k e. CC /\ x e. CC /\ y e. CC ) ) -> ( ( k x. x ) x. y ) = ( k x. ( x x. y ) ) )
31		simpr 110	. . . . . . 7 |- ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) -> j e. ( ZZ>= ` ( N + 1 ) ) )
32	19	adantr 276	. . . . . . 7 |- ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) -> N e. ( ZZ>= ` M ) )
33	4	eleq2i 2305	. . . . . . . . 9 |- ( k e. Z <-> k e. ( ZZ>= ` M ) )
34	33, 12	sylan2br 288	. . . . . . . 8 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. CC )
35	34	adantlr 481	. . . . . . 7 |- ( ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. CC )
36	28, 30, 31, 32, 35	seq3split 10925	. . . . . 6 |- ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) -> ( seq M ( x. , F ) ` j ) = ( ( seq M ( x. , F ) ` N ) x. ( seq ( N + 1 ) ( x. , F ) ` j ) ) )
37	36	eqcomd 2244	. . . . 5 |- ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) -> ( ( seq M ( x. , F ) ` N ) x. ( seq ( N + 1 ) ( x. , F ) ` j ) ) = ( seq M ( x. , F ) ` j ) )
38	14	adantr 276	. . . . . 6 |- ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) -> ( seq M ( x. , F ) ` N ) e. CC )
39	4	uztrn2 9940	. . . . . . . . . 10 |- ( ( ( N + 1 ) e. Z /\ k e. ( ZZ>= ` ( N + 1 ) ) ) -> k e. Z )
40	22, 39	sylan 283	. . . . . . . . 9 |- ( ( ph /\ k e. ( ZZ>= ` ( N + 1 ) ) ) -> k e. Z )
41	40, 12	syldan 282	. . . . . . . 8 |- ( ( ph /\ k e. ( ZZ>= ` ( N + 1 ) ) ) -> ( F ` k ) e. CC )
42	1, 7, 41	prodf 12305	. . . . . . 7 |- ( ph -> seq ( N + 1 ) ( x. , F ) : ( ZZ>= ` ( N + 1 ) ) --> CC )
43	42	ffvelcdmda 5843	. . . . . 6 |- ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) -> ( seq ( N + 1 ) ( x. , F ) ` j ) e. CC )
44	15	adantr 276	. . . . . 6 |- ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) -> ( seq M ( x. , F ) ` N ) # 0 )
45	26, 38, 43, 44	divmulapd 9142	. . . . 5 |- ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) -> ( ( ( seq M ( x. , F ) ` j ) / ( seq M ( x. , F ) ` N ) ) = ( seq ( N + 1 ) ( x. , F ) ` j ) <-> ( ( seq M ( x. , F ) ` N ) x. ( seq ( N + 1 ) ( x. , F ) ` j ) ) = ( seq M ( x. , F ) ` j ) ) )
46	37, 45	mpbird 167	. . . 4 |- ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) -> ( ( seq M ( x. , F ) ` j ) / ( seq M ( x. , F ) ` N ) ) = ( seq ( N + 1 ) ( x. , F ) ` j ) )
47	26, 38, 44	divrecap2d 9124	. . . 4 |- ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) -> ( ( seq M ( x. , F ) ` j ) / ( seq M ( x. , F ) ` N ) ) = ( ( 1 / ( seq M ( x. , F ) ` N ) ) x. ( seq M ( x. , F ) ` j ) ) )
48	46, 47	eqtr3d 2273	. . 3 |- ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) -> ( seq ( N + 1 ) ( x. , F ) ` j ) = ( ( 1 / ( seq M ( x. , F ) ` N ) ) x. ( seq M ( x. , F ) ` j ) ) )
49	1, 7, 8, 16, 18, 26, 48	climmulc2 12097	. 2 |- ( ph -> seq ( N + 1 ) ( x. , F ) ~~> ( ( 1 / ( seq M ( x. , F ) ` N ) ) x. A ) )
50		climcl 12048	. . . 4 |- ( seq M ( x. , F ) ~~> A -> A e. CC )
51	8, 50	syl 14	. . 3 |- ( ph -> A e. CC )
52	51, 14, 15	divrecap2d 9124	. 2 |- ( ph -> ( A / ( seq M ( x. , F ) ` N ) ) = ( ( 1 / ( seq M ( x. , F ) ` N ) ) x. A ) )
53	49, 52	breqtrrd 4158	1 |- ( ph -> seq ( N + 1 ) ( x. , F ) ~~> ( A / ( seq M ( x. , F ) ` N ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   /\ w3a 1009   = wceq 1402   e. wcel 2209   _Vcvv 2821   class class class wbr 4130   ` cfv 5377  (class class class)co 6085   CCcc 8177   0cc0 8179   1c1 8180   + caddc 8182   x. cmul 8184   # cap 8909   / cdiv 9002   ZZcz 9644   ZZ>=cuz 9921   seqcseq 10884   ~~> cli 12044

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297  ax-arch 8298  ax-caucvg 8299

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8903  df-ap 8910  df-div 9003  df-inn 9305  df-2 9363  df-3 9364  df-4 9365  df-n0 9564  df-z 9645  df-uz 9922  df-rp 10055  df-fz 10412  df-seqfrec 10885  df-exp 10976  df-cj 11607  df-re 11608  df-im 11609  df-rsqrt 11764  df-abs 11765  df-clim 12045

This theorem is used by: (None)