Theorem clim2divap 12164

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 2231	. . 3 |- ( ZZ>= ` ( N + 1 ) ) = ( ZZ>= ` ( N + 1 ) )
2		clim2div.2	. . . . 5 |- ( ph -> N e. Z )
3		eluzelz 9809	. . . . . 6 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
4		clim2div.1	. . . . . 6 |- Z = ( ZZ>= ` M )
5	3, 4	eleq2s 2326	. . . . 5 |- ( N e. Z -> N e. ZZ )
6	2, 5	syl 14	. . . 4 |- ( ph -> N e. ZZ )
7	6	peano2zd 9649	. . 3 |- ( ph -> ( N + 1 ) e. ZZ )
8		clim2div.4	. . 3 |- ( ph -> seq M ( x. , F ) ~~> A )
9		eluzel2 9804	. . . . . . . 8 |- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
10	9, 4	eleq2s 2326	. . . . . . 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 12162	. . . . 5 |- ( ph -> seq M ( x. , F ) : Z --> CC )
14	13, 2	ffvelcdmd 5791	. . . 4 |- ( ph -> ( seq M ( x. , F ) ` N ) e. CC )
15		clim2divap.5	. . . 4 |- ( ph -> ( seq M ( x. , F ) ` N ) # 0 )
16	14, 15	recclapd 9003	. . 3 |- ( ph -> ( 1 / ( seq M ( x. , F ) ` N ) ) e. CC )
17		seqex 10757	. . . 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 2324	. . . . . . 7 |- ( ph -> N e. ( ZZ>= ` M ) )
20		peano2uz 9861	. . . . . . 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 2325	. . . . 5 |- ( ph -> ( N + 1 ) e. Z )
23	4	uztrn2 9818	. . . . 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 5790	. . . 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 8202	. . . . . . . 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 8206	. . . . . . . 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 2298	. . . . . . . . 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 477	. . . . . . 7 |- ( ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. CC )
36	28, 30, 31, 32, 35	seq3split 10796	. . . . . 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 2237	. . . . 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 9818	. . . . . . . . . 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 12162	. . . . . . 7 |- ( ph -> seq ( N + 1 ) ( x. , F ) : ( ZZ>= ` ( N + 1 ) ) --> CC )
43	42	ffvelcdmda 5790	. . . . . 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 9034	. . . . 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 9016	. . . 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 2266	. . 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 11954	. 2 |- ( ph -> seq ( N + 1 ) ( x. , F ) ~~> ( ( 1 / ( seq M ( x. , F ) ` N ) ) x. A ) )
50		climcl 11905	. . . 4 |- ( seq M ( x. , F ) ~~> A -> A e. CC )
51	8, 50	syl 14	. . 3 |- ( ph -> A e. CC )
52	51, 14, 15	divrecap2d 9016	. 2 |- ( ph -> ( A / ( seq M ( x. , F ) ` N ) ) = ( ( 1 / ( seq M ( x. , F ) ` N ) ) x. A ) )
53	49, 52	breqtrrd 4121	1 |- ( ph -> seq ( N + 1 ) ( x. , F ) ~~> ( A / ( seq M ( x. , F ) ` N ) ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1005   = wceq 1398   e. wcel 2202   _Vcvv 2803   class class class wbr 4093   ` cfv 5333  (class class class)co 6028   CCcc 8073   0cc0 8075   1c1 8076   + caddc 8078   x. cmul 8080   # cap 8803   / cdiv 8894   ZZcz 9523   ZZ>=cuz 9799   seqcseq 10755   ~~> cli 11901

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-mulrcl 8174  ax-addcom 8175  ax-mulcom 8176  ax-addass 8177  ax-mulass 8178  ax-distr 8179  ax-i2m1 8180  ax-0lt1 8181  ax-1rid 8182  ax-0id 8183  ax-rnegex 8184  ax-precex 8185  ax-cnre 8186  ax-pre-ltirr 8187  ax-pre-ltwlin 8188  ax-pre-lttrn 8189  ax-pre-apti 8190  ax-pre-ltadd 8191  ax-pre-mulgt0 8192  ax-pre-mulext 8193  ax-arch 8194  ax-caucvg 8195

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-frec 6600  df-pnf 8258  df-mnf 8259  df-xr 8260  df-ltxr 8261  df-le 8262  df-sub 8394  df-neg 8395  df-reap 8797  df-ap 8804  df-div 8895  df-inn 9186  df-2 9244  df-3 9245  df-4 9246  df-n0 9445  df-z 9524  df-uz 9800  df-rp 9933  df-fz 10289  df-seqfrec 10756  df-exp 10847  df-cj 11465  df-re 11466  df-im 11467  df-rsqrt 11621  df-abs 11622  df-clim 11902

This theorem is referenced by: (None)