Theorem ntrivcvgap 11691

Description: A non-trivially converging infinite product converges. (Contributed by Scott Fenton, 18-Dec-2017.)

Hypotheses

Ref	Expression
ntrivcvg.1	|- Z = ( ZZ>= ` M )
ntrivcvgap.2	|- ( ph -> E. n e. Z E. y ( y # 0 /\ seq n ( x. , F ) ~~> y ) )
ntrivcvg.3	|- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )

Assertion

Ref	Expression
ntrivcvgap	|- ( ph -> seq M ( x. , F ) e. dom ~~> )

Distinct variable groups:   k, F, n, y   k, M, n, y   k, Z, y   ph, k, n, y

Allowed substitution hint:   Z( n)

Proof of Theorem ntrivcvgap

Step	Hyp	Ref	Expression
1		ntrivcvgap.2	. 2 |- ( ph -> E. n e. Z E. y ( y # 0 /\ seq n ( x. , F ) ~~> y ) )
2		uzm1 9623	. . . . . . . . 9 |- ( n e. ( ZZ>= ` M ) -> ( n = M \/ ( n - 1 ) e. ( ZZ>= ` M ) ) )
3		ntrivcvg.1	. . . . . . . . 9 |- Z = ( ZZ>= ` M )
4	2, 3	eleq2s 2288	. . . . . . . 8 |- ( n e. Z -> ( n = M \/ ( n - 1 ) e. ( ZZ>= ` M ) ) )
5	4	ad2antlr 489	. . . . . . 7 |- ( ( ( ph /\ n e. Z ) /\ seq n ( x. , F ) ~~> y ) -> ( n = M \/ ( n - 1 ) e. ( ZZ>= ` M ) ) )
6		seqeq1 10521	. . . . . . . . . . 11 |- ( n = M -> seq n ( x. , F ) = seq M ( x. , F ) )
7	6	breq1d 4039	. . . . . . . . . 10 |- ( n = M -> ( seq n ( x. , F ) ~~> y <-> seq M ( x. , F ) ~~> y ) )
8		seqex 10520	. . . . . . . . . . 11 |- seq M ( x. , F ) e. _V
9		vex 2763	. . . . . . . . . . 11 |- y e. _V
10	8, 9	breldm 4866	. . . . . . . . . 10 |- ( seq M ( x. , F ) ~~> y -> seq M ( x. , F ) e. dom ~~> )
11	7, 10	biimtrdi 163	. . . . . . . . 9 |- ( n = M -> ( seq n ( x. , F ) ~~> y -> seq M ( x. , F ) e. dom ~~> ) )
12	11	adantld 278	. . . . . . . 8 |- ( n = M -> ( ( ( ph /\ n e. Z ) /\ seq n ( x. , F ) ~~> y ) -> seq M ( x. , F ) e. dom ~~> ) )
13		eluzel2 9597	. . . . . . . . . . . . . . . . 17 |- ( n e. ( ZZ>= ` M ) -> M e. ZZ )
14	13, 3	eleq2s 2288	. . . . . . . . . . . . . . . 16 |- ( n e. Z -> M e. ZZ )
15	14	ad3antlr 493	. . . . . . . . . . . . . . 15 |- ( ( ( ( ph /\ n e. Z ) /\ ( n - 1 ) e. Z ) /\ seq n ( x. , F ) ~~> y ) -> M e. ZZ )
16		ntrivcvg.3	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )
17	16	ad5ant15 521	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ph /\ n e. Z ) /\ ( n - 1 ) e. Z ) /\ seq n ( x. , F ) ~~> y ) /\ k e. Z ) -> ( F ` k ) e. CC )
18	3, 15, 17	prodf 11681	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ n e. Z ) /\ ( n - 1 ) e. Z ) /\ seq n ( x. , F ) ~~> y ) -> seq M ( x. , F ) : Z --> CC )
19		simplr 528	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ n e. Z ) /\ ( n - 1 ) e. Z ) /\ seq n ( x. , F ) ~~> y ) -> ( n - 1 ) e. Z )
20	18, 19	ffvelcdmd 5694	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ n e. Z ) /\ ( n - 1 ) e. Z ) /\ seq n ( x. , F ) ~~> y ) -> ( seq M ( x. , F ) ` ( n - 1 ) ) e. CC )
21		climcl 11425	. . . . . . . . . . . . . 14 |- ( seq n ( x. , F ) ~~> y -> y e. CC )
22	21	adantl 277	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ n e. Z ) /\ ( n - 1 ) e. Z ) /\ seq n ( x. , F ) ~~> y ) -> y e. CC )
23	20, 22	mulcld 8040	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ n e. Z ) /\ ( n - 1 ) e. Z ) /\ seq n ( x. , F ) ~~> y ) -> ( ( seq M ( x. , F ) ` ( n - 1 ) ) x. y ) e. CC )
24		uzssz 9612	. . . . . . . . . . . . . . . . . . . 20 |- ( ZZ>= ` M ) C_ ZZ
25	3, 24	eqsstri 3211	. . . . . . . . . . . . . . . . . . 19 |- Z C_ ZZ
26		simplr 528	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ph /\ n e. Z ) /\ ( n - 1 ) e. Z ) -> n e. Z )
27	25, 26	sselid 3177	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ph /\ n e. Z ) /\ ( n - 1 ) e. Z ) -> n e. ZZ )
28	27	zcnd 9440	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ n e. Z ) /\ ( n - 1 ) e. Z ) -> n e. CC )
29		1cnd 8035	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ n e. Z ) /\ ( n - 1 ) e. Z ) -> 1 e. CC )
30	28, 29	npcand 8334	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ n e. Z ) /\ ( n - 1 ) e. Z ) -> ( ( n - 1 ) + 1 ) = n )
31	30	seqeq1d 10524	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ n e. Z ) /\ ( n - 1 ) e. Z ) -> seq ( ( n - 1 ) + 1 ) ( x. , F ) = seq n ( x. , F ) )
32	31	breq1d 4039	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ n e. Z ) /\ ( n - 1 ) e. Z ) -> ( seq ( ( n - 1 ) + 1 ) ( x. , F ) ~~> y <-> seq n ( x. , F ) ~~> y ) )
33	32	biimpar 297	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ n e. Z ) /\ ( n - 1 ) e. Z ) /\ seq n ( x. , F ) ~~> y ) -> seq ( ( n - 1 ) + 1 ) ( x. , F ) ~~> y )
34	3, 19, 17, 33	clim2prod 11682	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ n e. Z ) /\ ( n - 1 ) e. Z ) /\ seq n ( x. , F ) ~~> y ) -> seq M ( x. , F ) ~~> ( ( seq M ( x. , F ) ` ( n - 1 ) ) x. y ) )
35		breldmg 4868	. . . . . . . . . . . 12 |- ( ( seq M ( x. , F ) e. _V /\ ( ( seq M ( x. , F ) ` ( n - 1 ) ) x. y ) e. CC /\ seq M ( x. , F ) ~~> ( ( seq M ( x. , F ) ` ( n - 1 ) ) x. y ) ) -> seq M ( x. , F ) e. dom ~~> )
36	8, 23, 34, 35	mp3an2i 1353	. . . . . . . . . . 11 |- ( ( ( ( ph /\ n e. Z ) /\ ( n - 1 ) e. Z ) /\ seq n ( x. , F ) ~~> y ) -> seq M ( x. , F ) e. dom ~~> )
37	36	an32s 568	. . . . . . . . . 10 |- ( ( ( ( ph /\ n e. Z ) /\ seq n ( x. , F ) ~~> y ) /\ ( n - 1 ) e. Z ) -> seq M ( x. , F ) e. dom ~~> )
38	37	expcom 116	. . . . . . . . 9 |- ( ( n - 1 ) e. Z -> ( ( ( ph /\ n e. Z ) /\ seq n ( x. , F ) ~~> y ) -> seq M ( x. , F ) e. dom ~~> ) )
39	3	eqcomi 2197	. . . . . . . . 9 |- ( ZZ>= ` M ) = Z
40	38, 39	eleq2s 2288	. . . . . . . 8 |- ( ( n - 1 ) e. ( ZZ>= ` M ) -> ( ( ( ph /\ n e. Z ) /\ seq n ( x. , F ) ~~> y ) -> seq M ( x. , F ) e. dom ~~> ) )
41	12, 40	jaoi 717	. . . . . . 7 |- ( ( n = M \/ ( n - 1 ) e. ( ZZ>= ` M ) ) -> ( ( ( ph /\ n e. Z ) /\ seq n ( x. , F ) ~~> y ) -> seq M ( x. , F ) e. dom ~~> ) )
42	5, 41	mpcom 36	. . . . . 6 |- ( ( ( ph /\ n e. Z ) /\ seq n ( x. , F ) ~~> y ) -> seq M ( x. , F ) e. dom ~~> )
43	42	ex 115	. . . . 5 |- ( ( ph /\ n e. Z ) -> ( seq n ( x. , F ) ~~> y -> seq M ( x. , F ) e. dom ~~> ) )
44	43	adantld 278	. . . 4 |- ( ( ph /\ n e. Z ) -> ( ( y # 0 /\ seq n ( x. , F ) ~~> y ) -> seq M ( x. , F ) e. dom ~~> ) )
45	44	exlimdv 1830	. . 3 |- ( ( ph /\ n e. Z ) -> ( E. y ( y # 0 /\ seq n ( x. , F ) ~~> y ) -> seq M ( x. , F ) e. dom ~~> ) )
46	45	rexlimdva 2611	. 2 |- ( ph -> ( E. n e. Z E. y ( y # 0 /\ seq n ( x. , F ) ~~> y ) -> seq M ( x. , F ) e. dom ~~> ) )
47	1, 46	mpd 13	1 |- ( ph -> seq M ( x. , F ) e. dom ~~> )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 709   = wceq 1364   E.wex 1503   e. wcel 2164   E.wrex 2473   _Vcvv 2760   class class class wbr 4029   dom cdm 4659   ` cfv 5254  (class class class)co 5918   CCcc 7870   0cc0 7872   1c1 7873   + caddc 7875   x. cmul 7877   - cmin 8190   # cap 8600   ZZcz 9317   ZZ>=cuz 9592   seqcseq 10518   ~~> cli 11421

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-mulrcl 7971  ax-addcom 7972  ax-mulcom 7973  ax-addass 7974  ax-mulass 7975  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-1rid 7979  ax-0id 7980  ax-rnegex 7981  ax-precex 7982  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-apti 7987  ax-pre-ltadd 7988  ax-pre-mulgt0 7989  ax-pre-mulext 7990  ax-arch 7991  ax-caucvg 7992

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 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-ilim 4400  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-frec 6444  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-reap 8594  df-ap 8601  df-div 8692  df-inn 8983  df-2 9041  df-3 9042  df-4 9043  df-n0 9241  df-z 9318  df-uz 9593  df-rp 9720  df-seqfrec 10519  df-exp 10610  df-cj 10986  df-re 10987  df-im 10988  df-rsqrt 11142  df-abs 11143  df-clim 11422

This theorem is referenced by: (None)