Theorem ntrivcvgap 11603

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 9600	. . . . . . . . 9 |- ( n e. ( ZZ>= ` M ) -> ( n = M \/ ( n - 1 ) e. ( ZZ>= ` M ) ) )
3		ntrivcvg.1	. . . . . . . . 9 |- Z = ( ZZ>= ` M )
4	2, 3	eleq2s 2284	. . . . . . . 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 10493	. . . . . . . . . . 11 |- ( n = M -> seq n ( x. , F ) = seq M ( x. , F ) )
7	6	breq1d 4035	. . . . . . . . . 10 |- ( n = M -> ( seq n ( x. , F ) ~~> y <-> seq M ( x. , F ) ~~> y ) )
8		seqex 10492	. . . . . . . . . . 11 |- seq M ( x. , F ) e. _V
9		vex 2759	. . . . . . . . . . 11 |- y e. _V
10	8, 9	breldm 4856	. . . . . . . . . 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 9574	. . . . . . . . . . . . . . . . 17 |- ( n e. ( ZZ>= ` M ) -> M e. ZZ )
14	13, 3	eleq2s 2284	. . . . . . . . . . . . . . . 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 11593	. . . . . . . . . . . . . 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 5680	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ n e. Z ) /\ ( n - 1 ) e. Z ) /\ seq n ( x. , F ) ~~> y ) -> ( seq M ( x. , F ) ` ( n - 1 ) ) e. CC )
21		climcl 11337	. . . . . . . . . . . . . 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 8019	. . . . . . . . . . . 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 9589	. . . . . . . . . . . . . . . . . . . 20 |- ( ZZ>= ` M ) C_ ZZ
25	3, 24	eqsstri 3206	. . . . . . . . . . . . . . . . . . 19 |- Z C_ ZZ
26		simplr 528	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ph /\ n e. Z ) /\ ( n - 1 ) e. Z ) -> n e. Z )
27	25, 26	sselid 3172	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ph /\ n e. Z ) /\ ( n - 1 ) e. Z ) -> n e. ZZ )
28	27	zcnd 9417	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ n e. Z ) /\ ( n - 1 ) e. Z ) -> n e. CC )
29		1cnd 8014	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ n e. Z ) /\ ( n - 1 ) e. Z ) -> 1 e. CC )
30	28, 29	npcand 8313	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ n e. Z ) /\ ( n - 1 ) e. Z ) -> ( ( n - 1 ) + 1 ) = n )
31	30	seqeq1d 10496	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ n e. Z ) /\ ( n - 1 ) e. Z ) -> seq ( ( n - 1 ) + 1 ) ( x. , F ) = seq n ( x. , F ) )
32	31	breq1d 4035	. . . . . . . . . . . . . 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 11594	. . . . . . . . . . . 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 4858	. . . . . . . . . . . 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 2193	. . . . . . . . 9 |- ( ZZ>= ` M ) = Z
40	38, 39	eleq2s 2284	. . . . . . . 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 2607	. 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 2160   E.wrex 2469   _Vcvv 2756   class class class wbr 4025   dom cdm 4651   ` cfv 5242  (class class class)co 5904   CCcc 7849   0cc0 7851   1c1 7852   + caddc 7854   x. cmul 7856   - cmin 8169   # cap 8579   ZZcz 9294   ZZ>=cuz 9569   seqcseq 10490   ~~> cli 11333

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 2162  ax-14 2163  ax-ext 2171  ax-coll 4140  ax-sep 4143  ax-nul 4151  ax-pow 4199  ax-pr 4234  ax-un 4458  ax-setind 4561  ax-iinf 4612  ax-cnex 7942  ax-resscn 7943  ax-1cn 7944  ax-1re 7945  ax-icn 7946  ax-addcl 7947  ax-addrcl 7948  ax-mulcl 7949  ax-mulrcl 7950  ax-addcom 7951  ax-mulcom 7952  ax-addass 7953  ax-mulass 7954  ax-distr 7955  ax-i2m1 7956  ax-0lt1 7957  ax-1rid 7958  ax-0id 7959  ax-rnegex 7960  ax-precex 7961  ax-cnre 7962  ax-pre-ltirr 7963  ax-pre-ltwlin 7964  ax-pre-lttrn 7965  ax-pre-apti 7966  ax-pre-ltadd 7967  ax-pre-mulgt0 7968  ax-pre-mulext 7969  ax-arch 7970  ax-caucvg 7971

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 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2758  df-sbc 2982  df-csb 3077  df-dif 3150  df-un 3152  df-in 3154  df-ss 3161  df-nul 3442  df-if 3554  df-pw 3599  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3832  df-int 3867  df-iun 3910  df-br 4026  df-opab 4087  df-mpt 4088  df-tr 4124  df-id 4318  df-po 4321  df-iso 4322  df-iord 4391  df-on 4393  df-ilim 4394  df-suc 4396  df-iom 4615  df-xp 4657  df-rel 4658  df-cnv 4659  df-co 4660  df-dm 4661  df-rn 4662  df-res 4663  df-ima 4664  df-iota 5203  df-fun 5244  df-fn 5245  df-f 5246  df-f1 5247  df-fo 5248  df-f1o 5249  df-fv 5250  df-riota 5859  df-ov 5907  df-oprab 5908  df-mpo 5909  df-1st 6173  df-2nd 6174  df-recs 6338  df-frec 6424  df-pnf 8035  df-mnf 8036  df-xr 8037  df-ltxr 8038  df-le 8039  df-sub 8171  df-neg 8172  df-reap 8573  df-ap 8580  df-div 8671  df-inn 8961  df-2 9019  df-3 9020  df-4 9021  df-n0 9218  df-z 9295  df-uz 9570  df-rp 9696  df-seqfrec 10491  df-exp 10566  df-cj 10898  df-re 10899  df-im 10900  df-rsqrt 11054  df-abs 11055  df-clim 11334

This theorem is referenced by: (None)