Theorem ntrivcvgap 12315

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 9953	. . . . . . . . 9 |- ( n e. ( ZZ>= ` M ) -> ( n = M \/ ( n - 1 ) e. ( ZZ>= ` M ) ) )
3		ntrivcvg.1	. . . . . . . . 9 |- Z = ( ZZ>= ` M )
4	2, 3	eleq2s 2333	. . . . . . . 8 |- ( n e. Z -> ( n = M \/ ( n - 1 ) e. ( ZZ>= ` M ) ) )
5	4	ad2antlr 493	. . . . . . 7 |- ( ( ( ph /\ n e. Z ) /\ seq n ( x. , F ) ~~> y ) -> ( n = M \/ ( n - 1 ) e. ( ZZ>= ` M ) ) )
6		seqeq1 10887	. . . . . . . . . . 11 |- ( n = M -> seq n ( x. , F ) = seq M ( x. , F ) )
7	6	breq1d 4140	. . . . . . . . . 10 |- ( n = M -> ( seq n ( x. , F ) ~~> y <-> seq M ( x. , F ) ~~> y ) )
8		seqex 10886	. . . . . . . . . . 11 |- seq M ( x. , F ) e. _V
9		vex 2824	. . . . . . . . . . 11 |- y e. _V
10	8, 9	breldm 4985	. . . . . . . . . 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 9926	. . . . . . . . . . . . . . . . 17 |- ( n e. ( ZZ>= ` M ) -> M e. ZZ )
14	13, 3	eleq2s 2333	. . . . . . . . . . . . . . . 16 |- ( n e. Z -> M e. ZZ )
15	14	ad3antlr 497	. . . . . . . . . . . . . . 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 525	. . . . . . . . . . . . . . 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 12305	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ n e. Z ) /\ ( n - 1 ) e. Z ) /\ seq n ( x. , F ) ~~> y ) -> seq M ( x. , F ) : Z --> CC )
19		simplr 533	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ n e. Z ) /\ ( n - 1 ) e. Z ) /\ seq n ( x. , F ) ~~> y ) -> ( n - 1 ) e. Z )
20	18, 19	ffvelcdmd 5844	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ n e. Z ) /\ ( n - 1 ) e. Z ) /\ seq n ( x. , F ) ~~> y ) -> ( seq M ( x. , F ) ` ( n - 1 ) ) e. CC )
21		climcl 12048	. . . . . . . . . . . . . 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 8346	. . . . . . . . . . . 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 9942	. . . . . . . . . . . . . . . . . . . 20 |- ( ZZ>= ` M ) C_ ZZ
25	3, 24	eqsstri 3280	. . . . . . . . . . . . . . . . . . 19 |- Z C_ ZZ
26		simplr 533	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ph /\ n e. Z ) /\ ( n - 1 ) e. Z ) -> n e. Z )
27	25, 26	sselid 3246	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ph /\ n e. Z ) /\ ( n - 1 ) e. Z ) -> n e. ZZ )
28	27	zcnd 9769	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ n e. Z ) /\ ( n - 1 ) e. Z ) -> n e. CC )
29		1cnd 8342	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ n e. Z ) /\ ( n - 1 ) e. Z ) -> 1 e. CC )
30	28, 29	npcand 8641	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ n e. Z ) /\ ( n - 1 ) e. Z ) -> ( ( n - 1 ) + 1 ) = n )
31	30	seqeq1d 10890	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ n e. Z ) /\ ( n - 1 ) e. Z ) -> seq ( ( n - 1 ) + 1 ) ( x. , F ) = seq n ( x. , F ) )
32	31	breq1d 4140	. . . . . . . . . . . . . 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 12306	. . . . . . . . . . . 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 4987	. . . . . . . . . . . 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 1383	. . . . . . . . . . 11 |- ( ( ( ( ph /\ n e. Z ) /\ ( n - 1 ) e. Z ) /\ seq n ( x. , F ) ~~> y ) -> seq M ( x. , F ) e. dom ~~> )
37	36	an32s 574	. . . . . . . . . 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 2242	. . . . . . . . 9 |- ( ZZ>= ` M ) = Z
40	38, 39	eleq2s 2333	. . . . . . . 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 728	. . . . . . 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 1872	. . 3 |- ( ( ph /\ n e. Z ) -> ( E. y ( y # 0 /\ seq n ( x. , F ) ~~> y ) -> seq M ( x. , F ) e. dom ~~> ) )
46	45	rexlimdva 2668	. 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 ~~> )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   \/ wo 720   = wceq 1402   E.wex 1545   e. wcel 2209   E.wrex 2529   _Vcvv 2821   class class class wbr 4130   dom cdm 4774   ` cfv 5377  (class class class)co 6085   CCcc 8177   0cc0 8179   1c1 8180   + caddc 8182   x. cmul 8184   - cmin 8497   # cap 8909   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-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)