Theorem ntrivcvgap 12234

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 9885	. . . . . . . . 9 |- ( n e. ( ZZ>= ` M ) -> ( n = M \/ ( n - 1 ) e. ( ZZ>= ` M ) ) )
3		ntrivcvg.1	. . . . . . . . 9 |- Z = ( ZZ>= ` M )
4	2, 3	eleq2s 2327	. . . . . . . 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 10812	. . . . . . . . . . 11 |- ( n = M -> seq n ( x. , F ) = seq M ( x. , F ) )
7	6	breq1d 4119	. . . . . . . . . 10 |- ( n = M -> ( seq n ( x. , F ) ~~> y <-> seq M ( x. , F ) ~~> y ) )
8		seqex 10811	. . . . . . . . . . 11 |- seq M ( x. , F ) e. _V
9		vex 2816	. . . . . . . . . . 11 |- y e. _V
10	8, 9	breldm 4960	. . . . . . . . . 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 9858	. . . . . . . . . . . . . . . . 17 |- ( n e. ( ZZ>= ` M ) -> M e. ZZ )
14	13, 3	eleq2s 2327	. . . . . . . . . . . . . . . 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 12224	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ n e. Z ) /\ ( n - 1 ) e. Z ) /\ seq n ( x. , F ) ~~> y ) -> seq M ( x. , F ) : Z --> CC )
19		simplr 529	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ n e. Z ) /\ ( n - 1 ) e. Z ) /\ seq n ( x. , F ) ~~> y ) -> ( n - 1 ) e. Z )
20	18, 19	ffvelcdmd 5813	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ n e. Z ) /\ ( n - 1 ) e. Z ) /\ seq n ( x. , F ) ~~> y ) -> ( seq M ( x. , F ) ` ( n - 1 ) ) e. CC )
21		climcl 11967	. . . . . . . . . . . . . 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 8294	. . . . . . . . . . . 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 9874	. . . . . . . . . . . . . . . . . . . 20 |- ( ZZ>= ` M ) C_ ZZ
25	3, 24	eqsstri 3270	. . . . . . . . . . . . . . . . . . 19 |- Z C_ ZZ
26		simplr 529	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ph /\ n e. Z ) /\ ( n - 1 ) e. Z ) -> n e. Z )
27	25, 26	sselid 3236	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ph /\ n e. Z ) /\ ( n - 1 ) e. Z ) -> n e. ZZ )
28	27	zcnd 9701	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ n e. Z ) /\ ( n - 1 ) e. Z ) -> n e. CC )
29		1cnd 8290	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ n e. Z ) /\ ( n - 1 ) e. Z ) -> 1 e. CC )
30	28, 29	npcand 8588	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ n e. Z ) /\ ( n - 1 ) e. Z ) -> ( ( n - 1 ) + 1 ) = n )
31	30	seqeq1d 10815	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ n e. Z ) /\ ( n - 1 ) e. Z ) -> seq ( ( n - 1 ) + 1 ) ( x. , F ) = seq n ( x. , F ) )
32	31	breq1d 4119	. . . . . . . . . . . . . 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 12225	. . . . . . . . . . . 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 4962	. . . . . . . . . . . 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 1379	. . . . . . . . . . 11 |- ( ( ( ( ph /\ n e. Z ) /\ ( n - 1 ) e. Z ) /\ seq n ( x. , F ) ~~> y ) -> seq M ( x. , F ) e. dom ~~> )
37	36	an32s 570	. . . . . . . . . 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 2236	. . . . . . . . 9 |- ( ZZ>= ` M ) = Z
40	38, 39	eleq2s 2327	. . . . . . . 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 724	. . . . . . 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 1868	. . 3 |- ( ( ph /\ n e. Z ) -> ( E. y ( y # 0 /\ seq n ( x. , F ) ~~> y ) -> seq M ( x. , F ) e. dom ~~> ) )
46	45	rexlimdva 2660	. 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 716   = wceq 1398   E.wex 1541   e. wcel 2203   E.wrex 2521   _Vcvv 2813   class class class wbr 4109   dom cdm 4749   ` cfv 5352  (class class class)co 6050   CCcc 8125   0cc0 8127   1c1 8128   + caddc 8130   x. cmul 8132   - cmin 8444   # cap 8855   ZZcz 9577   ZZ>=cuz 9853   seqcseq 10809   ~~> cli 11963

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 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-mulrcl 8226  ax-addcom 8227  ax-mulcom 8228  ax-addass 8229  ax-mulass 8230  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-1rid 8234  ax-0id 8235  ax-rnegex 8236  ax-precex 8237  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-apti 8242  ax-pre-ltadd 8243  ax-pre-mulgt0 8244  ax-pre-mulext 8245  ax-arch 8246  ax-caucvg 8247

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 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-po 4417  df-iso 4418  df-iord 4487  df-on 4489  df-ilim 4490  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-frec 6622  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-reap 8849  df-ap 8856  df-div 8947  df-inn 9238  df-2 9296  df-3 9297  df-4 9298  df-n0 9497  df-z 9578  df-uz 9854  df-rp 9987  df-seqfrec 10810  df-exp 10901  df-cj 11527  df-re 11528  df-im 11529  df-rsqrt 11683  df-abs 11684  df-clim 11964

This theorem is referenced by: (None)