Theorem ntrivcvgap 11830

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 9678	. . . . . . . . 9 |- ( n e. ( ZZ>= ` M ) -> ( n = M \/ ( n - 1 ) e. ( ZZ>= ` M ) ) )
3		ntrivcvg.1	. . . . . . . . 9 |- Z = ( ZZ>= ` M )
4	2, 3	eleq2s 2299	. . . . . . . 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 10593	. . . . . . . . . . 11 |- ( n = M -> seq n ( x. , F ) = seq M ( x. , F ) )
7	6	breq1d 4053	. . . . . . . . . 10 |- ( n = M -> ( seq n ( x. , F ) ~~> y <-> seq M ( x. , F ) ~~> y ) )
8		seqex 10592	. . . . . . . . . . 11 |- seq M ( x. , F ) e. _V
9		vex 2774	. . . . . . . . . . 11 |- y e. _V
10	8, 9	breldm 4881	. . . . . . . . . 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 9652	. . . . . . . . . . . . . . . . 17 |- ( n e. ( ZZ>= ` M ) -> M e. ZZ )
14	13, 3	eleq2s 2299	. . . . . . . . . . . . . . . 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 11820	. . . . . . . . . . . . . 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 5715	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ n e. Z ) /\ ( n - 1 ) e. Z ) /\ seq n ( x. , F ) ~~> y ) -> ( seq M ( x. , F ) ` ( n - 1 ) ) e. CC )
21		climcl 11564	. . . . . . . . . . . . . 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 8092	. . . . . . . . . . . 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 9667	. . . . . . . . . . . . . . . . . . . 20 |- ( ZZ>= ` M ) C_ ZZ
25	3, 24	eqsstri 3224	. . . . . . . . . . . . . . . . . . 19 |- Z C_ ZZ
26		simplr 528	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ph /\ n e. Z ) /\ ( n - 1 ) e. Z ) -> n e. Z )
27	25, 26	sselid 3190	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ph /\ n e. Z ) /\ ( n - 1 ) e. Z ) -> n e. ZZ )
28	27	zcnd 9495	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ n e. Z ) /\ ( n - 1 ) e. Z ) -> n e. CC )
29		1cnd 8087	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ n e. Z ) /\ ( n - 1 ) e. Z ) -> 1 e. CC )
30	28, 29	npcand 8386	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ n e. Z ) /\ ( n - 1 ) e. Z ) -> ( ( n - 1 ) + 1 ) = n )
31	30	seqeq1d 10596	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ n e. Z ) /\ ( n - 1 ) e. Z ) -> seq ( ( n - 1 ) + 1 ) ( x. , F ) = seq n ( x. , F ) )
32	31	breq1d 4053	. . . . . . . . . . . . . 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 11821	. . . . . . . . . . . 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 4883	. . . . . . . . . . . 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 1354	. . . . . . . . . . 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 2208	. . . . . . . . 9 |- ( ZZ>= ` M ) = Z
40	38, 39	eleq2s 2299	. . . . . . . 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 1841	. . 3 |- ( ( ph /\ n e. Z ) -> ( E. y ( y # 0 /\ seq n ( x. , F ) ~~> y ) -> seq M ( x. , F ) e. dom ~~> ) )
46	45	rexlimdva 2622	. 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 1372   E.wex 1514   e. wcel 2175   E.wrex 2484   _Vcvv 2771   class class class wbr 4043   dom cdm 4674   ` cfv 5270  (class class class)co 5943   CCcc 7922   0cc0 7924   1c1 7925   + caddc 7927   x. cmul 7929   - cmin 8242   # cap 8653   ZZcz 9371   ZZ>=cuz 9647   seqcseq 10590   ~~> cli 11560

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-nul 4169  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584  ax-iinf 4635  ax-cnex 8015  ax-resscn 8016  ax-1cn 8017  ax-1re 8018  ax-icn 8019  ax-addcl 8020  ax-addrcl 8021  ax-mulcl 8022  ax-mulrcl 8023  ax-addcom 8024  ax-mulcom 8025  ax-addass 8026  ax-mulass 8027  ax-distr 8028  ax-i2m1 8029  ax-0lt1 8030  ax-1rid 8031  ax-0id 8032  ax-rnegex 8033  ax-precex 8034  ax-cnre 8035  ax-pre-ltirr 8036  ax-pre-ltwlin 8037  ax-pre-lttrn 8038  ax-pre-apti 8039  ax-pre-ltadd 8040  ax-pre-mulgt0 8041  ax-pre-mulext 8042  ax-arch 8043  ax-caucvg 8044

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-nel 2471  df-ral 2488  df-rex 2489  df-reu 2490  df-rmo 2491  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-if 3571  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-tr 4142  df-id 4339  df-po 4342  df-iso 4343  df-iord 4412  df-on 4414  df-ilim 4415  df-suc 4417  df-iom 4638  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-riota 5898  df-ov 5946  df-oprab 5947  df-mpo 5948  df-1st 6225  df-2nd 6226  df-recs 6390  df-frec 6476  df-pnf 8108  df-mnf 8109  df-xr 8110  df-ltxr 8111  df-le 8112  df-sub 8244  df-neg 8245  df-reap 8647  df-ap 8654  df-div 8745  df-inn 9036  df-2 9094  df-3 9095  df-4 9096  df-n0 9295  df-z 9372  df-uz 9648  df-rp 9775  df-seqfrec 10591  df-exp 10682  df-cj 11124  df-re 11125  df-im 11126  df-rsqrt 11280  df-abs 11281  df-clim 11561

This theorem is referenced by: (None)