Theorem ntrivcvgap0 11576

Description: A product that converges to a value apart from zero converges non-trivially. (Contributed by Scott Fenton, 18-Dec-2017.)

Hypotheses

Ref	Expression
ntrivcvgn0.1	|- Z = ( ZZ>= ` M )
ntrivcvgn0.2	|- ( ph -> M e. ZZ )
ntrivcvgn0.3	|- ( ph -> seq M ( x. , F ) ~~> X )
ntrivcvgap0.4	|- ( ph -> X # 0 )

Assertion

Ref	Expression
ntrivcvgap0	|- ( ph -> E. n e. Z E. y ( y # 0 /\ seq n ( x. , F ) ~~> y ) )

Distinct variable groups:   n, F, y   n, M, y   y, X   n, Z

Allowed substitution hints:   ph( y, n)   X( n)   Z( y)

Proof of Theorem ntrivcvgap0

Step	Hyp	Ref	Expression
1		ntrivcvgn0.2	. . . 4 |- ( ph -> M e. ZZ )
2		uzid 9561	. . . 4 |- ( M e. ZZ -> M e. ( ZZ>= ` M ) )
3	1, 2	syl 14	. . 3 |- ( ph -> M e. ( ZZ>= ` M ) )
4		ntrivcvgn0.1	. . 3 |- Z = ( ZZ>= ` M )
5	3, 4	eleqtrrdi 2283	. 2 |- ( ph -> M e. Z )
6		ntrivcvgn0.3	. . . 4 |- ( ph -> seq M ( x. , F ) ~~> X )
7		climrel 11307	. . . . 5 |- Rel ~~>
8	7	brrelex2i 4685	. . . 4 |- ( seq M ( x. , F ) ~~> X -> X e. _V )
9	6, 8	syl 14	. . 3 |- ( ph -> X e. _V )
10		ntrivcvgap0.4	. . . 4 |- ( ph -> X # 0 )
11	10, 6	jca 306	. . 3 |- ( ph -> ( X # 0 /\ seq M ( x. , F ) ~~> X ) )
12		breq1 4021	. . . 4 |- ( y = X -> ( y # 0 <-> X # 0 ) )
13		breq2 4022	. . . 4 |- ( y = X -> ( seq M ( x. , F ) ~~> y <-> seq M ( x. , F ) ~~> X ) )
14	12, 13	anbi12d 473	. . 3 |- ( y = X -> ( ( y # 0 /\ seq M ( x. , F ) ~~> y ) <-> ( X # 0 /\ seq M ( x. , F ) ~~> X ) ) )
15	9, 11, 14	elabd 2897	. 2 |- ( ph -> E. y ( y # 0 /\ seq M ( x. , F ) ~~> y ) )
16		seqeq1 10467	. . . . . 6 |- ( n = M -> seq n ( x. , F ) = seq M ( x. , F ) )
17	16	breq1d 4028	. . . . 5 |- ( n = M -> ( seq n ( x. , F ) ~~> y <-> seq M ( x. , F ) ~~> y ) )
18	17	anbi2d 464	. . . 4 |- ( n = M -> ( ( y # 0 /\ seq n ( x. , F ) ~~> y ) <-> ( y # 0 /\ seq M ( x. , F ) ~~> y ) ) )
19	18	exbidv 1836	. . 3 |- ( n = M -> ( E. y ( y # 0 /\ seq n ( x. , F ) ~~> y ) <-> E. y ( y # 0 /\ seq M ( x. , F ) ~~> y ) ) )
20	19	rspcev 2856	. 2 |- ( ( M e. Z /\ E. y ( y # 0 /\ seq M ( x. , F ) ~~> y ) ) -> E. n e. Z E. y ( y # 0 /\ seq n ( x. , F ) ~~> y ) )
21	5, 15, 20	syl2anc 411	1 |- ( ph -> E. n e. Z E. y ( y # 0 /\ seq n ( x. , F ) ~~> y ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   E.wex 1503   e. wcel 2160   E.wrex 2469   _Vcvv 2752   class class class wbr 4018   ` cfv 5231   0cc0 7830   x. cmul 7835   # cap 8557   ZZcz 9272   ZZ>=cuz 9547   seqcseq 10464   ~~> cli 11305

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-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-cnex 7921  ax-resscn 7922  ax-pre-ltirr 7942

This theorem depends on definitions:  df-bi 117  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-rab 2477  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-iota 5193  df-fun 5233  df-fv 5239  df-ov 5894  df-oprab 5895  df-mpo 5896  df-recs 6324  df-frec 6410  df-pnf 8013  df-mnf 8014  df-xr 8015  df-ltxr 8016  df-le 8017  df-neg 8150  df-z 9273  df-uz 9548  df-seqfrec 10465  df-clim 11306

This theorem is referenced by:  zprodap0  11608