Theorem ntrivcvgap0 11570

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 9555	. . . 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 2281	. 2 |- ( ph -> M e. Z )
6		ntrivcvgn0.3	. . . 4 |- ( ph -> seq M ( x. , F ) ~~> X )
7		climrel 11301	. . . . 5 |- Rel ~~>
8	7	brrelex2i 4682	. . . 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 4018	. . . 4 |- ( y = X -> ( y # 0 <-> X # 0 ) )
13		breq2 4019	. . . 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 2894	. 2 |- ( ph -> E. y ( y # 0 /\ seq M ( x. , F ) ~~> y ) )
16		seqeq1 10461	. . . . . 6 |- ( n = M -> seq n ( x. , F ) = seq M ( x. , F ) )
17	16	breq1d 4025	. . . . 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 1835	. . 3 |- ( n = M -> ( E. y ( y # 0 /\ seq n ( x. , F ) ~~> y ) <-> E. y ( y # 0 /\ seq M ( x. , F ) ~~> y ) ) )
20	19	rspcev 2853	. 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 1363   E.wex 1502   e. wcel 2158   E.wrex 2466   _Vcvv 2749   class class class wbr 4015   ` cfv 5228   0cc0 7824   x. cmul 7829   # cap 8551   ZZcz 9266   ZZ>=cuz 9541   seqcseq 10458   ~~> cli 11299

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-cnex 7915  ax-resscn 7916  ax-pre-ltirr 7936

This theorem depends on definitions:  df-bi 117  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-rab 2474  df-v 2751  df-sbc 2975  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-iota 5190  df-fun 5230  df-fv 5236  df-ov 5891  df-oprab 5892  df-mpo 5893  df-recs 6319  df-frec 6405  df-pnf 8007  df-mnf 8008  df-xr 8009  df-ltxr 8010  df-le 8011  df-neg 8144  df-z 9267  df-uz 9542  df-seqfrec 10459  df-clim 11300

This theorem is referenced by:  zprodap0  11602