Theorem ntrivcvgap0 11831

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 9661	. . . 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 2298	. 2 |- ( ph -> M e. Z )
6		ntrivcvgn0.3	. . . 4 |- ( ph -> seq M ( x. , F ) ~~> X )
7		climrel 11562	. . . . 5 |- Rel ~~>
8	7	brrelex2i 4718	. . . 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 4046	. . . 4 |- ( y = X -> ( y # 0 <-> X # 0 ) )
13		breq2 4047	. . . 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 2917	. 2 |- ( ph -> E. y ( y # 0 /\ seq M ( x. , F ) ~~> y ) )
16		seqeq1 10593	. . . . . 6 |- ( n = M -> seq n ( x. , F ) = seq M ( x. , F ) )
17	16	breq1d 4053	. . . . 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 1847	. . 3 |- ( n = M -> ( E. y ( y # 0 /\ seq n ( x. , F ) ~~> y ) <-> E. y ( y # 0 /\ seq M ( x. , F ) ~~> y ) ) )
20	19	rspcev 2876	. 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 1372   E.wex 1514   e. wcel 2175   E.wrex 2484   _Vcvv 2771   class class class wbr 4043   ` cfv 5270   0cc0 7924   x. cmul 7929   # 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-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584  ax-cnex 8015  ax-resscn 8016  ax-pre-ltirr 8036

This theorem depends on definitions:  df-bi 117  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-rab 2492  df-v 2773  df-sbc 2998  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-iota 5231  df-fun 5272  df-fv 5278  df-ov 5946  df-oprab 5947  df-mpo 5948  df-recs 6390  df-frec 6476  df-pnf 8108  df-mnf 8109  df-xr 8110  df-ltxr 8111  df-le 8112  df-neg 8245  df-z 9372  df-uz 9648  df-seqfrec 10591  df-clim 11561

This theorem is referenced by:  zprodap0  11863