Theorem ntrivcvgap0 11860

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 9662	. . . 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 2299	. 2 |- ( ph -> M e. Z )
6		ntrivcvgn0.3	. . . 4 |- ( ph -> seq M ( x. , F ) ~~> X )
7		climrel 11591	. . . . 5 |- Rel ~~>
8	7	brrelex2i 4719	. . . 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 4047	. . . 4 |- ( y = X -> ( y # 0 <-> X # 0 ) )
13		breq2 4048	. . . 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 2918	. 2 |- ( ph -> E. y ( y # 0 /\ seq M ( x. , F ) ~~> y ) )
16		seqeq1 10595	. . . . . 6 |- ( n = M -> seq n ( x. , F ) = seq M ( x. , F ) )
17	16	breq1d 4054	. . . . 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 1848	. . 3 |- ( n = M -> ( E. y ( y # 0 /\ seq n ( x. , F ) ~~> y ) <-> E. y ( y # 0 /\ seq M ( x. , F ) ~~> y ) ) )
20	19	rspcev 2877	. 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 1373   E.wex 1515   e. wcel 2176   E.wrex 2485   _Vcvv 2772   class class class wbr 4044   ` cfv 5271   0cc0 7925   x. cmul 7930   # cap 8654   ZZcz 9372   ZZ>=cuz 9648   seqcseq 10592   ~~> cli 11589

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-cnex 8016  ax-resscn 8017  ax-pre-ltirr 8037

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-sbc 2999  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-iota 5232  df-fun 5273  df-fv 5279  df-ov 5947  df-oprab 5948  df-mpo 5949  df-recs 6391  df-frec 6477  df-pnf 8109  df-mnf 8110  df-xr 8111  df-ltxr 8112  df-le 8113  df-neg 8246  df-z 9373  df-uz 9649  df-seqfrec 10593  df-clim 11590

This theorem is referenced by:  zprodap0  11892