Theorem ntrivcvgap0 12316

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 9936	. . . 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 2332	. 2 |- ( ph -> M e. Z )
6		ntrivcvgn0.3	. . . 4 |- ( ph -> seq M ( x. , F ) ~~> X )
7		climrel 12046	. . . . 5 |- Rel ~~>
8	7	brrelex2i 4819	. . . 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 4133	. . . 4 |- ( y = X -> ( y # 0 <-> X # 0 ) )
13		breq2 4134	. . . 4 |- ( y = X -> ( seq M ( x. , F ) ~~> y <-> seq M ( x. , F ) ~~> X ) )
14	12, 13	anbi12d 477	. . 3 |- ( y = X -> ( ( y # 0 /\ seq M ( x. , F ) ~~> y ) <-> ( X # 0 /\ seq M ( x. , F ) ~~> X ) ) )
15	9, 11, 14	elabd 2971	. 2 |- ( ph -> E. y ( y # 0 /\ seq M ( x. , F ) ~~> y ) )
16		seqeq1 10887	. . . . . 6 |- ( n = M -> seq n ( x. , F ) = seq M ( x. , F ) )
17	16	breq1d 4140	. . . . 5 |- ( n = M -> ( seq n ( x. , F ) ~~> y <-> seq M ( x. , F ) ~~> y ) )
18	17	anbi2d 468	. . . 4 |- ( n = M -> ( ( y # 0 /\ seq n ( x. , F ) ~~> y ) <-> ( y # 0 /\ seq M ( x. , F ) ~~> y ) ) )
19	18	exbidv 1878	. . 3 |- ( n = M -> ( E. y ( y # 0 /\ seq n ( x. , F ) ~~> y ) <-> E. y ( y # 0 /\ seq M ( x. , F ) ~~> y ) ) )
20	19	rspcev 2929	. 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 415	1 |- ( ph -> E. n e. Z E. y ( y # 0 /\ seq n ( x. , F ) ~~> y ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   E.wex 1545   e. wcel 2209   E.wrex 2529   _Vcvv 2821   class class class wbr 4130   ` cfv 5377   0cc0 8179   x. cmul 8184   # cap 8909   ZZcz 9644   ZZ>=cuz 9921   seqcseq 10884   ~~> cli 12044

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-pre-ltirr 8291

This proof depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-iota 5337  df-fun 5379  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-recs 6576  df-frec 6662  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-neg 8500  df-z 9645  df-uz 9922  df-seqfrec 10885  df-clim 12045

This theorem is used by:  zprodap0  12348