Theorem ntrivcvgn0 14630

Description: A product that converges to a nonzero value 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 )
ntrivcvgn0.4	|- ( ph -> X =/= 0 )

Assertion

Ref	Expression
ntrivcvgn0	|- ( 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 ntrivcvgn0

Step	Hyp	Ref	Expression
1		ntrivcvgn0.2	. . . 4 |- ( ph -> M e. ZZ )
2		uzid 11702	. . . 4 |- ( M e. ZZ -> M e. ( ZZ>= ` M ) )
3	1, 2	syl 17	. . 3 |- ( ph -> M e. ( ZZ>= ` M ) )
4		ntrivcvgn0.1	. . 3 |- Z = ( ZZ>= ` M )
5	3, 4	syl6eleqr 2712	. 2 |- ( ph -> M e. Z )
6		ntrivcvgn0.3	. . . 4 |- ( ph -> seq M ( x. , F ) ~~> X )
7		climrel 14223	. . . . 5 |- Rel ~~>
8	7	brrelex2i 5159	. . . 4 |- ( seq M ( x. , F ) ~~> X -> X e. _V )
9	6, 8	syl 17	. . 3 |- ( ph -> X e. _V )
10		ntrivcvgn0.4	. . . 4 |- ( ph -> X =/= 0 )
11	10, 6	jca 554	. . 3 |- ( ph -> ( X =/= 0 /\ seq M ( x. , F ) ~~> X ) )
12		neeq1 2856	. . . . 5 |- ( y = X -> ( y =/= 0 <-> X =/= 0 ) )
13		breq2 4657	. . . . 5 |- ( y = X -> ( seq M ( x. , F ) ~~> y <-> seq M ( x. , F ) ~~> X ) )
14	12, 13	anbi12d 747	. . . 4 |- ( y = X -> ( ( y =/= 0 /\ seq M ( x. , F ) ~~> y ) <-> ( X =/= 0 /\ seq M ( x. , F ) ~~> X ) ) )
15	14	spcegv 3294	. . 3 |- ( X e. _V -> ( ( X =/= 0 /\ seq M ( x. , F ) ~~> X ) -> E. y ( y =/= 0 /\ seq M ( x. , F ) ~~> y ) ) )
16	9, 11, 15	sylc 65	. 2 |- ( ph -> E. y ( y =/= 0 /\ seq M ( x. , F ) ~~> y ) )
17		seqeq1 12804	. . . . . 6 |- ( n = M -> seq n ( x. , F ) = seq M ( x. , F ) )
18	17	breq1d 4663	. . . . 5 |- ( n = M -> ( seq n ( x. , F ) ~~> y <-> seq M ( x. , F ) ~~> y ) )
19	18	anbi2d 740	. . . 4 |- ( n = M -> ( ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) <-> ( y =/= 0 /\ seq M ( x. , F ) ~~> y ) ) )
20	19	exbidv 1850	. . 3 |- ( n = M -> ( E. y ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) <-> E. y ( y =/= 0 /\ seq M ( x. , F ) ~~> y ) ) )
21	20	rspcev 3309	. 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 ) )
22	5, 16, 21	syl2anc 693	1 |- ( ph -> E. n e. Z E. y ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   E.wrex 2913   _Vcvv 3200   class class class wbr 4653   ` cfv 5888   0cc0 9936   x. cmul 9941   ZZcz 11377   ZZ>=cuz 11687   seqcseq 12801   ~~> cli 14215

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-pre-lttri 10010

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-neg 10269  df-z 11378  df-uz 11688  df-seq 12802  df-clim 14219

This theorem is referenced by:  zprodn0  14669