Metamath Proof Explorer


Theorem ntrivcvgfvn0

Description: Any value of a product sequence that converges to a nonzero value is itself nonzero. (Contributed by Scott Fenton, 20-Dec-2017)

Ref Expression
Hypotheses ntrivcvgfvn0.1
|- Z = ( ZZ>= ` M )
ntrivcvgfvn0.2
|- ( ph -> N e. Z )
ntrivcvgfvn0.3
|- ( ph -> seq M ( x. , F ) ~~> X )
ntrivcvgfvn0.4
|- ( ph -> X =/= 0 )
ntrivcvgfvn0.5
|- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )
Assertion ntrivcvgfvn0
|- ( ph -> ( seq M ( x. , F ) ` N ) =/= 0 )

Proof

Step Hyp Ref Expression
1 ntrivcvgfvn0.1
 |-  Z = ( ZZ>= ` M )
2 ntrivcvgfvn0.2
 |-  ( ph -> N e. Z )
3 ntrivcvgfvn0.3
 |-  ( ph -> seq M ( x. , F ) ~~> X )
4 ntrivcvgfvn0.4
 |-  ( ph -> X =/= 0 )
5 ntrivcvgfvn0.5
 |-  ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )
6 fclim
 |-  ~~> : dom ~~> --> CC
7 ffun
 |-  ( ~~> : dom ~~> --> CC -> Fun ~~> )
8 6 7 ax-mp
 |-  Fun ~~>
9 funbrfv
 |-  ( Fun ~~> -> ( seq M ( x. , F ) ~~> X -> ( ~~> ` seq M ( x. , F ) ) = X ) )
10 8 3 9 mpsyl
 |-  ( ph -> ( ~~> ` seq M ( x. , F ) ) = X )
11 10 adantr
 |-  ( ( ph /\ ( seq M ( x. , F ) ` N ) = 0 ) -> ( ~~> ` seq M ( x. , F ) ) = X )
12 eqid
 |-  ( ZZ>= ` N ) = ( ZZ>= ` N )
13 uzssz
 |-  ( ZZ>= ` M ) C_ ZZ
14 1 13 eqsstri
 |-  Z C_ ZZ
15 14 2 sselid
 |-  ( ph -> N e. ZZ )
16 15 adantr
 |-  ( ( ph /\ ( seq M ( x. , F ) ` N ) = 0 ) -> N e. ZZ )
17 seqex
 |-  seq M ( x. , F ) e. _V
18 17 a1i
 |-  ( ( ph /\ ( seq M ( x. , F ) ` N ) = 0 ) -> seq M ( x. , F ) e. _V )
19 0cnd
 |-  ( ( ph /\ ( seq M ( x. , F ) ` N ) = 0 ) -> 0 e. CC )
20 fveqeq2
 |-  ( m = N -> ( ( seq M ( x. , F ) ` m ) = 0 <-> ( seq M ( x. , F ) ` N ) = 0 ) )
21 20 imbi2d
 |-  ( m = N -> ( ( ( ph /\ ( seq M ( x. , F ) ` N ) = 0 ) -> ( seq M ( x. , F ) ` m ) = 0 ) <-> ( ( ph /\ ( seq M ( x. , F ) ` N ) = 0 ) -> ( seq M ( x. , F ) ` N ) = 0 ) ) )
22 fveqeq2
 |-  ( m = n -> ( ( seq M ( x. , F ) ` m ) = 0 <-> ( seq M ( x. , F ) ` n ) = 0 ) )
23 22 imbi2d
 |-  ( m = n -> ( ( ( ph /\ ( seq M ( x. , F ) ` N ) = 0 ) -> ( seq M ( x. , F ) ` m ) = 0 ) <-> ( ( ph /\ ( seq M ( x. , F ) ` N ) = 0 ) -> ( seq M ( x. , F ) ` n ) = 0 ) ) )
24 fveqeq2
 |-  ( m = ( n + 1 ) -> ( ( seq M ( x. , F ) ` m ) = 0 <-> ( seq M ( x. , F ) ` ( n + 1 ) ) = 0 ) )
25 24 imbi2d
 |-  ( m = ( n + 1 ) -> ( ( ( ph /\ ( seq M ( x. , F ) ` N ) = 0 ) -> ( seq M ( x. , F ) ` m ) = 0 ) <-> ( ( ph /\ ( seq M ( x. , F ) ` N ) = 0 ) -> ( seq M ( x. , F ) ` ( n + 1 ) ) = 0 ) ) )
26 fveqeq2
 |-  ( m = k -> ( ( seq M ( x. , F ) ` m ) = 0 <-> ( seq M ( x. , F ) ` k ) = 0 ) )
27 26 imbi2d
 |-  ( m = k -> ( ( ( ph /\ ( seq M ( x. , F ) ` N ) = 0 ) -> ( seq M ( x. , F ) ` m ) = 0 ) <-> ( ( ph /\ ( seq M ( x. , F ) ` N ) = 0 ) -> ( seq M ( x. , F ) ` k ) = 0 ) ) )
28 simpr
 |-  ( ( ph /\ ( seq M ( x. , F ) ` N ) = 0 ) -> ( seq M ( x. , F ) ` N ) = 0 )
29 2 1 eleqtrdi
 |-  ( ph -> N e. ( ZZ>= ` M ) )
30 uztrn
 |-  ( ( n e. ( ZZ>= ` N ) /\ N e. ( ZZ>= ` M ) ) -> n e. ( ZZ>= ` M ) )
31 29 30 sylan2
 |-  ( ( n e. ( ZZ>= ` N ) /\ ph ) -> n e. ( ZZ>= ` M ) )
32 31 3adant3
 |-  ( ( n e. ( ZZ>= ` N ) /\ ph /\ ( seq M ( x. , F ) ` n ) = 0 ) -> n e. ( ZZ>= ` M ) )
33 seqp1
 |-  ( n e. ( ZZ>= ` M ) -> ( seq M ( x. , F ) ` ( n + 1 ) ) = ( ( seq M ( x. , F ) ` n ) x. ( F ` ( n + 1 ) ) ) )
34 32 33 syl
 |-  ( ( n e. ( ZZ>= ` N ) /\ ph /\ ( seq M ( x. , F ) ` n ) = 0 ) -> ( seq M ( x. , F ) ` ( n + 1 ) ) = ( ( seq M ( x. , F ) ` n ) x. ( F ` ( n + 1 ) ) ) )
35 oveq1
 |-  ( ( seq M ( x. , F ) ` n ) = 0 -> ( ( seq M ( x. , F ) ` n ) x. ( F ` ( n + 1 ) ) ) = ( 0 x. ( F ` ( n + 1 ) ) ) )
36 35 3ad2ant3
 |-  ( ( n e. ( ZZ>= ` N ) /\ ph /\ ( seq M ( x. , F ) ` n ) = 0 ) -> ( ( seq M ( x. , F ) ` n ) x. ( F ` ( n + 1 ) ) ) = ( 0 x. ( F ` ( n + 1 ) ) ) )
37 peano2uz
 |-  ( n e. ( ZZ>= ` N ) -> ( n + 1 ) e. ( ZZ>= ` N ) )
38 1 uztrn2
 |-  ( ( N e. Z /\ ( n + 1 ) e. ( ZZ>= ` N ) ) -> ( n + 1 ) e. Z )
39 2 37 38 syl2an
 |-  ( ( ph /\ n e. ( ZZ>= ` N ) ) -> ( n + 1 ) e. Z )
40 5 ralrimiva
 |-  ( ph -> A. k e. Z ( F ` k ) e. CC )
41 fveq2
 |-  ( k = ( n + 1 ) -> ( F ` k ) = ( F ` ( n + 1 ) ) )
42 41 eleq1d
 |-  ( k = ( n + 1 ) -> ( ( F ` k ) e. CC <-> ( F ` ( n + 1 ) ) e. CC ) )
43 42 rspcv
 |-  ( ( n + 1 ) e. Z -> ( A. k e. Z ( F ` k ) e. CC -> ( F ` ( n + 1 ) ) e. CC ) )
44 40 43 mpan9
 |-  ( ( ph /\ ( n + 1 ) e. Z ) -> ( F ` ( n + 1 ) ) e. CC )
45 39 44 syldan
 |-  ( ( ph /\ n e. ( ZZ>= ` N ) ) -> ( F ` ( n + 1 ) ) e. CC )
46 45 ancoms
 |-  ( ( n e. ( ZZ>= ` N ) /\ ph ) -> ( F ` ( n + 1 ) ) e. CC )
47 46 mul02d
 |-  ( ( n e. ( ZZ>= ` N ) /\ ph ) -> ( 0 x. ( F ` ( n + 1 ) ) ) = 0 )
48 47 3adant3
 |-  ( ( n e. ( ZZ>= ` N ) /\ ph /\ ( seq M ( x. , F ) ` n ) = 0 ) -> ( 0 x. ( F ` ( n + 1 ) ) ) = 0 )
49 34 36 48 3eqtrd
 |-  ( ( n e. ( ZZ>= ` N ) /\ ph /\ ( seq M ( x. , F ) ` n ) = 0 ) -> ( seq M ( x. , F ) ` ( n + 1 ) ) = 0 )
50 49 3exp
 |-  ( n e. ( ZZ>= ` N ) -> ( ph -> ( ( seq M ( x. , F ) ` n ) = 0 -> ( seq M ( x. , F ) ` ( n + 1 ) ) = 0 ) ) )
51 50 adantrd
 |-  ( n e. ( ZZ>= ` N ) -> ( ( ph /\ ( seq M ( x. , F ) ` N ) = 0 ) -> ( ( seq M ( x. , F ) ` n ) = 0 -> ( seq M ( x. , F ) ` ( n + 1 ) ) = 0 ) ) )
52 51 a2d
 |-  ( n e. ( ZZ>= ` N ) -> ( ( ( ph /\ ( seq M ( x. , F ) ` N ) = 0 ) -> ( seq M ( x. , F ) ` n ) = 0 ) -> ( ( ph /\ ( seq M ( x. , F ) ` N ) = 0 ) -> ( seq M ( x. , F ) ` ( n + 1 ) ) = 0 ) ) )
53 21 23 25 27 28 52 uzind4i
 |-  ( k e. ( ZZ>= ` N ) -> ( ( ph /\ ( seq M ( x. , F ) ` N ) = 0 ) -> ( seq M ( x. , F ) ` k ) = 0 ) )
54 53 impcom
 |-  ( ( ( ph /\ ( seq M ( x. , F ) ` N ) = 0 ) /\ k e. ( ZZ>= ` N ) ) -> ( seq M ( x. , F ) ` k ) = 0 )
55 12 16 18 19 54 climconst
 |-  ( ( ph /\ ( seq M ( x. , F ) ` N ) = 0 ) -> seq M ( x. , F ) ~~> 0 )
56 funbrfv
 |-  ( Fun ~~> -> ( seq M ( x. , F ) ~~> 0 -> ( ~~> ` seq M ( x. , F ) ) = 0 ) )
57 8 55 56 mpsyl
 |-  ( ( ph /\ ( seq M ( x. , F ) ` N ) = 0 ) -> ( ~~> ` seq M ( x. , F ) ) = 0 )
58 11 57 eqtr3d
 |-  ( ( ph /\ ( seq M ( x. , F ) ` N ) = 0 ) -> X = 0 )
59 58 ex
 |-  ( ph -> ( ( seq M ( x. , F ) ` N ) = 0 -> X = 0 ) )
60 59 necon3d
 |-  ( ph -> ( X =/= 0 -> ( seq M ( x. , F ) ` N ) =/= 0 ) )
61 4 60 mpd
 |-  ( ph -> ( seq M ( x. , F ) ` N ) =/= 0 )