MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ntrivcvgfvn0 Structured version   Visualization version   GIF version
Theorem ntrivcvgfvn0 15810
Description: Any value of a product sequence that converges to a nonzero value is itself nonzero. (Contributed by Scott Fenton, 20-Dec-2017.)
Hypotheses
Ref Expression
ntrivcvgfvn0.1 𝑍 = (β„€β‰₯β€˜π‘€)
ntrivcvgfvn0.2 (πœ‘ β†’ 𝑁 ∈ 𝑍)
ntrivcvgfvn0.3 (πœ‘ β†’ seq𝑀( Β· , 𝐹) ⇝ 𝑋)
ntrivcvgfvn0.4 (πœ‘ β†’ 𝑋 β‰  0)
ntrivcvgfvn0.5 ((πœ‘ ∧ π‘˜ ∈ 𝑍) β†’ (πΉβ€˜π‘˜) ∈ β„‚)
Assertion
Ref Expression
ntrivcvgfvn0 (πœ‘ β†’ (seq𝑀( Β· , 𝐹)β€˜π‘) β‰  0)
Distinct variable groups:   π‘˜,𝐹   πœ‘,π‘˜   π‘˜,𝑀   π‘˜,𝑁   π‘˜,𝑍
Allowed substitution hint:   𝑋(π‘˜)
Proof of Theorem ntrivcvgfvn0
Dummy variables π‘š 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ntrivcvgfvn0.4 . 2 (πœ‘ β†’ 𝑋 β‰  0)
2 fclim 15462 . . . . . . . 8 ⇝ :dom ⇝ βŸΆβ„‚
3 ffun 6691 . . . . . . . 8 ( ⇝ :dom ⇝ βŸΆβ„‚ β†’ Fun ⇝ )
42, 3ax-mp 5 . . . . . . 7 Fun ⇝
5 ntrivcvgfvn0.3 . . . . . . 7 (πœ‘ β†’ seq𝑀( Β· , 𝐹) ⇝ 𝑋)
6 funbrfv 6913 . . . . . . 7 (Fun ⇝ β†’ (seq𝑀( Β· , 𝐹) ⇝ 𝑋 β†’ ( ⇝ β€˜seq𝑀( Β· , 𝐹)) = 𝑋))
74, 5, 6mpsyl 68 . . . . . 6 (πœ‘ β†’ ( ⇝ β€˜seq𝑀( Β· , 𝐹)) = 𝑋)
87adantr 481 . . . . 5 ((πœ‘ ∧ (seq𝑀( Β· , 𝐹)β€˜π‘) = 0) β†’ ( ⇝ β€˜seq𝑀( Β· , 𝐹)) = 𝑋)
9 eqid 2731 . . . . . . 7 (β„€β‰₯β€˜π‘) = (β„€β‰₯β€˜π‘)
10 ntrivcvgfvn0.1 . . . . . . . . . 10 𝑍 = (β„€β‰₯β€˜π‘€)
11 uzssz 12808 . . . . . . . . . 10 (β„€β‰₯β€˜π‘€) βŠ† β„€
1210, 11eqsstri 3996 . . . . . . . . 9 𝑍 βŠ† β„€
13 ntrivcvgfvn0.2 . . . . . . . . 9 (πœ‘ β†’ 𝑁 ∈ 𝑍)
1412, 13sselid 3960 . . . . . . . 8 (πœ‘ β†’ 𝑁 ∈ β„€)
1514adantr 481 . . . . . . 7 ((πœ‘ ∧ (seq𝑀( Β· , 𝐹)β€˜π‘) = 0) β†’ 𝑁 ∈ β„€)
16 seqex 13933 . . . . . . . 8 seq𝑀( Β· , 𝐹) ∈ V
1716a1i 11 . . . . . . 7 ((πœ‘ ∧ (seq𝑀( Β· , 𝐹)β€˜π‘) = 0) β†’ seq𝑀( Β· , 𝐹) ∈ V)
18 0cnd 11172 . . . . . . 7 ((πœ‘ ∧ (seq𝑀( Β· , 𝐹)β€˜π‘) = 0) β†’ 0 ∈ β„‚)
19 fveqeq2 6871 . . . . . . . . . 10 (π‘š = 𝑁 β†’ ((seq𝑀( Β· , 𝐹)β€˜π‘š) = 0 ↔ (seq𝑀( Β· , 𝐹)β€˜π‘) = 0))
2019imbi2d 340 . . . . . . . . 9 (π‘š = 𝑁 β†’ (((πœ‘ ∧ (seq𝑀( Β· , 𝐹)β€˜π‘) = 0) β†’ (seq𝑀( Β· , 𝐹)β€˜π‘š) = 0) ↔ ((πœ‘ ∧ (seq𝑀( Β· , 𝐹)β€˜π‘) = 0) β†’ (seq𝑀( Β· , 𝐹)β€˜π‘) = 0)))
21 fveqeq2 6871 . . . . . . . . . 10 (π‘š = 𝑛 β†’ ((seq𝑀( Β· , 𝐹)β€˜π‘š) = 0 ↔ (seq𝑀( Β· , 𝐹)β€˜π‘›) = 0))
2221imbi2d 340 . . . . . . . . 9 (π‘š = 𝑛 β†’ (((πœ‘ ∧ (seq𝑀( Β· , 𝐹)β€˜π‘) = 0) β†’ (seq𝑀( Β· , 𝐹)β€˜π‘š) = 0) ↔ ((πœ‘ ∧ (seq𝑀( Β· , 𝐹)β€˜π‘) = 0) β†’ (seq𝑀( Β· , 𝐹)β€˜π‘›) = 0)))
23 fveqeq2 6871 . . . . . . . . . 10 (π‘š = (𝑛 + 1) β†’ ((seq𝑀( Β· , 𝐹)β€˜π‘š) = 0 ↔ (seq𝑀( Β· , 𝐹)β€˜(𝑛 + 1)) = 0))
2423imbi2d 340 . . . . . . . . 9 (π‘š = (𝑛 + 1) β†’ (((πœ‘ ∧ (seq𝑀( Β· , 𝐹)β€˜π‘) = 0) β†’ (seq𝑀( Β· , 𝐹)β€˜π‘š) = 0) ↔ ((πœ‘ ∧ (seq𝑀( Β· , 𝐹)β€˜π‘) = 0) β†’ (seq𝑀( Β· , 𝐹)β€˜(𝑛 + 1)) = 0)))
25 fveqeq2 6871 . . . . . . . . . 10 (π‘š = π‘˜ β†’ ((seq𝑀( Β· , 𝐹)β€˜π‘š) = 0 ↔ (seq𝑀( Β· , 𝐹)β€˜π‘˜) = 0))
2625imbi2d 340 . . . . . . . . 9 (π‘š = π‘˜ β†’ (((πœ‘ ∧ (seq𝑀( Β· , 𝐹)β€˜π‘) = 0) β†’ (seq𝑀( Β· , 𝐹)β€˜π‘š) = 0) ↔ ((πœ‘ ∧ (seq𝑀( Β· , 𝐹)β€˜π‘) = 0) β†’ (seq𝑀( Β· , 𝐹)β€˜π‘˜) = 0)))
27 simpr 485 . . . . . . . . 9 ((πœ‘ ∧ (seq𝑀( Β· , 𝐹)β€˜π‘) = 0) β†’ (seq𝑀( Β· , 𝐹)β€˜π‘) = 0)
2813, 10eleqtrdi 2842 . . . . . . . . . . . . . . . 16 (πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜π‘€))
29 uztrn 12805 . . . . . . . . . . . . . . . 16 ((𝑛 ∈ (β„€β‰₯β€˜π‘) ∧ 𝑁 ∈ (β„€β‰₯β€˜π‘€)) β†’ 𝑛 ∈ (β„€β‰₯β€˜π‘€))
3028, 29sylan2 593 . . . . . . . . . . . . . . 15 ((𝑛 ∈ (β„€β‰₯β€˜π‘) ∧ πœ‘) β†’ 𝑛 ∈ (β„€β‰₯β€˜π‘€))
31303adant3 1132 . . . . . . . . . . . . . 14 ((𝑛 ∈ (β„€β‰₯β€˜π‘) ∧ πœ‘ ∧ (seq𝑀( Β· , 𝐹)β€˜π‘›) = 0) β†’ 𝑛 ∈ (β„€β‰₯β€˜π‘€))
32 seqp1 13946 . . . . . . . . . . . . . 14 (𝑛 ∈ (β„€β‰₯β€˜π‘€) β†’ (seq𝑀( Β· , 𝐹)β€˜(𝑛 + 1)) = ((seq𝑀( Β· , 𝐹)β€˜π‘›) Β· (πΉβ€˜(𝑛 + 1))))
3331, 32syl 17 . . . . . . . . . . . . 13 ((𝑛 ∈ (β„€β‰₯β€˜π‘) ∧ πœ‘ ∧ (seq𝑀( Β· , 𝐹)β€˜π‘›) = 0) β†’ (seq𝑀( Β· , 𝐹)β€˜(𝑛 + 1)) = ((seq𝑀( Β· , 𝐹)β€˜π‘›) Β· (πΉβ€˜(𝑛 + 1))))
34 oveq1 7384 . . . . . . . . . . . . . 14 ((seq𝑀( Β· , 𝐹)β€˜π‘›) = 0 β†’ ((seq𝑀( Β· , 𝐹)β€˜π‘›) Β· (πΉβ€˜(𝑛 + 1))) = (0 Β· (πΉβ€˜(𝑛 + 1))))
35343ad2ant3 1135 . . . . . . . . . . . . 13 ((𝑛 ∈ (β„€β‰₯β€˜π‘) ∧ πœ‘ ∧ (seq𝑀( Β· , 𝐹)β€˜π‘›) = 0) β†’ ((seq𝑀( Β· , 𝐹)β€˜π‘›) Β· (πΉβ€˜(𝑛 + 1))) = (0 Β· (πΉβ€˜(𝑛 + 1))))
36 peano2uz 12850 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ (β„€β‰₯β€˜π‘) β†’ (𝑛 + 1) ∈ (β„€β‰₯β€˜π‘))
3710uztrn2 12806 . . . . . . . . . . . . . . . . . 18 ((𝑁 ∈ 𝑍 ∧ (𝑛 + 1) ∈ (β„€β‰₯β€˜π‘)) β†’ (𝑛 + 1) ∈ 𝑍)
3813, 36, 37syl2an 596 . . . . . . . . . . . . . . . . 17 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘)) β†’ (𝑛 + 1) ∈ 𝑍)
39 ntrivcvgfvn0.5 . . . . . . . . . . . . . . . . . . 19 ((πœ‘ ∧ π‘˜ ∈ 𝑍) β†’ (πΉβ€˜π‘˜) ∈ β„‚)
4039ralrimiva 3145 . . . . . . . . . . . . . . . . . 18 (πœ‘ β†’ βˆ€π‘˜ ∈ 𝑍 (πΉβ€˜π‘˜) ∈ β„‚)
41 fveq2 6862 . . . . . . . . . . . . . . . . . . . 20 (π‘˜ = (𝑛 + 1) β†’ (πΉβ€˜π‘˜) = (πΉβ€˜(𝑛 + 1)))
4241eleq1d 2817 . . . . . . . . . . . . . . . . . . 19 (π‘˜ = (𝑛 + 1) β†’ ((πΉβ€˜π‘˜) ∈ β„‚ ↔ (πΉβ€˜(𝑛 + 1)) ∈ β„‚))
4342rspcv 3591 . . . . . . . . . . . . . . . . . 18 ((𝑛 + 1) ∈ 𝑍 β†’ (βˆ€π‘˜ ∈ 𝑍 (πΉβ€˜π‘˜) ∈ β„‚ β†’ (πΉβ€˜(𝑛 + 1)) ∈ β„‚))
4440, 43mpan9 507 . . . . . . . . . . . . . . . . 17 ((πœ‘ ∧ (𝑛 + 1) ∈ 𝑍) β†’ (πΉβ€˜(𝑛 + 1)) ∈ β„‚)
4538, 44syldan 591 . . . . . . . . . . . . . . . 16 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘)) β†’ (πΉβ€˜(𝑛 + 1)) ∈ β„‚)
4645ancoms 459 . . . . . . . . . . . . . . 15 ((𝑛 ∈ (β„€β‰₯β€˜π‘) ∧ πœ‘) β†’ (πΉβ€˜(𝑛 + 1)) ∈ β„‚)
4746mul02d 11377 . . . . . . . . . . . . . 14 ((𝑛 ∈ (β„€β‰₯β€˜π‘) ∧ πœ‘) β†’ (0 Β· (πΉβ€˜(𝑛 + 1))) = 0)
48473adant3 1132 . . . . . . . . . . . . 13 ((𝑛 ∈ (β„€β‰₯β€˜π‘) ∧ πœ‘ ∧ (seq𝑀( Β· , 𝐹)β€˜π‘›) = 0) β†’ (0 Β· (πΉβ€˜(𝑛 + 1))) = 0)
4933, 35, 483eqtrd 2775 . . . . . . . . . . . 12 ((𝑛 ∈ (β„€β‰₯β€˜π‘) ∧ πœ‘ ∧ (seq𝑀( Β· , 𝐹)β€˜π‘›) = 0) β†’ (seq𝑀( Β· , 𝐹)β€˜(𝑛 + 1)) = 0)
50493exp 1119 . . . . . . . . . . 11 (𝑛 ∈ (β„€β‰₯β€˜π‘) β†’ (πœ‘ β†’ ((seq𝑀( Β· , 𝐹)β€˜π‘›) = 0 β†’ (seq𝑀( Β· , 𝐹)β€˜(𝑛 + 1)) = 0)))
5150adantrd 492 . . . . . . . . . 10 (𝑛 ∈ (β„€β‰₯β€˜π‘) β†’ ((πœ‘ ∧ (seq𝑀( Β· , 𝐹)β€˜π‘) = 0) β†’ ((seq𝑀( Β· , 𝐹)β€˜π‘›) = 0 β†’ (seq𝑀( Β· , 𝐹)β€˜(𝑛 + 1)) = 0)))
5251a2d 29 . . . . . . . . 9 (𝑛 ∈ (β„€β‰₯β€˜π‘) β†’ (((πœ‘ ∧ (seq𝑀( Β· , 𝐹)β€˜π‘) = 0) β†’ (seq𝑀( Β· , 𝐹)β€˜π‘›) = 0) β†’ ((πœ‘ ∧ (seq𝑀( Β· , 𝐹)β€˜π‘) = 0) β†’ (seq𝑀( Β· , 𝐹)β€˜(𝑛 + 1)) = 0)))
5320, 22, 24, 26, 27, 52uzind4i 12859 . . . . . . . 8 (π‘˜ ∈ (β„€β‰₯β€˜π‘) β†’ ((πœ‘ ∧ (seq𝑀( Β· , 𝐹)β€˜π‘) = 0) β†’ (seq𝑀( Β· , 𝐹)β€˜π‘˜) = 0))
5453impcom 408 . . . . . . 7 (((πœ‘ ∧ (seq𝑀( Β· , 𝐹)β€˜π‘) = 0) ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘)) β†’ (seq𝑀( Β· , 𝐹)β€˜π‘˜) = 0)
559, 15, 17, 18, 54climconst 15452 . . . . . 6 ((πœ‘ ∧ (seq𝑀( Β· , 𝐹)β€˜π‘) = 0) β†’ seq𝑀( Β· , 𝐹) ⇝ 0)
56 funbrfv 6913 . . . . . 6 (Fun ⇝ β†’ (seq𝑀( Β· , 𝐹) ⇝ 0 β†’ ( ⇝ β€˜seq𝑀( Β· , 𝐹)) = 0))
574, 55, 56mpsyl 68 . . . . 5 ((πœ‘ ∧ (seq𝑀( Β· , 𝐹)β€˜π‘) = 0) β†’ ( ⇝ β€˜seq𝑀( Β· , 𝐹)) = 0)
588, 57eqtr3d 2773 . . . 4 ((πœ‘ ∧ (seq𝑀( Β· , 𝐹)β€˜π‘) = 0) β†’ 𝑋 = 0)
5958ex 413 . . 3 (πœ‘ β†’ ((seq𝑀( Β· , 𝐹)β€˜π‘) = 0 β†’ 𝑋 = 0))
6059necon3d 2960 . 2 (πœ‘ β†’ (𝑋 β‰  0 β†’ (seq𝑀( Β· , 𝐹)β€˜π‘) β‰  0))
611, 60mpd 15 1 (πœ‘ β†’ (seq𝑀( Β· , 𝐹)β€˜π‘) β‰  0)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   β‰  wne 2939  βˆ€wral 3060  Vcvv 3459   class class class wbr 5125  dom cdm 5653  Fun wfun 6510  βŸΆwf 6512  β€˜cfv 6516  (class class class)co 7377  β„‚cc 11073  0cc0 11075  1c1 11076   + caddc 11078   Β· cmul 11080  β„€cz 12523  β„€β‰₯cuz 12787  seqcseq 13931   ⇝ cli 15393
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5262  ax-sep 5276  ax-nul 5283  ax-pow 5340  ax-pr 5404  ax-un 7692  ax-inf2 9601  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152  ax-pre-sup 11153
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3364  df-reu 3365  df-rab 3419  df-v 3461  df-sbc 3758  df-csb 3874  df-dif 3931  df-un 3933  df-in 3935  df-ss 3945  df-pss 3947  df-nul 4303  df-if 4507  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4886  df-iun 4976  df-br 5126  df-opab 5188  df-mpt 5209  df-tr 5243  df-id 5551  df-eprel 5557  df-po 5565  df-so 5566  df-fr 5608  df-we 5610  df-xp 5659  df-rel 5660  df-cnv 5661  df-co 5662  df-dm 5663  df-rn 5664  df-res 5665  df-ima 5666  df-pred 6273  df-ord 6340  df-on 6341  df-lim 6342  df-suc 6343  df-iota 6468  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-riota 7333  df-ov 7380  df-oprab 7381  df-mpo 7382  df-om 7823  df-2nd 7942  df-frecs 8232  df-wrecs 8263  df-recs 8337  df-rdg 8376  df-er 8670  df-en 8906  df-dom 8907  df-sdom 8908  df-sup 9402  df-pnf 11215  df-mnf 11216  df-xr 11217  df-ltxr 11218  df-le 11219  df-sub 11411  df-neg 11412  df-div 11837  df-nn 12178  df-2 12240  df-3 12241  df-n0 12438  df-z 12524  df-uz 12788  df-rp 12940  df-seq 13932  df-exp 13993  df-cj 15011  df-re 15012  df-im 15013  df-sqrt 15147  df-abs 15148  df-clim 15397
This theorem is referenced by:  ntrivcvgtail  15811
  Copyright terms: Public domain W3C validator