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Theorem ntrivcvgfvn0 14556
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 = (ℤ𝑀)
ntrivcvgfvn0.2 (𝜑𝑁𝑍)
ntrivcvgfvn0.3 (𝜑 → seq𝑀( · , 𝐹) ⇝ 𝑋)
ntrivcvgfvn0.4 (𝜑𝑋 ≠ 0)
ntrivcvgfvn0.5 ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℂ)
Assertion
Ref Expression
ntrivcvgfvn0 (𝜑 → (seq𝑀( · , 𝐹)‘𝑁) ≠ 0)
Distinct variable groups:   𝑘,𝐹   𝜑,𝑘   𝑘,𝑀   𝑘,𝑁   𝑘,𝑍
Allowed substitution hint:   𝑋(𝑘)

Proof of Theorem ntrivcvgfvn0
Dummy variables 𝑚 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ntrivcvgfvn0.4 . 2 (𝜑𝑋 ≠ 0)
2 fclim 14218 . . . . . . . 8 ⇝ :dom ⇝ ⟶ℂ
3 ffun 6005 . . . . . . . 8 ( ⇝ :dom ⇝ ⟶ℂ → Fun ⇝ )
42, 3ax-mp 5 . . . . . . 7 Fun ⇝
5 ntrivcvgfvn0.3 . . . . . . 7 (𝜑 → seq𝑀( · , 𝐹) ⇝ 𝑋)
6 funbrfv 6191 . . . . . . 7 (Fun ⇝ → (seq𝑀( · , 𝐹) ⇝ 𝑋 → ( ⇝ ‘seq𝑀( · , 𝐹)) = 𝑋))
74, 5, 6mpsyl 68 . . . . . 6 (𝜑 → ( ⇝ ‘seq𝑀( · , 𝐹)) = 𝑋)
87adantr 481 . . . . 5 ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → ( ⇝ ‘seq𝑀( · , 𝐹)) = 𝑋)
9 eqid 2621 . . . . . . 7 (ℤ𝑁) = (ℤ𝑁)
10 ntrivcvgfvn0.1 . . . . . . . . . 10 𝑍 = (ℤ𝑀)
11 uzssz 11651 . . . . . . . . . 10 (ℤ𝑀) ⊆ ℤ
1210, 11eqsstri 3614 . . . . . . . . 9 𝑍 ⊆ ℤ
13 ntrivcvgfvn0.2 . . . . . . . . 9 (𝜑𝑁𝑍)
1412, 13sseldi 3581 . . . . . . . 8 (𝜑𝑁 ∈ ℤ)
1514adantr 481 . . . . . . 7 ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → 𝑁 ∈ ℤ)
16 seqex 12743 . . . . . . . 8 seq𝑀( · , 𝐹) ∈ V
1716a1i 11 . . . . . . 7 ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → seq𝑀( · , 𝐹) ∈ V)
18 0cnd 9977 . . . . . . 7 ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → 0 ∈ ℂ)
19 fveq2 6148 . . . . . . . . . . 11 (𝑚 = 𝑁 → (seq𝑀( · , 𝐹)‘𝑚) = (seq𝑀( · , 𝐹)‘𝑁))
2019eqeq1d 2623 . . . . . . . . . 10 (𝑚 = 𝑁 → ((seq𝑀( · , 𝐹)‘𝑚) = 0 ↔ (seq𝑀( · , 𝐹)‘𝑁) = 0))
2120imbi2d 330 . . . . . . . . 9 (𝑚 = 𝑁 → (((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → (seq𝑀( · , 𝐹)‘𝑚) = 0) ↔ ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → (seq𝑀( · , 𝐹)‘𝑁) = 0)))
22 fveq2 6148 . . . . . . . . . . 11 (𝑚 = 𝑛 → (seq𝑀( · , 𝐹)‘𝑚) = (seq𝑀( · , 𝐹)‘𝑛))
2322eqeq1d 2623 . . . . . . . . . 10 (𝑚 = 𝑛 → ((seq𝑀( · , 𝐹)‘𝑚) = 0 ↔ (seq𝑀( · , 𝐹)‘𝑛) = 0))
2423imbi2d 330 . . . . . . . . 9 (𝑚 = 𝑛 → (((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → (seq𝑀( · , 𝐹)‘𝑚) = 0) ↔ ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → (seq𝑀( · , 𝐹)‘𝑛) = 0)))
25 fveq2 6148 . . . . . . . . . . 11 (𝑚 = (𝑛 + 1) → (seq𝑀( · , 𝐹)‘𝑚) = (seq𝑀( · , 𝐹)‘(𝑛 + 1)))
2625eqeq1d 2623 . . . . . . . . . 10 (𝑚 = (𝑛 + 1) → ((seq𝑀( · , 𝐹)‘𝑚) = 0 ↔ (seq𝑀( · , 𝐹)‘(𝑛 + 1)) = 0))
2726imbi2d 330 . . . . . . . . 9 (𝑚 = (𝑛 + 1) → (((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → (seq𝑀( · , 𝐹)‘𝑚) = 0) ↔ ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → (seq𝑀( · , 𝐹)‘(𝑛 + 1)) = 0)))
28 fveq2 6148 . . . . . . . . . . 11 (𝑚 = 𝑘 → (seq𝑀( · , 𝐹)‘𝑚) = (seq𝑀( · , 𝐹)‘𝑘))
2928eqeq1d 2623 . . . . . . . . . 10 (𝑚 = 𝑘 → ((seq𝑀( · , 𝐹)‘𝑚) = 0 ↔ (seq𝑀( · , 𝐹)‘𝑘) = 0))
3029imbi2d 330 . . . . . . . . 9 (𝑚 = 𝑘 → (((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → (seq𝑀( · , 𝐹)‘𝑚) = 0) ↔ ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → (seq𝑀( · , 𝐹)‘𝑘) = 0)))
31 simpr 477 . . . . . . . . . 10 ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → (seq𝑀( · , 𝐹)‘𝑁) = 0)
3231a1i 11 . . . . . . . . 9 (𝑁 ∈ ℤ → ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → (seq𝑀( · , 𝐹)‘𝑁) = 0))
3313, 10syl6eleq 2708 . . . . . . . . . . . . . . . 16 (𝜑𝑁 ∈ (ℤ𝑀))
34 uztrn 11648 . . . . . . . . . . . . . . . 16 ((𝑛 ∈ (ℤ𝑁) ∧ 𝑁 ∈ (ℤ𝑀)) → 𝑛 ∈ (ℤ𝑀))
3533, 34sylan2 491 . . . . . . . . . . . . . . 15 ((𝑛 ∈ (ℤ𝑁) ∧ 𝜑) → 𝑛 ∈ (ℤ𝑀))
36353adant3 1079 . . . . . . . . . . . . . 14 ((𝑛 ∈ (ℤ𝑁) ∧ 𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑛) = 0) → 𝑛 ∈ (ℤ𝑀))
37 seqp1 12756 . . . . . . . . . . . . . 14 (𝑛 ∈ (ℤ𝑀) → (seq𝑀( · , 𝐹)‘(𝑛 + 1)) = ((seq𝑀( · , 𝐹)‘𝑛) · (𝐹‘(𝑛 + 1))))
3836, 37syl 17 . . . . . . . . . . . . 13 ((𝑛 ∈ (ℤ𝑁) ∧ 𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑛) = 0) → (seq𝑀( · , 𝐹)‘(𝑛 + 1)) = ((seq𝑀( · , 𝐹)‘𝑛) · (𝐹‘(𝑛 + 1))))
39 oveq1 6611 . . . . . . . . . . . . . 14 ((seq𝑀( · , 𝐹)‘𝑛) = 0 → ((seq𝑀( · , 𝐹)‘𝑛) · (𝐹‘(𝑛 + 1))) = (0 · (𝐹‘(𝑛 + 1))))
40393ad2ant3 1082 . . . . . . . . . . . . 13 ((𝑛 ∈ (ℤ𝑁) ∧ 𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑛) = 0) → ((seq𝑀( · , 𝐹)‘𝑛) · (𝐹‘(𝑛 + 1))) = (0 · (𝐹‘(𝑛 + 1))))
41 peano2uz 11685 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ (ℤ𝑁) → (𝑛 + 1) ∈ (ℤ𝑁))
4210uztrn2 11649 . . . . . . . . . . . . . . . . . 18 ((𝑁𝑍 ∧ (𝑛 + 1) ∈ (ℤ𝑁)) → (𝑛 + 1) ∈ 𝑍)
4313, 41, 42syl2an 494 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛 ∈ (ℤ𝑁)) → (𝑛 + 1) ∈ 𝑍)
44 ntrivcvgfvn0.5 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℂ)
4544ralrimiva 2960 . . . . . . . . . . . . . . . . . 18 (𝜑 → ∀𝑘𝑍 (𝐹𝑘) ∈ ℂ)
46 fveq2 6148 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = (𝑛 + 1) → (𝐹𝑘) = (𝐹‘(𝑛 + 1)))
4746eleq1d 2683 . . . . . . . . . . . . . . . . . . 19 (𝑘 = (𝑛 + 1) → ((𝐹𝑘) ∈ ℂ ↔ (𝐹‘(𝑛 + 1)) ∈ ℂ))
4847rspcv 3291 . . . . . . . . . . . . . . . . . 18 ((𝑛 + 1) ∈ 𝑍 → (∀𝑘𝑍 (𝐹𝑘) ∈ ℂ → (𝐹‘(𝑛 + 1)) ∈ ℂ))
4945, 48mpan9 486 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑛 + 1) ∈ 𝑍) → (𝐹‘(𝑛 + 1)) ∈ ℂ)
5043, 49syldan 487 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ (ℤ𝑁)) → (𝐹‘(𝑛 + 1)) ∈ ℂ)
5150ancoms 469 . . . . . . . . . . . . . . 15 ((𝑛 ∈ (ℤ𝑁) ∧ 𝜑) → (𝐹‘(𝑛 + 1)) ∈ ℂ)
5251mul02d 10178 . . . . . . . . . . . . . 14 ((𝑛 ∈ (ℤ𝑁) ∧ 𝜑) → (0 · (𝐹‘(𝑛 + 1))) = 0)
53523adant3 1079 . . . . . . . . . . . . 13 ((𝑛 ∈ (ℤ𝑁) ∧ 𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑛) = 0) → (0 · (𝐹‘(𝑛 + 1))) = 0)
5438, 40, 533eqtrd 2659 . . . . . . . . . . . 12 ((𝑛 ∈ (ℤ𝑁) ∧ 𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑛) = 0) → (seq𝑀( · , 𝐹)‘(𝑛 + 1)) = 0)
55543exp 1261 . . . . . . . . . . 11 (𝑛 ∈ (ℤ𝑁) → (𝜑 → ((seq𝑀( · , 𝐹)‘𝑛) = 0 → (seq𝑀( · , 𝐹)‘(𝑛 + 1)) = 0)))
5655adantrd 484 . . . . . . . . . 10 (𝑛 ∈ (ℤ𝑁) → ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → ((seq𝑀( · , 𝐹)‘𝑛) = 0 → (seq𝑀( · , 𝐹)‘(𝑛 + 1)) = 0)))
5756a2d 29 . . . . . . . . 9 (𝑛 ∈ (ℤ𝑁) → (((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → (seq𝑀( · , 𝐹)‘𝑛) = 0) → ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → (seq𝑀( · , 𝐹)‘(𝑛 + 1)) = 0)))
5821, 24, 27, 30, 32, 57uzind4 11690 . . . . . . . 8 (𝑘 ∈ (ℤ𝑁) → ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → (seq𝑀( · , 𝐹)‘𝑘) = 0))
5958impcom 446 . . . . . . 7 (((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) ∧ 𝑘 ∈ (ℤ𝑁)) → (seq𝑀( · , 𝐹)‘𝑘) = 0)
609, 15, 17, 18, 59climconst 14208 . . . . . 6 ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → seq𝑀( · , 𝐹) ⇝ 0)
61 funbrfv 6191 . . . . . 6 (Fun ⇝ → (seq𝑀( · , 𝐹) ⇝ 0 → ( ⇝ ‘seq𝑀( · , 𝐹)) = 0))
624, 60, 61mpsyl 68 . . . . 5 ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → ( ⇝ ‘seq𝑀( · , 𝐹)) = 0)
638, 62eqtr3d 2657 . . . 4 ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → 𝑋 = 0)
6463ex 450 . . 3 (𝜑 → ((seq𝑀( · , 𝐹)‘𝑁) = 0 → 𝑋 = 0))
6564necon3d 2811 . 2 (𝜑 → (𝑋 ≠ 0 → (seq𝑀( · , 𝐹)‘𝑁) ≠ 0))
661, 65mpd 15 1 (𝜑 → (seq𝑀( · , 𝐹)‘𝑁) ≠ 0)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wral 2907  Vcvv 3186   class class class wbr 4613  dom cdm 5074  Fun wfun 5841  wf 5843  cfv 5847  (class class class)co 6604  cc 9878  0cc0 9880  1c1 9881   + caddc 9883   · cmul 9885  cz 11321  cuz 11631  seqcseq 12741  cli 14149
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-inf2 8482  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957  ax-pre-sup 9958
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-er 7687  df-en 7900  df-dom 7901  df-sdom 7902  df-sup 8292  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-div 10629  df-nn 10965  df-2 11023  df-3 11024  df-n0 11237  df-z 11322  df-uz 11632  df-rp 11777  df-seq 12742  df-exp 12801  df-cj 13773  df-re 13774  df-im 13775  df-sqrt 13909  df-abs 13910  df-clim 14153
This theorem is referenced by:  ntrivcvgtail  14557
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