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Theorem ntrivcvgfvn0 15249
 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 14904 . . . . . . . 8 ⇝ :dom ⇝ ⟶ℂ
3 ffun 6490 . . . . . . . 8 ( ⇝ :dom ⇝ ⟶ℂ → Fun ⇝ )
42, 3ax-mp 5 . . . . . . 7 Fun ⇝
5 ntrivcvgfvn0.3 . . . . . . 7 (𝜑 → seq𝑀( · , 𝐹) ⇝ 𝑋)
6 funbrfv 6691 . . . . . . 7 (Fun ⇝ → (seq𝑀( · , 𝐹) ⇝ 𝑋 → ( ⇝ ‘seq𝑀( · , 𝐹)) = 𝑋))
74, 5, 6mpsyl 68 . . . . . 6 (𝜑 → ( ⇝ ‘seq𝑀( · , 𝐹)) = 𝑋)
87adantr 484 . . . . 5 ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → ( ⇝ ‘seq𝑀( · , 𝐹)) = 𝑋)
9 eqid 2798 . . . . . . 7 (ℤ𝑁) = (ℤ𝑁)
10 ntrivcvgfvn0.1 . . . . . . . . . 10 𝑍 = (ℤ𝑀)
11 uzssz 12254 . . . . . . . . . 10 (ℤ𝑀) ⊆ ℤ
1210, 11eqsstri 3949 . . . . . . . . 9 𝑍 ⊆ ℤ
13 ntrivcvgfvn0.2 . . . . . . . . 9 (𝜑𝑁𝑍)
1412, 13sseldi 3913 . . . . . . . 8 (𝜑𝑁 ∈ ℤ)
1514adantr 484 . . . . . . 7 ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → 𝑁 ∈ ℤ)
16 seqex 13368 . . . . . . . 8 seq𝑀( · , 𝐹) ∈ V
1716a1i 11 . . . . . . 7 ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → seq𝑀( · , 𝐹) ∈ V)
18 0cnd 10625 . . . . . . 7 ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → 0 ∈ ℂ)
19 fveqeq2 6654 . . . . . . . . . 10 (𝑚 = 𝑁 → ((seq𝑀( · , 𝐹)‘𝑚) = 0 ↔ (seq𝑀( · , 𝐹)‘𝑁) = 0))
2019imbi2d 344 . . . . . . . . 9 (𝑚 = 𝑁 → (((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → (seq𝑀( · , 𝐹)‘𝑚) = 0) ↔ ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → (seq𝑀( · , 𝐹)‘𝑁) = 0)))
21 fveqeq2 6654 . . . . . . . . . 10 (𝑚 = 𝑛 → ((seq𝑀( · , 𝐹)‘𝑚) = 0 ↔ (seq𝑀( · , 𝐹)‘𝑛) = 0))
2221imbi2d 344 . . . . . . . . 9 (𝑚 = 𝑛 → (((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → (seq𝑀( · , 𝐹)‘𝑚) = 0) ↔ ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → (seq𝑀( · , 𝐹)‘𝑛) = 0)))
23 fveqeq2 6654 . . . . . . . . . 10 (𝑚 = (𝑛 + 1) → ((seq𝑀( · , 𝐹)‘𝑚) = 0 ↔ (seq𝑀( · , 𝐹)‘(𝑛 + 1)) = 0))
2423imbi2d 344 . . . . . . . . 9 (𝑚 = (𝑛 + 1) → (((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → (seq𝑀( · , 𝐹)‘𝑚) = 0) ↔ ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → (seq𝑀( · , 𝐹)‘(𝑛 + 1)) = 0)))
25 fveqeq2 6654 . . . . . . . . . 10 (𝑚 = 𝑘 → ((seq𝑀( · , 𝐹)‘𝑚) = 0 ↔ (seq𝑀( · , 𝐹)‘𝑘) = 0))
2625imbi2d 344 . . . . . . . . 9 (𝑚 = 𝑘 → (((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → (seq𝑀( · , 𝐹)‘𝑚) = 0) ↔ ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → (seq𝑀( · , 𝐹)‘𝑘) = 0)))
27 simpr 488 . . . . . . . . 9 ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → (seq𝑀( · , 𝐹)‘𝑁) = 0)
2813, 10eleqtrdi 2900 . . . . . . . . . . . . . . . 16 (𝜑𝑁 ∈ (ℤ𝑀))
29 uztrn 12251 . . . . . . . . . . . . . . . 16 ((𝑛 ∈ (ℤ𝑁) ∧ 𝑁 ∈ (ℤ𝑀)) → 𝑛 ∈ (ℤ𝑀))
3028, 29sylan2 595 . . . . . . . . . . . . . . 15 ((𝑛 ∈ (ℤ𝑁) ∧ 𝜑) → 𝑛 ∈ (ℤ𝑀))
31303adant3 1129 . . . . . . . . . . . . . 14 ((𝑛 ∈ (ℤ𝑁) ∧ 𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑛) = 0) → 𝑛 ∈ (ℤ𝑀))
32 seqp1 13381 . . . . . . . . . . . . . 14 (𝑛 ∈ (ℤ𝑀) → (seq𝑀( · , 𝐹)‘(𝑛 + 1)) = ((seq𝑀( · , 𝐹)‘𝑛) · (𝐹‘(𝑛 + 1))))
3331, 32syl 17 . . . . . . . . . . . . 13 ((𝑛 ∈ (ℤ𝑁) ∧ 𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑛) = 0) → (seq𝑀( · , 𝐹)‘(𝑛 + 1)) = ((seq𝑀( · , 𝐹)‘𝑛) · (𝐹‘(𝑛 + 1))))
34 oveq1 7142 . . . . . . . . . . . . . 14 ((seq𝑀( · , 𝐹)‘𝑛) = 0 → ((seq𝑀( · , 𝐹)‘𝑛) · (𝐹‘(𝑛 + 1))) = (0 · (𝐹‘(𝑛 + 1))))
35343ad2ant3 1132 . . . . . . . . . . . . 13 ((𝑛 ∈ (ℤ𝑁) ∧ 𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑛) = 0) → ((seq𝑀( · , 𝐹)‘𝑛) · (𝐹‘(𝑛 + 1))) = (0 · (𝐹‘(𝑛 + 1))))
36 peano2uz 12291 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ (ℤ𝑁) → (𝑛 + 1) ∈ (ℤ𝑁))
3710uztrn2 12252 . . . . . . . . . . . . . . . . . 18 ((𝑁𝑍 ∧ (𝑛 + 1) ∈ (ℤ𝑁)) → (𝑛 + 1) ∈ 𝑍)
3813, 36, 37syl2an 598 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛 ∈ (ℤ𝑁)) → (𝑛 + 1) ∈ 𝑍)
39 ntrivcvgfvn0.5 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℂ)
4039ralrimiva 3149 . . . . . . . . . . . . . . . . . 18 (𝜑 → ∀𝑘𝑍 (𝐹𝑘) ∈ ℂ)
41 fveq2 6645 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = (𝑛 + 1) → (𝐹𝑘) = (𝐹‘(𝑛 + 1)))
4241eleq1d 2874 . . . . . . . . . . . . . . . . . . 19 (𝑘 = (𝑛 + 1) → ((𝐹𝑘) ∈ ℂ ↔ (𝐹‘(𝑛 + 1)) ∈ ℂ))
4342rspcv 3566 . . . . . . . . . . . . . . . . . 18 ((𝑛 + 1) ∈ 𝑍 → (∀𝑘𝑍 (𝐹𝑘) ∈ ℂ → (𝐹‘(𝑛 + 1)) ∈ ℂ))
4440, 43mpan9 510 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑛 + 1) ∈ 𝑍) → (𝐹‘(𝑛 + 1)) ∈ ℂ)
4538, 44syldan 594 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ (ℤ𝑁)) → (𝐹‘(𝑛 + 1)) ∈ ℂ)
4645ancoms 462 . . . . . . . . . . . . . . 15 ((𝑛 ∈ (ℤ𝑁) ∧ 𝜑) → (𝐹‘(𝑛 + 1)) ∈ ℂ)
4746mul02d 10829 . . . . . . . . . . . . . 14 ((𝑛 ∈ (ℤ𝑁) ∧ 𝜑) → (0 · (𝐹‘(𝑛 + 1))) = 0)
48473adant3 1129 . . . . . . . . . . . . 13 ((𝑛 ∈ (ℤ𝑁) ∧ 𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑛) = 0) → (0 · (𝐹‘(𝑛 + 1))) = 0)
4933, 35, 483eqtrd 2837 . . . . . . . . . . . 12 ((𝑛 ∈ (ℤ𝑁) ∧ 𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑛) = 0) → (seq𝑀( · , 𝐹)‘(𝑛 + 1)) = 0)
50493exp 1116 . . . . . . . . . . 11 (𝑛 ∈ (ℤ𝑁) → (𝜑 → ((seq𝑀( · , 𝐹)‘𝑛) = 0 → (seq𝑀( · , 𝐹)‘(𝑛 + 1)) = 0)))
5150adantrd 495 . . . . . . . . . 10 (𝑛 ∈ (ℤ𝑁) → ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → ((seq𝑀( · , 𝐹)‘𝑛) = 0 → (seq𝑀( · , 𝐹)‘(𝑛 + 1)) = 0)))
5251a2d 29 . . . . . . . . 9 (𝑛 ∈ (ℤ𝑁) → (((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → (seq𝑀( · , 𝐹)‘𝑛) = 0) → ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → (seq𝑀( · , 𝐹)‘(𝑛 + 1)) = 0)))
5320, 22, 24, 26, 27, 52uzind4i 12300 . . . . . . . 8 (𝑘 ∈ (ℤ𝑁) → ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → (seq𝑀( · , 𝐹)‘𝑘) = 0))
5453impcom 411 . . . . . . 7 (((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) ∧ 𝑘 ∈ (ℤ𝑁)) → (seq𝑀( · , 𝐹)‘𝑘) = 0)
559, 15, 17, 18, 54climconst 14894 . . . . . 6 ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → seq𝑀( · , 𝐹) ⇝ 0)
56 funbrfv 6691 . . . . . 6 (Fun ⇝ → (seq𝑀( · , 𝐹) ⇝ 0 → ( ⇝ ‘seq𝑀( · , 𝐹)) = 0))
574, 55, 56mpsyl 68 . . . . 5 ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → ( ⇝ ‘seq𝑀( · , 𝐹)) = 0)
588, 57eqtr3d 2835 . . . 4 ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → 𝑋 = 0)
5958ex 416 . . 3 (𝜑 → ((seq𝑀( · , 𝐹)‘𝑁) = 0 → 𝑋 = 0))
6059necon3d 3008 . 2 (𝜑 → (𝑋 ≠ 0 → (seq𝑀( · , 𝐹)‘𝑁) ≠ 0))
611, 60mpd 15 1 (𝜑 → (seq𝑀( · , 𝐹)‘𝑁) ≠ 0)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  ∀wral 3106  Vcvv 3441   class class class wbr 5030  dom cdm 5519  Fun wfun 6318  ⟶wf 6320  ‘cfv 6324  (class class class)co 7135  ℂcc 10526  0cc0 10528  1c1 10529   + caddc 10531   · cmul 10533  ℤcz 11971  ℤ≥cuz 12233  seqcseq 13366   ⇝ cli 14835 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7443  ax-inf2 9090  ax-cnex 10584  ax-resscn 10585  ax-1cn 10586  ax-icn 10587  ax-addcl 10588  ax-addrcl 10589  ax-mulcl 10590  ax-mulrcl 10591  ax-mulcom 10592  ax-addass 10593  ax-mulass 10594  ax-distr 10595  ax-i2m1 10596  ax-1ne0 10597  ax-1rid 10598  ax-rnegex 10599  ax-rrecex 10600  ax-cnre 10601  ax-pre-lttri 10602  ax-pre-lttrn 10603  ax-pre-ltadd 10604  ax-pre-mulgt0 10605  ax-pre-sup 10606 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7563  df-2nd 7674  df-wrecs 7932  df-recs 7993  df-rdg 8031  df-er 8274  df-en 8495  df-dom 8496  df-sdom 8497  df-sup 8892  df-pnf 10668  df-mnf 10669  df-xr 10670  df-ltxr 10671  df-le 10672  df-sub 10863  df-neg 10864  df-div 11289  df-nn 11628  df-2 11690  df-3 11691  df-n0 11888  df-z 11972  df-uz 12234  df-rp 12380  df-seq 13367  df-exp 13428  df-cj 14452  df-re 14453  df-im 14454  df-sqrt 14588  df-abs 14589  df-clim 14839 This theorem is referenced by:  ntrivcvgtail  15250
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