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Theorem ntrivcvgfvn0 15243
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 14898 . . . . . . . 8 ⇝ :dom ⇝ ⟶ℂ
3 ffun 6510 . . . . . . . 8 ( ⇝ :dom ⇝ ⟶ℂ → Fun ⇝ )
42, 3ax-mp 5 . . . . . . 7 Fun ⇝
5 ntrivcvgfvn0.3 . . . . . . 7 (𝜑 → seq𝑀( · , 𝐹) ⇝ 𝑋)
6 funbrfv 6709 . . . . . . 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 2818 . . . . . . 7 (ℤ𝑁) = (ℤ𝑁)
10 ntrivcvgfvn0.1 . . . . . . . . . 10 𝑍 = (ℤ𝑀)
11 uzssz 12252 . . . . . . . . . 10 (ℤ𝑀) ⊆ ℤ
1210, 11eqsstri 3998 . . . . . . . . 9 𝑍 ⊆ ℤ
13 ntrivcvgfvn0.2 . . . . . . . . 9 (𝜑𝑁𝑍)
1412, 13sseldi 3962 . . . . . . . 8 (𝜑𝑁 ∈ ℤ)
1514adantr 481 . . . . . . 7 ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → 𝑁 ∈ ℤ)
16 seqex 13359 . . . . . . . 8 seq𝑀( · , 𝐹) ∈ V
1716a1i 11 . . . . . . 7 ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → seq𝑀( · , 𝐹) ∈ V)
18 0cnd 10622 . . . . . . 7 ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → 0 ∈ ℂ)
19 fveqeq2 6672 . . . . . . . . . 10 (𝑚 = 𝑁 → ((seq𝑀( · , 𝐹)‘𝑚) = 0 ↔ (seq𝑀( · , 𝐹)‘𝑁) = 0))
2019imbi2d 342 . . . . . . . . 9 (𝑚 = 𝑁 → (((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → (seq𝑀( · , 𝐹)‘𝑚) = 0) ↔ ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → (seq𝑀( · , 𝐹)‘𝑁) = 0)))
21 fveqeq2 6672 . . . . . . . . . 10 (𝑚 = 𝑛 → ((seq𝑀( · , 𝐹)‘𝑚) = 0 ↔ (seq𝑀( · , 𝐹)‘𝑛) = 0))
2221imbi2d 342 . . . . . . . . 9 (𝑚 = 𝑛 → (((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → (seq𝑀( · , 𝐹)‘𝑚) = 0) ↔ ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → (seq𝑀( · , 𝐹)‘𝑛) = 0)))
23 fveqeq2 6672 . . . . . . . . . 10 (𝑚 = (𝑛 + 1) → ((seq𝑀( · , 𝐹)‘𝑚) = 0 ↔ (seq𝑀( · , 𝐹)‘(𝑛 + 1)) = 0))
2423imbi2d 342 . . . . . . . . 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 6672 . . . . . . . . . 10 (𝑚 = 𝑘 → ((seq𝑀( · , 𝐹)‘𝑚) = 0 ↔ (seq𝑀( · , 𝐹)‘𝑘) = 0))
2625imbi2d 342 . . . . . . . . 9 (𝑚 = 𝑘 → (((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → (seq𝑀( · , 𝐹)‘𝑚) = 0) ↔ ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → (seq𝑀( · , 𝐹)‘𝑘) = 0)))
27 simpr 485 . . . . . . . . 9 ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → (seq𝑀( · , 𝐹)‘𝑁) = 0)
2813, 10eleqtrdi 2920 . . . . . . . . . . . . . . . 16 (𝜑𝑁 ∈ (ℤ𝑀))
29 uztrn 12249 . . . . . . . . . . . . . . . 16 ((𝑛 ∈ (ℤ𝑁) ∧ 𝑁 ∈ (ℤ𝑀)) → 𝑛 ∈ (ℤ𝑀))
3028, 29sylan2 592 . . . . . . . . . . . . . . 15 ((𝑛 ∈ (ℤ𝑁) ∧ 𝜑) → 𝑛 ∈ (ℤ𝑀))
31303adant3 1124 . . . . . . . . . . . . . 14 ((𝑛 ∈ (ℤ𝑁) ∧ 𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑛) = 0) → 𝑛 ∈ (ℤ𝑀))
32 seqp1 13372 . . . . . . . . . . . . . 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 7152 . . . . . . . . . . . . . 14 ((seq𝑀( · , 𝐹)‘𝑛) = 0 → ((seq𝑀( · , 𝐹)‘𝑛) · (𝐹‘(𝑛 + 1))) = (0 · (𝐹‘(𝑛 + 1))))
35343ad2ant3 1127 . . . . . . . . . . . . 13 ((𝑛 ∈ (ℤ𝑁) ∧ 𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑛) = 0) → ((seq𝑀( · , 𝐹)‘𝑛) · (𝐹‘(𝑛 + 1))) = (0 · (𝐹‘(𝑛 + 1))))
36 peano2uz 12289 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ (ℤ𝑁) → (𝑛 + 1) ∈ (ℤ𝑁))
3710uztrn2 12250 . . . . . . . . . . . . . . . . . 18 ((𝑁𝑍 ∧ (𝑛 + 1) ∈ (ℤ𝑁)) → (𝑛 + 1) ∈ 𝑍)
3813, 36, 37syl2an 595 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛 ∈ (ℤ𝑁)) → (𝑛 + 1) ∈ 𝑍)
39 ntrivcvgfvn0.5 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℂ)
4039ralrimiva 3179 . . . . . . . . . . . . . . . . . 18 (𝜑 → ∀𝑘𝑍 (𝐹𝑘) ∈ ℂ)
41 fveq2 6663 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = (𝑛 + 1) → (𝐹𝑘) = (𝐹‘(𝑛 + 1)))
4241eleq1d 2894 . . . . . . . . . . . . . . . . . . 19 (𝑘 = (𝑛 + 1) → ((𝐹𝑘) ∈ ℂ ↔ (𝐹‘(𝑛 + 1)) ∈ ℂ))
4342rspcv 3615 . . . . . . . . . . . . . . . . . 18 ((𝑛 + 1) ∈ 𝑍 → (∀𝑘𝑍 (𝐹𝑘) ∈ ℂ → (𝐹‘(𝑛 + 1)) ∈ ℂ))
4440, 43mpan9 507 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑛 + 1) ∈ 𝑍) → (𝐹‘(𝑛 + 1)) ∈ ℂ)
4538, 44syldan 591 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ (ℤ𝑁)) → (𝐹‘(𝑛 + 1)) ∈ ℂ)
4645ancoms 459 . . . . . . . . . . . . . . 15 ((𝑛 ∈ (ℤ𝑁) ∧ 𝜑) → (𝐹‘(𝑛 + 1)) ∈ ℂ)
4746mul02d 10826 . . . . . . . . . . . . . 14 ((𝑛 ∈ (ℤ𝑁) ∧ 𝜑) → (0 · (𝐹‘(𝑛 + 1))) = 0)
48473adant3 1124 . . . . . . . . . . . . 13 ((𝑛 ∈ (ℤ𝑁) ∧ 𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑛) = 0) → (0 · (𝐹‘(𝑛 + 1))) = 0)
4933, 35, 483eqtrd 2857 . . . . . . . . . . . 12 ((𝑛 ∈ (ℤ𝑁) ∧ 𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑛) = 0) → (seq𝑀( · , 𝐹)‘(𝑛 + 1)) = 0)
50493exp 1111 . . . . . . . . . . 11 (𝑛 ∈ (ℤ𝑁) → (𝜑 → ((seq𝑀( · , 𝐹)‘𝑛) = 0 → (seq𝑀( · , 𝐹)‘(𝑛 + 1)) = 0)))
5150adantrd 492 . . . . . . . . . 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 12298 . . . . . . . 8 (𝑘 ∈ (ℤ𝑁) → ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → (seq𝑀( · , 𝐹)‘𝑘) = 0))
5453impcom 408 . . . . . . 7 (((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) ∧ 𝑘 ∈ (ℤ𝑁)) → (seq𝑀( · , 𝐹)‘𝑘) = 0)
559, 15, 17, 18, 54climconst 14888 . . . . . 6 ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → seq𝑀( · , 𝐹) ⇝ 0)
56 funbrfv 6709 . . . . . 6 (Fun ⇝ → (seq𝑀( · , 𝐹) ⇝ 0 → ( ⇝ ‘seq𝑀( · , 𝐹)) = 0))
574, 55, 56mpsyl 68 . . . . 5 ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → ( ⇝ ‘seq𝑀( · , 𝐹)) = 0)
588, 57eqtr3d 2855 . . . 4 ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → 𝑋 = 0)
5958ex 413 . . 3 (𝜑 → ((seq𝑀( · , 𝐹)‘𝑁) = 0 → 𝑋 = 0))
6059necon3d 3034 . 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 396  w3a 1079   = wceq 1528  wcel 2105  wne 3013  wral 3135  Vcvv 3492   class class class wbr 5057  dom cdm 5548  Fun wfun 6342  wf 6344  cfv 6348  (class class class)co 7145  cc 10523  0cc0 10525  1c1 10526   + caddc 10528   · cmul 10530  cz 11969  cuz 12231  seqcseq 13357  cli 14829
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-inf2 9092  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602  ax-pre-sup 10603
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500  df-sup 8894  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-div 11286  df-nn 11627  df-2 11688  df-3 11689  df-n0 11886  df-z 11970  df-uz 12232  df-rp 12378  df-seq 13358  df-exp 13418  df-cj 14446  df-re 14447  df-im 14448  df-sqrt 14582  df-abs 14583  df-clim 14833
This theorem is referenced by:  ntrivcvgtail  15244
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