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Theorem ntrivcvgfvn0 15885
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 15537 . . . . . . . 8 ⇝ :dom ⇝ ⟶ℂ
3 ffun 6730 . . . . . . . 8 ( ⇝ :dom ⇝ ⟶ℂ → Fun ⇝ )
42, 3ax-mp 5 . . . . . . 7 Fun ⇝
5 ntrivcvgfvn0.3 . . . . . . 7 (𝜑 → seq𝑀( · , 𝐹) ⇝ 𝑋)
6 funbrfv 6953 . . . . . . 7 (Fun ⇝ → (seq𝑀( · , 𝐹) ⇝ 𝑋 → ( ⇝ ‘seq𝑀( · , 𝐹)) = 𝑋))
74, 5, 6mpsyl 68 . . . . . 6 (𝜑 → ( ⇝ ‘seq𝑀( · , 𝐹)) = 𝑋)
87adantr 479 . . . . 5 ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → ( ⇝ ‘seq𝑀( · , 𝐹)) = 𝑋)
9 eqid 2728 . . . . . . 7 (ℤ𝑁) = (ℤ𝑁)
10 ntrivcvgfvn0.1 . . . . . . . . . 10 𝑍 = (ℤ𝑀)
11 uzssz 12881 . . . . . . . . . 10 (ℤ𝑀) ⊆ ℤ
1210, 11eqsstri 4016 . . . . . . . . 9 𝑍 ⊆ ℤ
13 ntrivcvgfvn0.2 . . . . . . . . 9 (𝜑𝑁𝑍)
1412, 13sselid 3980 . . . . . . . 8 (𝜑𝑁 ∈ ℤ)
1514adantr 479 . . . . . . 7 ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → 𝑁 ∈ ℤ)
16 seqex 14008 . . . . . . . 8 seq𝑀( · , 𝐹) ∈ V
1716a1i 11 . . . . . . 7 ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → seq𝑀( · , 𝐹) ∈ V)
18 0cnd 11245 . . . . . . 7 ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → 0 ∈ ℂ)
19 fveqeq2 6911 . . . . . . . . . 10 (𝑚 = 𝑁 → ((seq𝑀( · , 𝐹)‘𝑚) = 0 ↔ (seq𝑀( · , 𝐹)‘𝑁) = 0))
2019imbi2d 339 . . . . . . . . 9 (𝑚 = 𝑁 → (((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → (seq𝑀( · , 𝐹)‘𝑚) = 0) ↔ ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → (seq𝑀( · , 𝐹)‘𝑁) = 0)))
21 fveqeq2 6911 . . . . . . . . . 10 (𝑚 = 𝑛 → ((seq𝑀( · , 𝐹)‘𝑚) = 0 ↔ (seq𝑀( · , 𝐹)‘𝑛) = 0))
2221imbi2d 339 . . . . . . . . 9 (𝑚 = 𝑛 → (((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → (seq𝑀( · , 𝐹)‘𝑚) = 0) ↔ ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → (seq𝑀( · , 𝐹)‘𝑛) = 0)))
23 fveqeq2 6911 . . . . . . . . . 10 (𝑚 = (𝑛 + 1) → ((seq𝑀( · , 𝐹)‘𝑚) = 0 ↔ (seq𝑀( · , 𝐹)‘(𝑛 + 1)) = 0))
2423imbi2d 339 . . . . . . . . 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 6911 . . . . . . . . . 10 (𝑚 = 𝑘 → ((seq𝑀( · , 𝐹)‘𝑚) = 0 ↔ (seq𝑀( · , 𝐹)‘𝑘) = 0))
2625imbi2d 339 . . . . . . . . 9 (𝑚 = 𝑘 → (((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → (seq𝑀( · , 𝐹)‘𝑚) = 0) ↔ ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → (seq𝑀( · , 𝐹)‘𝑘) = 0)))
27 simpr 483 . . . . . . . . 9 ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → (seq𝑀( · , 𝐹)‘𝑁) = 0)
2813, 10eleqtrdi 2839 . . . . . . . . . . . . . . . 16 (𝜑𝑁 ∈ (ℤ𝑀))
29 uztrn 12878 . . . . . . . . . . . . . . . 16 ((𝑛 ∈ (ℤ𝑁) ∧ 𝑁 ∈ (ℤ𝑀)) → 𝑛 ∈ (ℤ𝑀))
3028, 29sylan2 591 . . . . . . . . . . . . . . 15 ((𝑛 ∈ (ℤ𝑁) ∧ 𝜑) → 𝑛 ∈ (ℤ𝑀))
31303adant3 1129 . . . . . . . . . . . . . 14 ((𝑛 ∈ (ℤ𝑁) ∧ 𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑛) = 0) → 𝑛 ∈ (ℤ𝑀))
32 seqp1 14021 . . . . . . . . . . . . . 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 7433 . . . . . . . . . . . . . 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 12923 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ (ℤ𝑁) → (𝑛 + 1) ∈ (ℤ𝑁))
3710uztrn2 12879 . . . . . . . . . . . . . . . . . 18 ((𝑁𝑍 ∧ (𝑛 + 1) ∈ (ℤ𝑁)) → (𝑛 + 1) ∈ 𝑍)
3813, 36, 37syl2an 594 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛 ∈ (ℤ𝑁)) → (𝑛 + 1) ∈ 𝑍)
39 ntrivcvgfvn0.5 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℂ)
4039ralrimiva 3143 . . . . . . . . . . . . . . . . . 18 (𝜑 → ∀𝑘𝑍 (𝐹𝑘) ∈ ℂ)
41 fveq2 6902 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = (𝑛 + 1) → (𝐹𝑘) = (𝐹‘(𝑛 + 1)))
4241eleq1d 2814 . . . . . . . . . . . . . . . . . . 19 (𝑘 = (𝑛 + 1) → ((𝐹𝑘) ∈ ℂ ↔ (𝐹‘(𝑛 + 1)) ∈ ℂ))
4342rspcv 3607 . . . . . . . . . . . . . . . . . 18 ((𝑛 + 1) ∈ 𝑍 → (∀𝑘𝑍 (𝐹𝑘) ∈ ℂ → (𝐹‘(𝑛 + 1)) ∈ ℂ))
4440, 43mpan9 505 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑛 + 1) ∈ 𝑍) → (𝐹‘(𝑛 + 1)) ∈ ℂ)
4538, 44syldan 589 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ (ℤ𝑁)) → (𝐹‘(𝑛 + 1)) ∈ ℂ)
4645ancoms 457 . . . . . . . . . . . . . . 15 ((𝑛 ∈ (ℤ𝑁) ∧ 𝜑) → (𝐹‘(𝑛 + 1)) ∈ ℂ)
4746mul02d 11450 . . . . . . . . . . . . . 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 2772 . . . . . . . . . . . 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 490 . . . . . . . . . 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 12932 . . . . . . . 8 (𝑘 ∈ (ℤ𝑁) → ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → (seq𝑀( · , 𝐹)‘𝑘) = 0))
5453impcom 406 . . . . . . 7 (((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) ∧ 𝑘 ∈ (ℤ𝑁)) → (seq𝑀( · , 𝐹)‘𝑘) = 0)
559, 15, 17, 18, 54climconst 15527 . . . . . 6 ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → seq𝑀( · , 𝐹) ⇝ 0)
56 funbrfv 6953 . . . . . 6 (Fun ⇝ → (seq𝑀( · , 𝐹) ⇝ 0 → ( ⇝ ‘seq𝑀( · , 𝐹)) = 0))
574, 55, 56mpsyl 68 . . . . 5 ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → ( ⇝ ‘seq𝑀( · , 𝐹)) = 0)
588, 57eqtr3d 2770 . . . 4 ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → 𝑋 = 0)
5958ex 411 . . 3 (𝜑 → ((seq𝑀( · , 𝐹)‘𝑁) = 0 → 𝑋 = 0))
6059necon3d 2958 . 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 394  w3a 1084   = wceq 1533  wcel 2098  wne 2937  wral 3058  Vcvv 3473   class class class wbr 5152  dom cdm 5682  Fun wfun 6547  wf 6549  cfv 6553  (class class class)co 7426  cc 11144  0cc0 11146  1c1 11147   + caddc 11149   · cmul 11151  cz 12596  cuz 12860  seqcseq 14006  cli 15468
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-inf2 9672  ax-cnex 11202  ax-resscn 11203  ax-1cn 11204  ax-icn 11205  ax-addcl 11206  ax-addrcl 11207  ax-mulcl 11208  ax-mulrcl 11209  ax-mulcom 11210  ax-addass 11211  ax-mulass 11212  ax-distr 11213  ax-i2m1 11214  ax-1ne0 11215  ax-1rid 11216  ax-rnegex 11217  ax-rrecex 11218  ax-cnre 11219  ax-pre-lttri 11220  ax-pre-lttrn 11221  ax-pre-ltadd 11222  ax-pre-mulgt0 11223  ax-pre-sup 11224
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7877  df-2nd 8000  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-rdg 8437  df-er 8731  df-en 8971  df-dom 8972  df-sdom 8973  df-sup 9473  df-pnf 11288  df-mnf 11289  df-xr 11290  df-ltxr 11291  df-le 11292  df-sub 11484  df-neg 11485  df-div 11910  df-nn 12251  df-2 12313  df-3 12314  df-n0 12511  df-z 12597  df-uz 12861  df-rp 13015  df-seq 14007  df-exp 14067  df-cj 15086  df-re 15087  df-im 15088  df-sqrt 15222  df-abs 15223  df-clim 15472
This theorem is referenced by:  ntrivcvgtail  15886
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