Theorem ntrivcvgfvn0 15836

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 = (ℤ≥‘𝑀)
ntrivcvgfvn0.2	⊢ (𝜑 → 𝑁 ∈ 𝑍)
ntrivcvgfvn0.3	⊢ (𝜑 → seq𝑀( · , 𝐹) ⇝ 𝑋)
ntrivcvgfvn0.4	⊢ (𝜑 → 𝑋 ≠ 0)
ntrivcvgfvn0.5	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ 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.

Step	Hyp	Ref	Expression
1		ntrivcvgfvn0.4	. 2 ⊢ (𝜑 → 𝑋 ≠ 0)
2		fclim 15490	. . . . . . . 8 ⊢ ⇝ :dom ⇝ ⟶ℂ
3		ffun 6675	. . . . . . . 8 ⊢ ( ⇝ :dom ⇝ ⟶ℂ → Fun ⇝ )
4	2, 3	ax-mp 5	. . . . . . 7 ⊢ Fun ⇝
5		ntrivcvgfvn0.3	. . . . . . 7 ⊢ (𝜑 → seq𝑀( · , 𝐹) ⇝ 𝑋)
6		funbrfv 6892	. . . . . . 7 ⊢ (Fun ⇝ → (seq𝑀( · , 𝐹) ⇝ 𝑋 → ( ⇝ ‘seq𝑀( · , 𝐹)) = 𝑋))
7	4, 5, 6	mpsyl 68	. . . . . 6 ⊢ (𝜑 → ( ⇝ ‘seq𝑀( · , 𝐹)) = 𝑋)
8	7	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → ( ⇝ ‘seq𝑀( · , 𝐹)) = 𝑋)
9		eqid 2737	. . . . . . 7 ⊢ (ℤ≥‘𝑁) = (ℤ≥‘𝑁)
10		ntrivcvgfvn0.1	. . . . . . . . . 10 ⊢ 𝑍 = (ℤ≥‘𝑀)
11		uzssz 12786	. . . . . . . . . 10 ⊢ (ℤ≥‘𝑀) ⊆ ℤ
12	10, 11	eqsstri 3982	. . . . . . . . 9 ⊢ 𝑍 ⊆ ℤ
13		ntrivcvgfvn0.2	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ 𝑍)
14	12, 13	sselid 3933	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℤ)
15	14	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → 𝑁 ∈ ℤ)
16		seqex 13940	. . . . . . . 8 ⊢ seq𝑀( · , 𝐹) ∈ V
17	16	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → seq𝑀( · , 𝐹) ∈ V)
18		0cnd 11139	. . . . . . 7 ⊢ ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → 0 ∈ ℂ)
19		fveqeq2 6853	. . . . . . . . . 10 ⊢ (𝑚 = 𝑁 → ((seq𝑀( · , 𝐹)‘𝑚) = 0 ↔ (seq𝑀( · , 𝐹)‘𝑁) = 0))
20	19	imbi2d 340	. . . . . . . . 9 ⊢ (𝑚 = 𝑁 → (((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → (seq𝑀( · , 𝐹)‘𝑚) = 0) ↔ ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → (seq𝑀( · , 𝐹)‘𝑁) = 0)))
21		fveqeq2 6853	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → ((seq𝑀( · , 𝐹)‘𝑚) = 0 ↔ (seq𝑀( · , 𝐹)‘𝑛) = 0))
22	21	imbi2d 340	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → (((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → (seq𝑀( · , 𝐹)‘𝑚) = 0) ↔ ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → (seq𝑀( · , 𝐹)‘𝑛) = 0)))
23		fveqeq2 6853	. . . . . . . . . 10 ⊢ (𝑚 = (𝑛 + 1) → ((seq𝑀( · , 𝐹)‘𝑚) = 0 ↔ (seq𝑀( · , 𝐹)‘(𝑛 + 1)) = 0))
24	23	imbi2d 340	. . . . . . . . 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 6853	. . . . . . . . . 10 ⊢ (𝑚 = 𝑘 → ((seq𝑀( · , 𝐹)‘𝑚) = 0 ↔ (seq𝑀( · , 𝐹)‘𝑘) = 0))
26	25	imbi2d 340	. . . . . . . . 9 ⊢ (𝑚 = 𝑘 → (((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → (seq𝑀( · , 𝐹)‘𝑚) = 0) ↔ ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → (seq𝑀( · , 𝐹)‘𝑘) = 0)))
27		simpr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → (seq𝑀( · , 𝐹)‘𝑁) = 0)
28	13, 10	eleqtrdi 2847	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
29		uztrn 12783	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 ∈ (ℤ≥‘𝑁) ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → 𝑛 ∈ (ℤ≥‘𝑀))
30	28, 29	sylan2 594	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ (ℤ≥‘𝑁) ∧ 𝜑) → 𝑛 ∈ (ℤ≥‘𝑀))
31	30	3adant3 1133	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ (ℤ≥‘𝑁) ∧ 𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑛) = 0) → 𝑛 ∈ (ℤ≥‘𝑀))
32		seqp1 13953	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (ℤ≥‘𝑀) → (seq𝑀( · , 𝐹)‘(𝑛 + 1)) = ((seq𝑀( · , 𝐹)‘𝑛) · (𝐹‘(𝑛 + 1))))
33	31, 32	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ (ℤ≥‘𝑁) ∧ 𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑛) = 0) → (seq𝑀( · , 𝐹)‘(𝑛 + 1)) = ((seq𝑀( · , 𝐹)‘𝑛) · (𝐹‘(𝑛 + 1))))
34		oveq1 7377	. . . . . . . . . . . . . 14 ⊢ ((seq𝑀( · , 𝐹)‘𝑛) = 0 → ((seq𝑀( · , 𝐹)‘𝑛) · (𝐹‘(𝑛 + 1))) = (0 · (𝐹‘(𝑛 + 1))))
35	34	3ad2ant3 1136	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ (ℤ≥‘𝑁) ∧ 𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑛) = 0) → ((seq𝑀( · , 𝐹)‘𝑛) · (𝐹‘(𝑛 + 1))) = (0 · (𝐹‘(𝑛 + 1))))
36		peano2uz 12828	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ (ℤ≥‘𝑁) → (𝑛 + 1) ∈ (ℤ≥‘𝑁))
37	10	uztrn2 12784	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ 𝑍 ∧ (𝑛 + 1) ∈ (ℤ≥‘𝑁)) → (𝑛 + 1) ∈ 𝑍)
38	13, 36, 37	syl2an 597	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑁)) → (𝑛 + 1) ∈ 𝑍)
39		ntrivcvgfvn0.5	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)
40	39	ralrimiva 3130	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ)
41		fveq2 6844	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = (𝑛 + 1) → (𝐹‘𝑘) = (𝐹‘(𝑛 + 1)))
42	41	eleq1d 2822	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = (𝑛 + 1) → ((𝐹‘𝑘) ∈ ℂ ↔ (𝐹‘(𝑛 + 1)) ∈ ℂ))
43	42	rspcv 3574	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑛 + 1) ∈ 𝑍 → (∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ → (𝐹‘(𝑛 + 1)) ∈ ℂ))
44	40, 43	mpan9 506	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑛 + 1) ∈ 𝑍) → (𝐹‘(𝑛 + 1)) ∈ ℂ)
45	38, 44	syldan 592	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑁)) → (𝐹‘(𝑛 + 1)) ∈ ℂ)
46	45	ancoms 458	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ (ℤ≥‘𝑁) ∧ 𝜑) → (𝐹‘(𝑛 + 1)) ∈ ℂ)
47	46	mul02d 11345	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ (ℤ≥‘𝑁) ∧ 𝜑) → (0 · (𝐹‘(𝑛 + 1))) = 0)
48	47	3adant3 1133	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ (ℤ≥‘𝑁) ∧ 𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑛) = 0) → (0 · (𝐹‘(𝑛 + 1))) = 0)
49	33, 35, 48	3eqtrd 2776	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ (ℤ≥‘𝑁) ∧ 𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑛) = 0) → (seq𝑀( · , 𝐹)‘(𝑛 + 1)) = 0)
50	49	3exp 1120	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (ℤ≥‘𝑁) → (𝜑 → ((seq𝑀( · , 𝐹)‘𝑛) = 0 → (seq𝑀( · , 𝐹)‘(𝑛 + 1)) = 0)))
51	50	adantrd 491	. . . . . . . . . 10 ⊢ (𝑛 ∈ (ℤ≥‘𝑁) → ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → ((seq𝑀( · , 𝐹)‘𝑛) = 0 → (seq𝑀( · , 𝐹)‘(𝑛 + 1)) = 0)))
52	51	a2d 29	. . . . . . . . 9 ⊢ (𝑛 ∈ (ℤ≥‘𝑁) → (((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → (seq𝑀( · , 𝐹)‘𝑛) = 0) → ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → (seq𝑀( · , 𝐹)‘(𝑛 + 1)) = 0)))
53	20, 22, 24, 26, 27, 52	uzind4i 12837	. . . . . . . 8 ⊢ (𝑘 ∈ (ℤ≥‘𝑁) → ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → (seq𝑀( · , 𝐹)‘𝑘) = 0))
54	53	impcom 407	. . . . . . 7 ⊢ (((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → (seq𝑀( · , 𝐹)‘𝑘) = 0)
55	9, 15, 17, 18, 54	climconst 15480	. . . . . 6 ⊢ ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → seq𝑀( · , 𝐹) ⇝ 0)
56		funbrfv 6892	. . . . . 6 ⊢ (Fun ⇝ → (seq𝑀( · , 𝐹) ⇝ 0 → ( ⇝ ‘seq𝑀( · , 𝐹)) = 0))
57	4, 55, 56	mpsyl 68	. . . . 5 ⊢ ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → ( ⇝ ‘seq𝑀( · , 𝐹)) = 0)
58	8, 57	eqtr3d 2774	. . . 4 ⊢ ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → 𝑋 = 0)
59	58	ex 412	. . 3 ⊢ (𝜑 → ((seq𝑀( · , 𝐹)‘𝑁) = 0 → 𝑋 = 0))
60	59	necon3d 2954	. 2 ⊢ (𝜑 → (𝑋 ≠ 0 → (seq𝑀( · , 𝐹)‘𝑁) ≠ 0))
61	1, 60	mpd 15	1 ⊢ (𝜑 → (seq𝑀( · , 𝐹)‘𝑁) ≠ 0)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  Vcvv 3442   class class class wbr 5100  dom cdm 5634  Fun wfun 6496  ⟶wf 6498  ‘cfv 6502  (class class class)co 7370  ℂcc 11038  0cc0 11040  1c1 11041   + caddc 11043   · cmul 11045  ℤcz 12502  ℤ_≥cuz 12765  seqcseq 13938   ⇝ cli 15421

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5314  ax-pr 5381  ax-un 7692  ax-inf2 9564  ax-cnex 11096  ax-resscn 11097  ax-1cn 11098  ax-icn 11099  ax-addcl 11100  ax-addrcl 11101  ax-mulcl 11102  ax-mulrcl 11103  ax-mulcom 11104  ax-addass 11105  ax-mulass 11106  ax-distr 11107  ax-i2m1 11108  ax-1ne0 11109  ax-1rid 11110  ax-rnegex 11111  ax-rrecex 11112  ax-cnre 11113  ax-pre-lttri 11114  ax-pre-lttrn 11115  ax-pre-ltadd 11116  ax-pre-mulgt0 11117  ax-pre-sup 11118

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5529  df-eprel 5534  df-po 5542  df-so 5543  df-fr 5587  df-we 5589  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6269  df-ord 6330  df-on 6331  df-lim 6332  df-suc 6333  df-iota 6458  df-fun 6504  df-fn 6505  df-f 6506  df-f1 6507  df-fo 6508  df-f1o 6509  df-fv 6510  df-riota 7327  df-ov 7373  df-oprab 7374  df-mpo 7375  df-om 7821  df-2nd 7946  df-frecs 8235  df-wrecs 8266  df-recs 8315  df-rdg 8353  df-er 8647  df-en 8898  df-dom 8899  df-sdom 8900  df-sup 9359  df-pnf 11182  df-mnf 11183  df-xr 11184  df-ltxr 11185  df-le 11186  df-sub 11380  df-neg 11381  df-div 11809  df-nn 12160  df-2 12222  df-3 12223  df-n0 12416  df-z 12503  df-uz 12766  df-rp 12920  df-seq 13939  df-exp 13999  df-cj 15036  df-re 15037  df-im 15038  df-sqrt 15172  df-abs 15173  df-clim 15425

This theorem is referenced by:  ntrivcvgtail  15837