Theorem ntrivcvgfvn0 15935

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 15589	. . . . . . . 8 ⊢ ⇝ :dom ⇝ ⟶ℂ
3		ffun 6739	. . . . . . . 8 ⊢ ( ⇝ :dom ⇝ ⟶ℂ → Fun ⇝ )
4	2, 3	ax-mp 5	. . . . . . 7 ⊢ Fun ⇝
5		ntrivcvgfvn0.3	. . . . . . 7 ⊢ (𝜑 → seq𝑀( · , 𝐹) ⇝ 𝑋)
6		funbrfv 6957	. . . . . . 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 12899	. . . . . . . . . 10 ⊢ (ℤ≥‘𝑀) ⊆ ℤ
12	10, 11	eqsstri 4030	. . . . . . . . 9 ⊢ 𝑍 ⊆ ℤ
13		ntrivcvgfvn0.2	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ 𝑍)
14	12, 13	sselid 3981	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℤ)
15	14	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → 𝑁 ∈ ℤ)
16		seqex 14044	. . . . . . . 8 ⊢ seq𝑀( · , 𝐹) ∈ V
17	16	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → seq𝑀( · , 𝐹) ∈ V)
18		0cnd 11254	. . . . . . 7 ⊢ ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → 0 ∈ ℂ)
19		fveqeq2 6915	. . . . . . . . . 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 6915	. . . . . . . . . 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 6915	. . . . . . . . . 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 6915	. . . . . . . . . 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 2851	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
29		uztrn 12896	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 ∈ (ℤ≥‘𝑁) ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → 𝑛 ∈ (ℤ≥‘𝑀))
30	28, 29	sylan2 593	. . . . . . . . . . . . . . 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 14057	. . . . . . . . . . . . . 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 7438	. . . . . . . . . . . . . 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 12943	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ (ℤ≥‘𝑁) → (𝑛 + 1) ∈ (ℤ≥‘𝑁))
37	10	uztrn2 12897	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ 𝑍 ∧ (𝑛 + 1) ∈ (ℤ≥‘𝑁)) → (𝑛 + 1) ∈ 𝑍)
38	13, 36, 37	syl2an 596	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑁)) → (𝑛 + 1) ∈ 𝑍)
39		ntrivcvgfvn0.5	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)
40	39	ralrimiva 3146	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ)
41		fveq2 6906	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = (𝑛 + 1) → (𝐹‘𝑘) = (𝐹‘(𝑛 + 1)))
42	41	eleq1d 2826	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = (𝑛 + 1) → ((𝐹‘𝑘) ∈ ℂ ↔ (𝐹‘(𝑛 + 1)) ∈ ℂ))
43	42	rspcv 3618	. . . . . . . . . . . . . . . . . 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 591	. . . . . . . . . . . . . . . 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 11459	. . . . . . . . . . . . . 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 2781	. . . . . . . . . . . 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 12952	. . . . . . . 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 15579	. . . . . 6 ⊢ ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → seq𝑀( · , 𝐹) ⇝ 0)
56		funbrfv 6957	. . . . . 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 2779	. . . 4 ⊢ ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → 𝑋 = 0)
59	58	ex 412	. . 3 ⊢ (𝜑 → ((seq𝑀( · , 𝐹)‘𝑁) = 0 → 𝑋 = 0))
60	59	necon3d 2961	. 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 1540   ∈ wcel 2108   ≠ wne 2940  ∀wral 3061  Vcvv 3480   class class class wbr 5143  dom cdm 5685  Fun wfun 6555  ⟶wf 6557  ‘cfv 6561  (class class class)co 7431  ℂcc 11153  0cc0 11155  1c1 11156   + caddc 11158   · cmul 11160  ℤcz 12613  ℤ_≥cuz 12878  seqcseq 14042   ⇝ cli 15520

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-sup 9482  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-n0 12527  df-z 12614  df-uz 12879  df-rp 13035  df-seq 14043  df-exp 14103  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-clim 15524

This theorem is referenced by:  ntrivcvgtail  15936