Theorem ntrivcvgfvn0 15953

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 15604	. . . . . . . 8 ⊢ ⇝ :dom ⇝ ⟶ℂ
3		ffun 6709	. . . . . . . 8 ⊢ ( ⇝ :dom ⇝ ⟶ℂ → Fun ⇝ )
4	2, 3	ax-mp 5	. . . . . . 7 ⊢ Fun ⇝
5		ntrivcvgfvn0.3	. . . . . . 7 ⊢ (𝜑 → seq𝑀( · , 𝐹) ⇝ 𝑋)
6		funbrfv 6930	. . . . . . 7 ⊢ (Fun ⇝ → (seq𝑀( · , 𝐹) ⇝ 𝑋 → ( ⇝ ‘seq𝑀( · , 𝐹)) = 𝑋))
7	4, 5, 6	mpsyl 69	. . . . . 6 ⊢ (𝜑 → ( ⇝ ‘seq𝑀( · , 𝐹)) = 𝑋)
8	7	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → ( ⇝ ‘seq𝑀( · , 𝐹)) = 𝑋)
9		eqid 2769	. . . . . . 7 ⊢ (ℤ≥‘𝑁) = (ℤ≥‘𝑁)
10		ntrivcvgfvn0.1	. . . . . . . . . 10 ⊢ 𝑍 = (ℤ≥‘𝑀)
11		uzssz 12883	. . . . . . . . . 10 ⊢ (ℤ≥‘𝑀) ⊆ ℤ
12	10, 11	eqsstri 3991	. . . . . . . . 9 ⊢ 𝑍 ⊆ ℤ
13		ntrivcvgfvn0.2	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ 𝑍)
14	12, 13	sselid 3943	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℤ)
15	14	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → 𝑁 ∈ ℤ)
16		seqex 14039	. . . . . . . 8 ⊢ seq𝑀( · , 𝐹) ∈ V
17	16	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → seq𝑀( · , 𝐹) ∈ V)
18		0cnd 11199	. . . . . . 7 ⊢ ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → 0 ∈ ℂ)
19		fveqeq2 6891	. . . . . . . . . 10 ⊢ (𝑚 = 𝑁 → ((seq𝑀( · , 𝐹)‘𝑚) = 0 ↔ (seq𝑀( · , 𝐹)‘𝑁) = 0))
20	19	imbi2d 343	. . . . . . . . 9 ⊢ (𝑚 = 𝑁 → (((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → (seq𝑀( · , 𝐹)‘𝑚) = 0) ↔ ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → (seq𝑀( · , 𝐹)‘𝑁) = 0)))
21		fveqeq2 6891	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → ((seq𝑀( · , 𝐹)‘𝑚) = 0 ↔ (seq𝑀( · , 𝐹)‘𝑛) = 0))
22	21	imbi2d 343	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → (((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → (seq𝑀( · , 𝐹)‘𝑚) = 0) ↔ ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → (seq𝑀( · , 𝐹)‘𝑛) = 0)))
23		fveqeq2 6891	. . . . . . . . . 10 ⊢ (𝑚 = (𝑛 + 1) → ((seq𝑀( · , 𝐹)‘𝑚) = 0 ↔ (seq𝑀( · , 𝐹)‘(𝑛 + 1)) = 0))
24	23	imbi2d 343	. . . . . . . . 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 6891	. . . . . . . . . 10 ⊢ (𝑚 = 𝑘 → ((seq𝑀( · , 𝐹)‘𝑚) = 0 ↔ (seq𝑀( · , 𝐹)‘𝑘) = 0))
26	25	imbi2d 343	. . . . . . . . 9 ⊢ (𝑚 = 𝑘 → (((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → (seq𝑀( · , 𝐹)‘𝑚) = 0) ↔ ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → (seq𝑀( · , 𝐹)‘𝑘) = 0)))
27		simpr 489	. . . . . . . . 9 ⊢ ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → (seq𝑀( · , 𝐹)‘𝑁) = 0)
28	13, 10	eleqtrdi 2879	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
29		uztrn 12880	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 ∈ (ℤ≥‘𝑁) ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → 𝑛 ∈ (ℤ≥‘𝑀))
30	28, 29	sylan2 604	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ (ℤ≥‘𝑁) ∧ 𝜑) → 𝑛 ∈ (ℤ≥‘𝑀))
31	30	3adant3 1148	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ (ℤ≥‘𝑁) ∧ 𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑛) = 0) → 𝑛 ∈ (ℤ≥‘𝑀))
32		seqp1 14052	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (ℤ≥‘𝑀) → (seq𝑀( · , 𝐹)‘(𝑛 + 1)) = ((seq𝑀( · , 𝐹)‘𝑛) · (𝐹‘(𝑛 + 1))))
33	31, 32	syl 18	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ (ℤ≥‘𝑁) ∧ 𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑛) = 0) → (seq𝑀( · , 𝐹)‘(𝑛 + 1)) = ((seq𝑀( · , 𝐹)‘𝑛) · (𝐹‘(𝑛 + 1))))
34		oveq1 7418	. . . . . . . . . . . . . 14 ⊢ ((seq𝑀( · , 𝐹)‘𝑛) = 0 → ((seq𝑀( · , 𝐹)‘𝑛) · (𝐹‘(𝑛 + 1))) = (0 · (𝐹‘(𝑛 + 1))))
35	34	3ad2ant3 1151	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ (ℤ≥‘𝑁) ∧ 𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑛) = 0) → ((seq𝑀( · , 𝐹)‘𝑛) · (𝐹‘(𝑛 + 1))) = (0 · (𝐹‘(𝑛 + 1))))
36		peano2uz 12925	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ (ℤ≥‘𝑁) → (𝑛 + 1) ∈ (ℤ≥‘𝑁))
37	10	uztrn2 12881	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ 𝑍 ∧ (𝑛 + 1) ∈ (ℤ≥‘𝑁)) → (𝑛 + 1) ∈ 𝑍)
38	13, 36, 37	syl2an 607	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑁)) → (𝑛 + 1) ∈ 𝑍)
39		ntrivcvgfvn0.5	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)
40	39	ralrimiva 3163	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ)
41		fveq2 6882	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = (𝑛 + 1) → (𝐹‘𝑘) = (𝐹‘(𝑛 + 1)))
42	41	eleq1d 2854	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = (𝑛 + 1) → ((𝐹‘𝑘) ∈ ℂ ↔ (𝐹‘(𝑛 + 1)) ∈ ℂ))
43	42	rspcv 3586	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑛 + 1) ∈ 𝑍 → (∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ → (𝐹‘(𝑛 + 1)) ∈ ℂ))
44	40, 43	mpan9 515	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑛 + 1) ∈ 𝑍) → (𝐹‘(𝑛 + 1)) ∈ ℂ)
45	38, 44	syldan 602	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑁)) → (𝐹‘(𝑛 + 1)) ∈ ℂ)
46	45	ancoms 463	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ (ℤ≥‘𝑁) ∧ 𝜑) → (𝐹‘(𝑛 + 1)) ∈ ℂ)
47	46	mul02d 11408	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ (ℤ≥‘𝑁) ∧ 𝜑) → (0 · (𝐹‘(𝑛 + 1))) = 0)
48	47	3adant3 1148	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ (ℤ≥‘𝑁) ∧ 𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑛) = 0) → (0 · (𝐹‘(𝑛 + 1))) = 0)
49	33, 35, 48	3eqtrd 2808	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ (ℤ≥‘𝑁) ∧ 𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑛) = 0) → (seq𝑀( · , 𝐹)‘(𝑛 + 1)) = 0)
50	49	3exp 1135	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (ℤ≥‘𝑁) → (𝜑 → ((seq𝑀( · , 𝐹)‘𝑛) = 0 → (seq𝑀( · , 𝐹)‘(𝑛 + 1)) = 0)))
51	50	adantrd 496	. . . . . . . . . 10 ⊢ (𝑛 ∈ (ℤ≥‘𝑁) → ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → ((seq𝑀( · , 𝐹)‘𝑛) = 0 → (seq𝑀( · , 𝐹)‘(𝑛 + 1)) = 0)))
52	51	a2d 30	. . . . . . . . 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 12934	. . . . . . . 8 ⊢ (𝑘 ∈ (ℤ≥‘𝑁) → ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → (seq𝑀( · , 𝐹)‘𝑘) = 0))
54	53	impcom 412	. . . . . . 7 ⊢ (((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → (seq𝑀( · , 𝐹)‘𝑘) = 0)
55	9, 15, 17, 18, 54	climconst 15594	. . . . . 6 ⊢ ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → seq𝑀( · , 𝐹) ⇝ 0)
56		funbrfv 6930	. . . . . 6 ⊢ (Fun ⇝ → (seq𝑀( · , 𝐹) ⇝ 0 → ( ⇝ ‘seq𝑀( · , 𝐹)) = 0))
57	4, 55, 56	mpsyl 69	. . . . 5 ⊢ ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → ( ⇝ ‘seq𝑀( · , 𝐹)) = 0)
58	8, 57	eqtr3d 2806	. . . 4 ⊢ ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → 𝑋 = 0)
59	58	ex 417	. . 3 ⊢ (𝜑 → ((seq𝑀( · , 𝐹)‘𝑁) = 0 → 𝑋 = 0))
60	59	necon3d 2985	. 2 ⊢ (𝜑 → (𝑋 ≠ 0 → (seq𝑀( · , 𝐹)‘𝑁) ≠ 0))
61	1, 60	mpd 16	1 ⊢ (𝜑 → (seq𝑀( · , 𝐹)‘𝑁) ≠ 0)

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149   ≠ wne 2964  ∀wral 3085  Vcvv 3463   class class class wbr 5113  dom cdm 5662  Fun wfun 6531  ⟶wf 6533  ‘cfv 6537  (class class class)co 7411  ℂcc 11098  0cc0 11100  1c1 11101   + caddc 11103   · cmul 11105  ℤcz 12591  ℤ_≥cuz 12862  seqcseq 14037   ⇝ cli 15535

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-inf2 9610  ax-cnex 11156  ax-resscn 11157  ax-1cn 11158  ax-icn 11159  ax-addcl 11160  ax-addrcl 11161  ax-mulcl 11162  ax-mulrcl 11163  ax-mulcom 11164  ax-addass 11165  ax-mulass 11166  ax-distr 11167  ax-i2m1 11168  ax-1ne0 11169  ax-1rid 11170  ax-rnegex 11171  ax-rrecex 11172  ax-cnre 11173  ax-pre-lttri 11174  ax-pre-lttrn 11175  ax-pre-ltadd 11176  ax-pre-mulgt0 11177  ax-pre-sup 11178

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7863  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-er 8694  df-en 8944  df-dom 8945  df-sdom 8946  df-sup 9402  df-pnf 11245  df-mnf 11246  df-xr 11247  df-ltxr 11248  df-le 11249  df-sub 11443  df-neg 11444  df-div 11872  df-nn 12234  df-2 12303  df-3 12304  df-n0 12505  df-z 12592  df-uz 12863  df-rp 13017  df-seq 14038  df-exp 14098  df-cj 15150  df-re 15151  df-im 15152  df-sqrt 15286  df-abs 15287  df-clim 15539

This theorem is referenced by:  ntrivcvgtail  15954