Theorem ntrivcvgfvn0 15683

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 15334	. . . . . . . 8 ⊢ ⇝ :dom ⇝ ⟶ℂ
3		ffun 6640	. . . . . . . 8 ⊢ ( ⇝ :dom ⇝ ⟶ℂ → Fun ⇝ )
4	2, 3	ax-mp 5	. . . . . . 7 ⊢ Fun ⇝
5		ntrivcvgfvn0.3	. . . . . . 7 ⊢ (𝜑 → seq𝑀( · , 𝐹) ⇝ 𝑋)
6		funbrfv 6859	. . . . . . 7 ⊢ (Fun ⇝ → (seq𝑀( · , 𝐹) ⇝ 𝑋 → ( ⇝ ‘seq𝑀( · , 𝐹)) = 𝑋))
7	4, 5, 6	mpsyl 68	. . . . . 6 ⊢ (𝜑 → ( ⇝ ‘seq𝑀( · , 𝐹)) = 𝑋)
8	7	adantr 481	. . . . 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 12676	. . . . . . . . . 10 ⊢ (ℤ≥‘𝑀) ⊆ ℤ
12	10, 11	eqsstri 3965	. . . . . . . . 9 ⊢ 𝑍 ⊆ ℤ
13		ntrivcvgfvn0.2	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ 𝑍)
14	12, 13	sselid 3929	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℤ)
15	14	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → 𝑁 ∈ ℤ)
16		seqex 13796	. . . . . . . 8 ⊢ seq𝑀( · , 𝐹) ∈ V
17	16	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → seq𝑀( · , 𝐹) ∈ V)
18		0cnd 11041	. . . . . . 7 ⊢ ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → 0 ∈ ℂ)
19		fveqeq2 6820	. . . . . . . . . 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 6820	. . . . . . . . . 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 6820	. . . . . . . . . 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 6820	. . . . . . . . . 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 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → (seq𝑀( · , 𝐹)‘𝑁) = 0)
28	13, 10	eleqtrdi 2848	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
29		uztrn 12673	. . . . . . . . . . . . . . . 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 1131	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ (ℤ≥‘𝑁) ∧ 𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑛) = 0) → 𝑛 ∈ (ℤ≥‘𝑀))
32		seqp1 13809	. . . . . . . . . . . . . 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 7322	. . . . . . . . . . . . . 14 ⊢ ((seq𝑀( · , 𝐹)‘𝑛) = 0 → ((seq𝑀( · , 𝐹)‘𝑛) · (𝐹‘(𝑛 + 1))) = (0 · (𝐹‘(𝑛 + 1))))
35	34	3ad2ant3 1134	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ (ℤ≥‘𝑁) ∧ 𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑛) = 0) → ((seq𝑀( · , 𝐹)‘𝑛) · (𝐹‘(𝑛 + 1))) = (0 · (𝐹‘(𝑛 + 1))))
36		peano2uz 12714	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ (ℤ≥‘𝑁) → (𝑛 + 1) ∈ (ℤ≥‘𝑁))
37	10	uztrn2 12674	. . . . . . . . . . . . . . . . . 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 3140	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ)
41		fveq2 6811	. . . . . . . . . . . . . . . . . . . 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 3566	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑛 + 1) ∈ 𝑍 → (∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ → (𝐹‘(𝑛 + 1)) ∈ ℂ))
44	40, 43	mpan9 507	. . . . . . . . . . . . . . . . 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 459	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ (ℤ≥‘𝑁) ∧ 𝜑) → (𝐹‘(𝑛 + 1)) ∈ ℂ)
47	46	mul02d 11246	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ (ℤ≥‘𝑁) ∧ 𝜑) → (0 · (𝐹‘(𝑛 + 1))) = 0)
48	47	3adant3 1131	. . . . . . . . . . . . 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 1118	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (ℤ≥‘𝑁) → (𝜑 → ((seq𝑀( · , 𝐹)‘𝑛) = 0 → (seq𝑀( · , 𝐹)‘(𝑛 + 1)) = 0)))
51	50	adantrd 492	. . . . . . . . . 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 12723	. . . . . . . 8 ⊢ (𝑘 ∈ (ℤ≥‘𝑁) → ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → (seq𝑀( · , 𝐹)‘𝑘) = 0))
54	53	impcom 408	. . . . . . 7 ⊢ (((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → (seq𝑀( · , 𝐹)‘𝑘) = 0)
55	9, 15, 17, 18, 54	climconst 15324	. . . . . 6 ⊢ ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → seq𝑀( · , 𝐹) ⇝ 0)
56		funbrfv 6859	. . . . . 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 413	. . 3 ⊢ (𝜑 → ((seq𝑀( · , 𝐹)‘𝑁) = 0 → 𝑋 = 0))
60	59	necon3d 2962	. 2 ⊢ (𝜑 → (𝑋 ≠ 0 → (seq𝑀( · , 𝐹)‘𝑁) ≠ 0))
61	1, 60	mpd 15	1 ⊢ (𝜑 → (seq𝑀( · , 𝐹)‘𝑁) ≠ 0)

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105   ≠ wne 2941  ∀wral 3062  Vcvv 3441   class class class wbr 5087  dom cdm 5607  Fun wfun 6459  ⟶wf 6461  ‘cfv 6465  (class class class)co 7315  ℂcc 10942  0cc0 10944  1c1 10945   + caddc 10947   · cmul 10949  ℤcz 12392  ℤ_≥cuz 12655  seqcseq 13794   ⇝ cli 15265

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2708  ax-rep 5224  ax-sep 5238  ax-nul 5245  ax-pow 5303  ax-pr 5367  ax-un 7628  ax-inf2 9470  ax-cnex 11000  ax-resscn 11001  ax-1cn 11002  ax-icn 11003  ax-addcl 11004  ax-addrcl 11005  ax-mulcl 11006  ax-mulrcl 11007  ax-mulcom 11008  ax-addass 11009  ax-mulass 11010  ax-distr 11011  ax-i2m1 11012  ax-1ne0 11013  ax-1rid 11014  ax-rnegex 11015  ax-rrecex 11016  ax-cnre 11017  ax-pre-lttri 11018  ax-pre-lttrn 11019  ax-pre-ltadd 11020  ax-pre-mulgt0 11021  ax-pre-sup 11022

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3350  df-reu 3351  df-rab 3405  df-v 3443  df-sbc 3727  df-csb 3843  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3916  df-nul 4268  df-if 4472  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4851  df-iun 4939  df-br 5088  df-opab 5150  df-mpt 5171  df-tr 5205  df-id 5507  df-eprel 5513  df-po 5521  df-so 5522  df-fr 5562  df-we 5564  df-xp 5613  df-rel 5614  df-cnv 5615  df-co 5616  df-dm 5617  df-rn 5618  df-res 5619  df-ima 5620  df-pred 6224  df-ord 6291  df-on 6292  df-lim 6293  df-suc 6294  df-iota 6417  df-fun 6467  df-fn 6468  df-f 6469  df-f1 6470  df-fo 6471  df-f1o 6472  df-fv 6473  df-riota 7272  df-ov 7318  df-oprab 7319  df-mpo 7320  df-om 7758  df-2nd 7877  df-frecs 8144  df-wrecs 8175  df-recs 8249  df-rdg 8288  df-er 8546  df-en 8782  df-dom 8783  df-sdom 8784  df-sup 9271  df-pnf 11084  df-mnf 11085  df-xr 11086  df-ltxr 11087  df-le 11088  df-sub 11280  df-neg 11281  df-div 11706  df-nn 12047  df-2 12109  df-3 12110  df-n0 12307  df-z 12393  df-uz 12656  df-rp 12804  df-seq 13795  df-exp 13856  df-cj 14882  df-re 14883  df-im 14884  df-sqrt 15018  df-abs 15019  df-clim 15269

This theorem is referenced by:  ntrivcvgtail  15684