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Theorem ntrivcvgtail 15851

Description: A tail of a non-trivially convergent sequence converges non-trivially. (Contributed by Scott Fenton, 20-Dec-2017.)

**Hypotheses**
Ref	Expression
ntrivcvgtail.1	⊢ 𝑍 = (ℤ_≥‘𝑀)
ntrivcvgtail.2	⊢ (𝜑 → 𝑁 ∈ 𝑍)
ntrivcvgtail.3	⊢ (𝜑 → seq𝑀( · , 𝐹) ⇝ 𝑋)
ntrivcvgtail.4	⊢ (𝜑 → 𝑋 ≠ 0)
ntrivcvgtail.5	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)

**Assertion**
Ref	Expression
ntrivcvgtail	⊢ (𝜑 → (( ⇝ ‘seq𝑁( · , 𝐹)) ≠ 0 ∧ seq𝑁( · , 𝐹) ⇝ ( ⇝ ‘seq𝑁( · , 𝐹))))

Distinct variable groups: 𝑘,𝐹 𝜑,𝑘 𝑘,𝑀 𝑘,𝑁 𝑘,𝑍 Allowed substitution hint: 𝑋(𝑘)

**Proof of Theorem ntrivcvgtail**
Step	Hyp	Ref	Expression
1		fclim 15502	. . . . . . . 8 ⊢ ⇝ :dom ⇝ ⟶ℂ
2		ffun 6721	. . . . . . . 8 ⊢ ( ⇝ :dom ⇝ ⟶ℂ → Fun ⇝ )
3	1, 2	ax-mp 5	. . . . . . 7 ⊢ Fun ⇝
4		ntrivcvgtail.3	. . . . . . 7 ⊢ (𝜑 → seq𝑀( · , 𝐹) ⇝ 𝑋)
5		funbrfv 6943	. . . . . . 7 ⊢ (Fun ⇝ → (seq𝑀( · , 𝐹) ⇝ 𝑋 → ( ⇝ ‘seq𝑀( · , 𝐹)) = 𝑋))
6	3, 4, 5	mpsyl 68	. . . . . 6 ⊢ (𝜑 → ( ⇝ ‘seq𝑀( · , 𝐹)) = 𝑋)
7		ntrivcvgtail.4	. . . . . 6 ⊢ (𝜑 → 𝑋 ≠ 0)
8	6, 7	eqnetrd 3007	. . . . 5 ⊢ (𝜑 → ( ⇝ ‘seq𝑀( · , 𝐹)) ≠ 0)
9	4, 6	breqtrrd 5177	. . . . 5 ⊢ (𝜑 → seq𝑀( · , 𝐹) ⇝ ( ⇝ ‘seq𝑀( · , 𝐹)))
10	8, 9	jca 511	. . . 4 ⊢ (𝜑 → (( ⇝ ‘seq𝑀( · , 𝐹)) ≠ 0 ∧ seq𝑀( · , 𝐹) ⇝ ( ⇝ ‘seq𝑀( · , 𝐹))))
11	10	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑁 = 𝑀) → (( ⇝ ‘seq𝑀( · , 𝐹)) ≠ 0 ∧ seq𝑀( · , 𝐹) ⇝ ( ⇝ ‘seq𝑀( · , 𝐹))))
12		seqeq1 13974	. . . . . . 7 ⊢ (𝑁 = 𝑀 → seq𝑁( · , 𝐹) = seq𝑀( · , 𝐹))
13	12	fveq2d 6896	. . . . . 6 ⊢ (𝑁 = 𝑀 → ( ⇝ ‘seq𝑁( · , 𝐹)) = ( ⇝ ‘seq𝑀( · , 𝐹)))
14	13	neeq1d 2999	. . . . 5 ⊢ (𝑁 = 𝑀 → (( ⇝ ‘seq𝑁( · , 𝐹)) ≠ 0 ↔ ( ⇝ ‘seq𝑀( · , 𝐹)) ≠ 0))
15	12, 13	breq12d 5162	. . . . 5 ⊢ (𝑁 = 𝑀 → (seq𝑁( · , 𝐹) ⇝ ( ⇝ ‘seq𝑁( · , 𝐹)) ↔ seq𝑀( · , 𝐹) ⇝ ( ⇝ ‘seq𝑀( · , 𝐹))))
16	14, 15	anbi12d 630	. . . 4 ⊢ (𝑁 = 𝑀 → ((( ⇝ ‘seq𝑁( · , 𝐹)) ≠ 0 ∧ seq𝑁( · , 𝐹) ⇝ ( ⇝ ‘seq𝑁( · , 𝐹))) ↔ (( ⇝ ‘seq𝑀( · , 𝐹)) ≠ 0 ∧ seq𝑀( · , 𝐹) ⇝ ( ⇝ ‘seq𝑀( · , 𝐹)))))
17	16	adantl 481	. . 3 ⊢ ((𝜑 ∧ 𝑁 = 𝑀) → ((( ⇝ ‘seq𝑁( · , 𝐹)) ≠ 0 ∧ seq𝑁( · , 𝐹) ⇝ ( ⇝ ‘seq𝑁( · , 𝐹))) ↔ (( ⇝ ‘seq𝑀( · , 𝐹)) ≠ 0 ∧ seq𝑀( · , 𝐹) ⇝ ( ⇝ ‘seq𝑀( · , 𝐹)))))
18	11, 17	mpbird 256	. 2 ⊢ ((𝜑 ∧ 𝑁 = 𝑀) → (( ⇝ ‘seq𝑁( · , 𝐹)) ≠ 0 ∧ seq𝑁( · , 𝐹) ⇝ ( ⇝ ‘seq𝑁( · , 𝐹))))
19		ntrivcvgtail.1	. . . . . 6 ⊢ 𝑍 = (ℤ_≥‘𝑀)
20		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑁 − 1) ∈ (ℤ_≥‘𝑀)) → (𝑁 − 1) ∈ (ℤ_≥‘𝑀))
21	20, 19	eleqtrrdi 2843	. . . . . 6 ⊢ ((𝜑 ∧ (𝑁 − 1) ∈ (ℤ_≥‘𝑀)) → (𝑁 − 1) ∈ 𝑍)
22		ntrivcvgtail.5	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)
23	22	adantlr 712	. . . . . 6 ⊢ (((𝜑 ∧ (𝑁 − 1) ∈ (ℤ_≥‘𝑀)) ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)
24	4	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑁 − 1) ∈ (ℤ_≥‘𝑀)) → seq𝑀( · , 𝐹) ⇝ 𝑋)
25	7	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑁 − 1) ∈ (ℤ_≥‘𝑀)) → 𝑋 ≠ 0)
26	19, 21, 24, 25, 23	ntrivcvgfvn0 15850	. . . . . 6 ⊢ ((𝜑 ∧ (𝑁 − 1) ∈ (ℤ_≥‘𝑀)) → (seq𝑀( · , 𝐹)‘(𝑁 − 1)) ≠ 0)
27	19, 21, 23, 24, 26	clim2div 15840	. . . . 5 ⊢ ((𝜑 ∧ (𝑁 − 1) ∈ (ℤ_≥‘𝑀)) → seq((𝑁 − 1) + 1)( · , 𝐹) ⇝ (𝑋 / (seq𝑀( · , 𝐹)‘(𝑁 − 1))))
28		funbrfv 6943	. . . . 5 ⊢ (Fun ⇝ → (seq((𝑁 − 1) + 1)( · , 𝐹) ⇝ (𝑋 / (seq𝑀( · , 𝐹)‘(𝑁 − 1))) → ( ⇝ ‘seq((𝑁 − 1) + 1)( · , 𝐹)) = (𝑋 / (seq𝑀( · , 𝐹)‘(𝑁 − 1)))))
29	3, 27, 28	mpsyl 68	. . . 4 ⊢ ((𝜑 ∧ (𝑁 − 1) ∈ (ℤ_≥‘𝑀)) → ( ⇝ ‘seq((𝑁 − 1) + 1)( · , 𝐹)) = (𝑋 / (seq𝑀( · , 𝐹)‘(𝑁 − 1))))
30		climcl 15448	. . . . . . 7 ⊢ (seq𝑀( · , 𝐹) ⇝ 𝑋 → 𝑋 ∈ ℂ)
31	4, 30	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ ℂ)
32	31	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑁 − 1) ∈ (ℤ_≥‘𝑀)) → 𝑋 ∈ ℂ)
33		ntrivcvgtail.2	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ 𝑍)
34		eluzel2 12832	. . . . . . . . . 10 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑀 ∈ ℤ)
35	34, 19	eleq2s 2850	. . . . . . . . 9 ⊢ (𝑁 ∈ 𝑍 → 𝑀 ∈ ℤ)
36	33, 35	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℤ)
37	19, 36, 22	prodf 15838	. . . . . . 7 ⊢ (𝜑 → seq𝑀( · , 𝐹):𝑍⟶ℂ)
38	19	feq2i 6710	. . . . . . 7 ⊢ (seq𝑀( · , 𝐹):𝑍⟶ℂ ↔ seq𝑀( · , 𝐹):(ℤ_≥‘𝑀)⟶ℂ)
39	37, 38	sylib 217	. . . . . 6 ⊢ (𝜑 → seq𝑀( · , 𝐹):(ℤ_≥‘𝑀)⟶ℂ)
40	39	ffvelcdmda 7087	. . . . 5 ⊢ ((𝜑 ∧ (𝑁 − 1) ∈ (ℤ_≥‘𝑀)) → (seq𝑀( · , 𝐹)‘(𝑁 − 1)) ∈ ℂ)
41	32, 40, 25, 26	divne0d 12011	. . . 4 ⊢ ((𝜑 ∧ (𝑁 − 1) ∈ (ℤ_≥‘𝑀)) → (𝑋 / (seq𝑀( · , 𝐹)‘(𝑁 − 1))) ≠ 0)
42	29, 41	eqnetrd 3007	. . 3 ⊢ ((𝜑 ∧ (𝑁 − 1) ∈ (ℤ_≥‘𝑀)) → ( ⇝ ‘seq((𝑁 − 1) + 1)( · , 𝐹)) ≠ 0)
43	27, 29	breqtrrd 5177	. . 3 ⊢ ((𝜑 ∧ (𝑁 − 1) ∈ (ℤ_≥‘𝑀)) → seq((𝑁 − 1) + 1)( · , 𝐹) ⇝ ( ⇝ ‘seq((𝑁 − 1) + 1)( · , 𝐹)))
44		uzssz 12848	. . . . . . . . . . . 12 ⊢ (ℤ_≥‘𝑀) ⊆ ℤ
45	19, 44	eqsstri 4017	. . . . . . . . . . 11 ⊢ 𝑍 ⊆ ℤ
46	45, 33	sselid 3981	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℤ)
47	46	zcnd 12672	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℂ)
48	47	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑁 − 1) ∈ (ℤ_≥‘𝑀)) → 𝑁 ∈ ℂ)
49		1cnd 11214	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑁 − 1) ∈ (ℤ_≥‘𝑀)) → 1 ∈ ℂ)
50	48, 49	npcand 11580	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑁 − 1) ∈ (ℤ_≥‘𝑀)) → ((𝑁 − 1) + 1) = 𝑁)
51	50	seqeq1d 13977	. . . . . 6 ⊢ ((𝜑 ∧ (𝑁 − 1) ∈ (ℤ_≥‘𝑀)) → seq((𝑁 − 1) + 1)( · , 𝐹) = seq𝑁( · , 𝐹))
52	51	fveq2d 6896	. . . . 5 ⊢ ((𝜑 ∧ (𝑁 − 1) ∈ (ℤ_≥‘𝑀)) → ( ⇝ ‘seq((𝑁 − 1) + 1)( · , 𝐹)) = ( ⇝ ‘seq𝑁( · , 𝐹)))
53	52	neeq1d 2999	. . . 4 ⊢ ((𝜑 ∧ (𝑁 − 1) ∈ (ℤ_≥‘𝑀)) → (( ⇝ ‘seq((𝑁 − 1) + 1)( · , 𝐹)) ≠ 0 ↔ ( ⇝ ‘seq𝑁( · , 𝐹)) ≠ 0))
54	51, 52	breq12d 5162	. . . 4 ⊢ ((𝜑 ∧ (𝑁 − 1) ∈ (ℤ_≥‘𝑀)) → (seq((𝑁 − 1) + 1)( · , 𝐹) ⇝ ( ⇝ ‘seq((𝑁 − 1) + 1)( · , 𝐹)) ↔ seq𝑁( · , 𝐹) ⇝ ( ⇝ ‘seq𝑁( · , 𝐹))))
55	53, 54	anbi12d 630	. . 3 ⊢ ((𝜑 ∧ (𝑁 − 1) ∈ (ℤ_≥‘𝑀)) → ((( ⇝ ‘seq((𝑁 − 1) + 1)( · , 𝐹)) ≠ 0 ∧ seq((𝑁 − 1) + 1)( · , 𝐹) ⇝ ( ⇝ ‘seq((𝑁 − 1) + 1)( · , 𝐹))) ↔ (( ⇝ ‘seq𝑁( · , 𝐹)) ≠ 0 ∧ seq𝑁( · , 𝐹) ⇝ ( ⇝ ‘seq𝑁( · , 𝐹)))))
56	42, 43, 55	mpbi2and 709	. 2 ⊢ ((𝜑 ∧ (𝑁 − 1) ∈ (ℤ_≥‘𝑀)) → (( ⇝ ‘seq𝑁( · , 𝐹)) ≠ 0 ∧ seq𝑁( · , 𝐹) ⇝ ( ⇝ ‘seq𝑁( · , 𝐹))))
57	33, 19	eleqtrdi 2842	. . 3 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))
58		uzm1 12865	. . 3 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → (𝑁 = 𝑀 ∨ (𝑁 − 1) ∈ (ℤ_≥‘𝑀)))
59	57, 58	syl 17	. 2 ⊢ (𝜑 → (𝑁 = 𝑀 ∨ (𝑁 − 1) ∈ (ℤ_≥‘𝑀)))
60	18, 56, 59	mpjaodan 956	1 ⊢ (𝜑 → (( ⇝ ‘seq𝑁( · , 𝐹)) ≠ 0 ∧ seq𝑁( · , 𝐹) ⇝ ( ⇝ ‘seq𝑁( · , 𝐹))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∨ wo 844 = wceq 1540 ∈ wcel 2105 ≠ wne 2939 class class class wbr 5149 dom cdm 5677 Fun wfun 6538 ⟶wf 6540 ‘cfv 6544 (class class class)co 7412 ℂcc 11111 0cc0 11113 1c1 11114 + caddc 11116 · cmul 11118 − cmin 11449 / cdiv 11876 ℤcz 12563 ℤ_≥cuz 12827 seqcseq 13971 ⇝ cli 15433

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 2702 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7728 ax-inf2 9639 ax-cnex 11169 ax-resscn 11170 ax-1cn 11171 ax-icn 11172 ax-addcl 11173 ax-addrcl 11174 ax-mulcl 11175 ax-mulrcl 11176 ax-mulcom 11177 ax-addass 11178 ax-mulass 11179 ax-distr 11180 ax-i2m1 11181 ax-1ne0 11182 ax-1rid 11183 ax-rnegex 11184 ax-rrecex 11185 ax-cnre 11186 ax-pre-lttri 11187 ax-pre-lttrn 11188 ax-pre-ltadd 11189 ax-pre-mulgt0 11190 ax-pre-sup 11191

This theorem depends on definitions: df-bi 206 df-an 396 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 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 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 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7859 df-1st 7978 df-2nd 7979 df-frecs 8269 df-wrecs 8300 df-recs 8374 df-rdg 8413 df-er 8706 df-en 8943 df-dom 8944 df-sdom 8945 df-sup 9440 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-div 11877 df-nn 12218 df-2 12280 df-3 12281 df-n0 12478 df-z 12564 df-uz 12828 df-rp 12980 df-fz 13490 df-seq 13972 df-exp 14033 df-cj 15051 df-re 15052 df-im 15053 df-sqrt 15187 df-abs 15188 df-clim 15437

This theorem is referenced by: ntrivcvgmullem 15852

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