xlimmnfv - Mathbox for Glauco Siliprandi

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Theorem xlimmnfv 45122

Description: A function converges to minus infinity if it eventually becomes (and stays) smaller than any given real number. (Contributed by Glauco Siliprandi, 5-Feb-2022.)

**Hypotheses**
Ref	Expression
xlimmnfv.m	⊢ (𝜑 → 𝑀 ∈ ℤ)
xlimmnfv.z	⊢ 𝑍 = (ℤ_≥‘𝑀)
xlimmnfv.f	⊢ (𝜑 → 𝐹:𝑍⟶ℝ^*)

**Assertion**
Ref	Expression
xlimmnfv	⊢ (𝜑 → (𝐹~~>*-∞ ↔ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) ≤ 𝑥))

Distinct variable groups: 𝑗,𝐹,𝑘,𝑥 𝑗,𝑀 𝑗,𝑍,𝑘,𝑥 𝜑,𝑗,𝑘,𝑥 Allowed substitution hints: 𝑀(𝑥,𝑘)

Proof of Theorem xlimmnfv Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		xlimmnfv.m	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ)
2	1	ad2antrr 723	. . . 4 ⊢ (((𝜑 ∧ 𝐹~~>*-∞) ∧ 𝑥 ∈ ℝ) → 𝑀 ∈ ℤ)
3		xlimmnfv.z	. . . 4 ⊢ 𝑍 = (ℤ_≥‘𝑀)
4		xlimmnfv.f	. . . . 5 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ^*)
5	4	ad2antrr 723	. . . 4 ⊢ (((𝜑 ∧ 𝐹~~>-∞) ∧ 𝑥 ∈ ℝ) → 𝐹:𝑍⟶ℝ^)
6		simplr 766	. . . 4 ⊢ (((𝜑 ∧ 𝐹~~>-∞) ∧ 𝑥 ∈ ℝ) → 𝐹~~>-∞)
7		simpr 484	. . . 4 ⊢ (((𝜑 ∧ 𝐹~~>*-∞) ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℝ)
8	2, 3, 5, 6, 7	xlimmnfvlem1 45120	. . 3 ⊢ (((𝜑 ∧ 𝐹~~>*-∞) ∧ 𝑥 ∈ ℝ) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) ≤ 𝑥)
9	8	ralrimiva 3140	. 2 ⊢ ((𝜑 ∧ 𝐹~~>*-∞) → ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) ≤ 𝑥)
10		nfv 1909	. . . 4 ⊢ Ⅎ𝑘𝜑
11		nfcv 2897	. . . . 5 ⊢ Ⅎ𝑘ℝ
12		nfcv 2897	. . . . . 6 ⊢ Ⅎ𝑘𝑍
13		nfra1 3275	. . . . . 6 ⊢ Ⅎ𝑘∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) ≤ 𝑥
14	12, 13	nfrexw 3304	. . . . 5 ⊢ Ⅎ𝑘∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) ≤ 𝑥
15	11, 14	nfralw 3302	. . . 4 ⊢ Ⅎ𝑘∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) ≤ 𝑥
16	10, 15	nfan 1894	. . 3 ⊢ Ⅎ𝑘(𝜑 ∧ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) ≤ 𝑥)
17		nfv 1909	. . . 4 ⊢ Ⅎ𝑗𝜑
18		nfcv 2897	. . . . 5 ⊢ Ⅎ𝑗ℝ
19		nfre1 3276	. . . . 5 ⊢ Ⅎ𝑗∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) ≤ 𝑥
20	18, 19	nfralw 3302	. . . 4 ⊢ Ⅎ𝑗∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) ≤ 𝑥
21	17, 20	nfan 1894	. . 3 ⊢ Ⅎ𝑗(𝜑 ∧ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) ≤ 𝑥)
22	1	adantr 480	. . 3 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) ≤ 𝑥) → 𝑀 ∈ ℤ)
23	4	adantr 480	. . 3 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) ≤ 𝑥) → 𝐹:𝑍⟶ℝ^*)
24		nfv 1909	. . . . . 6 ⊢ Ⅎ𝑗 𝑦 ∈ ℝ
25	21, 24	nfan 1894	. . . . 5 ⊢ Ⅎ𝑗((𝜑 ∧ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) ≤ 𝑥) ∧ 𝑦 ∈ ℝ)
26	4	3ad2ant1 1130	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝐹:𝑍⟶ℝ^*)
27	3	uztrn2 12845	. . . . . . . . . . . . . 14 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ 𝑍)
28	27	3adant1 1127	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ 𝑍)
29	26, 28	ffvelcdmd 7081	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝐹‘𝑘) ∈ ℝ^*)
30	29	ad5ant134 1364	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ (𝐹‘𝑘) ≤ (𝑦 − 1)) → (𝐹‘𝑘) ∈ ℝ^*)
31		simp-4r 781	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ (𝐹‘𝑘) ≤ (𝑦 − 1)) → 𝑦 ∈ ℝ)
32		peano2rem 11531	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℝ → (𝑦 − 1) ∈ ℝ)
33	32	rexrd 11268	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℝ → (𝑦 − 1) ∈ ℝ^*)
34	31, 33	syl 17	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ (𝐹‘𝑘) ≤ (𝑦 − 1)) → (𝑦 − 1) ∈ ℝ^*)
35		rexr 11264	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℝ → 𝑦 ∈ ℝ^*)
36	35	ad4antlr 730	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ (𝐹‘𝑘) ≤ (𝑦 − 1)) → 𝑦 ∈ ℝ^*)
37		simpr 484	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ (𝐹‘𝑘) ≤ (𝑦 − 1)) → (𝐹‘𝑘) ≤ (𝑦 − 1))
38	31	ltm1d 12150	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ (𝐹‘𝑘) ≤ (𝑦 − 1)) → (𝑦 − 1) < 𝑦)
39	30, 34, 36, 37, 38	xrlelttrd 13145	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ (𝐹‘𝑘) ≤ (𝑦 − 1)) → (𝐹‘𝑘) < 𝑦)
40	39	ex 412	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → ((𝐹‘𝑘) ≤ (𝑦 − 1) → (𝐹‘𝑘) < 𝑦))
41	40	ralimdva 3161	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) → (∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) ≤ (𝑦 − 1) → ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) < 𝑦))
42	41	imp 406	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) ≤ (𝑦 − 1)) → ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) < 𝑦)
43	42	adantl3r 747	. . . . . 6 ⊢ (((((𝜑 ∧ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) ≤ 𝑥) ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) ≤ (𝑦 − 1)) → ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) < 𝑦)
44	43	3impa 1107	. . . . 5 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) ≤ 𝑥) ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) ≤ (𝑦 − 1)) → ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) < 𝑦)
45	32	adantl 481	. . . . . . 7 ⊢ ((∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) ≤ 𝑥 ∧ 𝑦 ∈ ℝ) → (𝑦 − 1) ∈ ℝ)
46		simpl 482	. . . . . . 7 ⊢ ((∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) ≤ 𝑥 ∧ 𝑦 ∈ ℝ) → ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) ≤ 𝑥)
47		breq2 5145	. . . . . . . . . 10 ⊢ (𝑥 = (𝑦 − 1) → ((𝐹‘𝑘) ≤ 𝑥 ↔ (𝐹‘𝑘) ≤ (𝑦 − 1)))
48	47	ralbidv 3171	. . . . . . . . 9 ⊢ (𝑥 = (𝑦 − 1) → (∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) ≤ 𝑥 ↔ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) ≤ (𝑦 − 1)))
49	48	rexbidv 3172	. . . . . . . 8 ⊢ (𝑥 = (𝑦 − 1) → (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) ≤ 𝑥 ↔ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) ≤ (𝑦 − 1)))
50	49	rspcva 3604	. . . . . . 7 ⊢ (((𝑦 − 1) ∈ ℝ ∧ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) ≤ 𝑥) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) ≤ (𝑦 − 1))
51	45, 46, 50	syl2anc 583	. . . . . 6 ⊢ ((∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) ≤ 𝑥 ∧ 𝑦 ∈ ℝ) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) ≤ (𝑦 − 1))
52	51	adantll 711	. . . . 5 ⊢ (((𝜑 ∧ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) ≤ 𝑥) ∧ 𝑦 ∈ ℝ) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) ≤ (𝑦 − 1))
53	25, 44, 52	reximdd 44416	. . . 4 ⊢ (((𝜑 ∧ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) ≤ 𝑥) ∧ 𝑦 ∈ ℝ) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) < 𝑦)
54	53	ralrimiva 3140	. . 3 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) ≤ 𝑥) → ∀𝑦 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) < 𝑦)
55	16, 21, 22, 3, 23, 54	xlimmnfvlem2 45121	. 2 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) ≤ 𝑥) → 𝐹~~>*-∞)
56	9, 55	impbida 798	1 ⊢ (𝜑 → (𝐹~~>*-∞ ↔ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) ≤ 𝑥))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∀wral 3055 ∃wrex 3064 class class class wbr 5141 ⟶wf 6533 ‘cfv 6537 (class class class)co 7405 ℝcr 11111 1c1 11113 -∞cmnf 11250 ℝ^*cxr 11251 < clt 11252 ≤ cle 11253 − cmin 11448 ℤcz 12562 ℤ_≥cuz 12826 ~~>*clsxlim 45106

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-1o 8467 df-er 8705 df-pm 8825 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fi 9408 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-z 12563 df-uz 12827 df-ioo 13334 df-ioc 13335 df-ico 13336 df-icc 13337 df-topgen 17398 df-ordt 17456 df-ps 18531 df-tsr 18532 df-top 22751 df-topon 22768 df-bases 22804 df-lm 23088 df-xlim 45107

This theorem is referenced by: xlimmnf 45129 xlimmnflimsup2 45140 xlimmnflimsup 45144

W3C validator