xrralrecnnge - Mathbox for Glauco Siliprandi

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Theorem xrralrecnnge 45386

Description: Show that 𝐴 is less than 𝐵 by showing that there is no positive bound on the difference. (Contributed by Glauco Siliprandi, 26-Jun-2021.)

**Hypotheses**
Ref	Expression
xrralrecnnge.n	⊢ Ⅎ𝑛𝜑
xrralrecnnge.a	⊢ (𝜑 → 𝐴 ∈ ℝ)
xrralrecnnge.b	⊢ (𝜑 → 𝐵 ∈ ℝ^*)

**Assertion**
Ref	Expression
xrralrecnnge	⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ ∀𝑛 ∈ ℕ (𝐴 − (1 / 𝑛)) ≤ 𝐵))

Distinct variable groups: 𝐴,𝑛 𝐵,𝑛 Allowed substitution hint: 𝜑(𝑛)

**Proof of Theorem xrralrecnnge**
Step	Hyp	Ref	Expression
1		xrralrecnnge.n	. . . . 5 ⊢ Ⅎ𝑛𝜑
2		nfv 1914	. . . . 5 ⊢ Ⅎ𝑛 𝐴 ≤ 𝐵
3	1, 2	nfan 1899	. . . 4 ⊢ Ⅎ𝑛(𝜑 ∧ 𝐴 ≤ 𝐵)
4		xrralrecnnge.a	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ ℝ)
5	4	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴 ∈ ℝ)
6		nnrecre 12228	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (1 / 𝑛) ∈ ℝ)
7	6	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1 / 𝑛) ∈ ℝ)
8	5, 7	resubcld 11606	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴 − (1 / 𝑛)) ∈ ℝ)
9	8	rexrd 11224	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴 − (1 / 𝑛)) ∈ ℝ^*)
10	9	adantlr 715	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑛 ∈ ℕ) → (𝐴 − (1 / 𝑛)) ∈ ℝ^*)
11		xrralrecnnge.b	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
12	11	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑛 ∈ ℕ) → 𝐵 ∈ ℝ^*)
13	4	rexrd 11224	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
14	13	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑛 ∈ ℕ) → 𝐴 ∈ ℝ^*)
15		nnrp 12963	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ⁺)
16	15	rpreccld 13005	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (1 / 𝑛) ∈ ℝ⁺)
17	16	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1 / 𝑛) ∈ ℝ⁺)
18	5, 17	ltsubrpd 13027	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴 − (1 / 𝑛)) < 𝐴)
19	18	adantlr 715	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑛 ∈ ℕ) → (𝐴 − (1 / 𝑛)) < 𝐴)
20		simplr 768	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑛 ∈ ℕ) → 𝐴 ≤ 𝐵)
21	10, 14, 12, 19, 20	xrltletrd 13121	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑛 ∈ ℕ) → (𝐴 − (1 / 𝑛)) < 𝐵)
22	10, 12, 21	xrltled 13110	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑛 ∈ ℕ) → (𝐴 − (1 / 𝑛)) ≤ 𝐵)
23	22	ex 412	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → (𝑛 ∈ ℕ → (𝐴 − (1 / 𝑛)) ≤ 𝐵))
24	3, 23	ralrimi 3235	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ∀𝑛 ∈ ℕ (𝐴 − (1 / 𝑛)) ≤ 𝐵)
25	24	ex 412	. 2 ⊢ (𝜑 → (𝐴 ≤ 𝐵 → ∀𝑛 ∈ ℕ (𝐴 − (1 / 𝑛)) ≤ 𝐵))
26		pnfxr 11228	. . . . . . . 8 ⊢ +∞ ∈ ℝ^*
27	26	a1i 11	. . . . . . 7 ⊢ (𝜑 → +∞ ∈ ℝ^*)
28	4	ltpnfd 13081	. . . . . . 7 ⊢ (𝜑 → 𝐴 < +∞)
29	13, 27, 28	xrltled 13110	. . . . . 6 ⊢ (𝜑 → 𝐴 ≤ +∞)
30	29	ad2antrr 726	. . . . 5 ⊢ (((𝜑 ∧ ∀𝑛 ∈ ℕ (𝐴 − (1 / 𝑛)) ≤ 𝐵) ∧ 𝐵 = +∞) → 𝐴 ≤ +∞)
31		id 22	. . . . . . 7 ⊢ (𝐵 = +∞ → 𝐵 = +∞)
32	31	eqcomd 2735	. . . . . 6 ⊢ (𝐵 = +∞ → +∞ = 𝐵)
33	32	adantl 481	. . . . 5 ⊢ (((𝜑 ∧ ∀𝑛 ∈ ℕ (𝐴 − (1 / 𝑛)) ≤ 𝐵) ∧ 𝐵 = +∞) → +∞ = 𝐵)
34	30, 33	breqtrd 5133	. . . 4 ⊢ (((𝜑 ∧ ∀𝑛 ∈ ℕ (𝐴 − (1 / 𝑛)) ≤ 𝐵) ∧ 𝐵 = +∞) → 𝐴 ≤ 𝐵)
35	11	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑛 ∈ ℕ (𝐴 − (1 / 𝑛)) ≤ 𝐵) ∧ ¬ 𝐵 = +∞) → 𝐵 ∈ ℝ^*)
36		1nn 12197	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℕ
37	36	a1i 11	. . . . . . . . . . . . 13 ⊢ (∀𝑛 ∈ ℕ (𝐴 − (1 / 𝑛)) ≤ 𝐵 → 1 ∈ ℕ)
38		id 22	. . . . . . . . . . . . 13 ⊢ (∀𝑛 ∈ ℕ (𝐴 − (1 / 𝑛)) ≤ 𝐵 → ∀𝑛 ∈ ℕ (𝐴 − (1 / 𝑛)) ≤ 𝐵)
39		oveq2 7395	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 1 → (1 / 𝑛) = (1 / 1))
40	39	oveq2d 7403	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 1 → (𝐴 − (1 / 𝑛)) = (𝐴 − (1 / 1)))
41	40	breq1d 5117	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 1 → ((𝐴 − (1 / 𝑛)) ≤ 𝐵 ↔ (𝐴 − (1 / 1)) ≤ 𝐵))
42	41	rspcva 3586	. . . . . . . . . . . . 13 ⊢ ((1 ∈ ℕ ∧ ∀𝑛 ∈ ℕ (𝐴 − (1 / 𝑛)) ≤ 𝐵) → (𝐴 − (1 / 1)) ≤ 𝐵)
43	37, 38, 42	syl2anc 584	. . . . . . . . . . . 12 ⊢ (∀𝑛 ∈ ℕ (𝐴 − (1 / 𝑛)) ≤ 𝐵 → (𝐴 − (1 / 1)) ≤ 𝐵)
44	43	adantr 480	. . . . . . . . . . 11 ⊢ ((∀𝑛 ∈ ℕ (𝐴 − (1 / 𝑛)) ≤ 𝐵 ∧ 𝐵 = -∞) → (𝐴 − (1 / 1)) ≤ 𝐵)
45		simpr 484	. . . . . . . . . . 11 ⊢ ((∀𝑛 ∈ ℕ (𝐴 − (1 / 𝑛)) ≤ 𝐵 ∧ 𝐵 = -∞) → 𝐵 = -∞)
46	44, 45	breqtrd 5133	. . . . . . . . . 10 ⊢ ((∀𝑛 ∈ ℕ (𝐴 − (1 / 𝑛)) ≤ 𝐵 ∧ 𝐵 = -∞) → (𝐴 − (1 / 1)) ≤ -∞)
47	46	adantll 714	. . . . . . . . 9 ⊢ (((𝜑 ∧ ∀𝑛 ∈ ℕ (𝐴 − (1 / 𝑛)) ≤ 𝐵) ∧ 𝐵 = -∞) → (𝐴 − (1 / 1)) ≤ -∞)
48		1red 11175	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 1 ∈ ℝ)
49		ax-1ne0 11137	. . . . . . . . . . . . . . 15 ⊢ 1 ≠ 0
50	49	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 1 ≠ 0)
51	48, 48, 50	redivcld 12010	. . . . . . . . . . . . 13 ⊢ (𝜑 → (1 / 1) ∈ ℝ)
52	4, 51	resubcld 11606	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 − (1 / 1)) ∈ ℝ)
53	52	mnfltd 13084	. . . . . . . . . . 11 ⊢ (𝜑 → -∞ < (𝐴 − (1 / 1)))
54		mnfxr 11231	. . . . . . . . . . . . 13 ⊢ -∞ ∈ ℝ^*
55	54	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → -∞ ∈ ℝ^*)
56	52	rexrd 11224	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 − (1 / 1)) ∈ ℝ^*)
57	55, 56	xrltnled 45359	. . . . . . . . . . 11 ⊢ (𝜑 → (-∞ < (𝐴 − (1 / 1)) ↔ ¬ (𝐴 − (1 / 1)) ≤ -∞))
58	53, 57	mpbid 232	. . . . . . . . . 10 ⊢ (𝜑 → ¬ (𝐴 − (1 / 1)) ≤ -∞)
59	58	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ ∀𝑛 ∈ ℕ (𝐴 − (1 / 𝑛)) ≤ 𝐵) ∧ 𝐵 = -∞) → ¬ (𝐴 − (1 / 1)) ≤ -∞)
60	47, 59	pm2.65da 816	. . . . . . . 8 ⊢ ((𝜑 ∧ ∀𝑛 ∈ ℕ (𝐴 − (1 / 𝑛)) ≤ 𝐵) → ¬ 𝐵 = -∞)
61	60	neqned 2932	. . . . . . 7 ⊢ ((𝜑 ∧ ∀𝑛 ∈ ℕ (𝐴 − (1 / 𝑛)) ≤ 𝐵) → 𝐵 ≠ -∞)
62	61	adantr 480	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑛 ∈ ℕ (𝐴 − (1 / 𝑛)) ≤ 𝐵) ∧ ¬ 𝐵 = +∞) → 𝐵 ≠ -∞)
63		neqne 2933	. . . . . . 7 ⊢ (¬ 𝐵 = +∞ → 𝐵 ≠ +∞)
64	63	adantl 481	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑛 ∈ ℕ (𝐴 − (1 / 𝑛)) ≤ 𝐵) ∧ ¬ 𝐵 = +∞) → 𝐵 ≠ +∞)
65	35, 62, 64	xrred 45361	. . . . 5 ⊢ (((𝜑 ∧ ∀𝑛 ∈ ℕ (𝐴 − (1 / 𝑛)) ≤ 𝐵) ∧ ¬ 𝐵 = +∞) → 𝐵 ∈ ℝ)
66		nfv 1914	. . . . . . . . . . 11 ⊢ Ⅎ𝑛 𝐵 ∈ ℝ
67	1, 66	nfan 1899	. . . . . . . . . 10 ⊢ Ⅎ𝑛(𝜑 ∧ 𝐵 ∈ ℝ)
68	13	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐵 ∈ ℝ) → 𝐴 ∈ ℝ^*)
69		simpr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐵 ∈ ℝ) → 𝐵 ∈ ℝ)
70	67, 68, 69	xrralrecnnle 45379	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ ∀𝑛 ∈ ℕ 𝐴 ≤ (𝐵 + (1 / 𝑛))))
71	5	adantlr 715	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐵 ∈ ℝ) ∧ 𝑛 ∈ ℕ) → 𝐴 ∈ ℝ)
72	6	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐵 ∈ ℝ) ∧ 𝑛 ∈ ℕ) → (1 / 𝑛) ∈ ℝ)
73	69	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐵 ∈ ℝ) ∧ 𝑛 ∈ ℕ) → 𝐵 ∈ ℝ)
74	71, 72, 73	lesubaddd 11775	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐵 ∈ ℝ) ∧ 𝑛 ∈ ℕ) → ((𝐴 − (1 / 𝑛)) ≤ 𝐵 ↔ 𝐴 ≤ (𝐵 + (1 / 𝑛))))
75	74	bicomd 223	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐵 ∈ ℝ) ∧ 𝑛 ∈ ℕ) → (𝐴 ≤ (𝐵 + (1 / 𝑛)) ↔ (𝐴 − (1 / 𝑛)) ≤ 𝐵))
76	67, 75	ralbida 3248	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐵 ∈ ℝ) → (∀𝑛 ∈ ℕ 𝐴 ≤ (𝐵 + (1 / 𝑛)) ↔ ∀𝑛 ∈ ℕ (𝐴 − (1 / 𝑛)) ≤ 𝐵))
77	70, 76	bitr2d 280	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐵 ∈ ℝ) → (∀𝑛 ∈ ℕ (𝐴 − (1 / 𝑛)) ≤ 𝐵 ↔ 𝐴 ≤ 𝐵))
78	77	biimpd 229	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ∈ ℝ) → (∀𝑛 ∈ ℕ (𝐴 − (1 / 𝑛)) ≤ 𝐵 → 𝐴 ≤ 𝐵))
79	78	imp 406	. . . . . 6 ⊢ (((𝜑 ∧ 𝐵 ∈ ℝ) ∧ ∀𝑛 ∈ ℕ (𝐴 − (1 / 𝑛)) ≤ 𝐵) → 𝐴 ≤ 𝐵)
80	79	an32s 652	. . . . 5 ⊢ (((𝜑 ∧ ∀𝑛 ∈ ℕ (𝐴 − (1 / 𝑛)) ≤ 𝐵) ∧ 𝐵 ∈ ℝ) → 𝐴 ≤ 𝐵)
81	65, 80	syldan 591	. . . 4 ⊢ (((𝜑 ∧ ∀𝑛 ∈ ℕ (𝐴 − (1 / 𝑛)) ≤ 𝐵) ∧ ¬ 𝐵 = +∞) → 𝐴 ≤ 𝐵)
82	34, 81	pm2.61dan 812	. . 3 ⊢ ((𝜑 ∧ ∀𝑛 ∈ ℕ (𝐴 − (1 / 𝑛)) ≤ 𝐵) → 𝐴 ≤ 𝐵)
83	82	ex 412	. 2 ⊢ (𝜑 → (∀𝑛 ∈ ℕ (𝐴 − (1 / 𝑛)) ≤ 𝐵 → 𝐴 ≤ 𝐵))
84	25, 83	impbid 212	1 ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ ∀𝑛 ∈ ℕ (𝐴 − (1 / 𝑛)) ≤ 𝐵))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 Ⅎwnf 1783 ∈ wcel 2109 ≠ wne 2925 ∀wral 3044 class class class wbr 5107 (class class class)co 7387 ℝcr 11067 0cc0 11068 1c1 11069 + caddc 11071 +∞cpnf 11205 -∞cmnf 11206 ℝ^*cxr 11207 < clt 11208 ≤ cle 11209 − cmin 11405 / cdiv 11835 ℕcn 12186 ℝ⁺crp 12951

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-pre-sup 11146

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-om 7843 df-1st 7968 df-2nd 7969 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-er 8671 df-en 8919 df-dom 8920 df-sdom 8921 df-sup 9393 df-inf 9394 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-div 11836 df-nn 12187 df-n0 12443 df-z 12530 df-uz 12794 df-q 12908 df-rp 12952 df-fl 13754

This theorem is referenced by: preimageiingt 46718

W3C validator