xralrple3 - Mathbox for Glauco Siliprandi

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Theorem xralrple3 42803

Description: Show that 𝐴 is less than 𝐵 by showing that there is no positive bound on the difference. (Contributed by Glauco Siliprandi, 8-Apr-2021.)

**Hypotheses**
Ref	Expression
xralrple3.a	⊢ (𝜑 → 𝐴 ∈ ℝ^*)
xralrple3.b	⊢ (𝜑 → 𝐵 ∈ ℝ)
xralrple3.c	⊢ (𝜑 → 𝐶 ∈ ℝ)
xralrple3.g	⊢ (𝜑 → 0 ≤ 𝐶)

**Assertion**
Ref	Expression
xralrple3	⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ ∀𝑥 ∈ ℝ⁺ 𝐴 ≤ (𝐵 + (𝐶 · 𝑥))))

Distinct variable groups: 𝑥,𝐴 𝑥,𝐵 𝑥,𝐶 𝜑,𝑥

Proof of Theorem xralrple3 Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		xralrple3.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
2	1	ad2antrr 722	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑥 ∈ ℝ⁺) → 𝐴 ∈ ℝ^*)
3		xralrple3.b	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ)
4	3	rexrd 10956	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
5	4	ad2antrr 722	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑥 ∈ ℝ⁺) → 𝐵 ∈ ℝ^*)
6	3	ad2antrr 722	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑥 ∈ ℝ⁺) → 𝐵 ∈ ℝ)
7		xralrple3.c	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ ℝ)
8	7	ad2antrr 722	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑥 ∈ ℝ⁺) → 𝐶 ∈ ℝ)
9		rpre 12667	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
10	9	adantl 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ)
11	8, 10	remulcld 10936	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑥 ∈ ℝ⁺) → (𝐶 · 𝑥) ∈ ℝ)
12	6, 11	readdcld 10935	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑥 ∈ ℝ⁺) → (𝐵 + (𝐶 · 𝑥)) ∈ ℝ)
13	12	rexrd 10956	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑥 ∈ ℝ⁺) → (𝐵 + (𝐶 · 𝑥)) ∈ ℝ^*)
14		simplr 765	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑥 ∈ ℝ⁺) → 𝐴 ≤ 𝐵)
15	7	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝐶 ∈ ℝ)
16	9	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ)
17		xralrple3.g	. . . . . . . . 9 ⊢ (𝜑 → 0 ≤ 𝐶)
18	17	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 0 ≤ 𝐶)
19		rpge0 12672	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → 0 ≤ 𝑥)
20	19	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 0 ≤ 𝑥)
21	15, 16, 18, 20	mulge0d 11482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 0 ≤ (𝐶 · 𝑥))
22	3	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝐵 ∈ ℝ)
23	15, 16	remulcld 10936	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝐶 · 𝑥) ∈ ℝ)
24	22, 23	addge01d 11493	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (0 ≤ (𝐶 · 𝑥) ↔ 𝐵 ≤ (𝐵 + (𝐶 · 𝑥))))
25	21, 24	mpbid 231	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝐵 ≤ (𝐵 + (𝐶 · 𝑥)))
26	25	adantlr 711	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑥 ∈ ℝ⁺) → 𝐵 ≤ (𝐵 + (𝐶 · 𝑥)))
27	2, 5, 13, 14, 26	xrletrd 12825	. . . 4 ⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑥 ∈ ℝ⁺) → 𝐴 ≤ (𝐵 + (𝐶 · 𝑥)))
28	27	ralrimiva 3107	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ∀𝑥 ∈ ℝ⁺ 𝐴 ≤ (𝐵 + (𝐶 · 𝑥)))
29	28	ex 412	. 2 ⊢ (𝜑 → (𝐴 ≤ 𝐵 → ∀𝑥 ∈ ℝ⁺ 𝐴 ≤ (𝐵 + (𝐶 · 𝑥))))
30		1rp 12663	. . . . . . 7 ⊢ 1 ∈ ℝ⁺
31		oveq2 7263	. . . . . . . . . 10 ⊢ (𝑥 = 1 → (𝐶 · 𝑥) = (𝐶 · 1))
32	31	oveq2d 7271	. . . . . . . . 9 ⊢ (𝑥 = 1 → (𝐵 + (𝐶 · 𝑥)) = (𝐵 + (𝐶 · 1)))
33	32	breq2d 5082	. . . . . . . 8 ⊢ (𝑥 = 1 → (𝐴 ≤ (𝐵 + (𝐶 · 𝑥)) ↔ 𝐴 ≤ (𝐵 + (𝐶 · 1))))
34	33	rspcva 3550	. . . . . . 7 ⊢ ((1 ∈ ℝ⁺ ∧ ∀𝑥 ∈ ℝ⁺ 𝐴 ≤ (𝐵 + (𝐶 · 𝑥))) → 𝐴 ≤ (𝐵 + (𝐶 · 1)))
35	30, 34	mpan 686	. . . . . 6 ⊢ (∀𝑥 ∈ ℝ⁺ 𝐴 ≤ (𝐵 + (𝐶 · 𝑥)) → 𝐴 ≤ (𝐵 + (𝐶 · 1)))
36	35	ad2antlr 723	. . . . 5 ⊢ (((𝜑 ∧ ∀𝑥 ∈ ℝ⁺ 𝐴 ≤ (𝐵 + (𝐶 · 𝑥))) ∧ 𝐶 = 0) → 𝐴 ≤ (𝐵 + (𝐶 · 1)))
37		oveq1 7262	. . . . . . . . . 10 ⊢ (𝐶 = 0 → (𝐶 · 1) = (0 · 1))
38		0cn 10898	. . . . . . . . . . . 12 ⊢ 0 ∈ ℂ
39	38	mulid1i 10910	. . . . . . . . . . 11 ⊢ (0 · 1) = 0
40	39	a1i 11	. . . . . . . . . 10 ⊢ (𝐶 = 0 → (0 · 1) = 0)
41	37, 40	eqtrd 2778	. . . . . . . . 9 ⊢ (𝐶 = 0 → (𝐶 · 1) = 0)
42	41	oveq2d 7271	. . . . . . . 8 ⊢ (𝐶 = 0 → (𝐵 + (𝐶 · 1)) = (𝐵 + 0))
43	42	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 = 0) → (𝐵 + (𝐶 · 1)) = (𝐵 + 0))
44	3	recnd 10934	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ ℂ)
45	44	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 = 0) → 𝐵 ∈ ℂ)
46	45	addid1d 11105	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 = 0) → (𝐵 + 0) = 𝐵)
47	43, 46	eqtrd 2778	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶 = 0) → (𝐵 + (𝐶 · 1)) = 𝐵)
48	47	adantlr 711	. . . . 5 ⊢ (((𝜑 ∧ ∀𝑥 ∈ ℝ⁺ 𝐴 ≤ (𝐵 + (𝐶 · 𝑥))) ∧ 𝐶 = 0) → (𝐵 + (𝐶 · 1)) = 𝐵)
49	36, 48	breqtrd 5096	. . . 4 ⊢ (((𝜑 ∧ ∀𝑥 ∈ ℝ⁺ 𝐴 ≤ (𝐵 + (𝐶 · 𝑥))) ∧ 𝐶 = 0) → 𝐴 ≤ 𝐵)
50		neqne 2950	. . . . . . . 8 ⊢ (¬ 𝐶 = 0 → 𝐶 ≠ 0)
51	50	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐶 = 0) → 𝐶 ≠ 0)
52	7	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ≠ 0) → 𝐶 ∈ ℝ)
53		0red 10909	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 ≠ 0) → 0 ∈ ℝ)
54	17	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 ≠ 0) → 0 ≤ 𝐶)
55		simpr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 ≠ 0) → 𝐶 ≠ 0)
56	53, 52, 54, 55	leneltd 11059	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ≠ 0) → 0 < 𝐶)
57	52, 56	elrpd 12698	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 ≠ 0) → 𝐶 ∈ ℝ⁺)
58	51, 57	syldan 590	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐶 = 0) → 𝐶 ∈ ℝ⁺)
59	58	adantlr 711	. . . . 5 ⊢ (((𝜑 ∧ ∀𝑥 ∈ ℝ⁺ 𝐴 ≤ (𝐵 + (𝐶 · 𝑥))) ∧ ¬ 𝐶 = 0) → 𝐶 ∈ ℝ⁺)
60		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺) → 𝑦 ∈ ℝ⁺)
61		simpl 482	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺) → 𝐶 ∈ ℝ⁺)
62	60, 61	rpdivcld 12718	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺) → (𝑦 / 𝐶) ∈ ℝ⁺)
63	62	adantll 710	. . . . . . . . . 10 ⊢ (((∀𝑥 ∈ ℝ⁺ 𝐴 ≤ (𝐵 + (𝐶 · 𝑥)) ∧ 𝐶 ∈ ℝ⁺) ∧ 𝑦 ∈ ℝ⁺) → (𝑦 / 𝐶) ∈ ℝ⁺)
64		simpll 763	. . . . . . . . . 10 ⊢ (((∀𝑥 ∈ ℝ⁺ 𝐴 ≤ (𝐵 + (𝐶 · 𝑥)) ∧ 𝐶 ∈ ℝ⁺) ∧ 𝑦 ∈ ℝ⁺) → ∀𝑥 ∈ ℝ⁺ 𝐴 ≤ (𝐵 + (𝐶 · 𝑥)))
65		oveq2 7263	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝑦 / 𝐶) → (𝐶 · 𝑥) = (𝐶 · (𝑦 / 𝐶)))
66	65	oveq2d 7271	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑦 / 𝐶) → (𝐵 + (𝐶 · 𝑥)) = (𝐵 + (𝐶 · (𝑦 / 𝐶))))
67	66	breq2d 5082	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑦 / 𝐶) → (𝐴 ≤ (𝐵 + (𝐶 · 𝑥)) ↔ 𝐴 ≤ (𝐵 + (𝐶 · (𝑦 / 𝐶)))))
68	67	rspcva 3550	. . . . . . . . . 10 ⊢ (((𝑦 / 𝐶) ∈ ℝ⁺ ∧ ∀𝑥 ∈ ℝ⁺ 𝐴 ≤ (𝐵 + (𝐶 · 𝑥))) → 𝐴 ≤ (𝐵 + (𝐶 · (𝑦 / 𝐶))))
69	63, 64, 68	syl2anc 583	. . . . . . . . 9 ⊢ (((∀𝑥 ∈ ℝ⁺ 𝐴 ≤ (𝐵 + (𝐶 · 𝑥)) ∧ 𝐶 ∈ ℝ⁺) ∧ 𝑦 ∈ ℝ⁺) → 𝐴 ≤ (𝐵 + (𝐶 · (𝑦 / 𝐶))))
70	69	adantlll 714	. . . . . . . 8 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ ℝ⁺ 𝐴 ≤ (𝐵 + (𝐶 · 𝑥))) ∧ 𝐶 ∈ ℝ⁺) ∧ 𝑦 ∈ ℝ⁺) → 𝐴 ≤ (𝐵 + (𝐶 · (𝑦 / 𝐶))))
71	60	rpcnd 12703	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺) → 𝑦 ∈ ℂ)
72	61	rpcnd 12703	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺) → 𝐶 ∈ ℂ)
73	61	rpne0d 12706	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺) → 𝐶 ≠ 0)
74	71, 72, 73	divcan2d 11683	. . . . . . . . . 10 ⊢ ((𝐶 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺) → (𝐶 · (𝑦 / 𝐶)) = 𝑦)
75	74	adantll 710	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ ℝ⁺ 𝐴 ≤ (𝐵 + (𝐶 · 𝑥))) ∧ 𝐶 ∈ ℝ⁺) ∧ 𝑦 ∈ ℝ⁺) → (𝐶 · (𝑦 / 𝐶)) = 𝑦)
76	75	oveq2d 7271	. . . . . . . 8 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ ℝ⁺ 𝐴 ≤ (𝐵 + (𝐶 · 𝑥))) ∧ 𝐶 ∈ ℝ⁺) ∧ 𝑦 ∈ ℝ⁺) → (𝐵 + (𝐶 · (𝑦 / 𝐶))) = (𝐵 + 𝑦))
77	70, 76	breqtrd 5096	. . . . . . 7 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ ℝ⁺ 𝐴 ≤ (𝐵 + (𝐶 · 𝑥))) ∧ 𝐶 ∈ ℝ⁺) ∧ 𝑦 ∈ ℝ⁺) → 𝐴 ≤ (𝐵 + 𝑦))
78	77	ralrimiva 3107	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑥 ∈ ℝ⁺ 𝐴 ≤ (𝐵 + (𝐶 · 𝑥))) ∧ 𝐶 ∈ ℝ⁺) → ∀𝑦 ∈ ℝ⁺ 𝐴 ≤ (𝐵 + 𝑦))
79		xralrple 12868	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ ∀𝑦 ∈ ℝ⁺ 𝐴 ≤ (𝐵 + 𝑦)))
80	1, 3, 79	syl2anc 583	. . . . . . 7 ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ ∀𝑦 ∈ ℝ⁺ 𝐴 ≤ (𝐵 + 𝑦)))
81	80	ad2antrr 722	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑥 ∈ ℝ⁺ 𝐴 ≤ (𝐵 + (𝐶 · 𝑥))) ∧ 𝐶 ∈ ℝ⁺) → (𝐴 ≤ 𝐵 ↔ ∀𝑦 ∈ ℝ⁺ 𝐴 ≤ (𝐵 + 𝑦)))
82	78, 81	mpbird 256	. . . . 5 ⊢ (((𝜑 ∧ ∀𝑥 ∈ ℝ⁺ 𝐴 ≤ (𝐵 + (𝐶 · 𝑥))) ∧ 𝐶 ∈ ℝ⁺) → 𝐴 ≤ 𝐵)
83	59, 82	syldan 590	. . . 4 ⊢ (((𝜑 ∧ ∀𝑥 ∈ ℝ⁺ 𝐴 ≤ (𝐵 + (𝐶 · 𝑥))) ∧ ¬ 𝐶 = 0) → 𝐴 ≤ 𝐵)
84	49, 83	pm2.61dan 809	. . 3 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℝ⁺ 𝐴 ≤ (𝐵 + (𝐶 · 𝑥))) → 𝐴 ≤ 𝐵)
85	84	ex 412	. 2 ⊢ (𝜑 → (∀𝑥 ∈ ℝ⁺ 𝐴 ≤ (𝐵 + (𝐶 · 𝑥)) → 𝐴 ≤ 𝐵))
86	29, 85	impbid 211	1 ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ ∀𝑥 ∈ ℝ⁺ 𝐴 ≤ (𝐵 + (𝐶 · 𝑥))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1539 ∈ wcel 2108 ≠ wne 2942 ∀wral 3063 class class class wbr 5070 (class class class)co 7255 ℂcc 10800 ℝcr 10801 0cc0 10802 1c1 10803 + caddc 10805 · cmul 10807 ℝ^*cxr 10939 ≤ cle 10941 / cdiv 11562 ℝ⁺crp 12659

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-er 8456 df-en 8692 df-dom 8693 df-sdom 8694 df-sup 9131 df-inf 9132 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-n0 12164 df-z 12250 df-uz 12512 df-q 12618 df-rp 12660

This theorem is referenced by: (None)

W3C validator