xralrple3 - Mathbox for Glauco Siliprandi

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Theorem xralrple3 41649

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 724	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑥 ∈ ℝ⁺) → 𝐴 ∈ ℝ^*)
3		xralrple3.b	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ)
4	3	rexrd 10693	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
5	4	ad2antrr 724	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑥 ∈ ℝ⁺) → 𝐵 ∈ ℝ^*)
6	3	ad2antrr 724	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑥 ∈ ℝ⁺) → 𝐵 ∈ ℝ)
7		xralrple3.c	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ ℝ)
8	7	ad2antrr 724	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑥 ∈ ℝ⁺) → 𝐶 ∈ ℝ)
9		rpre 12400	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
10	9	adantl 484	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ)
11	8, 10	remulcld 10673	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑥 ∈ ℝ⁺) → (𝐶 · 𝑥) ∈ ℝ)
12	6, 11	readdcld 10672	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑥 ∈ ℝ⁺) → (𝐵 + (𝐶 · 𝑥)) ∈ ℝ)
13	12	rexrd 10693	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑥 ∈ ℝ⁺) → (𝐵 + (𝐶 · 𝑥)) ∈ ℝ^*)
14		simplr 767	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑥 ∈ ℝ⁺) → 𝐴 ≤ 𝐵)
15	7	adantr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝐶 ∈ ℝ)
16	9	adantl 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ)
17		xralrple3.g	. . . . . . . . 9 ⊢ (𝜑 → 0 ≤ 𝐶)
18	17	adantr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 0 ≤ 𝐶)
19		rpge0 12405	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → 0 ≤ 𝑥)
20	19	adantl 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 0 ≤ 𝑥)
21	15, 16, 18, 20	mulge0d 11219	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 0 ≤ (𝐶 · 𝑥))
22	3	adantr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝐵 ∈ ℝ)
23	15, 16	remulcld 10673	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝐶 · 𝑥) ∈ ℝ)
24	22, 23	addge01d 11230	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (0 ≤ (𝐶 · 𝑥) ↔ 𝐵 ≤ (𝐵 + (𝐶 · 𝑥))))
25	21, 24	mpbid 234	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝐵 ≤ (𝐵 + (𝐶 · 𝑥)))
26	25	adantlr 713	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑥 ∈ ℝ⁺) → 𝐵 ≤ (𝐵 + (𝐶 · 𝑥)))
27	2, 5, 13, 14, 26	xrletrd 12558	. . . 4 ⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑥 ∈ ℝ⁺) → 𝐴 ≤ (𝐵 + (𝐶 · 𝑥)))
28	27	ralrimiva 3184	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ∀𝑥 ∈ ℝ⁺ 𝐴 ≤ (𝐵 + (𝐶 · 𝑥)))
29	28	ex 415	. 2 ⊢ (𝜑 → (𝐴 ≤ 𝐵 → ∀𝑥 ∈ ℝ⁺ 𝐴 ≤ (𝐵 + (𝐶 · 𝑥))))
30		1rp 12396	. . . . . . 7 ⊢ 1 ∈ ℝ⁺
31		oveq2 7166	. . . . . . . . . 10 ⊢ (𝑥 = 1 → (𝐶 · 𝑥) = (𝐶 · 1))
32	31	oveq2d 7174	. . . . . . . . 9 ⊢ (𝑥 = 1 → (𝐵 + (𝐶 · 𝑥)) = (𝐵 + (𝐶 · 1)))
33	32	breq2d 5080	. . . . . . . 8 ⊢ (𝑥 = 1 → (𝐴 ≤ (𝐵 + (𝐶 · 𝑥)) ↔ 𝐴 ≤ (𝐵 + (𝐶 · 1))))
34	33	rspcva 3623	. . . . . . 7 ⊢ ((1 ∈ ℝ⁺ ∧ ∀𝑥 ∈ ℝ⁺ 𝐴 ≤ (𝐵 + (𝐶 · 𝑥))) → 𝐴 ≤ (𝐵 + (𝐶 · 1)))
35	30, 34	mpan 688	. . . . . 6 ⊢ (∀𝑥 ∈ ℝ⁺ 𝐴 ≤ (𝐵 + (𝐶 · 𝑥)) → 𝐴 ≤ (𝐵 + (𝐶 · 1)))
36	35	ad2antlr 725	. . . . 5 ⊢ (((𝜑 ∧ ∀𝑥 ∈ ℝ⁺ 𝐴 ≤ (𝐵 + (𝐶 · 𝑥))) ∧ 𝐶 = 0) → 𝐴 ≤ (𝐵 + (𝐶 · 1)))
37		oveq1 7165	. . . . . . . . . 10 ⊢ (𝐶 = 0 → (𝐶 · 1) = (0 · 1))
38		0cn 10635	. . . . . . . . . . . 12 ⊢ 0 ∈ ℂ
39	38	mulid1i 10647	. . . . . . . . . . 11 ⊢ (0 · 1) = 0
40	39	a1i 11	. . . . . . . . . 10 ⊢ (𝐶 = 0 → (0 · 1) = 0)
41	37, 40	eqtrd 2858	. . . . . . . . 9 ⊢ (𝐶 = 0 → (𝐶 · 1) = 0)
42	41	oveq2d 7174	. . . . . . . 8 ⊢ (𝐶 = 0 → (𝐵 + (𝐶 · 1)) = (𝐵 + 0))
43	42	adantl 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 = 0) → (𝐵 + (𝐶 · 1)) = (𝐵 + 0))
44	3	recnd 10671	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ ℂ)
45	44	adantr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 = 0) → 𝐵 ∈ ℂ)
46	45	addid1d 10842	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 = 0) → (𝐵 + 0) = 𝐵)
47	43, 46	eqtrd 2858	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶 = 0) → (𝐵 + (𝐶 · 1)) = 𝐵)
48	47	adantlr 713	. . . . 5 ⊢ (((𝜑 ∧ ∀𝑥 ∈ ℝ⁺ 𝐴 ≤ (𝐵 + (𝐶 · 𝑥))) ∧ 𝐶 = 0) → (𝐵 + (𝐶 · 1)) = 𝐵)
49	36, 48	breqtrd 5094	. . . 4 ⊢ (((𝜑 ∧ ∀𝑥 ∈ ℝ⁺ 𝐴 ≤ (𝐵 + (𝐶 · 𝑥))) ∧ 𝐶 = 0) → 𝐴 ≤ 𝐵)
50		neqne 3026	. . . . . . . 8 ⊢ (¬ 𝐶 = 0 → 𝐶 ≠ 0)
51	50	adantl 484	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐶 = 0) → 𝐶 ≠ 0)
52	7	adantr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ≠ 0) → 𝐶 ∈ ℝ)
53		0red 10646	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 ≠ 0) → 0 ∈ ℝ)
54	17	adantr 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 ≠ 0) → 0 ≤ 𝐶)
55		simpr 487	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 ≠ 0) → 𝐶 ≠ 0)
56	53, 52, 54, 55	leneltd 10796	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ≠ 0) → 0 < 𝐶)
57	52, 56	elrpd 12431	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 ≠ 0) → 𝐶 ∈ ℝ⁺)
58	51, 57	syldan 593	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐶 = 0) → 𝐶 ∈ ℝ⁺)
59	58	adantlr 713	. . . . 5 ⊢ (((𝜑 ∧ ∀𝑥 ∈ ℝ⁺ 𝐴 ≤ (𝐵 + (𝐶 · 𝑥))) ∧ ¬ 𝐶 = 0) → 𝐶 ∈ ℝ⁺)
60		simpr 487	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺) → 𝑦 ∈ ℝ⁺)
61		simpl 485	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺) → 𝐶 ∈ ℝ⁺)
62	60, 61	rpdivcld 12451	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺) → (𝑦 / 𝐶) ∈ ℝ⁺)
63	62	adantll 712	. . . . . . . . . 10 ⊢ (((∀𝑥 ∈ ℝ⁺ 𝐴 ≤ (𝐵 + (𝐶 · 𝑥)) ∧ 𝐶 ∈ ℝ⁺) ∧ 𝑦 ∈ ℝ⁺) → (𝑦 / 𝐶) ∈ ℝ⁺)
64		simpll 765	. . . . . . . . . 10 ⊢ (((∀𝑥 ∈ ℝ⁺ 𝐴 ≤ (𝐵 + (𝐶 · 𝑥)) ∧ 𝐶 ∈ ℝ⁺) ∧ 𝑦 ∈ ℝ⁺) → ∀𝑥 ∈ ℝ⁺ 𝐴 ≤ (𝐵 + (𝐶 · 𝑥)))
65		oveq2 7166	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝑦 / 𝐶) → (𝐶 · 𝑥) = (𝐶 · (𝑦 / 𝐶)))
66	65	oveq2d 7174	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑦 / 𝐶) → (𝐵 + (𝐶 · 𝑥)) = (𝐵 + (𝐶 · (𝑦 / 𝐶))))
67	66	breq2d 5080	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑦 / 𝐶) → (𝐴 ≤ (𝐵 + (𝐶 · 𝑥)) ↔ 𝐴 ≤ (𝐵 + (𝐶 · (𝑦 / 𝐶)))))
68	67	rspcva 3623	. . . . . . . . . 10 ⊢ (((𝑦 / 𝐶) ∈ ℝ⁺ ∧ ∀𝑥 ∈ ℝ⁺ 𝐴 ≤ (𝐵 + (𝐶 · 𝑥))) → 𝐴 ≤ (𝐵 + (𝐶 · (𝑦 / 𝐶))))
69	63, 64, 68	syl2anc 586	. . . . . . . . 9 ⊢ (((∀𝑥 ∈ ℝ⁺ 𝐴 ≤ (𝐵 + (𝐶 · 𝑥)) ∧ 𝐶 ∈ ℝ⁺) ∧ 𝑦 ∈ ℝ⁺) → 𝐴 ≤ (𝐵 + (𝐶 · (𝑦 / 𝐶))))
70	69	adantlll 716	. . . . . . . 8 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ ℝ⁺ 𝐴 ≤ (𝐵 + (𝐶 · 𝑥))) ∧ 𝐶 ∈ ℝ⁺) ∧ 𝑦 ∈ ℝ⁺) → 𝐴 ≤ (𝐵 + (𝐶 · (𝑦 / 𝐶))))
71	60	rpcnd 12436	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺) → 𝑦 ∈ ℂ)
72	61	rpcnd 12436	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺) → 𝐶 ∈ ℂ)
73	61	rpne0d 12439	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺) → 𝐶 ≠ 0)
74	71, 72, 73	divcan2d 11420	. . . . . . . . . 10 ⊢ ((𝐶 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺) → (𝐶 · (𝑦 / 𝐶)) = 𝑦)
75	74	adantll 712	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ ℝ⁺ 𝐴 ≤ (𝐵 + (𝐶 · 𝑥))) ∧ 𝐶 ∈ ℝ⁺) ∧ 𝑦 ∈ ℝ⁺) → (𝐶 · (𝑦 / 𝐶)) = 𝑦)
76	75	oveq2d 7174	. . . . . . . 8 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ ℝ⁺ 𝐴 ≤ (𝐵 + (𝐶 · 𝑥))) ∧ 𝐶 ∈ ℝ⁺) ∧ 𝑦 ∈ ℝ⁺) → (𝐵 + (𝐶 · (𝑦 / 𝐶))) = (𝐵 + 𝑦))
77	70, 76	breqtrd 5094	. . . . . . 7 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ ℝ⁺ 𝐴 ≤ (𝐵 + (𝐶 · 𝑥))) ∧ 𝐶 ∈ ℝ⁺) ∧ 𝑦 ∈ ℝ⁺) → 𝐴 ≤ (𝐵 + 𝑦))
78	77	ralrimiva 3184	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑥 ∈ ℝ⁺ 𝐴 ≤ (𝐵 + (𝐶 · 𝑥))) ∧ 𝐶 ∈ ℝ⁺) → ∀𝑦 ∈ ℝ⁺ 𝐴 ≤ (𝐵 + 𝑦))
79		xralrple 12601	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ ∀𝑦 ∈ ℝ⁺ 𝐴 ≤ (𝐵 + 𝑦)))
80	1, 3, 79	syl2anc 586	. . . . . . 7 ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ ∀𝑦 ∈ ℝ⁺ 𝐴 ≤ (𝐵 + 𝑦)))
81	80	ad2antrr 724	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑥 ∈ ℝ⁺ 𝐴 ≤ (𝐵 + (𝐶 · 𝑥))) ∧ 𝐶 ∈ ℝ⁺) → (𝐴 ≤ 𝐵 ↔ ∀𝑦 ∈ ℝ⁺ 𝐴 ≤ (𝐵 + 𝑦)))
82	78, 81	mpbird 259	. . . . 5 ⊢ (((𝜑 ∧ ∀𝑥 ∈ ℝ⁺ 𝐴 ≤ (𝐵 + (𝐶 · 𝑥))) ∧ 𝐶 ∈ ℝ⁺) → 𝐴 ≤ 𝐵)
83	59, 82	syldan 593	. . . 4 ⊢ (((𝜑 ∧ ∀𝑥 ∈ ℝ⁺ 𝐴 ≤ (𝐵 + (𝐶 · 𝑥))) ∧ ¬ 𝐶 = 0) → 𝐴 ≤ 𝐵)
84	49, 83	pm2.61dan 811	. . 3 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℝ⁺ 𝐴 ≤ (𝐵 + (𝐶 · 𝑥))) → 𝐴 ≤ 𝐵)
85	84	ex 415	. 2 ⊢ (𝜑 → (∀𝑥 ∈ ℝ⁺ 𝐴 ≤ (𝐵 + (𝐶 · 𝑥)) → 𝐴 ≤ 𝐵))
86	29, 85	impbid 214	1 ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ ∀𝑥 ∈ ℝ⁺ 𝐴 ≤ (𝐵 + (𝐶 · 𝑥))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 ∀wral 3140 class class class wbr 5068 (class class class)co 7158 ℂcc 10537 ℝcr 10538 0cc0 10539 1c1 10540 + caddc 10542 · cmul 10544 ℝ^*cxr 10676 ≤ cle 10678 / cdiv 11299 ℝ⁺crp 12392

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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-sup 8908 df-inf 8909 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-n0 11901 df-z 11985 df-uz 12247 df-q 12352 df-rp 12393

This theorem is referenced by: (None)

W3C validator