xralrple3 - Mathbox for Glauco Siliprandi

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Theorem xralrple3 38531

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 758	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑥 ∈ ℝ⁺) → 𝐴 ∈ ℝ^*)
3		xralrple3.b	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ)
4	3	rexrd 9968	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
5	4	ad2antrr 758	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑥 ∈ ℝ⁺) → 𝐵 ∈ ℝ^*)
6	3	ad2antrr 758	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑥 ∈ ℝ⁺) → 𝐵 ∈ ℝ)
7		xralrple3.c	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ ℝ)
8	7	ad2antrr 758	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑥 ∈ ℝ⁺) → 𝐶 ∈ ℝ)
9		rpre 11715	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
10	9	adantl 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ)
11	8, 10	remulcld 9949	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑥 ∈ ℝ⁺) → (𝐶 · 𝑥) ∈ ℝ)
12	6, 11	readdcld 9948	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑥 ∈ ℝ⁺) → (𝐵 + (𝐶 · 𝑥)) ∈ ℝ)
13	12	rexrd 9968	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑥 ∈ ℝ⁺) → (𝐵 + (𝐶 · 𝑥)) ∈ ℝ^*)
14		simplr 788	. . . . 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 11721	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → 0 ≤ 𝑥)
20	19	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 0 ≤ 𝑥)
21	15, 16, 18, 20	mulge0d 10483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 0 ≤ (𝐶 · 𝑥))
22	3	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝐵 ∈ ℝ)
23	15, 16	remulcld 9949	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝐶 · 𝑥) ∈ ℝ)
24	22, 23	addge01d 10494	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (0 ≤ (𝐶 · 𝑥) ↔ 𝐵 ≤ (𝐵 + (𝐶 · 𝑥))))
25	21, 24	mpbid 221	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝐵 ≤ (𝐵 + (𝐶 · 𝑥)))
26	25	adantlr 747	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑥 ∈ ℝ⁺) → 𝐵 ≤ (𝐵 + (𝐶 · 𝑥)))
27	2, 5, 13, 14, 26	xrletrd 11869	. . . 4 ⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑥 ∈ ℝ⁺) → 𝐴 ≤ (𝐵 + (𝐶 · 𝑥)))
28	27	ralrimiva 2949	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ∀𝑥 ∈ ℝ⁺ 𝐴 ≤ (𝐵 + (𝐶 · 𝑥)))
29	28	ex 449	. 2 ⊢ (𝜑 → (𝐴 ≤ 𝐵 → ∀𝑥 ∈ ℝ⁺ 𝐴 ≤ (𝐵 + (𝐶 · 𝑥))))
30		1rp 11712	. . . . . . 7 ⊢ 1 ∈ ℝ⁺
31		oveq2 6557	. . . . . . . . . 10 ⊢ (𝑥 = 1 → (𝐶 · 𝑥) = (𝐶 · 1))
32	31	oveq2d 6565	. . . . . . . . 9 ⊢ (𝑥 = 1 → (𝐵 + (𝐶 · 𝑥)) = (𝐵 + (𝐶 · 1)))
33	32	breq2d 4595	. . . . . . . 8 ⊢ (𝑥 = 1 → (𝐴 ≤ (𝐵 + (𝐶 · 𝑥)) ↔ 𝐴 ≤ (𝐵 + (𝐶 · 1))))
34	33	rspcva 3280	. . . . . . 7 ⊢ ((1 ∈ ℝ⁺ ∧ ∀𝑥 ∈ ℝ⁺ 𝐴 ≤ (𝐵 + (𝐶 · 𝑥))) → 𝐴 ≤ (𝐵 + (𝐶 · 1)))
35	30, 34	mpan 702	. . . . . 6 ⊢ (∀𝑥 ∈ ℝ⁺ 𝐴 ≤ (𝐵 + (𝐶 · 𝑥)) → 𝐴 ≤ (𝐵 + (𝐶 · 1)))
36	35	ad2antlr 759	. . . . 5 ⊢ (((𝜑 ∧ ∀𝑥 ∈ ℝ⁺ 𝐴 ≤ (𝐵 + (𝐶 · 𝑥))) ∧ 𝐶 = 0) → 𝐴 ≤ (𝐵 + (𝐶 · 1)))
37		oveq1 6556	. . . . . . . . . 10 ⊢ (𝐶 = 0 → (𝐶 · 1) = (0 · 1))
38		0cn 9911	. . . . . . . . . . . 12 ⊢ 0 ∈ ℂ
39	38	mulid1i 9921	. . . . . . . . . . 11 ⊢ (0 · 1) = 0
40	39	a1i 11	. . . . . . . . . 10 ⊢ (𝐶 = 0 → (0 · 1) = 0)
41	37, 40	eqtrd 2644	. . . . . . . . 9 ⊢ (𝐶 = 0 → (𝐶 · 1) = 0)
42	41	oveq2d 6565	. . . . . . . 8 ⊢ (𝐶 = 0 → (𝐵 + (𝐶 · 1)) = (𝐵 + 0))
43	42	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 = 0) → (𝐵 + (𝐶 · 1)) = (𝐵 + 0))
44	3	recnd 9947	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ ℂ)
45	44	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 = 0) → 𝐵 ∈ ℂ)
46	45	addid1d 10115	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 = 0) → (𝐵 + 0) = 𝐵)
47	43, 46	eqtrd 2644	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶 = 0) → (𝐵 + (𝐶 · 1)) = 𝐵)
48	47	adantlr 747	. . . . 5 ⊢ (((𝜑 ∧ ∀𝑥 ∈ ℝ⁺ 𝐴 ≤ (𝐵 + (𝐶 · 𝑥))) ∧ 𝐶 = 0) → (𝐵 + (𝐶 · 1)) = 𝐵)
49	36, 48	breqtrd 4609	. . . 4 ⊢ (((𝜑 ∧ ∀𝑥 ∈ ℝ⁺ 𝐴 ≤ (𝐵 + (𝐶 · 𝑥))) ∧ 𝐶 = 0) → 𝐴 ≤ 𝐵)
50		neqne 2790	. . . . . . . 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 9920	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 ≠ 0) → 0 ∈ ℝ)
54	17	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 ≠ 0) → 0 ≤ 𝐶)
55		simpr 476	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 ≠ 0) → 𝐶 ≠ 0)
56	53, 52, 54, 55	leneltd 10070	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ≠ 0) → 0 < 𝐶)
57	52, 56	elrpd 11745	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 ≠ 0) → 𝐶 ∈ ℝ⁺)
58	51, 57	syldan 486	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐶 = 0) → 𝐶 ∈ ℝ⁺)
59	58	adantlr 747	. . . . 5 ⊢ (((𝜑 ∧ ∀𝑥 ∈ ℝ⁺ 𝐴 ≤ (𝐵 + (𝐶 · 𝑥))) ∧ ¬ 𝐶 = 0) → 𝐶 ∈ ℝ⁺)
60		simpr 476	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺) → 𝑦 ∈ ℝ⁺)
61		simpl 472	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺) → 𝐶 ∈ ℝ⁺)
62	60, 61	rpdivcld 11765	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺) → (𝑦 / 𝐶) ∈ ℝ⁺)
63	62	adantll 746	. . . . . . . . . 10 ⊢ (((∀𝑥 ∈ ℝ⁺ 𝐴 ≤ (𝐵 + (𝐶 · 𝑥)) ∧ 𝐶 ∈ ℝ⁺) ∧ 𝑦 ∈ ℝ⁺) → (𝑦 / 𝐶) ∈ ℝ⁺)
64		simpll 786	. . . . . . . . . 10 ⊢ (((∀𝑥 ∈ ℝ⁺ 𝐴 ≤ (𝐵 + (𝐶 · 𝑥)) ∧ 𝐶 ∈ ℝ⁺) ∧ 𝑦 ∈ ℝ⁺) → ∀𝑥 ∈ ℝ⁺ 𝐴 ≤ (𝐵 + (𝐶 · 𝑥)))
65		oveq2 6557	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝑦 / 𝐶) → (𝐶 · 𝑥) = (𝐶 · (𝑦 / 𝐶)))
66	65	oveq2d 6565	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑦 / 𝐶) → (𝐵 + (𝐶 · 𝑥)) = (𝐵 + (𝐶 · (𝑦 / 𝐶))))
67	66	breq2d 4595	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑦 / 𝐶) → (𝐴 ≤ (𝐵 + (𝐶 · 𝑥)) ↔ 𝐴 ≤ (𝐵 + (𝐶 · (𝑦 / 𝐶)))))
68	67	rspcva 3280	. . . . . . . . . 10 ⊢ (((𝑦 / 𝐶) ∈ ℝ⁺ ∧ ∀𝑥 ∈ ℝ⁺ 𝐴 ≤ (𝐵 + (𝐶 · 𝑥))) → 𝐴 ≤ (𝐵 + (𝐶 · (𝑦 / 𝐶))))
69	63, 64, 68	syl2anc 691	. . . . . . . . 9 ⊢ (((∀𝑥 ∈ ℝ⁺ 𝐴 ≤ (𝐵 + (𝐶 · 𝑥)) ∧ 𝐶 ∈ ℝ⁺) ∧ 𝑦 ∈ ℝ⁺) → 𝐴 ≤ (𝐵 + (𝐶 · (𝑦 / 𝐶))))
70	69	adantlll 750	. . . . . . . 8 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ ℝ⁺ 𝐴 ≤ (𝐵 + (𝐶 · 𝑥))) ∧ 𝐶 ∈ ℝ⁺) ∧ 𝑦 ∈ ℝ⁺) → 𝐴 ≤ (𝐵 + (𝐶 · (𝑦 / 𝐶))))
71	60	rpcnd 11750	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺) → 𝑦 ∈ ℂ)
72	61	rpcnd 11750	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺) → 𝐶 ∈ ℂ)
73	61	rpne0d 11753	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺) → 𝐶 ≠ 0)
74	71, 72, 73	divcan2d 10682	. . . . . . . . . 10 ⊢ ((𝐶 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺) → (𝐶 · (𝑦 / 𝐶)) = 𝑦)
75	74	adantll 746	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ ℝ⁺ 𝐴 ≤ (𝐵 + (𝐶 · 𝑥))) ∧ 𝐶 ∈ ℝ⁺) ∧ 𝑦 ∈ ℝ⁺) → (𝐶 · (𝑦 / 𝐶)) = 𝑦)
76	75	oveq2d 6565	. . . . . . . 8 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ ℝ⁺ 𝐴 ≤ (𝐵 + (𝐶 · 𝑥))) ∧ 𝐶 ∈ ℝ⁺) ∧ 𝑦 ∈ ℝ⁺) → (𝐵 + (𝐶 · (𝑦 / 𝐶))) = (𝐵 + 𝑦))
77	70, 76	breqtrd 4609	. . . . . . 7 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ ℝ⁺ 𝐴 ≤ (𝐵 + (𝐶 · 𝑥))) ∧ 𝐶 ∈ ℝ⁺) ∧ 𝑦 ∈ ℝ⁺) → 𝐴 ≤ (𝐵 + 𝑦))
78	77	ralrimiva 2949	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑥 ∈ ℝ⁺ 𝐴 ≤ (𝐵 + (𝐶 · 𝑥))) ∧ 𝐶 ∈ ℝ⁺) → ∀𝑦 ∈ ℝ⁺ 𝐴 ≤ (𝐵 + 𝑦))
79		xralrple 11910	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ ∀𝑦 ∈ ℝ⁺ 𝐴 ≤ (𝐵 + 𝑦)))
80	1, 3, 79	syl2anc 691	. . . . . . 7 ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ ∀𝑦 ∈ ℝ⁺ 𝐴 ≤ (𝐵 + 𝑦)))
81	80	ad2antrr 758	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑥 ∈ ℝ⁺ 𝐴 ≤ (𝐵 + (𝐶 · 𝑥))) ∧ 𝐶 ∈ ℝ⁺) → (𝐴 ≤ 𝐵 ↔ ∀𝑦 ∈ ℝ⁺ 𝐴 ≤ (𝐵 + 𝑦)))
82	78, 81	mpbird 246	. . . . 5 ⊢ (((𝜑 ∧ ∀𝑥 ∈ ℝ⁺ 𝐴 ≤ (𝐵 + (𝐶 · 𝑥))) ∧ 𝐶 ∈ ℝ⁺) → 𝐴 ≤ 𝐵)
83	59, 82	syldan 486	. . . 4 ⊢ (((𝜑 ∧ ∀𝑥 ∈ ℝ⁺ 𝐴 ≤ (𝐵 + (𝐶 · 𝑥))) ∧ ¬ 𝐶 = 0) → 𝐴 ≤ 𝐵)
84	49, 83	pm2.61dan 828	. . 3 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℝ⁺ 𝐴 ≤ (𝐵 + (𝐶 · 𝑥))) → 𝐴 ≤ 𝐵)
85	84	ex 449	. 2 ⊢ (𝜑 → (∀𝑥 ∈ ℝ⁺ 𝐴 ≤ (𝐵 + (𝐶 · 𝑥)) → 𝐴 ≤ 𝐵))
86	29, 85	impbid 201	1 ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ ∀𝑥 ∈ ℝ⁺ 𝐴 ≤ (𝐵 + (𝐶 · 𝑥))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 class class class wbr 4583 (class class class)co 6549 ℂcc 9813 ℝcr 9814 0cc0 9815 1c1 9816 + caddc 9818 · cmul 9820 ℝ^*cxr 9952 ≤ cle 9954 / cdiv 10563 ℝ⁺crp 11708

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-sup 8231 df-inf 8232 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-q 11665 df-rp 11709

This theorem is referenced by: (None)

W3C validator