xrlexaddrp - Mathbox for Glauco Siliprandi

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Theorem xrlexaddrp 46109

Description: If an extended real number 𝐴 can be approximated from above, adding positive reals to 𝐵, then 𝐴 is less than or equal to 𝐵. (Contributed by Glauco Siliprandi, 11-Oct-2020.)

**Hypotheses**
Ref	Expression
xrlexaddrp.1	⊢ (𝜑 → 𝐴 ∈ ℝ^*)
xrlexaddrp.2	⊢ (𝜑 → 𝐵 ∈ ℝ^*)
xrlexaddrp.3	⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝐴 ≤ (𝐵 +_𝑒 𝑥))

**Assertion**
Ref	Expression
xrlexaddrp	⊢ (𝜑 → 𝐴 ≤ 𝐵)

Distinct variable groups: 𝑥,𝐴 𝑥,𝐵 𝜑,𝑥

**Proof of Theorem xrlexaddrp**
Step	Hyp	Ref	Expression
1		xrlexaddrp.1	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
2		pnfge 13173	. . . . 5 ⊢ (𝐴 ∈ ℝ^* → 𝐴 ≤ +∞)
3	1, 2	syl 18	. . . 4 ⊢ (𝜑 → 𝐴 ≤ +∞)
4	3	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝐵 = +∞) → 𝐴 ≤ +∞)
5		id 23	. . . . 5 ⊢ (𝐵 = +∞ → 𝐵 = +∞)
6	5	eqcomd 2772	. . . 4 ⊢ (𝐵 = +∞ → +∞ = 𝐵)
7	6	adantl 487	. . 3 ⊢ ((𝜑 ∧ 𝐵 = +∞) → +∞ = 𝐵)
8	4, 7	breqtrd 5142	. 2 ⊢ ((𝜑 ∧ 𝐵 = +∞) → 𝐴 ≤ 𝐵)
9		simpl 488	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐵 = +∞) → 𝜑)
10		neqne 2969	. . . 4 ⊢ (¬ 𝐵 = +∞ → 𝐵 ≠ +∞)
11	10	adantl 487	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐵 = +∞) → 𝐵 ≠ +∞)
12		simpr 490	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 = -∞) → 𝐴 = -∞)
13		xrlexaddrp.2	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
14		mnfle 13178	. . . . . . . 8 ⊢ (𝐵 ∈ ℝ^* → -∞ ≤ 𝐵)
15	13, 14	syl 18	. . . . . . 7 ⊢ (𝜑 → -∞ ≤ 𝐵)
16	15	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 = -∞) → -∞ ≤ 𝐵)
17	12, 16	eqbrtrd 5138	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 = -∞) → 𝐴 ≤ 𝐵)
18	17	adantlr 728	. . . 4 ⊢ (((𝜑 ∧ 𝐵 ≠ +∞) ∧ 𝐴 = -∞) → 𝐴 ≤ 𝐵)
19		simpl 488	. . . . 5 ⊢ (((𝜑 ∧ 𝐵 ≠ +∞) ∧ ¬ 𝐴 = -∞) → (𝜑 ∧ 𝐵 ≠ +∞))
20		neqne 2969	. . . . . 6 ⊢ (¬ 𝐴 = -∞ → 𝐴 ≠ -∞)
21	20	adantl 487	. . . . 5 ⊢ (((𝜑 ∧ 𝐵 ≠ +∞) ∧ ¬ 𝐴 = -∞) → 𝐴 ≠ -∞)
22		simpll 779	. . . . . 6 ⊢ (((𝜑 ∧ 𝐵 ≠ +∞) ∧ 𝐴 ≠ -∞) → 𝜑)
23	13	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐵 ≠ +∞) → 𝐵 ∈ ℝ^*)
24		simpr 490	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐵 ≠ +∞) → 𝐵 ≠ +∞)
25	23, 24	jca 521	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐵 ≠ +∞) → (𝐵 ∈ ℝ^* ∧ 𝐵 ≠ +∞))
26		xrnepnf 13161	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ +∞) ↔ (𝐵 ∈ ℝ ∨ 𝐵 = -∞))
27	25, 26	sylib 221	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐵 ≠ +∞) → (𝐵 ∈ ℝ ∨ 𝐵 = -∞))
28	27	adantr 486	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐵 ≠ +∞) ∧ ¬ 𝐵 ∈ ℝ) → (𝐵 ∈ ℝ ∨ 𝐵 = -∞))
29		simpr 490	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐵 ≠ +∞) ∧ ¬ 𝐵 ∈ ℝ) → ¬ 𝐵 ∈ ℝ)
30		pm2.53 865	. . . . . . . . 9 ⊢ ((𝐵 ∈ ℝ ∨ 𝐵 = -∞) → (¬ 𝐵 ∈ ℝ → 𝐵 = -∞))
31	28, 29, 30	sylc 66	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐵 ≠ +∞) ∧ ¬ 𝐵 ∈ ℝ) → 𝐵 = -∞)
32	31	adantlr 728	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐵 ≠ +∞) ∧ 𝐴 ≠ -∞) ∧ ¬ 𝐵 ∈ ℝ) → 𝐵 = -∞)
33		id 23	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝜑)
34		1rp 13038	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℝ⁺
35	34	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → 1 ∈ ℝ⁺)
36		1re 11226	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ ℝ
37	36	elexi 3480	. . . . . . . . . . . . . 14 ⊢ 1 ∈ V
38		eleq1 2854	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 1 → (𝑥 ∈ ℝ⁺ ↔ 1 ∈ ℝ⁺))
39	38	anbi2d 642	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 1 → ((𝜑 ∧ 𝑥 ∈ ℝ⁺) ↔ (𝜑 ∧ 1 ∈ ℝ⁺)))
40		oveq2 7431	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 1 → (𝐵 +_𝑒 𝑥) = (𝐵 +_𝑒 1))
41	40	breq2d 5126	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 1 → (𝐴 ≤ (𝐵 +_𝑒 𝑥) ↔ 𝐴 ≤ (𝐵 +_𝑒 1)))
42	39, 41	imbi12d 347	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 1 → (((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝐴 ≤ (𝐵 +_𝑒 𝑥)) ↔ ((𝜑 ∧ 1 ∈ ℝ⁺) → 𝐴 ≤ (𝐵 +_𝑒 1))))
43		xrlexaddrp.3	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝐴 ≤ (𝐵 +_𝑒 𝑥))
44	37, 42, 43	vtocl 3528	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 1 ∈ ℝ⁺) → 𝐴 ≤ (𝐵 +_𝑒 1))
45	33, 35, 44	syl2anc 596	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ≤ (𝐵 +_𝑒 1))
46	45	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 ≠ -∞) ∧ 𝐵 = -∞) → 𝐴 ≤ (𝐵 +_𝑒 1))
47		oveq1 7430	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 = -∞ → (𝐵 +_𝑒 1) = (-∞ +_𝑒 1))
48		1xr 11286	. . . . . . . . . . . . . . . . . 18 ⊢ 1 ∈ ℝ^*
49		ltpnf 13163	. . . . . . . . . . . . . . . . . . . 20 ⊢ (1 ∈ ℝ → 1 < +∞)
50	36, 49	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 < +∞
51	36, 50	ltneii 11341	. . . . . . . . . . . . . . . . . 18 ⊢ 1 ≠ +∞
52		xaddmnf2 13273	. . . . . . . . . . . . . . . . . 18 ⊢ ((1 ∈ ℝ^* ∧ 1 ≠ +∞) → (-∞ +_𝑒 1) = -∞)
53	48, 51, 52	mp2an 705	. . . . . . . . . . . . . . . . 17 ⊢ (-∞ +_𝑒 1) = -∞
54	53	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 = -∞ → (-∞ +_𝑒 1) = -∞)
55	47, 54	eqtr2d 2802	. . . . . . . . . . . . . . 15 ⊢ (𝐵 = -∞ → -∞ = (𝐵 +_𝑒 1))
56	55	adantl 487	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐴 ≠ -∞) ∧ 𝐵 = -∞) → -∞ = (𝐵 +_𝑒 1))
57	56	eqcomd 2772	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐴 ≠ -∞) ∧ 𝐵 = -∞) → (𝐵 +_𝑒 1) = -∞)
58	1	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐴 ≠ -∞) → 𝐴 ∈ ℝ^*)
59		simpr 490	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐴 ≠ -∞) → 𝐴 ≠ -∞)
60		nemnftgtmnft 46101	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≠ -∞) → -∞ < 𝐴)
61	58, 59, 60	syl2anc 596	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐴 ≠ -∞) → -∞ < 𝐴)
62	61	adantr 486	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐴 ≠ -∞) ∧ 𝐵 = -∞) → -∞ < 𝐴)
63	57, 62	eqbrtrd 5138	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐴 ≠ -∞) ∧ 𝐵 = -∞) → (𝐵 +_𝑒 1) < 𝐴)
64	13	ad2antrr 739	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐴 ≠ -∞) ∧ 𝐵 = -∞) → 𝐵 ∈ ℝ^*)
65	48	a1i 11	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐴 ≠ -∞) ∧ 𝐵 = -∞) → 1 ∈ ℝ^*)
66	64, 65	xaddcld 13345	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐴 ≠ -∞) ∧ 𝐵 = -∞) → (𝐵 +_𝑒 1) ∈ ℝ^*)
67	1	ad2antrr 739	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐴 ≠ -∞) ∧ 𝐵 = -∞) → 𝐴 ∈ ℝ^*)
68		xrltnle 11294	. . . . . . . . . . . . 13 ⊢ (((𝐵 +_𝑒 1) ∈ ℝ^* ∧ 𝐴 ∈ ℝ^*) → ((𝐵 +_𝑒 1) < 𝐴 ↔ ¬ 𝐴 ≤ (𝐵 +_𝑒 1)))
69	66, 67, 68	syl2anc 596	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐴 ≠ -∞) ∧ 𝐵 = -∞) → ((𝐵 +_𝑒 1) < 𝐴 ↔ ¬ 𝐴 ≤ (𝐵 +_𝑒 1)))
70	63, 69	mpbid 235	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 ≠ -∞) ∧ 𝐵 = -∞) → ¬ 𝐴 ≤ (𝐵 +_𝑒 1))
71	46, 70	pm2.65da 829	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 ≠ -∞) → ¬ 𝐵 = -∞)
72	71	neqned 2968	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ≠ -∞) → 𝐵 ≠ -∞)
73	72	ad4ant13 764	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐵 ≠ +∞) ∧ 𝐴 ≠ -∞) ∧ ¬ 𝐵 ∈ ℝ) → 𝐵 ≠ -∞)
74	73	neneqd 2966	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐵 ≠ +∞) ∧ 𝐴 ≠ -∞) ∧ ¬ 𝐵 ∈ ℝ) → ¬ 𝐵 = -∞)
75	32, 74	condan 830	. . . . . 6 ⊢ (((𝜑 ∧ 𝐵 ≠ +∞) ∧ 𝐴 ≠ -∞) → 𝐵 ∈ ℝ)
76	43	adantlr 728	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐵 ∈ ℝ) ∧ 𝑥 ∈ ℝ⁺) → 𝐴 ≤ (𝐵 +_𝑒 𝑥))
77		simpl 488	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ ℝ ∧ 𝑥 ∈ ℝ⁺) → 𝐵 ∈ ℝ)
78		rpre 13043	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
79	78	adantl 487	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ ℝ ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ)
80		rexadd 13276	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (𝐵 +_𝑒 𝑥) = (𝐵 + 𝑥))
81	77, 79, 80	syl2anc 596	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ℝ ∧ 𝑥 ∈ ℝ⁺) → (𝐵 +_𝑒 𝑥) = (𝐵 + 𝑥))
82	81	adantll 727	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐵 ∈ ℝ) ∧ 𝑥 ∈ ℝ⁺) → (𝐵 +_𝑒 𝑥) = (𝐵 + 𝑥))
83	76, 82	breqtrd 5142	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐵 ∈ ℝ) ∧ 𝑥 ∈ ℝ⁺) → 𝐴 ≤ (𝐵 + 𝑥))
84	83	ralrimiva 3160	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ∈ ℝ) → ∀𝑥 ∈ ℝ⁺ 𝐴 ≤ (𝐵 + 𝑥))
85	1	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐵 ∈ ℝ) → 𝐴 ∈ ℝ^*)
86		simpr 490	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐵 ∈ ℝ) → 𝐵 ∈ ℝ)
87		xralrple 13249	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ ∀𝑥 ∈ ℝ⁺ 𝐴 ≤ (𝐵 + 𝑥)))
88	85, 86, 87	syl2anc 596	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ ∀𝑥 ∈ ℝ⁺ 𝐴 ≤ (𝐵 + 𝑥)))
89	84, 88	mpbird 260	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ∈ ℝ) → 𝐴 ≤ 𝐵)
90	22, 75, 89	syl2anc 596	. . . . 5 ⊢ (((𝜑 ∧ 𝐵 ≠ +∞) ∧ 𝐴 ≠ -∞) → 𝐴 ≤ 𝐵)
91	19, 21, 90	syl2anc 596	. . . 4 ⊢ (((𝜑 ∧ 𝐵 ≠ +∞) ∧ ¬ 𝐴 = -∞) → 𝐴 ≤ 𝐵)
92	18, 91	pm2.61dan 825	. . 3 ⊢ ((𝜑 ∧ 𝐵 ≠ +∞) → 𝐴 ≤ 𝐵)
93	9, 11, 92	syl2anc 596	. 2 ⊢ ((𝜑 ∧ ¬ 𝐵 = +∞) → 𝐴 ≤ 𝐵)
94	8, 93	pm2.61dan 825	1 ⊢ (𝜑 → 𝐴 ≤ 𝐵)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 = wceq 1570 ∈ wcel 2146 ≠ wne 2961 ∀wral 3082 class class class wbr 5114 (class class class)co 7423 ℝcr 11117 1c1 11119 + caddc 11121 +∞cpnf 11258 -∞cmnf 11259 ℝ^*cxr 11260 < clt 11261 ≤ cle 11262 ℝ⁺crp 13034 +_𝑒 cxad 13153

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2738 ax-sep 5262 ax-nul 5274 ax-pow 5341 ax-pr 5409 ax-un 7745 ax-cnex 11174 ax-resscn 11175 ax-1cn 11176 ax-icn 11177 ax-addcl 11178 ax-addrcl 11179 ax-mulcl 11180 ax-mulrcl 11181 ax-mulcom 11182 ax-addass 11183 ax-mulass 11184 ax-distr 11185 ax-i2m1 11186 ax-1ne0 11187 ax-1rid 11188 ax-rnegex 11189 ax-rrecex 11190 ax-cnre 11191 ax-pre-lttri 11192 ax-pre-lttrn 11193 ax-pre-ltadd 11194 ax-pre-mulgt0 11195 ax-pre-sup 11196

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2570 df-eu 2600 df-clab 2745 df-cleq 2758 df-clel 2841 df-nfc 2915 df-ne 2962 df-nel 3068 df-ral 3083 df-rex 3093 df-rmo 3372 df-reu 3373 df-rab 3420 df-v 3460 df-sbc 3748 df-csb 3857 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3928 df-nul 4290 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4878 df-iun 4963 df-br 5115 df-opab 5179 df-mpt 5198 df-tr 5224 df-id 5561 df-eprel 5566 df-po 5574 df-so 5575 df-fr 5619 df-we 5621 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6309 df-ord 6370 df-on 6371 df-lim 6372 df-suc 6373 df-iota 6499 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7380 df-ov 7426 df-oprab 7427 df-mpo 7428 df-om 7872 df-1st 7995 df-2nd 7996 df-frecs 8287 df-wrecs 8318 df-recs 8367 df-rdg 8406 df-er 8703 df-en 8953 df-dom 8954 df-sdom 8955 df-sup 9412 df-inf 9413 df-pnf 11263 df-mnf 11264 df-xr 11265 df-ltxr 11266 df-le 11267 df-sub 11461 df-neg 11462 df-div 11890 df-nn 12252 df-n0 12523 df-z 12610 df-uz 12881 df-q 12991 df-rp 13035 df-xadd 13156

This theorem is used by: infleinf 46128 sge0xaddlem2 47189 ovnsubadd 47327

W3C validator