xrlexaddrp - Mathbox for Glauco Siliprandi

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Theorem xrlexaddrp 45475

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 13031	. . . . 5 ⊢ (𝐴 ∈ ℝ^* → 𝐴 ≤ +∞)
3	1, 2	syl 17	. . . 4 ⊢ (𝜑 → 𝐴 ≤ +∞)
4	3	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝐵 = +∞) → 𝐴 ≤ +∞)
5		id 22	. . . . 5 ⊢ (𝐵 = +∞ → 𝐵 = +∞)
6	5	eqcomd 2739	. . . 4 ⊢ (𝐵 = +∞ → +∞ = 𝐵)
7	6	adantl 481	. . 3 ⊢ ((𝜑 ∧ 𝐵 = +∞) → +∞ = 𝐵)
8	4, 7	breqtrd 5119	. 2 ⊢ ((𝜑 ∧ 𝐵 = +∞) → 𝐴 ≤ 𝐵)
9		simpl 482	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐵 = +∞) → 𝜑)
10		neqne 2937	. . . 4 ⊢ (¬ 𝐵 = +∞ → 𝐵 ≠ +∞)
11	10	adantl 481	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐵 = +∞) → 𝐵 ≠ +∞)
12		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 = -∞) → 𝐴 = -∞)
13		xrlexaddrp.2	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
14		mnfle 13036	. . . . . . . 8 ⊢ (𝐵 ∈ ℝ^* → -∞ ≤ 𝐵)
15	13, 14	syl 17	. . . . . . 7 ⊢ (𝜑 → -∞ ≤ 𝐵)
16	15	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 = -∞) → -∞ ≤ 𝐵)
17	12, 16	eqbrtrd 5115	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 = -∞) → 𝐴 ≤ 𝐵)
18	17	adantlr 715	. . . 4 ⊢ (((𝜑 ∧ 𝐵 ≠ +∞) ∧ 𝐴 = -∞) → 𝐴 ≤ 𝐵)
19		simpl 482	. . . . 5 ⊢ (((𝜑 ∧ 𝐵 ≠ +∞) ∧ ¬ 𝐴 = -∞) → (𝜑 ∧ 𝐵 ≠ +∞))
20		neqne 2937	. . . . . 6 ⊢ (¬ 𝐴 = -∞ → 𝐴 ≠ -∞)
21	20	adantl 481	. . . . 5 ⊢ (((𝜑 ∧ 𝐵 ≠ +∞) ∧ ¬ 𝐴 = -∞) → 𝐴 ≠ -∞)
22		simpll 766	. . . . . 6 ⊢ (((𝜑 ∧ 𝐵 ≠ +∞) ∧ 𝐴 ≠ -∞) → 𝜑)
23	13	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐵 ≠ +∞) → 𝐵 ∈ ℝ^*)
24		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐵 ≠ +∞) → 𝐵 ≠ +∞)
25	23, 24	jca 511	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐵 ≠ +∞) → (𝐵 ∈ ℝ^* ∧ 𝐵 ≠ +∞))
26		xrnepnf 13019	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ +∞) ↔ (𝐵 ∈ ℝ ∨ 𝐵 = -∞))
27	25, 26	sylib 218	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐵 ≠ +∞) → (𝐵 ∈ ℝ ∨ 𝐵 = -∞))
28	27	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐵 ≠ +∞) ∧ ¬ 𝐵 ∈ ℝ) → (𝐵 ∈ ℝ ∨ 𝐵 = -∞))
29		simpr 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐵 ≠ +∞) ∧ ¬ 𝐵 ∈ ℝ) → ¬ 𝐵 ∈ ℝ)
30		pm2.53 851	. . . . . . . . 9 ⊢ ((𝐵 ∈ ℝ ∨ 𝐵 = -∞) → (¬ 𝐵 ∈ ℝ → 𝐵 = -∞))
31	28, 29, 30	sylc 65	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐵 ≠ +∞) ∧ ¬ 𝐵 ∈ ℝ) → 𝐵 = -∞)
32	31	adantlr 715	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐵 ≠ +∞) ∧ 𝐴 ≠ -∞) ∧ ¬ 𝐵 ∈ ℝ) → 𝐵 = -∞)
33		id 22	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝜑)
34		1rp 12896	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℝ⁺
35	34	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → 1 ∈ ℝ⁺)
36		1re 11119	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ ℝ
37	36	elexi 3460	. . . . . . . . . . . . . 14 ⊢ 1 ∈ V
38		eleq1 2821	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 1 → (𝑥 ∈ ℝ⁺ ↔ 1 ∈ ℝ⁺))
39	38	anbi2d 630	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 1 → ((𝜑 ∧ 𝑥 ∈ ℝ⁺) ↔ (𝜑 ∧ 1 ∈ ℝ⁺)))
40		oveq2 7360	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 1 → (𝐵 +_𝑒 𝑥) = (𝐵 +_𝑒 1))
41	40	breq2d 5105	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 1 → (𝐴 ≤ (𝐵 +_𝑒 𝑥) ↔ 𝐴 ≤ (𝐵 +_𝑒 1)))
42	39, 41	imbi12d 344	. . . . . . . . . . . . . 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 3512	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 1 ∈ ℝ⁺) → 𝐴 ≤ (𝐵 +_𝑒 1))
45	33, 35, 44	syl2anc 584	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ≤ (𝐵 +_𝑒 1))
46	45	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 ≠ -∞) ∧ 𝐵 = -∞) → 𝐴 ≤ (𝐵 +_𝑒 1))
47		oveq1 7359	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 = -∞ → (𝐵 +_𝑒 1) = (-∞ +_𝑒 1))
48		1xr 11178	. . . . . . . . . . . . . . . . . 18 ⊢ 1 ∈ ℝ^*
49		ltpnf 13021	. . . . . . . . . . . . . . . . . . . 20 ⊢ (1 ∈ ℝ → 1 < +∞)
50	36, 49	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 < +∞
51	36, 50	ltneii 11233	. . . . . . . . . . . . . . . . . 18 ⊢ 1 ≠ +∞
52		xaddmnf2 13130	. . . . . . . . . . . . . . . . . 18 ⊢ ((1 ∈ ℝ^* ∧ 1 ≠ +∞) → (-∞ +_𝑒 1) = -∞)
53	48, 51, 52	mp2an 692	. . . . . . . . . . . . . . . . 17 ⊢ (-∞ +_𝑒 1) = -∞
54	53	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 = -∞ → (-∞ +_𝑒 1) = -∞)
55	47, 54	eqtr2d 2769	. . . . . . . . . . . . . . 15 ⊢ (𝐵 = -∞ → -∞ = (𝐵 +_𝑒 1))
56	55	adantl 481	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐴 ≠ -∞) ∧ 𝐵 = -∞) → -∞ = (𝐵 +_𝑒 1))
57	56	eqcomd 2739	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐴 ≠ -∞) ∧ 𝐵 = -∞) → (𝐵 +_𝑒 1) = -∞)
58	1	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐴 ≠ -∞) → 𝐴 ∈ ℝ^*)
59		simpr 484	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐴 ≠ -∞) → 𝐴 ≠ -∞)
60		nemnftgtmnft 45467	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≠ -∞) → -∞ < 𝐴)
61	58, 59, 60	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐴 ≠ -∞) → -∞ < 𝐴)
62	61	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐴 ≠ -∞) ∧ 𝐵 = -∞) → -∞ < 𝐴)
63	57, 62	eqbrtrd 5115	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐴 ≠ -∞) ∧ 𝐵 = -∞) → (𝐵 +_𝑒 1) < 𝐴)
64	13	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐴 ≠ -∞) ∧ 𝐵 = -∞) → 𝐵 ∈ ℝ^*)
65	48	a1i 11	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐴 ≠ -∞) ∧ 𝐵 = -∞) → 1 ∈ ℝ^*)
66	64, 65	xaddcld 13202	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐴 ≠ -∞) ∧ 𝐵 = -∞) → (𝐵 +_𝑒 1) ∈ ℝ^*)
67	1	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐴 ≠ -∞) ∧ 𝐵 = -∞) → 𝐴 ∈ ℝ^*)
68		xrltnle 11186	. . . . . . . . . . . . 13 ⊢ (((𝐵 +_𝑒 1) ∈ ℝ^* ∧ 𝐴 ∈ ℝ^*) → ((𝐵 +_𝑒 1) < 𝐴 ↔ ¬ 𝐴 ≤ (𝐵 +_𝑒 1)))
69	66, 67, 68	syl2anc 584	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐴 ≠ -∞) ∧ 𝐵 = -∞) → ((𝐵 +_𝑒 1) < 𝐴 ↔ ¬ 𝐴 ≤ (𝐵 +_𝑒 1)))
70	63, 69	mpbid 232	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 ≠ -∞) ∧ 𝐵 = -∞) → ¬ 𝐴 ≤ (𝐵 +_𝑒 1))
71	46, 70	pm2.65da 816	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 ≠ -∞) → ¬ 𝐵 = -∞)
72	71	neqned 2936	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ≠ -∞) → 𝐵 ≠ -∞)
73	72	ad4ant13 751	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐵 ≠ +∞) ∧ 𝐴 ≠ -∞) ∧ ¬ 𝐵 ∈ ℝ) → 𝐵 ≠ -∞)
74	73	neneqd 2934	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐵 ≠ +∞) ∧ 𝐴 ≠ -∞) ∧ ¬ 𝐵 ∈ ℝ) → ¬ 𝐵 = -∞)
75	32, 74	condan 817	. . . . . 6 ⊢ (((𝜑 ∧ 𝐵 ≠ +∞) ∧ 𝐴 ≠ -∞) → 𝐵 ∈ ℝ)
76	43	adantlr 715	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐵 ∈ ℝ) ∧ 𝑥 ∈ ℝ⁺) → 𝐴 ≤ (𝐵 +_𝑒 𝑥))
77		simpl 482	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ ℝ ∧ 𝑥 ∈ ℝ⁺) → 𝐵 ∈ ℝ)
78		rpre 12901	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
79	78	adantl 481	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ ℝ ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ)
80		rexadd 13133	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (𝐵 +_𝑒 𝑥) = (𝐵 + 𝑥))
81	77, 79, 80	syl2anc 584	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ℝ ∧ 𝑥 ∈ ℝ⁺) → (𝐵 +_𝑒 𝑥) = (𝐵 + 𝑥))
82	81	adantll 714	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐵 ∈ ℝ) ∧ 𝑥 ∈ ℝ⁺) → (𝐵 +_𝑒 𝑥) = (𝐵 + 𝑥))
83	76, 82	breqtrd 5119	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐵 ∈ ℝ) ∧ 𝑥 ∈ ℝ⁺) → 𝐴 ≤ (𝐵 + 𝑥))
84	83	ralrimiva 3125	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ∈ ℝ) → ∀𝑥 ∈ ℝ⁺ 𝐴 ≤ (𝐵 + 𝑥))
85	1	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐵 ∈ ℝ) → 𝐴 ∈ ℝ^*)
86		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐵 ∈ ℝ) → 𝐵 ∈ ℝ)
87		xralrple 13106	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ ∀𝑥 ∈ ℝ⁺ 𝐴 ≤ (𝐵 + 𝑥)))
88	85, 86, 87	syl2anc 584	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ ∀𝑥 ∈ ℝ⁺ 𝐴 ≤ (𝐵 + 𝑥)))
89	84, 88	mpbird 257	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ∈ ℝ) → 𝐴 ≤ 𝐵)
90	22, 75, 89	syl2anc 584	. . . . 5 ⊢ (((𝜑 ∧ 𝐵 ≠ +∞) ∧ 𝐴 ≠ -∞) → 𝐴 ≤ 𝐵)
91	19, 21, 90	syl2anc 584	. . . 4 ⊢ (((𝜑 ∧ 𝐵 ≠ +∞) ∧ ¬ 𝐴 = -∞) → 𝐴 ≤ 𝐵)
92	18, 91	pm2.61dan 812	. . 3 ⊢ ((𝜑 ∧ 𝐵 ≠ +∞) → 𝐴 ≤ 𝐵)
93	9, 11, 92	syl2anc 584	. 2 ⊢ ((𝜑 ∧ ¬ 𝐵 = +∞) → 𝐴 ≤ 𝐵)
94	8, 93	pm2.61dan 812	1 ⊢ (𝜑 → 𝐴 ≤ 𝐵)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 = wceq 1541 ∈ wcel 2113 ≠ wne 2929 ∀wral 3048 class class class wbr 5093 (class class class)co 7352 ℝcr 11012 1c1 11014 + caddc 11016 +∞cpnf 11150 -∞cmnf 11151 ℝ^*cxr 11152 < clt 11153 ≤ cle 11154 ℝ⁺crp 12892 +_𝑒 cxad 13011

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 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-sep 5236 ax-nul 5246 ax-pow 5305 ax-pr 5372 ax-un 7674 ax-cnex 11069 ax-resscn 11070 ax-1cn 11071 ax-icn 11072 ax-addcl 11073 ax-addrcl 11074 ax-mulcl 11075 ax-mulrcl 11076 ax-mulcom 11077 ax-addass 11078 ax-mulass 11079 ax-distr 11080 ax-i2m1 11081 ax-1ne0 11082 ax-1rid 11083 ax-rnegex 11084 ax-rrecex 11085 ax-cnre 11086 ax-pre-lttri 11087 ax-pre-lttrn 11088 ax-pre-ltadd 11089 ax-pre-mulgt0 11090 ax-pre-sup 11091

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-nel 3034 df-ral 3049 df-rex 3058 df-rmo 3347 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4283 df-if 4475 df-pw 4551 df-sn 4576 df-pr 4578 df-op 4582 df-uni 4859 df-iun 4943 df-br 5094 df-opab 5156 df-mpt 5175 df-tr 5201 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-riota 7309 df-ov 7355 df-oprab 7356 df-mpo 7357 df-om 7803 df-1st 7927 df-2nd 7928 df-frecs 8217 df-wrecs 8248 df-recs 8297 df-rdg 8335 df-er 8628 df-en 8876 df-dom 8877 df-sdom 8878 df-sup 9333 df-inf 9334 df-pnf 11155 df-mnf 11156 df-xr 11157 df-ltxr 11158 df-le 11159 df-sub 11353 df-neg 11354 df-div 11782 df-nn 12133 df-n0 12389 df-z 12476 df-uz 12739 df-q 12849 df-rp 12893 df-xadd 13014

This theorem is referenced by: infleinf 45494 sge0xaddlem2 46556 ovnsubadd 46694

W3C validator