xrlexaddrp - Mathbox for Glauco Siliprandi

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Theorem xrlexaddrp 44869

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 13145	. . . . 5 ⊢ (𝐴 ∈ ℝ^* → 𝐴 ≤ +∞)
3	1, 2	syl 17	. . . 4 ⊢ (𝜑 → 𝐴 ≤ +∞)
4	3	adantr 479	. . 3 ⊢ ((𝜑 ∧ 𝐵 = +∞) → 𝐴 ≤ +∞)
5		id 22	. . . . 5 ⊢ (𝐵 = +∞ → 𝐵 = +∞)
6	5	eqcomd 2731	. . . 4 ⊢ (𝐵 = +∞ → +∞ = 𝐵)
7	6	adantl 480	. . 3 ⊢ ((𝜑 ∧ 𝐵 = +∞) → +∞ = 𝐵)
8	4, 7	breqtrd 5175	. 2 ⊢ ((𝜑 ∧ 𝐵 = +∞) → 𝐴 ≤ 𝐵)
9		simpl 481	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐵 = +∞) → 𝜑)
10		neqne 2937	. . . 4 ⊢ (¬ 𝐵 = +∞ → 𝐵 ≠ +∞)
11	10	adantl 480	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐵 = +∞) → 𝐵 ≠ +∞)
12		simpr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 = -∞) → 𝐴 = -∞)
13		xrlexaddrp.2	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
14		mnfle 13149	. . . . . . . 8 ⊢ (𝐵 ∈ ℝ^* → -∞ ≤ 𝐵)
15	13, 14	syl 17	. . . . . . 7 ⊢ (𝜑 → -∞ ≤ 𝐵)
16	15	adantr 479	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 = -∞) → -∞ ≤ 𝐵)
17	12, 16	eqbrtrd 5171	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 = -∞) → 𝐴 ≤ 𝐵)
18	17	adantlr 713	. . . 4 ⊢ (((𝜑 ∧ 𝐵 ≠ +∞) ∧ 𝐴 = -∞) → 𝐴 ≤ 𝐵)
19		simpl 481	. . . . 5 ⊢ (((𝜑 ∧ 𝐵 ≠ +∞) ∧ ¬ 𝐴 = -∞) → (𝜑 ∧ 𝐵 ≠ +∞))
20		neqne 2937	. . . . . 6 ⊢ (¬ 𝐴 = -∞ → 𝐴 ≠ -∞)
21	20	adantl 480	. . . . 5 ⊢ (((𝜑 ∧ 𝐵 ≠ +∞) ∧ ¬ 𝐴 = -∞) → 𝐴 ≠ -∞)
22		simpll 765	. . . . . 6 ⊢ (((𝜑 ∧ 𝐵 ≠ +∞) ∧ 𝐴 ≠ -∞) → 𝜑)
23	13	adantr 479	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐵 ≠ +∞) → 𝐵 ∈ ℝ^*)
24		simpr 483	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐵 ≠ +∞) → 𝐵 ≠ +∞)
25	23, 24	jca 510	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐵 ≠ +∞) → (𝐵 ∈ ℝ^* ∧ 𝐵 ≠ +∞))
26		xrnepnf 13133	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ +∞) ↔ (𝐵 ∈ ℝ ∨ 𝐵 = -∞))
27	25, 26	sylib 217	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐵 ≠ +∞) → (𝐵 ∈ ℝ ∨ 𝐵 = -∞))
28	27	adantr 479	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐵 ≠ +∞) ∧ ¬ 𝐵 ∈ ℝ) → (𝐵 ∈ ℝ ∨ 𝐵 = -∞))
29		simpr 483	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐵 ≠ +∞) ∧ ¬ 𝐵 ∈ ℝ) → ¬ 𝐵 ∈ ℝ)
30		pm2.53 849	. . . . . . . . 9 ⊢ ((𝐵 ∈ ℝ ∨ 𝐵 = -∞) → (¬ 𝐵 ∈ ℝ → 𝐵 = -∞))
31	28, 29, 30	sylc 65	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐵 ≠ +∞) ∧ ¬ 𝐵 ∈ ℝ) → 𝐵 = -∞)
32	31	adantlr 713	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐵 ≠ +∞) ∧ 𝐴 ≠ -∞) ∧ ¬ 𝐵 ∈ ℝ) → 𝐵 = -∞)
33		id 22	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝜑)
34		1rp 13013	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℝ⁺
35	34	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → 1 ∈ ℝ⁺)
36		1re 11246	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ ℝ
37	36	elexi 3482	. . . . . . . . . . . . . 14 ⊢ 1 ∈ V
38		eleq1 2813	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 1 → (𝑥 ∈ ℝ⁺ ↔ 1 ∈ ℝ⁺))
39	38	anbi2d 628	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 1 → ((𝜑 ∧ 𝑥 ∈ ℝ⁺) ↔ (𝜑 ∧ 1 ∈ ℝ⁺)))
40		oveq2 7427	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 1 → (𝐵 +_𝑒 𝑥) = (𝐵 +_𝑒 1))
41	40	breq2d 5161	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 1 → (𝐴 ≤ (𝐵 +_𝑒 𝑥) ↔ 𝐴 ≤ (𝐵 +_𝑒 1)))
42	39, 41	imbi12d 343	. . . . . . . . . . . . . 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 3536	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 1 ∈ ℝ⁺) → 𝐴 ≤ (𝐵 +_𝑒 1))
45	33, 35, 44	syl2anc 582	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ≤ (𝐵 +_𝑒 1))
46	45	ad2antrr 724	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 ≠ -∞) ∧ 𝐵 = -∞) → 𝐴 ≤ (𝐵 +_𝑒 1))
47		oveq1 7426	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 = -∞ → (𝐵 +_𝑒 1) = (-∞ +_𝑒 1))
48		1xr 11305	. . . . . . . . . . . . . . . . . 18 ⊢ 1 ∈ ℝ^*
49		ltpnf 13135	. . . . . . . . . . . . . . . . . . . 20 ⊢ (1 ∈ ℝ → 1 < +∞)
50	36, 49	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 < +∞
51	36, 50	ltneii 11359	. . . . . . . . . . . . . . . . . 18 ⊢ 1 ≠ +∞
52		xaddmnf2 13243	. . . . . . . . . . . . . . . . . 18 ⊢ ((1 ∈ ℝ^* ∧ 1 ≠ +∞) → (-∞ +_𝑒 1) = -∞)
53	48, 51, 52	mp2an 690	. . . . . . . . . . . . . . . . 17 ⊢ (-∞ +_𝑒 1) = -∞
54	53	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 = -∞ → (-∞ +_𝑒 1) = -∞)
55	47, 54	eqtr2d 2766	. . . . . . . . . . . . . . 15 ⊢ (𝐵 = -∞ → -∞ = (𝐵 +_𝑒 1))
56	55	adantl 480	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐴 ≠ -∞) ∧ 𝐵 = -∞) → -∞ = (𝐵 +_𝑒 1))
57	56	eqcomd 2731	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐴 ≠ -∞) ∧ 𝐵 = -∞) → (𝐵 +_𝑒 1) = -∞)
58	1	adantr 479	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐴 ≠ -∞) → 𝐴 ∈ ℝ^*)
59		simpr 483	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐴 ≠ -∞) → 𝐴 ≠ -∞)
60		nemnftgtmnft 44861	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≠ -∞) → -∞ < 𝐴)
61	58, 59, 60	syl2anc 582	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐴 ≠ -∞) → -∞ < 𝐴)
62	61	adantr 479	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐴 ≠ -∞) ∧ 𝐵 = -∞) → -∞ < 𝐴)
63	57, 62	eqbrtrd 5171	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐴 ≠ -∞) ∧ 𝐵 = -∞) → (𝐵 +_𝑒 1) < 𝐴)
64	13	ad2antrr 724	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐴 ≠ -∞) ∧ 𝐵 = -∞) → 𝐵 ∈ ℝ^*)
65	48	a1i 11	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐴 ≠ -∞) ∧ 𝐵 = -∞) → 1 ∈ ℝ^*)
66	64, 65	xaddcld 13315	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐴 ≠ -∞) ∧ 𝐵 = -∞) → (𝐵 +_𝑒 1) ∈ ℝ^*)
67	1	ad2antrr 724	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐴 ≠ -∞) ∧ 𝐵 = -∞) → 𝐴 ∈ ℝ^*)
68		xrltnle 11313	. . . . . . . . . . . . 13 ⊢ (((𝐵 +_𝑒 1) ∈ ℝ^* ∧ 𝐴 ∈ ℝ^*) → ((𝐵 +_𝑒 1) < 𝐴 ↔ ¬ 𝐴 ≤ (𝐵 +_𝑒 1)))
69	66, 67, 68	syl2anc 582	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐴 ≠ -∞) ∧ 𝐵 = -∞) → ((𝐵 +_𝑒 1) < 𝐴 ↔ ¬ 𝐴 ≤ (𝐵 +_𝑒 1)))
70	63, 69	mpbid 231	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 ≠ -∞) ∧ 𝐵 = -∞) → ¬ 𝐴 ≤ (𝐵 +_𝑒 1))
71	46, 70	pm2.65da 815	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 ≠ -∞) → ¬ 𝐵 = -∞)
72	71	neqned 2936	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ≠ -∞) → 𝐵 ≠ -∞)
73	72	ad4ant13 749	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐵 ≠ +∞) ∧ 𝐴 ≠ -∞) ∧ ¬ 𝐵 ∈ ℝ) → 𝐵 ≠ -∞)
74	73	neneqd 2934	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐵 ≠ +∞) ∧ 𝐴 ≠ -∞) ∧ ¬ 𝐵 ∈ ℝ) → ¬ 𝐵 = -∞)
75	32, 74	condan 816	. . . . . 6 ⊢ (((𝜑 ∧ 𝐵 ≠ +∞) ∧ 𝐴 ≠ -∞) → 𝐵 ∈ ℝ)
76	43	adantlr 713	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐵 ∈ ℝ) ∧ 𝑥 ∈ ℝ⁺) → 𝐴 ≤ (𝐵 +_𝑒 𝑥))
77		simpl 481	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ ℝ ∧ 𝑥 ∈ ℝ⁺) → 𝐵 ∈ ℝ)
78		rpre 13017	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
79	78	adantl 480	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ ℝ ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ)
80		rexadd 13246	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (𝐵 +_𝑒 𝑥) = (𝐵 + 𝑥))
81	77, 79, 80	syl2anc 582	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ℝ ∧ 𝑥 ∈ ℝ⁺) → (𝐵 +_𝑒 𝑥) = (𝐵 + 𝑥))
82	81	adantll 712	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐵 ∈ ℝ) ∧ 𝑥 ∈ ℝ⁺) → (𝐵 +_𝑒 𝑥) = (𝐵 + 𝑥))
83	76, 82	breqtrd 5175	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐵 ∈ ℝ) ∧ 𝑥 ∈ ℝ⁺) → 𝐴 ≤ (𝐵 + 𝑥))
84	83	ralrimiva 3135	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ∈ ℝ) → ∀𝑥 ∈ ℝ⁺ 𝐴 ≤ (𝐵 + 𝑥))
85	1	adantr 479	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐵 ∈ ℝ) → 𝐴 ∈ ℝ^*)
86		simpr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐵 ∈ ℝ) → 𝐵 ∈ ℝ)
87		xralrple 13219	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ ∀𝑥 ∈ ℝ⁺ 𝐴 ≤ (𝐵 + 𝑥)))
88	85, 86, 87	syl2anc 582	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ ∀𝑥 ∈ ℝ⁺ 𝐴 ≤ (𝐵 + 𝑥)))
89	84, 88	mpbird 256	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ∈ ℝ) → 𝐴 ≤ 𝐵)
90	22, 75, 89	syl2anc 582	. . . . 5 ⊢ (((𝜑 ∧ 𝐵 ≠ +∞) ∧ 𝐴 ≠ -∞) → 𝐴 ≤ 𝐵)
91	19, 21, 90	syl2anc 582	. . . 4 ⊢ (((𝜑 ∧ 𝐵 ≠ +∞) ∧ ¬ 𝐴 = -∞) → 𝐴 ≤ 𝐵)
92	18, 91	pm2.61dan 811	. . 3 ⊢ ((𝜑 ∧ 𝐵 ≠ +∞) → 𝐴 ≤ 𝐵)
93	9, 11, 92	syl2anc 582	. 2 ⊢ ((𝜑 ∧ ¬ 𝐵 = +∞) → 𝐴 ≤ 𝐵)
94	8, 93	pm2.61dan 811	1 ⊢ (𝜑 → 𝐴 ≤ 𝐵)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 ∨ wo 845 = wceq 1533 ∈ wcel 2098 ≠ wne 2929 ∀wral 3050 class class class wbr 5149 (class class class)co 7419 ℝcr 11139 1c1 11141 + caddc 11143 +∞cpnf 11277 -∞cmnf 11278 ℝ^*cxr 11279 < clt 11280 ≤ cle 11281 ℝ⁺crp 13009 +_𝑒 cxad 13125

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 ax-pre-sup 11218

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-sup 9467 df-inf 9468 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-div 11904 df-nn 12246 df-n0 12506 df-z 12592 df-uz 12856 df-q 12966 df-rp 13010 df-xadd 13128

This theorem is referenced by: infleinf 44889 sge0xaddlem2 45957 ovnsubadd 46095

W3C validator