xsubge0 - Intuitionistic Logic Explorer

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Theorem xsubge0 9950

Description: Extended real version of subge0 8496. (Contributed by Mario Carneiro, 24-Aug-2015.)

**Assertion**
Ref	Expression
xsubge0	⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (0 ≤ (𝐴 +_𝑒 -_𝑒𝐵) ↔ 𝐵 ≤ 𝐴))

**Proof of Theorem xsubge0**
Step	Hyp	Ref	Expression
1		elxr 9845	. 2 ⊢ (𝐵 ∈ ℝ^* ↔ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞))
2		0xr 8068	. . . . 5 ⊢ 0 ∈ ℝ^*
3		rexr 8067	. . . . . 6 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℝ^*)
4		xnegcl 9901	. . . . . . 7 ⊢ (𝐵 ∈ ℝ^* → -_𝑒𝐵 ∈ ℝ^*)
5		xaddcl 9929	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ -_𝑒𝐵 ∈ ℝ^) → (𝐴 +_𝑒 -_𝑒𝐵) ∈ ℝ^)
6	4, 5	sylan2 286	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐴 +_𝑒 -_𝑒𝐵) ∈ ℝ^)
7	3, 6	sylan2 286	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → (𝐴 +_𝑒 -_𝑒𝐵) ∈ ℝ^*)
8		simpr 110	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → 𝐵 ∈ ℝ)
9		xleadd1 9944	. . . . 5 ⊢ ((0 ∈ ℝ^* ∧ (𝐴 +_𝑒 -_𝑒𝐵) ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → (0 ≤ (𝐴 +_𝑒 -_𝑒𝐵) ↔ (0 +_𝑒 𝐵) ≤ ((𝐴 +_𝑒 -_𝑒𝐵) +_𝑒 𝐵)))
10	2, 7, 8, 9	mp3an2i 1353	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → (0 ≤ (𝐴 +_𝑒 -_𝑒𝐵) ↔ (0 +_𝑒 𝐵) ≤ ((𝐴 +_𝑒 -_𝑒𝐵) +_𝑒 𝐵)))
11	3	adantl 277	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → 𝐵 ∈ ℝ^*)
12		xaddid2 9932	. . . . . 6 ⊢ (𝐵 ∈ ℝ^* → (0 +_𝑒 𝐵) = 𝐵)
13	11, 12	syl 14	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → (0 +_𝑒 𝐵) = 𝐵)
14		xnpcan 9941	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → ((𝐴 +_𝑒 -_𝑒𝐵) +_𝑒 𝐵) = 𝐴)
15	13, 14	breq12d 4043	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → ((0 +_𝑒 𝐵) ≤ ((𝐴 +_𝑒 -_𝑒𝐵) +_𝑒 𝐵) ↔ 𝐵 ≤ 𝐴))
16	10, 15	bitrd 188	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → (0 ≤ (𝐴 +_𝑒 -_𝑒𝐵) ↔ 𝐵 ≤ 𝐴))
17		pnfxr 8074	. . . . . . 7 ⊢ +∞ ∈ ℝ^*
18		xrletri3 9873	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → (𝐴 = +∞ ↔ (𝐴 ≤ +∞ ∧ +∞ ≤ 𝐴)))
19	17, 18	mpan2 425	. . . . . 6 ⊢ (𝐴 ∈ ℝ^* → (𝐴 = +∞ ↔ (𝐴 ≤ +∞ ∧ +∞ ≤ 𝐴)))
20		rexr 8067	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ^*)
21		renepnf 8069	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ +∞)
22		xaddmnf1 9917	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≠ +∞) → (𝐴 +_𝑒 -∞) = -∞)
23	20, 21, 22	syl2anc 411	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ → (𝐴 +_𝑒 -∞) = -∞)
24		mnflt0 9853	. . . . . . . . . . . . 13 ⊢ -∞ < 0
25		mnfxr 8078	. . . . . . . . . . . . . . 15 ⊢ -∞ ∈ ℝ^*
26		xrlenlt 8086	. . . . . . . . . . . . . . 15 ⊢ ((0 ∈ ℝ^* ∧ -∞ ∈ ℝ^*) → (0 ≤ -∞ ↔ ¬ -∞ < 0))
27	2, 25, 26	mp2an 426	. . . . . . . . . . . . . 14 ⊢ (0 ≤ -∞ ↔ ¬ -∞ < 0)
28	27	biimpi 120	. . . . . . . . . . . . 13 ⊢ (0 ≤ -∞ → ¬ -∞ < 0)
29	24, 28	mt2 641	. . . . . . . . . . . 12 ⊢ ¬ 0 ≤ -∞
30		breq2 4034	. . . . . . . . . . . 12 ⊢ ((𝐴 +_𝑒 -∞) = -∞ → (0 ≤ (𝐴 +_𝑒 -∞) ↔ 0 ≤ -∞))
31	29, 30	mtbiri 676	. . . . . . . . . . 11 ⊢ ((𝐴 +_𝑒 -∞) = -∞ → ¬ 0 ≤ (𝐴 +_𝑒 -∞))
32	31	pm2.21d 620	. . . . . . . . . 10 ⊢ ((𝐴 +_𝑒 -∞) = -∞ → (0 ≤ (𝐴 +_𝑒 -∞) → 𝐴 = +∞))
33	23, 32	syl 14	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → (0 ≤ (𝐴 +_𝑒 -∞) → 𝐴 = +∞))
34	33	adantl 277	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ∈ ℝ) → (0 ≤ (𝐴 +_𝑒 -∞) → 𝐴 = +∞))
35		simpr 110	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 = +∞) → 𝐴 = +∞)
36	35	a1d 22	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 = +∞) → (0 ≤ (𝐴 +_𝑒 -∞) → 𝐴 = +∞))
37		eleq1 2256	. . . . . . . . . . . 12 ⊢ (𝐴 = -∞ → (𝐴 ∈ ℝ^* ↔ -∞ ∈ ℝ^*))
38	25, 37	mpbiri 168	. . . . . . . . . . 11 ⊢ (𝐴 = -∞ → 𝐴 ∈ ℝ^*)
39		mnfnepnf 8077	. . . . . . . . . . . 12 ⊢ -∞ ≠ +∞
40		neeq1 2377	. . . . . . . . . . . 12 ⊢ (𝐴 = -∞ → (𝐴 ≠ +∞ ↔ -∞ ≠ +∞))
41	39, 40	mpbiri 168	. . . . . . . . . . 11 ⊢ (𝐴 = -∞ → 𝐴 ≠ +∞)
42	38, 41, 22	syl2anc 411	. . . . . . . . . 10 ⊢ (𝐴 = -∞ → (𝐴 +_𝑒 -∞) = -∞)
43	42, 32	syl 14	. . . . . . . . 9 ⊢ (𝐴 = -∞ → (0 ≤ (𝐴 +_𝑒 -∞) → 𝐴 = +∞))
44	43	adantl 277	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 = -∞) → (0 ≤ (𝐴 +_𝑒 -∞) → 𝐴 = +∞))
45		elxr 9845	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ^* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
46	45	biimpi 120	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ^* → (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
47	34, 36, 44, 46	mpjao3dan 1318	. . . . . . 7 ⊢ (𝐴 ∈ ℝ^* → (0 ≤ (𝐴 +_𝑒 -∞) → 𝐴 = +∞))
48		0le0 9073	. . . . . . . 8 ⊢ 0 ≤ 0
49		oveq1 5926	. . . . . . . . 9 ⊢ (𝐴 = +∞ → (𝐴 +_𝑒 -∞) = (+∞ +_𝑒 -∞))
50		pnfaddmnf 9919	. . . . . . . . 9 ⊢ (+∞ +_𝑒 -∞) = 0
51	49, 50	eqtrdi 2242	. . . . . . . 8 ⊢ (𝐴 = +∞ → (𝐴 +_𝑒 -∞) = 0)
52	48, 51	breqtrrid 4068	. . . . . . 7 ⊢ (𝐴 = +∞ → 0 ≤ (𝐴 +_𝑒 -∞))
53	47, 52	impbid1 142	. . . . . 6 ⊢ (𝐴 ∈ ℝ^* → (0 ≤ (𝐴 +_𝑒 -∞) ↔ 𝐴 = +∞))
54		pnfge 9858	. . . . . . 7 ⊢ (𝐴 ∈ ℝ^* → 𝐴 ≤ +∞)
55	54	biantrurd 305	. . . . . 6 ⊢ (𝐴 ∈ ℝ^* → (+∞ ≤ 𝐴 ↔ (𝐴 ≤ +∞ ∧ +∞ ≤ 𝐴)))
56	19, 53, 55	3bitr4d 220	. . . . 5 ⊢ (𝐴 ∈ ℝ^* → (0 ≤ (𝐴 +_𝑒 -∞) ↔ +∞ ≤ 𝐴))
57	56	adantr 276	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 = +∞) → (0 ≤ (𝐴 +_𝑒 -∞) ↔ +∞ ≤ 𝐴))
58		xnegeq 9896	. . . . . . . 8 ⊢ (𝐵 = +∞ → -_𝑒𝐵 = -_𝑒+∞)
59		xnegpnf 9897	. . . . . . . 8 ⊢ -_𝑒+∞ = -∞
60	58, 59	eqtrdi 2242	. . . . . . 7 ⊢ (𝐵 = +∞ → -_𝑒𝐵 = -∞)
61	60	adantl 277	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 = +∞) → -_𝑒𝐵 = -∞)
62	61	oveq2d 5935	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 = +∞) → (𝐴 +_𝑒 -_𝑒𝐵) = (𝐴 +_𝑒 -∞))
63	62	breq2d 4042	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 = +∞) → (0 ≤ (𝐴 +_𝑒 -_𝑒𝐵) ↔ 0 ≤ (𝐴 +_𝑒 -∞)))
64		breq1 4033	. . . . 5 ⊢ (𝐵 = +∞ → (𝐵 ≤ 𝐴 ↔ +∞ ≤ 𝐴))
65	64	adantl 277	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 = +∞) → (𝐵 ≤ 𝐴 ↔ +∞ ≤ 𝐴))
66	57, 63, 65	3bitr4d 220	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 = +∞) → (0 ≤ (𝐴 +_𝑒 -_𝑒𝐵) ↔ 𝐵 ≤ 𝐴))
67		oveq1 5926	. . . . . . . . . 10 ⊢ (𝐴 = -∞ → (𝐴 +_𝑒 +∞) = (-∞ +_𝑒 +∞))
68		mnfaddpnf 9920	. . . . . . . . . 10 ⊢ (-∞ +_𝑒 +∞) = 0
69	67, 68	eqtrdi 2242	. . . . . . . . 9 ⊢ (𝐴 = -∞ → (𝐴 +_𝑒 +∞) = 0)
70	69	adantl 277	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 = -∞) → (𝐴 +_𝑒 +∞) = 0)
71	48, 70	breqtrrid 4068	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 = -∞) → 0 ≤ (𝐴 +_𝑒 +∞))
72		df-ne 2365	. . . . . . . 8 ⊢ (𝐴 ≠ -∞ ↔ ¬ 𝐴 = -∞)
73		0lepnf 9859	. . . . . . . . 9 ⊢ 0 ≤ +∞
74		xaddpnf1 9915	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≠ -∞) → (𝐴 +_𝑒 +∞) = +∞)
75	73, 74	breqtrrid 4068	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≠ -∞) → 0 ≤ (𝐴 +_𝑒 +∞))
76	72, 75	sylan2br 288	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ ¬ 𝐴 = -∞) → 0 ≤ (𝐴 +_𝑒 +∞))
77		xrmnfdc 9912	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ^* → DECID 𝐴 = -∞)
78		exmiddc 837	. . . . . . . 8 ⊢ (DECID 𝐴 = -∞ → (𝐴 = -∞ ∨ ¬ 𝐴 = -∞))
79	77, 78	syl 14	. . . . . . 7 ⊢ (𝐴 ∈ ℝ^* → (𝐴 = -∞ ∨ ¬ 𝐴 = -∞))
80	71, 76, 79	mpjaodan 799	. . . . . 6 ⊢ (𝐴 ∈ ℝ^* → 0 ≤ (𝐴 +_𝑒 +∞))
81		mnfle 9861	. . . . . 6 ⊢ (𝐴 ∈ ℝ^* → -∞ ≤ 𝐴)
82	80, 81	2thd 175	. . . . 5 ⊢ (𝐴 ∈ ℝ^* → (0 ≤ (𝐴 +_𝑒 +∞) ↔ -∞ ≤ 𝐴))
83	82	adantr 276	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 = -∞) → (0 ≤ (𝐴 +_𝑒 +∞) ↔ -∞ ≤ 𝐴))
84		xnegeq 9896	. . . . . . . 8 ⊢ (𝐵 = -∞ → -_𝑒𝐵 = -_𝑒-∞)
85		xnegmnf 9898	. . . . . . . 8 ⊢ -_𝑒-∞ = +∞
86	84, 85	eqtrdi 2242	. . . . . . 7 ⊢ (𝐵 = -∞ → -_𝑒𝐵 = +∞)
87	86	adantl 277	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 = -∞) → -_𝑒𝐵 = +∞)
88	87	oveq2d 5935	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 = -∞) → (𝐴 +_𝑒 -_𝑒𝐵) = (𝐴 +_𝑒 +∞))
89	88	breq2d 4042	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 = -∞) → (0 ≤ (𝐴 +_𝑒 -_𝑒𝐵) ↔ 0 ≤ (𝐴 +_𝑒 +∞)))
90		breq1 4033	. . . . 5 ⊢ (𝐵 = -∞ → (𝐵 ≤ 𝐴 ↔ -∞ ≤ 𝐴))
91	90	adantl 277	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 = -∞) → (𝐵 ≤ 𝐴 ↔ -∞ ≤ 𝐴))
92	83, 89, 91	3bitr4d 220	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 = -∞) → (0 ≤ (𝐴 +_𝑒 -_𝑒𝐵) ↔ 𝐵 ≤ 𝐴))
93	16, 66, 92	3jaodan 1317	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → (0 ≤ (𝐴 +_𝑒 -_𝑒𝐵) ↔ 𝐵 ≤ 𝐴))
94	1, 93	sylan2b 287	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (0 ≤ (𝐴 +_𝑒 -_𝑒𝐵) ↔ 𝐵 ≤ 𝐴))

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 709 DECID wdc 835 ∨ w3o 979 = wceq 1364 ∈ wcel 2164 ≠ wne 2364 class class class wbr 4030 (class class class)co 5919 ℝcr 7873 0cc0 7874 +∞cpnf 8053 -∞cmnf 8054 ℝ^*cxr 8055 < clt 8056 ≤ cle 8057 -_𝑒cxne 9838 +_𝑒 cxad 9839

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-sep 4148 ax-pow 4204 ax-pr 4239 ax-un 4465 ax-setind 4570 ax-cnex 7965 ax-resscn 7966 ax-1cn 7967 ax-1re 7968 ax-icn 7969 ax-addcl 7970 ax-addrcl 7971 ax-mulcl 7972 ax-addcom 7974 ax-addass 7976 ax-distr 7978 ax-i2m1 7979 ax-0id 7982 ax-rnegex 7983 ax-cnre 7985 ax-pre-ltirr 7986 ax-pre-apti 7989 ax-pre-ltadd 7990

This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rab 2481 df-v 2762 df-sbc 2987 df-csb 3082 df-dif 3156 df-un 3158 df-in 3160 df-ss 3167 df-if 3559 df-pw 3604 df-sn 3625 df-pr 3626 df-op 3628 df-uni 3837 df-iun 3915 df-br 4031 df-opab 4092 df-mpt 4093 df-id 4325 df-xp 4666 df-rel 4667 df-cnv 4668 df-co 4669 df-dm 4670 df-rn 4671 df-res 4672 df-ima 4673 df-iota 5216 df-fun 5257 df-fn 5258 df-f 5259 df-fv 5263 df-riota 5874 df-ov 5922 df-oprab 5923 df-mpo 5924 df-1st 6195 df-2nd 6196 df-pnf 8058 df-mnf 8059 df-xr 8060 df-ltxr 8061 df-le 8062 df-sub 8194 df-neg 8195 df-xneg 9841 df-xadd 9842

This theorem is referenced by: ssblps 14604 ssbl 14605

W3C validator