xsubge0 - Intuitionistic Logic Explorer

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Theorem xsubge0 9808

Description: Extended real version of subge0 8364. (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 9703	. 2 ⊢ (𝐵 ∈ ℝ^* ↔ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞))
2		0xr 7936	. . . . 5 ⊢ 0 ∈ ℝ^*
3		rexr 7935	. . . . . 6 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℝ^*)
4		xnegcl 9759	. . . . . . 7 ⊢ (𝐵 ∈ ℝ^* → -_𝑒𝐵 ∈ ℝ^*)
5		xaddcl 9787	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ -_𝑒𝐵 ∈ ℝ^) → (𝐴 +_𝑒 -_𝑒𝐵) ∈ ℝ^)
6	4, 5	sylan2 284	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐴 +_𝑒 -_𝑒𝐵) ∈ ℝ^)
7	3, 6	sylan2 284	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → (𝐴 +_𝑒 -_𝑒𝐵) ∈ ℝ^*)
8		simpr 109	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → 𝐵 ∈ ℝ)
9		xleadd1 9802	. . . . 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 1331	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → (0 ≤ (𝐴 +_𝑒 -_𝑒𝐵) ↔ (0 +_𝑒 𝐵) ≤ ((𝐴 +_𝑒 -_𝑒𝐵) +_𝑒 𝐵)))
11	3	adantl 275	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → 𝐵 ∈ ℝ^*)
12		xaddid2 9790	. . . . . 6 ⊢ (𝐵 ∈ ℝ^* → (0 +_𝑒 𝐵) = 𝐵)
13	11, 12	syl 14	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → (0 +_𝑒 𝐵) = 𝐵)
14		xnpcan 9799	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → ((𝐴 +_𝑒 -_𝑒𝐵) +_𝑒 𝐵) = 𝐴)
15	13, 14	breq12d 3989	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → ((0 +_𝑒 𝐵) ≤ ((𝐴 +_𝑒 -_𝑒𝐵) +_𝑒 𝐵) ↔ 𝐵 ≤ 𝐴))
16	10, 15	bitrd 187	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → (0 ≤ (𝐴 +_𝑒 -_𝑒𝐵) ↔ 𝐵 ≤ 𝐴))
17		pnfxr 7942	. . . . . . 7 ⊢ +∞ ∈ ℝ^*
18		xrletri3 9731	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → (𝐴 = +∞ ↔ (𝐴 ≤ +∞ ∧ +∞ ≤ 𝐴)))
19	17, 18	mpan2 422	. . . . . 6 ⊢ (𝐴 ∈ ℝ^* → (𝐴 = +∞ ↔ (𝐴 ≤ +∞ ∧ +∞ ≤ 𝐴)))
20		rexr 7935	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ^*)
21		renepnf 7937	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ +∞)
22		xaddmnf1 9775	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≠ +∞) → (𝐴 +_𝑒 -∞) = -∞)
23	20, 21, 22	syl2anc 409	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ → (𝐴 +_𝑒 -∞) = -∞)
24		mnflt0 9711	. . . . . . . . . . . . 13 ⊢ -∞ < 0
25		mnfxr 7946	. . . . . . . . . . . . . . 15 ⊢ -∞ ∈ ℝ^*
26		xrlenlt 7954	. . . . . . . . . . . . . . 15 ⊢ ((0 ∈ ℝ^* ∧ -∞ ∈ ℝ^*) → (0 ≤ -∞ ↔ ¬ -∞ < 0))
27	2, 25, 26	mp2an 423	. . . . . . . . . . . . . 14 ⊢ (0 ≤ -∞ ↔ ¬ -∞ < 0)
28	27	biimpi 119	. . . . . . . . . . . . 13 ⊢ (0 ≤ -∞ → ¬ -∞ < 0)
29	24, 28	mt2 630	. . . . . . . . . . . 12 ⊢ ¬ 0 ≤ -∞
30		breq2 3980	. . . . . . . . . . . 12 ⊢ ((𝐴 +_𝑒 -∞) = -∞ → (0 ≤ (𝐴 +_𝑒 -∞) ↔ 0 ≤ -∞))
31	29, 30	mtbiri 665	. . . . . . . . . . 11 ⊢ ((𝐴 +_𝑒 -∞) = -∞ → ¬ 0 ≤ (𝐴 +_𝑒 -∞))
32	31	pm2.21d 609	. . . . . . . . . 10 ⊢ ((𝐴 +_𝑒 -∞) = -∞ → (0 ≤ (𝐴 +_𝑒 -∞) → 𝐴 = +∞))
33	23, 32	syl 14	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → (0 ≤ (𝐴 +_𝑒 -∞) → 𝐴 = +∞))
34	33	adantl 275	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ∈ ℝ) → (0 ≤ (𝐴 +_𝑒 -∞) → 𝐴 = +∞))
35		simpr 109	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 = +∞) → 𝐴 = +∞)
36	35	a1d 22	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 = +∞) → (0 ≤ (𝐴 +_𝑒 -∞) → 𝐴 = +∞))
37		eleq1 2227	. . . . . . . . . . . 12 ⊢ (𝐴 = -∞ → (𝐴 ∈ ℝ^* ↔ -∞ ∈ ℝ^*))
38	25, 37	mpbiri 167	. . . . . . . . . . 11 ⊢ (𝐴 = -∞ → 𝐴 ∈ ℝ^*)
39		mnfnepnf 7945	. . . . . . . . . . . 12 ⊢ -∞ ≠ +∞
40		neeq1 2347	. . . . . . . . . . . 12 ⊢ (𝐴 = -∞ → (𝐴 ≠ +∞ ↔ -∞ ≠ +∞))
41	39, 40	mpbiri 167	. . . . . . . . . . 11 ⊢ (𝐴 = -∞ → 𝐴 ≠ +∞)
42	38, 41, 22	syl2anc 409	. . . . . . . . . 10 ⊢ (𝐴 = -∞ → (𝐴 +_𝑒 -∞) = -∞)
43	42, 32	syl 14	. . . . . . . . 9 ⊢ (𝐴 = -∞ → (0 ≤ (𝐴 +_𝑒 -∞) → 𝐴 = +∞))
44	43	adantl 275	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 = -∞) → (0 ≤ (𝐴 +_𝑒 -∞) → 𝐴 = +∞))
45		elxr 9703	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ^* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
46	45	biimpi 119	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ^* → (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
47	34, 36, 44, 46	mpjao3dan 1296	. . . . . . 7 ⊢ (𝐴 ∈ ℝ^* → (0 ≤ (𝐴 +_𝑒 -∞) → 𝐴 = +∞))
48		0le0 8937	. . . . . . . 8 ⊢ 0 ≤ 0
49		oveq1 5843	. . . . . . . . 9 ⊢ (𝐴 = +∞ → (𝐴 +_𝑒 -∞) = (+∞ +_𝑒 -∞))
50		pnfaddmnf 9777	. . . . . . . . 9 ⊢ (+∞ +_𝑒 -∞) = 0
51	49, 50	eqtrdi 2213	. . . . . . . 8 ⊢ (𝐴 = +∞ → (𝐴 +_𝑒 -∞) = 0)
52	48, 51	breqtrrid 4014	. . . . . . 7 ⊢ (𝐴 = +∞ → 0 ≤ (𝐴 +_𝑒 -∞))
53	47, 52	impbid1 141	. . . . . 6 ⊢ (𝐴 ∈ ℝ^* → (0 ≤ (𝐴 +_𝑒 -∞) ↔ 𝐴 = +∞))
54		pnfge 9716	. . . . . . 7 ⊢ (𝐴 ∈ ℝ^* → 𝐴 ≤ +∞)
55	54	biantrurd 303	. . . . . 6 ⊢ (𝐴 ∈ ℝ^* → (+∞ ≤ 𝐴 ↔ (𝐴 ≤ +∞ ∧ +∞ ≤ 𝐴)))
56	19, 53, 55	3bitr4d 219	. . . . 5 ⊢ (𝐴 ∈ ℝ^* → (0 ≤ (𝐴 +_𝑒 -∞) ↔ +∞ ≤ 𝐴))
57	56	adantr 274	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 = +∞) → (0 ≤ (𝐴 +_𝑒 -∞) ↔ +∞ ≤ 𝐴))
58		xnegeq 9754	. . . . . . . 8 ⊢ (𝐵 = +∞ → -_𝑒𝐵 = -_𝑒+∞)
59		xnegpnf 9755	. . . . . . . 8 ⊢ -_𝑒+∞ = -∞
60	58, 59	eqtrdi 2213	. . . . . . 7 ⊢ (𝐵 = +∞ → -_𝑒𝐵 = -∞)
61	60	adantl 275	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 = +∞) → -_𝑒𝐵 = -∞)
62	61	oveq2d 5852	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 = +∞) → (𝐴 +_𝑒 -_𝑒𝐵) = (𝐴 +_𝑒 -∞))
63	62	breq2d 3988	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 = +∞) → (0 ≤ (𝐴 +_𝑒 -_𝑒𝐵) ↔ 0 ≤ (𝐴 +_𝑒 -∞)))
64		breq1 3979	. . . . 5 ⊢ (𝐵 = +∞ → (𝐵 ≤ 𝐴 ↔ +∞ ≤ 𝐴))
65	64	adantl 275	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 = +∞) → (𝐵 ≤ 𝐴 ↔ +∞ ≤ 𝐴))
66	57, 63, 65	3bitr4d 219	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 = +∞) → (0 ≤ (𝐴 +_𝑒 -_𝑒𝐵) ↔ 𝐵 ≤ 𝐴))
67		oveq1 5843	. . . . . . . . . 10 ⊢ (𝐴 = -∞ → (𝐴 +_𝑒 +∞) = (-∞ +_𝑒 +∞))
68		mnfaddpnf 9778	. . . . . . . . . 10 ⊢ (-∞ +_𝑒 +∞) = 0
69	67, 68	eqtrdi 2213	. . . . . . . . 9 ⊢ (𝐴 = -∞ → (𝐴 +_𝑒 +∞) = 0)
70	69	adantl 275	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 = -∞) → (𝐴 +_𝑒 +∞) = 0)
71	48, 70	breqtrrid 4014	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 = -∞) → 0 ≤ (𝐴 +_𝑒 +∞))
72		df-ne 2335	. . . . . . . 8 ⊢ (𝐴 ≠ -∞ ↔ ¬ 𝐴 = -∞)
73		0lepnf 9717	. . . . . . . . 9 ⊢ 0 ≤ +∞
74		xaddpnf1 9773	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≠ -∞) → (𝐴 +_𝑒 +∞) = +∞)
75	73, 74	breqtrrid 4014	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≠ -∞) → 0 ≤ (𝐴 +_𝑒 +∞))
76	72, 75	sylan2br 286	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ ¬ 𝐴 = -∞) → 0 ≤ (𝐴 +_𝑒 +∞))
77		xrmnfdc 9770	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ^* → DECID 𝐴 = -∞)
78		exmiddc 826	. . . . . . . 8 ⊢ (DECID 𝐴 = -∞ → (𝐴 = -∞ ∨ ¬ 𝐴 = -∞))
79	77, 78	syl 14	. . . . . . 7 ⊢ (𝐴 ∈ ℝ^* → (𝐴 = -∞ ∨ ¬ 𝐴 = -∞))
80	71, 76, 79	mpjaodan 788	. . . . . 6 ⊢ (𝐴 ∈ ℝ^* → 0 ≤ (𝐴 +_𝑒 +∞))
81		mnfle 9719	. . . . . 6 ⊢ (𝐴 ∈ ℝ^* → -∞ ≤ 𝐴)
82	80, 81	2thd 174	. . . . 5 ⊢ (𝐴 ∈ ℝ^* → (0 ≤ (𝐴 +_𝑒 +∞) ↔ -∞ ≤ 𝐴))
83	82	adantr 274	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 = -∞) → (0 ≤ (𝐴 +_𝑒 +∞) ↔ -∞ ≤ 𝐴))
84		xnegeq 9754	. . . . . . . 8 ⊢ (𝐵 = -∞ → -_𝑒𝐵 = -_𝑒-∞)
85		xnegmnf 9756	. . . . . . . 8 ⊢ -_𝑒-∞ = +∞
86	84, 85	eqtrdi 2213	. . . . . . 7 ⊢ (𝐵 = -∞ → -_𝑒𝐵 = +∞)
87	86	adantl 275	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 = -∞) → -_𝑒𝐵 = +∞)
88	87	oveq2d 5852	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 = -∞) → (𝐴 +_𝑒 -_𝑒𝐵) = (𝐴 +_𝑒 +∞))
89	88	breq2d 3988	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 = -∞) → (0 ≤ (𝐴 +_𝑒 -_𝑒𝐵) ↔ 0 ≤ (𝐴 +_𝑒 +∞)))
90		breq1 3979	. . . . 5 ⊢ (𝐵 = -∞ → (𝐵 ≤ 𝐴 ↔ -∞ ≤ 𝐴))
91	90	adantl 275	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 = -∞) → (𝐵 ≤ 𝐴 ↔ -∞ ≤ 𝐴))
92	83, 89, 91	3bitr4d 219	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 = -∞) → (0 ≤ (𝐴 +_𝑒 -_𝑒𝐵) ↔ 𝐵 ≤ 𝐴))
93	16, 66, 92	3jaodan 1295	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → (0 ≤ (𝐴 +_𝑒 -_𝑒𝐵) ↔ 𝐵 ≤ 𝐴))
94	1, 93	sylan2b 285	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (0 ≤ (𝐴 +_𝑒 -_𝑒𝐵) ↔ 𝐵 ≤ 𝐴))

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ↔ wb 104 ∨ wo 698 DECID wdc 824 ∨ w3o 966 = wceq 1342 ∈ wcel 2135 ≠ wne 2334 class class class wbr 3976 (class class class)co 5836 ℝcr 7743 0cc0 7744 +∞cpnf 7921 -∞cmnf 7922 ℝ^*cxr 7923 < clt 7924 ≤ cle 7925 -_𝑒cxne 9696 +_𝑒 cxad 9697

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-addcom 7844 ax-addass 7846 ax-distr 7848 ax-i2m1 7849 ax-0id 7852 ax-rnegex 7853 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-apti 7859 ax-pre-ltadd 7860

This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-if 3516 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-xneg 9699 df-xadd 9700

This theorem is referenced by: ssblps 12966 ssbl 12967

W3C validator