xsubge0 - Intuitionistic Logic Explorer

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Theorem xsubge0 9868

Description: Extended real version of subge0 8422. (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 9763	. 2 ⊢ (𝐵 ∈ ℝ^* ↔ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞))
2		0xr 7994	. . . . 5 ⊢ 0 ∈ ℝ^*
3		rexr 7993	. . . . . 6 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℝ^*)
4		xnegcl 9819	. . . . . . 7 ⊢ (𝐵 ∈ ℝ^* → -_𝑒𝐵 ∈ ℝ^*)
5		xaddcl 9847	. . . . . . 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 9862	. . . . 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 1342	. . . 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 9850	. . . . . 6 ⊢ (𝐵 ∈ ℝ^* → (0 +_𝑒 𝐵) = 𝐵)
13	11, 12	syl 14	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → (0 +_𝑒 𝐵) = 𝐵)
14		xnpcan 9859	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → ((𝐴 +_𝑒 -_𝑒𝐵) +_𝑒 𝐵) = 𝐴)
15	13, 14	breq12d 4013	. . . 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 8000	. . . . . . 7 ⊢ +∞ ∈ ℝ^*
18		xrletri3 9791	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → (𝐴 = +∞ ↔ (𝐴 ≤ +∞ ∧ +∞ ≤ 𝐴)))
19	17, 18	mpan2 425	. . . . . 6 ⊢ (𝐴 ∈ ℝ^* → (𝐴 = +∞ ↔ (𝐴 ≤ +∞ ∧ +∞ ≤ 𝐴)))
20		rexr 7993	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ^*)
21		renepnf 7995	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ +∞)
22		xaddmnf1 9835	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≠ +∞) → (𝐴 +_𝑒 -∞) = -∞)
23	20, 21, 22	syl2anc 411	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ → (𝐴 +_𝑒 -∞) = -∞)
24		mnflt0 9771	. . . . . . . . . . . . 13 ⊢ -∞ < 0
25		mnfxr 8004	. . . . . . . . . . . . . . 15 ⊢ -∞ ∈ ℝ^*
26		xrlenlt 8012	. . . . . . . . . . . . . . 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 640	. . . . . . . . . . . 12 ⊢ ¬ 0 ≤ -∞
30		breq2 4004	. . . . . . . . . . . 12 ⊢ ((𝐴 +_𝑒 -∞) = -∞ → (0 ≤ (𝐴 +_𝑒 -∞) ↔ 0 ≤ -∞))
31	29, 30	mtbiri 675	. . . . . . . . . . 11 ⊢ ((𝐴 +_𝑒 -∞) = -∞ → ¬ 0 ≤ (𝐴 +_𝑒 -∞))
32	31	pm2.21d 619	. . . . . . . . . 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 2240	. . . . . . . . . . . 12 ⊢ (𝐴 = -∞ → (𝐴 ∈ ℝ^* ↔ -∞ ∈ ℝ^*))
38	25, 37	mpbiri 168	. . . . . . . . . . 11 ⊢ (𝐴 = -∞ → 𝐴 ∈ ℝ^*)
39		mnfnepnf 8003	. . . . . . . . . . . 12 ⊢ -∞ ≠ +∞
40		neeq1 2360	. . . . . . . . . . . 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 9763	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ^* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
46	45	biimpi 120	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ^* → (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
47	34, 36, 44, 46	mpjao3dan 1307	. . . . . . 7 ⊢ (𝐴 ∈ ℝ^* → (0 ≤ (𝐴 +_𝑒 -∞) → 𝐴 = +∞))
48		0le0 8997	. . . . . . . 8 ⊢ 0 ≤ 0
49		oveq1 5876	. . . . . . . . 9 ⊢ (𝐴 = +∞ → (𝐴 +_𝑒 -∞) = (+∞ +_𝑒 -∞))
50		pnfaddmnf 9837	. . . . . . . . 9 ⊢ (+∞ +_𝑒 -∞) = 0
51	49, 50	eqtrdi 2226	. . . . . . . 8 ⊢ (𝐴 = +∞ → (𝐴 +_𝑒 -∞) = 0)
52	48, 51	breqtrrid 4038	. . . . . . 7 ⊢ (𝐴 = +∞ → 0 ≤ (𝐴 +_𝑒 -∞))
53	47, 52	impbid1 142	. . . . . 6 ⊢ (𝐴 ∈ ℝ^* → (0 ≤ (𝐴 +_𝑒 -∞) ↔ 𝐴 = +∞))
54		pnfge 9776	. . . . . . 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 9814	. . . . . . . 8 ⊢ (𝐵 = +∞ → -_𝑒𝐵 = -_𝑒+∞)
59		xnegpnf 9815	. . . . . . . 8 ⊢ -_𝑒+∞ = -∞
60	58, 59	eqtrdi 2226	. . . . . . 7 ⊢ (𝐵 = +∞ → -_𝑒𝐵 = -∞)
61	60	adantl 277	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 = +∞) → -_𝑒𝐵 = -∞)
62	61	oveq2d 5885	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 = +∞) → (𝐴 +_𝑒 -_𝑒𝐵) = (𝐴 +_𝑒 -∞))
63	62	breq2d 4012	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 = +∞) → (0 ≤ (𝐴 +_𝑒 -_𝑒𝐵) ↔ 0 ≤ (𝐴 +_𝑒 -∞)))
64		breq1 4003	. . . . 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 5876	. . . . . . . . . 10 ⊢ (𝐴 = -∞ → (𝐴 +_𝑒 +∞) = (-∞ +_𝑒 +∞))
68		mnfaddpnf 9838	. . . . . . . . . 10 ⊢ (-∞ +_𝑒 +∞) = 0
69	67, 68	eqtrdi 2226	. . . . . . . . 9 ⊢ (𝐴 = -∞ → (𝐴 +_𝑒 +∞) = 0)
70	69	adantl 277	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 = -∞) → (𝐴 +_𝑒 +∞) = 0)
71	48, 70	breqtrrid 4038	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 = -∞) → 0 ≤ (𝐴 +_𝑒 +∞))
72		df-ne 2348	. . . . . . . 8 ⊢ (𝐴 ≠ -∞ ↔ ¬ 𝐴 = -∞)
73		0lepnf 9777	. . . . . . . . 9 ⊢ 0 ≤ +∞
74		xaddpnf1 9833	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≠ -∞) → (𝐴 +_𝑒 +∞) = +∞)
75	73, 74	breqtrrid 4038	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≠ -∞) → 0 ≤ (𝐴 +_𝑒 +∞))
76	72, 75	sylan2br 288	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ ¬ 𝐴 = -∞) → 0 ≤ (𝐴 +_𝑒 +∞))
77		xrmnfdc 9830	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ^* → DECID 𝐴 = -∞)
78		exmiddc 836	. . . . . . . 8 ⊢ (DECID 𝐴 = -∞ → (𝐴 = -∞ ∨ ¬ 𝐴 = -∞))
79	77, 78	syl 14	. . . . . . 7 ⊢ (𝐴 ∈ ℝ^* → (𝐴 = -∞ ∨ ¬ 𝐴 = -∞))
80	71, 76, 79	mpjaodan 798	. . . . . 6 ⊢ (𝐴 ∈ ℝ^* → 0 ≤ (𝐴 +_𝑒 +∞))
81		mnfle 9779	. . . . . 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 9814	. . . . . . . 8 ⊢ (𝐵 = -∞ → -_𝑒𝐵 = -_𝑒-∞)
85		xnegmnf 9816	. . . . . . . 8 ⊢ -_𝑒-∞ = +∞
86	84, 85	eqtrdi 2226	. . . . . . 7 ⊢ (𝐵 = -∞ → -_𝑒𝐵 = +∞)
87	86	adantl 277	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 = -∞) → -_𝑒𝐵 = +∞)
88	87	oveq2d 5885	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 = -∞) → (𝐴 +_𝑒 -_𝑒𝐵) = (𝐴 +_𝑒 +∞))
89	88	breq2d 4012	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 = -∞) → (0 ≤ (𝐴 +_𝑒 -_𝑒𝐵) ↔ 0 ≤ (𝐴 +_𝑒 +∞)))
90		breq1 4003	. . . . 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 1306	. 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 708 DECID wdc 834 ∨ w3o 977 = wceq 1353 ∈ wcel 2148 ≠ wne 2347 class class class wbr 4000 (class class class)co 5869 ℝcr 7801 0cc0 7802 +∞cpnf 7979 -∞cmnf 7980 ℝ^*cxr 7981 < clt 7982 ≤ cle 7983 -_𝑒cxne 9756 +_𝑒 cxad 9757

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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-setind 4533 ax-cnex 7893 ax-resscn 7894 ax-1cn 7895 ax-1re 7896 ax-icn 7897 ax-addcl 7898 ax-addrcl 7899 ax-mulcl 7900 ax-addcom 7902 ax-addass 7904 ax-distr 7906 ax-i2m1 7907 ax-0id 7910 ax-rnegex 7911 ax-cnre 7913 ax-pre-ltirr 7914 ax-pre-apti 7917 ax-pre-ltadd 7918

This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-if 3535 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-id 4290 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-rn 4634 df-res 4635 df-ima 4636 df-iota 5174 df-fun 5214 df-fn 5215 df-f 5216 df-fv 5220 df-riota 5825 df-ov 5872 df-oprab 5873 df-mpo 5874 df-1st 6135 df-2nd 6136 df-pnf 7984 df-mnf 7985 df-xr 7986 df-ltxr 7987 df-le 7988 df-sub 8120 df-neg 8121 df-xneg 9759 df-xadd 9760

This theorem is referenced by: ssblps 13592 ssbl 13593

W3C validator