xsubge0 - Intuitionistic Logic Explorer

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Theorem xsubge0 9898

Description: Extended real version of subge0 8449. (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 9793	. 2 ⊢ (𝐵 ∈ ℝ^* ↔ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞))
2		0xr 8021	. . . . 5 ⊢ 0 ∈ ℝ^*
3		rexr 8020	. . . . . 6 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℝ^*)
4		xnegcl 9849	. . . . . . 7 ⊢ (𝐵 ∈ ℝ^* → -_𝑒𝐵 ∈ ℝ^*)
5		xaddcl 9877	. . . . . . 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 9892	. . . . 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 1352	. . . 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 9880	. . . . . 6 ⊢ (𝐵 ∈ ℝ^* → (0 +_𝑒 𝐵) = 𝐵)
13	11, 12	syl 14	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → (0 +_𝑒 𝐵) = 𝐵)
14		xnpcan 9889	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → ((𝐴 +_𝑒 -_𝑒𝐵) +_𝑒 𝐵) = 𝐴)
15	13, 14	breq12d 4030	. . . 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 8027	. . . . . . 7 ⊢ +∞ ∈ ℝ^*
18		xrletri3 9821	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → (𝐴 = +∞ ↔ (𝐴 ≤ +∞ ∧ +∞ ≤ 𝐴)))
19	17, 18	mpan2 425	. . . . . 6 ⊢ (𝐴 ∈ ℝ^* → (𝐴 = +∞ ↔ (𝐴 ≤ +∞ ∧ +∞ ≤ 𝐴)))
20		rexr 8020	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ^*)
21		renepnf 8022	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ +∞)
22		xaddmnf1 9865	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≠ +∞) → (𝐴 +_𝑒 -∞) = -∞)
23	20, 21, 22	syl2anc 411	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ → (𝐴 +_𝑒 -∞) = -∞)
24		mnflt0 9801	. . . . . . . . . . . . 13 ⊢ -∞ < 0
25		mnfxr 8031	. . . . . . . . . . . . . . 15 ⊢ -∞ ∈ ℝ^*
26		xrlenlt 8039	. . . . . . . . . . . . . . 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 4021	. . . . . . . . . . . 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 2251	. . . . . . . . . . . 12 ⊢ (𝐴 = -∞ → (𝐴 ∈ ℝ^* ↔ -∞ ∈ ℝ^*))
38	25, 37	mpbiri 168	. . . . . . . . . . 11 ⊢ (𝐴 = -∞ → 𝐴 ∈ ℝ^*)
39		mnfnepnf 8030	. . . . . . . . . . . 12 ⊢ -∞ ≠ +∞
40		neeq1 2372	. . . . . . . . . . . 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 9793	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ^* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
46	45	biimpi 120	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ^* → (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
47	34, 36, 44, 46	mpjao3dan 1317	. . . . . . 7 ⊢ (𝐴 ∈ ℝ^* → (0 ≤ (𝐴 +_𝑒 -∞) → 𝐴 = +∞))
48		0le0 9025	. . . . . . . 8 ⊢ 0 ≤ 0
49		oveq1 5897	. . . . . . . . 9 ⊢ (𝐴 = +∞ → (𝐴 +_𝑒 -∞) = (+∞ +_𝑒 -∞))
50		pnfaddmnf 9867	. . . . . . . . 9 ⊢ (+∞ +_𝑒 -∞) = 0
51	49, 50	eqtrdi 2237	. . . . . . . 8 ⊢ (𝐴 = +∞ → (𝐴 +_𝑒 -∞) = 0)
52	48, 51	breqtrrid 4055	. . . . . . 7 ⊢ (𝐴 = +∞ → 0 ≤ (𝐴 +_𝑒 -∞))
53	47, 52	impbid1 142	. . . . . 6 ⊢ (𝐴 ∈ ℝ^* → (0 ≤ (𝐴 +_𝑒 -∞) ↔ 𝐴 = +∞))
54		pnfge 9806	. . . . . . 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 9844	. . . . . . . 8 ⊢ (𝐵 = +∞ → -_𝑒𝐵 = -_𝑒+∞)
59		xnegpnf 9845	. . . . . . . 8 ⊢ -_𝑒+∞ = -∞
60	58, 59	eqtrdi 2237	. . . . . . 7 ⊢ (𝐵 = +∞ → -_𝑒𝐵 = -∞)
61	60	adantl 277	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 = +∞) → -_𝑒𝐵 = -∞)
62	61	oveq2d 5906	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 = +∞) → (𝐴 +_𝑒 -_𝑒𝐵) = (𝐴 +_𝑒 -∞))
63	62	breq2d 4029	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 = +∞) → (0 ≤ (𝐴 +_𝑒 -_𝑒𝐵) ↔ 0 ≤ (𝐴 +_𝑒 -∞)))
64		breq1 4020	. . . . 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 5897	. . . . . . . . . 10 ⊢ (𝐴 = -∞ → (𝐴 +_𝑒 +∞) = (-∞ +_𝑒 +∞))
68		mnfaddpnf 9868	. . . . . . . . . 10 ⊢ (-∞ +_𝑒 +∞) = 0
69	67, 68	eqtrdi 2237	. . . . . . . . 9 ⊢ (𝐴 = -∞ → (𝐴 +_𝑒 +∞) = 0)
70	69	adantl 277	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 = -∞) → (𝐴 +_𝑒 +∞) = 0)
71	48, 70	breqtrrid 4055	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 = -∞) → 0 ≤ (𝐴 +_𝑒 +∞))
72		df-ne 2360	. . . . . . . 8 ⊢ (𝐴 ≠ -∞ ↔ ¬ 𝐴 = -∞)
73		0lepnf 9807	. . . . . . . . 9 ⊢ 0 ≤ +∞
74		xaddpnf1 9863	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≠ -∞) → (𝐴 +_𝑒 +∞) = +∞)
75	73, 74	breqtrrid 4055	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≠ -∞) → 0 ≤ (𝐴 +_𝑒 +∞))
76	72, 75	sylan2br 288	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ ¬ 𝐴 = -∞) → 0 ≤ (𝐴 +_𝑒 +∞))
77		xrmnfdc 9860	. . . . . . . 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 9809	. . . . . 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 9844	. . . . . . . 8 ⊢ (𝐵 = -∞ → -_𝑒𝐵 = -_𝑒-∞)
85		xnegmnf 9846	. . . . . . . 8 ⊢ -_𝑒-∞ = +∞
86	84, 85	eqtrdi 2237	. . . . . . 7 ⊢ (𝐵 = -∞ → -_𝑒𝐵 = +∞)
87	86	adantl 277	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 = -∞) → -_𝑒𝐵 = +∞)
88	87	oveq2d 5906	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 = -∞) → (𝐴 +_𝑒 -_𝑒𝐵) = (𝐴 +_𝑒 +∞))
89	88	breq2d 4029	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 = -∞) → (0 ≤ (𝐴 +_𝑒 -_𝑒𝐵) ↔ 0 ≤ (𝐴 +_𝑒 +∞)))
90		breq1 4020	. . . . 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 1316	. 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 978 = wceq 1363 ∈ wcel 2159 ≠ wne 2359 class class class wbr 4017 (class class class)co 5890 ℝcr 7827 0cc0 7828 +∞cpnf 8006 -∞cmnf 8007 ℝ^*cxr 8008 < clt 8009 ≤ cle 8010 -_𝑒cxne 9786 +_𝑒 cxad 9787

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 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2161 ax-14 2162 ax-ext 2170 ax-sep 4135 ax-pow 4188 ax-pr 4223 ax-un 4447 ax-setind 4550 ax-cnex 7919 ax-resscn 7920 ax-1cn 7921 ax-1re 7922 ax-icn 7923 ax-addcl 7924 ax-addrcl 7925 ax-mulcl 7926 ax-addcom 7928 ax-addass 7930 ax-distr 7932 ax-i2m1 7933 ax-0id 7936 ax-rnegex 7937 ax-cnre 7939 ax-pre-ltirr 7940 ax-pre-apti 7943 ax-pre-ltadd 7944

This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 980 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2040 df-mo 2041 df-clab 2175 df-cleq 2181 df-clel 2184 df-nfc 2320 df-ne 2360 df-nel 2455 df-ral 2472 df-rex 2473 df-reu 2474 df-rab 2476 df-v 2753 df-sbc 2977 df-csb 3072 df-dif 3145 df-un 3147 df-in 3149 df-ss 3156 df-if 3549 df-pw 3591 df-sn 3612 df-pr 3613 df-op 3615 df-uni 3824 df-iun 3902 df-br 4018 df-opab 4079 df-mpt 4080 df-id 4307 df-xp 4646 df-rel 4647 df-cnv 4648 df-co 4649 df-dm 4650 df-rn 4651 df-res 4652 df-ima 4653 df-iota 5192 df-fun 5232 df-fn 5233 df-f 5234 df-fv 5238 df-riota 5846 df-ov 5893 df-oprab 5894 df-mpo 5895 df-1st 6158 df-2nd 6159 df-pnf 8011 df-mnf 8012 df-xr 8013 df-ltxr 8014 df-le 8015 df-sub 8147 df-neg 8148 df-xneg 9789 df-xadd 9790

This theorem is referenced by: ssblps 14308 ssbl 14309

W3C validator