xsubge0 - Metamath Proof Explorer

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Theorem xsubge0 13240

Description: Extended real version of subge0 11727. (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 13096	. 2 ⊢ (𝐵 ∈ ℝ^* ↔ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞))
2		0xr 11261	. . . . 5 ⊢ 0 ∈ ℝ^*
3		rexr 11260	. . . . . 6 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℝ^*)
4		xnegcl 13192	. . . . . . 7 ⊢ (𝐵 ∈ ℝ^* → -_𝑒𝐵 ∈ ℝ^*)
5		xaddcl 13218	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ -_𝑒𝐵 ∈ ℝ^) → (𝐴 +_𝑒 -_𝑒𝐵) ∈ ℝ^)
6	4, 5	sylan2 594	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐴 +_𝑒 -_𝑒𝐵) ∈ ℝ^)
7	3, 6	sylan2 594	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → (𝐴 +_𝑒 -_𝑒𝐵) ∈ ℝ^*)
8		simpr 486	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → 𝐵 ∈ ℝ)
9		xleadd1 13234	. . . . 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 1467	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → (0 ≤ (𝐴 +_𝑒 -_𝑒𝐵) ↔ (0 +_𝑒 𝐵) ≤ ((𝐴 +_𝑒 -_𝑒𝐵) +_𝑒 𝐵)))
11	3	adantl 483	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → 𝐵 ∈ ℝ^*)
12		xaddlid 13221	. . . . . 6 ⊢ (𝐵 ∈ ℝ^* → (0 +_𝑒 𝐵) = 𝐵)
13	11, 12	syl 17	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → (0 +_𝑒 𝐵) = 𝐵)
14		xnpcan 13231	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → ((𝐴 +_𝑒 -_𝑒𝐵) +_𝑒 𝐵) = 𝐴)
15	13, 14	breq12d 5162	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → ((0 +_𝑒 𝐵) ≤ ((𝐴 +_𝑒 -_𝑒𝐵) +_𝑒 𝐵) ↔ 𝐵 ≤ 𝐴))
16	10, 15	bitrd 279	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → (0 ≤ (𝐴 +_𝑒 -_𝑒𝐵) ↔ 𝐵 ≤ 𝐴))
17		pnfxr 11268	. . . . . . 7 ⊢ +∞ ∈ ℝ^*
18		xrletri3 13133	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → (𝐴 = +∞ ↔ (𝐴 ≤ +∞ ∧ +∞ ≤ 𝐴)))
19	17, 18	mpan2 690	. . . . . 6 ⊢ (𝐴 ∈ ℝ^* → (𝐴 = +∞ ↔ (𝐴 ≤ +∞ ∧ +∞ ≤ 𝐴)))
20		mnflt0 13105	. . . . . . . . . . 11 ⊢ -∞ < 0
21		mnfxr 11271	. . . . . . . . . . . 12 ⊢ -∞ ∈ ℝ^*
22		xrltnle 11281	. . . . . . . . . . . 12 ⊢ ((-∞ ∈ ℝ^* ∧ 0 ∈ ℝ^*) → (-∞ < 0 ↔ ¬ 0 ≤ -∞))
23	21, 2, 22	mp2an 691	. . . . . . . . . . 11 ⊢ (-∞ < 0 ↔ ¬ 0 ≤ -∞)
24	20, 23	mpbi 229	. . . . . . . . . 10 ⊢ ¬ 0 ≤ -∞
25		xaddmnf1 13207	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≠ +∞) → (𝐴 +_𝑒 -∞) = -∞)
26	25	breq2d 5161	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≠ +∞) → (0 ≤ (𝐴 +_𝑒 -∞) ↔ 0 ≤ -∞))
27	24, 26	mtbiri 327	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≠ +∞) → ¬ 0 ≤ (𝐴 +_𝑒 -∞))
28	27	ex 414	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ^* → (𝐴 ≠ +∞ → ¬ 0 ≤ (𝐴 +_𝑒 -∞)))
29	28	necon4ad 2960	. . . . . . 7 ⊢ (𝐴 ∈ ℝ^* → (0 ≤ (𝐴 +_𝑒 -∞) → 𝐴 = +∞))
30		0le0 12313	. . . . . . . 8 ⊢ 0 ≤ 0
31		oveq1 7416	. . . . . . . . 9 ⊢ (𝐴 = +∞ → (𝐴 +_𝑒 -∞) = (+∞ +_𝑒 -∞))
32		pnfaddmnf 13209	. . . . . . . . 9 ⊢ (+∞ +_𝑒 -∞) = 0
33	31, 32	eqtrdi 2789	. . . . . . . 8 ⊢ (𝐴 = +∞ → (𝐴 +_𝑒 -∞) = 0)
34	30, 33	breqtrrid 5187	. . . . . . 7 ⊢ (𝐴 = +∞ → 0 ≤ (𝐴 +_𝑒 -∞))
35	29, 34	impbid1 224	. . . . . 6 ⊢ (𝐴 ∈ ℝ^* → (0 ≤ (𝐴 +_𝑒 -∞) ↔ 𝐴 = +∞))
36		pnfge 13110	. . . . . . 7 ⊢ (𝐴 ∈ ℝ^* → 𝐴 ≤ +∞)
37	36	biantrurd 534	. . . . . 6 ⊢ (𝐴 ∈ ℝ^* → (+∞ ≤ 𝐴 ↔ (𝐴 ≤ +∞ ∧ +∞ ≤ 𝐴)))
38	19, 35, 37	3bitr4d 311	. . . . 5 ⊢ (𝐴 ∈ ℝ^* → (0 ≤ (𝐴 +_𝑒 -∞) ↔ +∞ ≤ 𝐴))
39	38	adantr 482	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 = +∞) → (0 ≤ (𝐴 +_𝑒 -∞) ↔ +∞ ≤ 𝐴))
40		xnegeq 13186	. . . . . . . 8 ⊢ (𝐵 = +∞ → -_𝑒𝐵 = -_𝑒+∞)
41		xnegpnf 13188	. . . . . . . 8 ⊢ -_𝑒+∞ = -∞
42	40, 41	eqtrdi 2789	. . . . . . 7 ⊢ (𝐵 = +∞ → -_𝑒𝐵 = -∞)
43	42	adantl 483	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 = +∞) → -_𝑒𝐵 = -∞)
44	43	oveq2d 7425	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 = +∞) → (𝐴 +_𝑒 -_𝑒𝐵) = (𝐴 +_𝑒 -∞))
45	44	breq2d 5161	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 = +∞) → (0 ≤ (𝐴 +_𝑒 -_𝑒𝐵) ↔ 0 ≤ (𝐴 +_𝑒 -∞)))
46		breq1 5152	. . . . 5 ⊢ (𝐵 = +∞ → (𝐵 ≤ 𝐴 ↔ +∞ ≤ 𝐴))
47	46	adantl 483	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 = +∞) → (𝐵 ≤ 𝐴 ↔ +∞ ≤ 𝐴))
48	39, 45, 47	3bitr4d 311	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 = +∞) → (0 ≤ (𝐴 +_𝑒 -_𝑒𝐵) ↔ 𝐵 ≤ 𝐴))
49		oveq1 7416	. . . . . . . . . 10 ⊢ (𝐴 = -∞ → (𝐴 +_𝑒 +∞) = (-∞ +_𝑒 +∞))
50		mnfaddpnf 13210	. . . . . . . . . 10 ⊢ (-∞ +_𝑒 +∞) = 0
51	49, 50	eqtrdi 2789	. . . . . . . . 9 ⊢ (𝐴 = -∞ → (𝐴 +_𝑒 +∞) = 0)
52	51	adantl 483	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 = -∞) → (𝐴 +_𝑒 +∞) = 0)
53	30, 52	breqtrrid 5187	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 = -∞) → 0 ≤ (𝐴 +_𝑒 +∞))
54		0lepnf 13112	. . . . . . . 8 ⊢ 0 ≤ +∞
55		xaddpnf1 13205	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≠ -∞) → (𝐴 +_𝑒 +∞) = +∞)
56	54, 55	breqtrrid 5187	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≠ -∞) → 0 ≤ (𝐴 +_𝑒 +∞))
57	53, 56	pm2.61dane 3030	. . . . . 6 ⊢ (𝐴 ∈ ℝ^* → 0 ≤ (𝐴 +_𝑒 +∞))
58		mnfle 13114	. . . . . 6 ⊢ (𝐴 ∈ ℝ^* → -∞ ≤ 𝐴)
59	57, 58	2thd 265	. . . . 5 ⊢ (𝐴 ∈ ℝ^* → (0 ≤ (𝐴 +_𝑒 +∞) ↔ -∞ ≤ 𝐴))
60	59	adantr 482	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 = -∞) → (0 ≤ (𝐴 +_𝑒 +∞) ↔ -∞ ≤ 𝐴))
61		xnegeq 13186	. . . . . . . 8 ⊢ (𝐵 = -∞ → -_𝑒𝐵 = -_𝑒-∞)
62		xnegmnf 13189	. . . . . . . 8 ⊢ -_𝑒-∞ = +∞
63	61, 62	eqtrdi 2789	. . . . . . 7 ⊢ (𝐵 = -∞ → -_𝑒𝐵 = +∞)
64	63	adantl 483	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 = -∞) → -_𝑒𝐵 = +∞)
65	64	oveq2d 7425	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 = -∞) → (𝐴 +_𝑒 -_𝑒𝐵) = (𝐴 +_𝑒 +∞))
66	65	breq2d 5161	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 = -∞) → (0 ≤ (𝐴 +_𝑒 -_𝑒𝐵) ↔ 0 ≤ (𝐴 +_𝑒 +∞)))
67		breq1 5152	. . . . 5 ⊢ (𝐵 = -∞ → (𝐵 ≤ 𝐴 ↔ -∞ ≤ 𝐴))
68	67	adantl 483	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 = -∞) → (𝐵 ≤ 𝐴 ↔ -∞ ≤ 𝐴))
69	60, 66, 68	3bitr4d 311	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 = -∞) → (0 ≤ (𝐴 +_𝑒 -_𝑒𝐵) ↔ 𝐵 ≤ 𝐴))
70	16, 48, 69	3jaodan 1431	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → (0 ≤ (𝐴 +_𝑒 -_𝑒𝐵) ↔ 𝐵 ≤ 𝐴))
71	1, 70	sylan2b 595	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (0 ≤ (𝐴 +_𝑒 -_𝑒𝐵) ↔ 𝐵 ≤ 𝐴))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 ∨ w3o 1087 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 class class class wbr 5149 (class class class)co 7409 ℝcr 11109 0cc0 11110 +∞cpnf 11245 -∞cmnf 11246 ℝ^*cxr 11247 < clt 11248 ≤ cle 11249 -_𝑒cxne 13089 +_𝑒 cxad 13090

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-po 5589 df-so 5590 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-1st 7975 df-2nd 7976 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-xneg 13092 df-xadd 13093

This theorem is referenced by: xposdif 13241 ssblps 23928 ssbl 23929 xrsxmet 24325 xrge0subcld 31976 esumle 33056 esumlef 33060

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