xsubge0 - Metamath Proof Explorer

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Theorem xsubge0 13418

Description: Extended real version of subge0 11856. (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 13272	. 2 ⊢ (𝐵 ∈ ℝ^* ↔ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞))
2		0xr 11383	. . . . 5 ⊢ 0 ∈ ℝ^*
3		rexr 11382	. . . . . 6 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℝ^*)
4		xnegcl 13370	. . . . . . 7 ⊢ (𝐵 ∈ ℝ^* → -_e𝐵 ∈ ℝ^*)
5		xaddcl 13396	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ -_e𝐵 ∈ ℝ^) → (𝐴 +_e -_e𝐵) ∈ ℝ^)
6	4, 5	sylan2 605	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐴 +_e -_e𝐵) ∈ ℝ^)
7	3, 6	sylan2 605	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → (𝐴 +_e -_e𝐵) ∈ ℝ^*)
8		simpr 490	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → 𝐵 ∈ ℝ)
9		xleadd1 13412	. . . . 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 1496	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → (0 ≤ (𝐴 +_e -_e𝐵) ↔ (0 +_e 𝐵) ≤ ((𝐴 +_e -_e𝐵) +_e 𝐵)))
11	3	adantl 487	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → 𝐵 ∈ ℝ^*)
12		xaddlid 13399	. . . . . 6 ⊢ (𝐵 ∈ ℝ^* → (0 +_e 𝐵) = 𝐵)
13	11, 12	syl 18	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → (0 +_e 𝐵) = 𝐵)
14		xnpcan 13409	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → ((𝐴 +_e -_e𝐵) +_e 𝐵) = 𝐴)
15	13, 14	breq12d 5119	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → ((0 +_e 𝐵) ≤ ((𝐴 +_e -_e𝐵) +_e 𝐵) ↔ 𝐵 ≤ 𝐴))
16	10, 15	bitrd 282	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → (0 ≤ (𝐴 +_e -_e𝐵) ↔ 𝐵 ≤ 𝐴))
17		pnfxr 11390	. . . . . . 7 ⊢ +∞ ∈ ℝ^*
18		xrletri3 13310	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → (𝐴 = +∞ ↔ (𝐴 ≤ +∞ ∧ +∞ ≤ 𝐴)))
19	17, 18	mpan2 704	. . . . . 6 ⊢ (𝐴 ∈ ℝ^* → (𝐴 = +∞ ↔ (𝐴 ≤ +∞ ∧ +∞ ≤ 𝐴)))
20		mnflt0 13281	. . . . . . . . . . 11 ⊢ -∞ < 0
21		mnfxr 11393	. . . . . . . . . . . 12 ⊢ -∞ ∈ ℝ^*
22		xrltnle 11403	. . . . . . . . . . . 12 ⊢ ((-∞ ∈ ℝ^* ∧ 0 ∈ ℝ^*) → (-∞ < 0 ↔ ¬ 0 ≤ -∞))
23	21, 2, 22	mp2an 705	. . . . . . . . . . 11 ⊢ (-∞ < 0 ↔ ¬ 0 ≤ -∞)
24	20, 23	mpbi 233	. . . . . . . . . 10 ⊢ ¬ 0 ≤ -∞
25		xaddmnf1 13385	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≠ +∞) → (𝐴 +_e -∞) = -∞)
26	25	breq2d 5118	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≠ +∞) → (0 ≤ (𝐴 +_e -∞) ↔ 0 ≤ -∞))
27	24, 26	mtbiri 330	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≠ +∞) → ¬ 0 ≤ (𝐴 +_e -∞))
28	27	ex 418	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ^* → (𝐴 ≠ +∞ → ¬ 0 ≤ (𝐴 +_e -∞)))
29	28	necon4ad 2978	. . . . . . 7 ⊢ (𝐴 ∈ ℝ^* → (0 ≤ (𝐴 +_e -∞) → 𝐴 = +∞))
30		0le0 12471	. . . . . . . 8 ⊢ 0 ≤ 0
31		oveq1 7450	. . . . . . . . 9 ⊢ (𝐴 = +∞ → (𝐴 +_e -∞) = (+∞ +_e -∞))
32		pnfaddmnf 13387	. . . . . . . . 9 ⊢ (+∞ +_e -∞) = 0
33	31, 32	eqtrdi 2814	. . . . . . . 8 ⊢ (𝐴 = +∞ → (𝐴 +_e -∞) = 0)
34	30, 33	breqtrrid 5146	. . . . . . 7 ⊢ (𝐴 = +∞ → 0 ≤ (𝐴 +_e -∞))
35	29, 34	impbid1 228	. . . . . 6 ⊢ (𝐴 ∈ ℝ^* → (0 ≤ (𝐴 +_e -∞) ↔ 𝐴 = +∞))
36		pnfge 13286	. . . . . . 7 ⊢ (𝐴 ∈ ℝ^* → 𝐴 ≤ +∞)
37	36	biantrurd 542	. . . . . 6 ⊢ (𝐴 ∈ ℝ^* → (+∞ ≤ 𝐴 ↔ (𝐴 ≤ +∞ ∧ +∞ ≤ 𝐴)))
38	19, 35, 37	3bitr4d 314	. . . . 5 ⊢ (𝐴 ∈ ℝ^* → (0 ≤ (𝐴 +_e -∞) ↔ +∞ ≤ 𝐴))
39	38	adantr 486	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 = +∞) → (0 ≤ (𝐴 +_e -∞) ↔ +∞ ≤ 𝐴))
40		xnegeq 13364	. . . . . . . 8 ⊢ (𝐵 = +∞ → -_e𝐵 = -_e+∞)
41		xnegpnf 13366	. . . . . . . 8 ⊢ -_e+∞ = -∞
42	40, 41	eqtrdi 2814	. . . . . . 7 ⊢ (𝐵 = +∞ → -_e𝐵 = -∞)
43	42	adantl 487	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 = +∞) → -_e𝐵 = -∞)
44	43	oveq2d 7459	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 = +∞) → (𝐴 +_e -_e𝐵) = (𝐴 +_e -∞))
45	44	breq2d 5118	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 = +∞) → (0 ≤ (𝐴 +_e -_e𝐵) ↔ 0 ≤ (𝐴 +_e -∞)))
46		breq1 5109	. . . . 5 ⊢ (𝐵 = +∞ → (𝐵 ≤ 𝐴 ↔ +∞ ≤ 𝐴))
47	46	adantl 487	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 = +∞) → (𝐵 ≤ 𝐴 ↔ +∞ ≤ 𝐴))
48	39, 45, 47	3bitr4d 314	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 = +∞) → (0 ≤ (𝐴 +_e -_e𝐵) ↔ 𝐵 ≤ 𝐴))
49		oveq1 7450	. . . . . . . . . 10 ⊢ (𝐴 = -∞ → (𝐴 +_e +∞) = (-∞ +_e +∞))
50		mnfaddpnf 13388	. . . . . . . . . 10 ⊢ (-∞ +_e +∞) = 0
51	49, 50	eqtrdi 2814	. . . . . . . . 9 ⊢ (𝐴 = -∞ → (𝐴 +_e +∞) = 0)
52	51	adantl 487	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 = -∞) → (𝐴 +_e +∞) = 0)
53	30, 52	breqtrrid 5146	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 = -∞) → 0 ≤ (𝐴 +_e +∞))
54		0lepnf 13289	. . . . . . . 8 ⊢ 0 ≤ +∞
55		xaddpnf1 13383	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≠ -∞) → (𝐴 +_e +∞) = +∞)
56	54, 55	breqtrrid 5146	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≠ -∞) → 0 ≤ (𝐴 +_e +∞))
57	53, 56	pm2.61dane 3046	. . . . . 6 ⊢ (𝐴 ∈ ℝ^* → 0 ≤ (𝐴 +_e +∞))
58		mnfle 13291	. . . . . 6 ⊢ (𝐴 ∈ ℝ^* → -∞ ≤ 𝐴)
59	57, 58	2thd 268	. . . . 5 ⊢ (𝐴 ∈ ℝ^* → (0 ≤ (𝐴 +_e +∞) ↔ -∞ ≤ 𝐴))
60	59	adantr 486	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 = -∞) → (0 ≤ (𝐴 +_e +∞) ↔ -∞ ≤ 𝐴))
61		xnegeq 13364	. . . . . . . 8 ⊢ (𝐵 = -∞ → -_e𝐵 = -_e-∞)
62		xnegmnf 13367	. . . . . . . 8 ⊢ -_e-∞ = +∞
63	61, 62	eqtrdi 2814	. . . . . . 7 ⊢ (𝐵 = -∞ → -_e𝐵 = +∞)
64	63	adantl 487	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 = -∞) → -_e𝐵 = +∞)
65	64	oveq2d 7459	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 = -∞) → (𝐴 +_e -_e𝐵) = (𝐴 +_e +∞))
66	65	breq2d 5118	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 = -∞) → (0 ≤ (𝐴 +_e -_e𝐵) ↔ 0 ≤ (𝐴 +_e +∞)))
67		breq1 5109	. . . . 5 ⊢ (𝐵 = -∞ → (𝐵 ≤ 𝐴 ↔ -∞ ≤ 𝐴))
68	67	adantl 487	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 = -∞) → (𝐵 ≤ 𝐴 ↔ -∞ ≤ 𝐴))
69	60, 66, 68	3bitr4d 314	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 = -∞) → (0 ≤ (𝐴 +_e -_e𝐵) ↔ 𝐵 ≤ 𝐴))
70	16, 48, 69	3jaodan 1459	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → (0 ≤ (𝐴 +_e -_e𝐵) ↔ 𝐵 ≤ 𝐴))
71	1, 70	sylan2b 606	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (0 ≤ (𝐴 +_e -_e𝐵) ↔ 𝐵 ≤ 𝐴))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∨ w3o 1102 = wceq 1570 ∈ wcel 2147 ≠ wne 2959 class class class wbr 5106 (class class class)co 7443 ℝcr 11226 0cc0 11227 +∞cpnf 11367 -∞cmnf 11368 ℝ^*cxr 11369 < clt 11370 ≤ cle 11371 -_ecxne 13265 +_e cxad 13266

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7775 ax-cnex 11283 ax-resscn 11284 ax-1cn 11285 ax-icn 11286 ax-addcl 11287 ax-addrcl 11288 ax-mulcl 11289 ax-mulrcl 11290 ax-mulcom 11291 ax-addass 11292 ax-mulass 11293 ax-distr 11294 ax-i2m1 11295 ax-1ne0 11296 ax-1rid 11297 ax-rnegex 11298 ax-rrecex 11299 ax-cnre 11300 ax-pre-lttri 11301 ax-pre-lttrn 11302 ax-pre-ltadd 11303

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4839 df-uni 4871 df-iun 4956 df-br 5107 df-opab 5171 df-mpt 5190 df-id 5550 df-po 5563 df-so 5564 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-iota 6512 df-fun 6560 df-fn 6561 df-f 6562 df-f1 6563 df-fo 6564 df-f1o 6565 df-fv 6566 df-riota 7400 df-ov 7446 df-oprab 7447 df-mpo 7448 df-1st 8025 df-2nd 8026 df-er 8745 df-en 9002 df-dom 9003 df-sdom 9004 df-pnf 11372 df-mnf 11373 df-xr 11374 df-ltxr 11375 df-le 11376 df-sub 11570 df-neg 11571 df-xneg 13268 df-xadd 13269

This theorem is used by: xposdif 13419 ssblps 24771 ssbl 24772 xrsxmet 25159 xrge0subcld 33425 esumle 34760 esumlef 34764

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