xsubge0 - Metamath Proof Explorer

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Theorem xsubge0 13317

Description: Extended real version of subge0 11755. (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 13171	. 2 ⊢ (𝐵 ∈ ℝ^* ↔ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞))
2		0xr 11284	. . . . 5 ⊢ 0 ∈ ℝ^*
3		rexr 11283	. . . . . 6 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℝ^*)
4		xnegcl 13269	. . . . . . 7 ⊢ (𝐵 ∈ ℝ^* → -_𝑒𝐵 ∈ ℝ^*)
5		xaddcl 13295	. . . . . . 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 13311	. . . . 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 1495	. . . 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 13298	. . . . . 6 ⊢ (𝐵 ∈ ℝ^* → (0 +_𝑒 𝐵) = 𝐵)
13	11, 12	syl 18	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → (0 +_𝑒 𝐵) = 𝐵)
14		xnpcan 13308	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → ((𝐴 +_𝑒 -_𝑒𝐵) +_𝑒 𝐵) = 𝐴)
15	13, 14	breq12d 5120	. . . 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 11291	. . . . . . 7 ⊢ +∞ ∈ ℝ^*
18		xrletri3 13209	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → (𝐴 = +∞ ↔ (𝐴 ≤ +∞ ∧ +∞ ≤ 𝐴)))
19	17, 18	mpan2 704	. . . . . 6 ⊢ (𝐴 ∈ ℝ^* → (𝐴 = +∞ ↔ (𝐴 ≤ +∞ ∧ +∞ ≤ 𝐴)))
20		mnflt0 13180	. . . . . . . . . . 11 ⊢ -∞ < 0
21		mnfxr 11294	. . . . . . . . . . . 12 ⊢ -∞ ∈ ℝ^*
22		xrltnle 11304	. . . . . . . . . . . 12 ⊢ ((-∞ ∈ ℝ^* ∧ 0 ∈ ℝ^*) → (-∞ < 0 ↔ ¬ 0 ≤ -∞))
23	21, 2, 22	mp2an 705	. . . . . . . . . . 11 ⊢ (-∞ < 0 ↔ ¬ 0 ≤ -∞)
24	20, 23	mpbi 233	. . . . . . . . . 10 ⊢ ¬ 0 ≤ -∞
25		xaddmnf1 13284	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≠ +∞) → (𝐴 +_𝑒 -∞) = -∞)
26	25	breq2d 5119	. . . . . . . . . 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 2976	. . . . . . 7 ⊢ (𝐴 ∈ ℝ^* → (0 ≤ (𝐴 +_𝑒 -∞) → 𝐴 = +∞))
30		0le0 12370	. . . . . . . 8 ⊢ 0 ≤ 0
31		oveq1 7424	. . . . . . . . 9 ⊢ (𝐴 = +∞ → (𝐴 +_𝑒 -∞) = (+∞ +_𝑒 -∞))
32		pnfaddmnf 13286	. . . . . . . . 9 ⊢ (+∞ +_𝑒 -∞) = 0
33	31, 32	eqtrdi 2813	. . . . . . . 8 ⊢ (𝐴 = +∞ → (𝐴 +_𝑒 -∞) = 0)
34	30, 33	breqtrrid 5147	. . . . . . 7 ⊢ (𝐴 = +∞ → 0 ≤ (𝐴 +_𝑒 -∞))
35	29, 34	impbid1 228	. . . . . 6 ⊢ (𝐴 ∈ ℝ^* → (0 ≤ (𝐴 +_𝑒 -∞) ↔ 𝐴 = +∞))
36		pnfge 13185	. . . . . . 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 13263	. . . . . . . 8 ⊢ (𝐵 = +∞ → -_𝑒𝐵 = -_𝑒+∞)
41		xnegpnf 13265	. . . . . . . 8 ⊢ -_𝑒+∞ = -∞
42	40, 41	eqtrdi 2813	. . . . . . 7 ⊢ (𝐵 = +∞ → -_𝑒𝐵 = -∞)
43	42	adantl 487	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 = +∞) → -_𝑒𝐵 = -∞)
44	43	oveq2d 7433	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 = +∞) → (𝐴 +_𝑒 -_𝑒𝐵) = (𝐴 +_𝑒 -∞))
45	44	breq2d 5119	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 = +∞) → (0 ≤ (𝐴 +_𝑒 -_𝑒𝐵) ↔ 0 ≤ (𝐴 +_𝑒 -∞)))
46		breq1 5110	. . . . 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 7424	. . . . . . . . . 10 ⊢ (𝐴 = -∞ → (𝐴 +_𝑒 +∞) = (-∞ +_𝑒 +∞))
50		mnfaddpnf 13287	. . . . . . . . . 10 ⊢ (-∞ +_𝑒 +∞) = 0
51	49, 50	eqtrdi 2813	. . . . . . . . 9 ⊢ (𝐴 = -∞ → (𝐴 +_𝑒 +∞) = 0)
52	51	adantl 487	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 = -∞) → (𝐴 +_𝑒 +∞) = 0)
53	30, 52	breqtrrid 5147	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 = -∞) → 0 ≤ (𝐴 +_𝑒 +∞))
54		0lepnf 13188	. . . . . . . 8 ⊢ 0 ≤ +∞
55		xaddpnf1 13282	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≠ -∞) → (𝐴 +_𝑒 +∞) = +∞)
56	54, 55	breqtrrid 5147	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≠ -∞) → 0 ≤ (𝐴 +_𝑒 +∞))
57	53, 56	pm2.61dane 3044	. . . . . 6 ⊢ (𝐴 ∈ ℝ^* → 0 ≤ (𝐴 +_𝑒 +∞))
58		mnfle 13190	. . . . . 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 13263	. . . . . . . 8 ⊢ (𝐵 = -∞ → -_𝑒𝐵 = -_𝑒-∞)
62		xnegmnf 13266	. . . . . . . 8 ⊢ -_𝑒-∞ = +∞
63	61, 62	eqtrdi 2813	. . . . . . 7 ⊢ (𝐵 = -∞ → -_𝑒𝐵 = +∞)
64	63	adantl 487	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 = -∞) → -_𝑒𝐵 = +∞)
65	64	oveq2d 7433	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 = -∞) → (𝐴 +_𝑒 -_𝑒𝐵) = (𝐴 +_𝑒 +∞))
66	65	breq2d 5119	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 = -∞) → (0 ≤ (𝐴 +_𝑒 -_𝑒𝐵) ↔ 0 ≤ (𝐴 +_𝑒 +∞)))
67		breq1 5110	. . . . 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 1458	. 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 2145 ≠ wne 2957 class class class wbr 5107 (class class class)co 7417 ℝcr 11127 0cc0 11128 +∞cpnf 11268 -∞cmnf 11269 ℝ^*cxr 11270 < clt 11271 ≤ cle 11272 -_𝑒cxne 13164 +_𝑒 cxad 13165

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 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7740 ax-cnex 11184 ax-resscn 11185 ax-1cn 11186 ax-icn 11187 ax-addcl 11188 ax-addrcl 11189 ax-mulcl 11190 ax-mulrcl 11191 ax-mulcom 11192 ax-addass 11193 ax-mulass 11194 ax-distr 11195 ax-i2m1 11196 ax-1ne0 11197 ax-1rid 11198 ax-rnegex 11199 ax-rrecex 11200 ax-cnre 11201 ax-pre-lttri 11202 ax-pre-lttrn 11203 ax-pre-ltadd 11204

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 2100 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4871 df-iun 4956 df-br 5108 df-opab 5172 df-mpt 5191 df-id 5554 df-po 5567 df-so 5568 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7374 df-ov 7420 df-oprab 7421 df-mpo 7422 df-1st 7990 df-2nd 7991 df-er 8700 df-en 8957 df-dom 8958 df-sdom 8959 df-pnf 11273 df-mnf 11274 df-xr 11275 df-ltxr 11276 df-le 11277 df-sub 11471 df-neg 11472 df-xneg 13167 df-xadd 13168

This theorem is used by: xposdif 13318 ssblps 24654 ssbl 24655 xrsxmet 25042 xrge0subcld 33242 esumle 34576 esumlef 34580

W3C validator