Theorem ltxr 12498

Description: The 'less than' binary relation on the set of extended reals. Definition 12-3.1 of [Gleason] p. 173. (Contributed by NM, 14-Oct-2005.)

Assertion

Ref	Expression
ltxr	⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 < 𝐵 ↔ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐴 <ℝ 𝐵) ∨ (𝐴 = -∞ ∧ 𝐵 = +∞)) ∨ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) ∨ (𝐴 = -∞ ∧ 𝐵 ∈ ℝ)))))

Proof of Theorem ltxr

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq12 5035	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 <ℝ 𝑦 ↔ 𝐴 <ℝ 𝐵))
2		df-3an 1086	. . . . . 6 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦) ↔ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝑥 <ℝ 𝑦))
3	2	opabbii 5097	. . . . 5 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝑥 <ℝ 𝑦)}
4	1, 3	brab2a 5608	. . . 4 ⊢ (𝐴{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)}𝐵 ↔ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐴 <ℝ 𝐵))
5	4	a1i 11	. . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)}𝐵 ↔ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐴 <ℝ 𝐵)))
6		brun 5081	. . . 4 ⊢ (𝐴(((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))𝐵 ↔ (𝐴((ℝ ∪ {-∞}) × {+∞})𝐵 ∨ 𝐴({-∞} × ℝ)𝐵))
7		brxp 5565	. . . . . . 7 ⊢ (𝐴((ℝ ∪ {-∞}) × {+∞})𝐵 ↔ (𝐴 ∈ (ℝ ∪ {-∞}) ∧ 𝐵 ∈ {+∞}))
8		elun 4076	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (ℝ ∪ {-∞}) ↔ (𝐴 ∈ ℝ ∨ 𝐴 ∈ {-∞}))
9		orcom 867	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 ∈ {-∞}) ↔ (𝐴 ∈ {-∞} ∨ 𝐴 ∈ ℝ))
10	8, 9	bitri 278	. . . . . . . . . 10 ⊢ (𝐴 ∈ (ℝ ∪ {-∞}) ↔ (𝐴 ∈ {-∞} ∨ 𝐴 ∈ ℝ))
11		elsng 4539	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ* → (𝐴 ∈ {-∞} ↔ 𝐴 = -∞))
12	11	orbi1d 914	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ* → ((𝐴 ∈ {-∞} ∨ 𝐴 ∈ ℝ) ↔ (𝐴 = -∞ ∨ 𝐴 ∈ ℝ)))
13	10, 12	syl5bb 286	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ* → (𝐴 ∈ (ℝ ∪ {-∞}) ↔ (𝐴 = -∞ ∨ 𝐴 ∈ ℝ)))
14		elsng 4539	. . . . . . . . 9 ⊢ (𝐵 ∈ ℝ* → (𝐵 ∈ {+∞} ↔ 𝐵 = +∞))
15	13, 14	bi2anan9 638	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((𝐴 ∈ (ℝ ∪ {-∞}) ∧ 𝐵 ∈ {+∞}) ↔ ((𝐴 = -∞ ∨ 𝐴 ∈ ℝ) ∧ 𝐵 = +∞)))
16		andir 1006	. . . . . . . 8 ⊢ (((𝐴 = -∞ ∨ 𝐴 ∈ ℝ) ∧ 𝐵 = +∞) ↔ ((𝐴 = -∞ ∧ 𝐵 = +∞) ∨ (𝐴 ∈ ℝ ∧ 𝐵 = +∞)))
17	15, 16	syl6bb 290	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((𝐴 ∈ (ℝ ∪ {-∞}) ∧ 𝐵 ∈ {+∞}) ↔ ((𝐴 = -∞ ∧ 𝐵 = +∞) ∨ (𝐴 ∈ ℝ ∧ 𝐵 = +∞))))
18	7, 17	syl5bb 286	. . . . . 6 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴((ℝ ∪ {-∞}) × {+∞})𝐵 ↔ ((𝐴 = -∞ ∧ 𝐵 = +∞) ∨ (𝐴 ∈ ℝ ∧ 𝐵 = +∞))))
19		brxp 5565	. . . . . . 7 ⊢ (𝐴({-∞} × ℝ)𝐵 ↔ (𝐴 ∈ {-∞} ∧ 𝐵 ∈ ℝ))
20	11	anbi1d 632	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ* → ((𝐴 ∈ {-∞} ∧ 𝐵 ∈ ℝ) ↔ (𝐴 = -∞ ∧ 𝐵 ∈ ℝ)))
21	20	adantr 484	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((𝐴 ∈ {-∞} ∧ 𝐵 ∈ ℝ) ↔ (𝐴 = -∞ ∧ 𝐵 ∈ ℝ)))
22	19, 21	syl5bb 286	. . . . . 6 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴({-∞} × ℝ)𝐵 ↔ (𝐴 = -∞ ∧ 𝐵 ∈ ℝ)))
23	18, 22	orbi12d 916	. . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((𝐴((ℝ ∪ {-∞}) × {+∞})𝐵 ∨ 𝐴({-∞} × ℝ)𝐵) ↔ (((𝐴 = -∞ ∧ 𝐵 = +∞) ∨ (𝐴 ∈ ℝ ∧ 𝐵 = +∞)) ∨ (𝐴 = -∞ ∧ 𝐵 ∈ ℝ))))
24		orass 919	. . . . 5 ⊢ ((((𝐴 = -∞ ∧ 𝐵 = +∞) ∨ (𝐴 ∈ ℝ ∧ 𝐵 = +∞)) ∨ (𝐴 = -∞ ∧ 𝐵 ∈ ℝ)) ↔ ((𝐴 = -∞ ∧ 𝐵 = +∞) ∨ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) ∨ (𝐴 = -∞ ∧ 𝐵 ∈ ℝ))))
25	23, 24	syl6bb 290	. . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((𝐴((ℝ ∪ {-∞}) × {+∞})𝐵 ∨ 𝐴({-∞} × ℝ)𝐵) ↔ ((𝐴 = -∞ ∧ 𝐵 = +∞) ∨ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) ∨ (𝐴 = -∞ ∧ 𝐵 ∈ ℝ)))))
26	6, 25	syl5bb 286	. . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴(((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))𝐵 ↔ ((𝐴 = -∞ ∧ 𝐵 = +∞) ∨ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) ∨ (𝐴 = -∞ ∧ 𝐵 ∈ ℝ)))))
27	5, 26	orbi12d 916	. 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((𝐴{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)}𝐵 ∨ 𝐴(((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))𝐵) ↔ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐴 <ℝ 𝐵) ∨ ((𝐴 = -∞ ∧ 𝐵 = +∞) ∨ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) ∨ (𝐴 = -∞ ∧ 𝐵 ∈ ℝ))))))
28		df-ltxr 10669	. . . 4 ⊢ < = ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)} ∪ (((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ)))
29	28	breqi 5036	. . 3 ⊢ (𝐴 < 𝐵 ↔ 𝐴({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)} ∪ (((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ)))𝐵)
30		brun 5081	. . 3 ⊢ (𝐴({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)} ∪ (((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ)))𝐵 ↔ (𝐴{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)}𝐵 ∨ 𝐴(((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))𝐵))
31	29, 30	bitri 278	. 2 ⊢ (𝐴 < 𝐵 ↔ (𝐴{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)}𝐵 ∨ 𝐴(((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))𝐵))
32		orass 919	. 2 ⊢ (((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐴 <ℝ 𝐵) ∨ (𝐴 = -∞ ∧ 𝐵 = +∞)) ∨ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) ∨ (𝐴 = -∞ ∧ 𝐵 ∈ ℝ))) ↔ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐴 <ℝ 𝐵) ∨ ((𝐴 = -∞ ∧ 𝐵 = +∞) ∨ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) ∨ (𝐴 = -∞ ∧ 𝐵 ∈ ℝ)))))
33	27, 31, 32	3bitr4g 317	1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 < 𝐵 ↔ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐴 <ℝ 𝐵) ∨ (𝐴 = -∞ ∧ 𝐵 = +∞)) ∨ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) ∨ (𝐴 = -∞ ∧ 𝐵 ∈ ℝ)))))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ∪ cun 3879  {csn 4525   class class class wbr 5030  {copab 5092   × cxp 5517  ℝcr 10525   <_ℝ cltrr 10530  +∞cpnf 10661  -∞cmnf 10662  ℝ^*cxr 10663   < clt 10664

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031  df-opab 5093  df-xp 5525  df-ltxr 10669

This theorem is referenced by:  xrltnr  12502  ltpnf  12503  mnflt  12506  mnfltpnf  12509  pnfnlt  12511  nltmnf  12512