Theorem ltxrlt 8287

Description: The standard less-than <_ℝ and the extended real less-than < are identical when restricted to the non-extended reals ℝ. (Contributed by NM, 13-Oct-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)

Assertion

Ref	Expression
ltxrlt	⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ 𝐴 <ℝ 𝐵))

Proof of Theorem ltxrlt

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

Step	Hyp	Ref	Expression
1		df-ltxr 8261	. . . . 5 ⊢ < = ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)} ∪ (((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ)))
2	1	breqi 4099	. . . 4 ⊢ (𝐴 < 𝐵 ↔ 𝐴({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)} ∪ (((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ)))𝐵)
3		brun 4145	. . . 4 ⊢ (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))
4	2, 3	bitri 184	. . 3 ⊢ (𝐴 < 𝐵 ↔ (𝐴{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)}𝐵 ∨ 𝐴(((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))𝐵))
5		eleq1 2294	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ ℝ ↔ 𝐴 ∈ ℝ))
6		breq1 4096	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑥 <ℝ 𝑦 ↔ 𝐴 <ℝ 𝑦))
7	5, 6	3anbi13d 1351	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦) ↔ (𝐴 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝐴 <ℝ 𝑦)))
8		eleq1 2294	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝑦 ∈ ℝ ↔ 𝐵 ∈ ℝ))
9		breq2 4097	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝐴 <ℝ 𝑦 ↔ 𝐴 <ℝ 𝐵))
10	8, 9	3anbi23d 1352	. . . . . 6 ⊢ (𝑦 = 𝐵 → ((𝐴 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝐴 <ℝ 𝑦) ↔ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 <ℝ 𝐵)))
11		eqid 2231	. . . . . 6 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)}
12	7, 10, 11	brabg 4369	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)}𝐵 ↔ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 <ℝ 𝐵)))
13		simp3 1026	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 <ℝ 𝐵) → 𝐴 <ℝ 𝐵)
14	12, 13	biimtrdi 163	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)}𝐵 → 𝐴 <ℝ 𝐵))
15		brun 4145	. . . . 5 ⊢ (𝐴(((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))𝐵 ↔ (𝐴((ℝ ∪ {-∞}) × {+∞})𝐵 ∨ 𝐴({-∞} × ℝ)𝐵))
16		brxp 4762	. . . . . . . . . . 11 ⊢ (𝐴((ℝ ∪ {-∞}) × {+∞})𝐵 ↔ (𝐴 ∈ (ℝ ∪ {-∞}) ∧ 𝐵 ∈ {+∞}))
17	16	simprbi 275	. . . . . . . . . 10 ⊢ (𝐴((ℝ ∪ {-∞}) × {+∞})𝐵 → 𝐵 ∈ {+∞})
18		elsni 3691	. . . . . . . . . 10 ⊢ (𝐵 ∈ {+∞} → 𝐵 = +∞)
19	17, 18	syl 14	. . . . . . . . 9 ⊢ (𝐴((ℝ ∪ {-∞}) × {+∞})𝐵 → 𝐵 = +∞)
20	19	a1i 9	. . . . . . . 8 ⊢ (𝐵 ∈ ℝ → (𝐴((ℝ ∪ {-∞}) × {+∞})𝐵 → 𝐵 = +∞))
21		renepnf 8269	. . . . . . . . 9 ⊢ (𝐵 ∈ ℝ → 𝐵 ≠ +∞)
22	21	neneqd 2424	. . . . . . . 8 ⊢ (𝐵 ∈ ℝ → ¬ 𝐵 = +∞)
23		pm2.24 626	. . . . . . . 8 ⊢ (𝐵 = +∞ → (¬ 𝐵 = +∞ → 𝐴 <ℝ 𝐵))
24	20, 22, 23	syl6ci 1491	. . . . . . 7 ⊢ (𝐵 ∈ ℝ → (𝐴((ℝ ∪ {-∞}) × {+∞})𝐵 → 𝐴 <ℝ 𝐵))
25	24	adantl 277	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴((ℝ ∪ {-∞}) × {+∞})𝐵 → 𝐴 <ℝ 𝐵))
26		brxp 4762	. . . . . . . . . . 11 ⊢ (𝐴({-∞} × ℝ)𝐵 ↔ (𝐴 ∈ {-∞} ∧ 𝐵 ∈ ℝ))
27	26	simplbi 274	. . . . . . . . . 10 ⊢ (𝐴({-∞} × ℝ)𝐵 → 𝐴 ∈ {-∞})
28		elsni 3691	. . . . . . . . . 10 ⊢ (𝐴 ∈ {-∞} → 𝐴 = -∞)
29	27, 28	syl 14	. . . . . . . . 9 ⊢ (𝐴({-∞} × ℝ)𝐵 → 𝐴 = -∞)
30	29	a1i 9	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → (𝐴({-∞} × ℝ)𝐵 → 𝐴 = -∞))
31		renemnf 8270	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ -∞)
32	31	neneqd 2424	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → ¬ 𝐴 = -∞)
33		pm2.24 626	. . . . . . . 8 ⊢ (𝐴 = -∞ → (¬ 𝐴 = -∞ → 𝐴 <ℝ 𝐵))
34	30, 32, 33	syl6ci 1491	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (𝐴({-∞} × ℝ)𝐵 → 𝐴 <ℝ 𝐵))
35	34	adantr 276	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴({-∞} × ℝ)𝐵 → 𝐴 <ℝ 𝐵))
36	25, 35	jaod 725	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴((ℝ ∪ {-∞}) × {+∞})𝐵 ∨ 𝐴({-∞} × ℝ)𝐵) → 𝐴 <ℝ 𝐵))
37	15, 36	biimtrid 152	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴(((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))𝐵 → 𝐴 <ℝ 𝐵))
38	14, 37	jaod 725	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)}𝐵 ∨ 𝐴(((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))𝐵) → 𝐴 <ℝ 𝐵))
39	4, 38	biimtrid 152	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 → 𝐴 <ℝ 𝐵))
40	12	3adant3 1044	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 <ℝ 𝐵) → (𝐴{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)}𝐵 ↔ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 <ℝ 𝐵)))
41	40	ibir 177	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 <ℝ 𝐵) → 𝐴{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)}𝐵)
42	41	orcd 741	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 <ℝ 𝐵) → (𝐴{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)}𝐵 ∨ 𝐴(((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))𝐵))
43	42, 4	sylibr 134	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 <ℝ 𝐵) → 𝐴 < 𝐵)
44	43	3expia 1232	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 <ℝ 𝐵 → 𝐴 < 𝐵))
45	39, 44	impbid 129	1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ 𝐴 <ℝ 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 716   ∧ w3a 1005   = wceq 1398   ∈ wcel 2202   ∪ cun 3199  {csn 3673   class class class wbr 4093  {copab 4154   × cxp 4729  ℝcr 8074   <_ℝ cltrr 8079  +∞cpnf 8253  -∞cmnf 8254   < clt 8256

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8166  ax-resscn 8167

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-xp 4737  df-pnf 8258  df-mnf 8259  df-ltxr 8261

This theorem is referenced by:  axltirr  8288  axltwlin  8289  axlttrn  8290  axltadd  8291  axapti  8292  axmulgt0  8293  axsuploc  8294  0lt1  8348  recexre  8800  recexgt0  8802  remulext1  8821  arch  9441  caucvgrelemcau  11603  caucvgre  11604