Theorem ltxrlt 7830

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 7805	. . . . 5 ⊢ < = ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)} ∪ (((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ)))
2	1	breqi 3935	. . . 4 ⊢ (𝐴 < 𝐵 ↔ 𝐴({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)} ∪ (((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ)))𝐵)
3		brun 3979	. . . 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 183	. . 3 ⊢ (𝐴 < 𝐵 ↔ (𝐴{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)}𝐵 ∨ 𝐴(((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))𝐵))
5		eleq1 2202	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ ℝ ↔ 𝐴 ∈ ℝ))
6		breq1 3932	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑥 <ℝ 𝑦 ↔ 𝐴 <ℝ 𝑦))
7	5, 6	3anbi13d 1292	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦) ↔ (𝐴 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝐴 <ℝ 𝑦)))
8		eleq1 2202	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝑦 ∈ ℝ ↔ 𝐵 ∈ ℝ))
9		breq2 3933	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝐴 <ℝ 𝑦 ↔ 𝐴 <ℝ 𝐵))
10	8, 9	3anbi23d 1293	. . . . . 6 ⊢ (𝑦 = 𝐵 → ((𝐴 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝐴 <ℝ 𝑦) ↔ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 <ℝ 𝐵)))
11		eqid 2139	. . . . . 6 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)}
12	7, 10, 11	brabg 4191	. . . . 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 983	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 <ℝ 𝐵) → 𝐴 <ℝ 𝐵)
14	12, 13	syl6bi 162	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)}𝐵 → 𝐴 <ℝ 𝐵))
15		brun 3979	. . . . 5 ⊢ (𝐴(((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))𝐵 ↔ (𝐴((ℝ ∪ {-∞}) × {+∞})𝐵 ∨ 𝐴({-∞} × ℝ)𝐵))
16		brxp 4570	. . . . . . . . . . 11 ⊢ (𝐴((ℝ ∪ {-∞}) × {+∞})𝐵 ↔ (𝐴 ∈ (ℝ ∪ {-∞}) ∧ 𝐵 ∈ {+∞}))
17	16	simprbi 273	. . . . . . . . . 10 ⊢ (𝐴((ℝ ∪ {-∞}) × {+∞})𝐵 → 𝐵 ∈ {+∞})
18		elsni 3545	. . . . . . . . . 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 7813	. . . . . . . . 9 ⊢ (𝐵 ∈ ℝ → 𝐵 ≠ +∞)
22	21	neneqd 2329	. . . . . . . 8 ⊢ (𝐵 ∈ ℝ → ¬ 𝐵 = +∞)
23		pm2.24 610	. . . . . . . 8 ⊢ (𝐵 = +∞ → (¬ 𝐵 = +∞ → 𝐴 <ℝ 𝐵))
24	20, 22, 23	syl6ci 1421	. . . . . . 7 ⊢ (𝐵 ∈ ℝ → (𝐴((ℝ ∪ {-∞}) × {+∞})𝐵 → 𝐴 <ℝ 𝐵))
25	24	adantl 275	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴((ℝ ∪ {-∞}) × {+∞})𝐵 → 𝐴 <ℝ 𝐵))
26		brxp 4570	. . . . . . . . . . 11 ⊢ (𝐴({-∞} × ℝ)𝐵 ↔ (𝐴 ∈ {-∞} ∧ 𝐵 ∈ ℝ))
27	26	simplbi 272	. . . . . . . . . 10 ⊢ (𝐴({-∞} × ℝ)𝐵 → 𝐴 ∈ {-∞})
28		elsni 3545	. . . . . . . . . 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 7814	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ -∞)
32	31	neneqd 2329	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → ¬ 𝐴 = -∞)
33		pm2.24 610	. . . . . . . 8 ⊢ (𝐴 = -∞ → (¬ 𝐴 = -∞ → 𝐴 <ℝ 𝐵))
34	30, 32, 33	syl6ci 1421	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (𝐴({-∞} × ℝ)𝐵 → 𝐴 <ℝ 𝐵))
35	34	adantr 274	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴({-∞} × ℝ)𝐵 → 𝐴 <ℝ 𝐵))
36	25, 35	jaod 706	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴((ℝ ∪ {-∞}) × {+∞})𝐵 ∨ 𝐴({-∞} × ℝ)𝐵) → 𝐴 <ℝ 𝐵))
37	15, 36	syl5bi 151	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴(((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))𝐵 → 𝐴 <ℝ 𝐵))
38	14, 37	jaod 706	. . 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	syl5bi 151	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 → 𝐴 <ℝ 𝐵))
40	12	3adant3 1001	. . . . . 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 176	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 <ℝ 𝐵) → 𝐴{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)}𝐵)
42	41	orcd 722	. . . 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 133	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 <ℝ 𝐵) → 𝐴 < 𝐵)
44	43	3expia 1183	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 <ℝ 𝐵 → 𝐴 < 𝐵))
45	39, 44	impbid 128	1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ 𝐴 <ℝ 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 697   ∧ w3a 962   = wceq 1331   ∈ wcel 1480   ∪ cun 3069  {csn 3527   class class class wbr 3929  {copab 3988   × cxp 4537  ℝcr 7619   <_ℝ cltrr 7624  +∞cpnf 7797  -∞cmnf 7798   < clt 7800

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7711  ax-resscn 7712

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-xp 4545  df-pnf 7802  df-mnf 7803  df-ltxr 7805

This theorem is referenced by:  axltirr  7831  axltwlin  7832  axlttrn  7833  axltadd  7834  axapti  7835  axmulgt0  7836  axsuploc  7837  0lt1  7889  recexre  8340  recexgt0  8342  remulext1  8361  arch  8974  caucvgrelemcau  10752  caucvgre  10753