Theorem ltxrlt 7178

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 7158	. . . . 5 ⊢ < = ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)} ∪ (((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ)))
2	1	breqi 3791	. . . 4 ⊢ (𝐴 < 𝐵 ↔ 𝐴({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)} ∪ (((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ)))𝐵)
3		brun 3831	. . . 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 182	. . 3 ⊢ (𝐴 < 𝐵 ↔ (𝐴{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)}𝐵 ∨ 𝐴(((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))𝐵))
5		eleq1 2141	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ ℝ ↔ 𝐴 ∈ ℝ))
6		breq1 3788	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑥 <ℝ 𝑦 ↔ 𝐴 <ℝ 𝑦))
7	5, 6	3anbi13d 1245	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦) ↔ (𝐴 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝐴 <ℝ 𝑦)))
8		eleq1 2141	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝑦 ∈ ℝ ↔ 𝐵 ∈ ℝ))
9		breq2 3789	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝐴 <ℝ 𝑦 ↔ 𝐴 <ℝ 𝐵))
10	8, 9	3anbi23d 1246	. . . . . 6 ⊢ (𝑦 = 𝐵 → ((𝐴 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝐴 <ℝ 𝑦) ↔ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 <ℝ 𝐵)))
11		eqid 2081	. . . . . 6 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)}
12	7, 10, 11	brabg 4024	. . . . 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 940	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 <ℝ 𝐵) → 𝐴 <ℝ 𝐵)
14	12, 13	syl6bi 161	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)}𝐵 → 𝐴 <ℝ 𝐵))
15		brun 3831	. . . . 5 ⊢ (𝐴(((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))𝐵 ↔ (𝐴((ℝ ∪ {-∞}) × {+∞})𝐵 ∨ 𝐴({-∞} × ℝ)𝐵))
16		brxp 4393	. . . . . . . . . . 11 ⊢ (𝐴((ℝ ∪ {-∞}) × {+∞})𝐵 ↔ (𝐴 ∈ (ℝ ∪ {-∞}) ∧ 𝐵 ∈ {+∞}))
17	16	simprbi 269	. . . . . . . . . 10 ⊢ (𝐴((ℝ ∪ {-∞}) × {+∞})𝐵 → 𝐵 ∈ {+∞})
18		elsni 3416	. . . . . . . . . 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 7166	. . . . . . . . 9 ⊢ (𝐵 ∈ ℝ → 𝐵 ≠ +∞)
22	21	neneqd 2266	. . . . . . . 8 ⊢ (𝐵 ∈ ℝ → ¬ 𝐵 = +∞)
23		pm2.24 583	. . . . . . . 8 ⊢ (𝐵 = +∞ → (¬ 𝐵 = +∞ → 𝐴 <ℝ 𝐵))
24	20, 22, 23	syl6ci 1374	. . . . . . 7 ⊢ (𝐵 ∈ ℝ → (𝐴((ℝ ∪ {-∞}) × {+∞})𝐵 → 𝐴 <ℝ 𝐵))
25	24	adantl 271	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴((ℝ ∪ {-∞}) × {+∞})𝐵 → 𝐴 <ℝ 𝐵))
26		brxp 4393	. . . . . . . . . . 11 ⊢ (𝐴({-∞} × ℝ)𝐵 ↔ (𝐴 ∈ {-∞} ∧ 𝐵 ∈ ℝ))
27	26	simplbi 268	. . . . . . . . . 10 ⊢ (𝐴({-∞} × ℝ)𝐵 → 𝐴 ∈ {-∞})
28		elsni 3416	. . . . . . . . . 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 7167	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ -∞)
32	31	neneqd 2266	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → ¬ 𝐴 = -∞)
33		pm2.24 583	. . . . . . . 8 ⊢ (𝐴 = -∞ → (¬ 𝐴 = -∞ → 𝐴 <ℝ 𝐵))
34	30, 32, 33	syl6ci 1374	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (𝐴({-∞} × ℝ)𝐵 → 𝐴 <ℝ 𝐵))
35	34	adantr 270	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴({-∞} × ℝ)𝐵 → 𝐴 <ℝ 𝐵))
36	25, 35	jaod 669	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴((ℝ ∪ {-∞}) × {+∞})𝐵 ∨ 𝐴({-∞} × ℝ)𝐵) → 𝐴 <ℝ 𝐵))
37	15, 36	syl5bi 150	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴(((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))𝐵 → 𝐴 <ℝ 𝐵))
38	14, 37	jaod 669	. . 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 150	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 → 𝐴 <ℝ 𝐵))
40	12	3adant3 958	. . . . . 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 175	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 <ℝ 𝐵) → 𝐴{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)}𝐵)
42	41	orcd 684	. . . 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 132	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 <ℝ 𝐵) → 𝐴 < 𝐵)
44	43	3expia 1140	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 <ℝ 𝐵 → 𝐴 < 𝐵))
45	39, 44	impbid 127	1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ 𝐴 <ℝ 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 102   ↔ wb 103   ∨ wo 661   ∧ w3a 919   = wceq 1284   ∈ wcel 1433   ∪ cun 2971  {csn 3398   class class class wbr 3785  {copab 3838   × cxp 4361  ℝcr 6980   <_ℝ cltrr 6985  +∞cpnf 7150  -∞cmnf 7151   < clt 7153

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-cnex 7067  ax-resscn 7068

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-nel 2340  df-ral 2353  df-rex 2354  df-rab 2357  df-v 2603  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-xp 4369  df-pnf 7155  df-mnf 7156  df-ltxr 7158

This theorem is referenced by:  axltirr  7179  axltwlin  7180  axlttrn  7181  axltadd  7182  axapti  7183  axmulgt0  7184  0lt1  7236  recexre  7678  recexgt0  7680  remulext1  7699  arch  8285  caucvgrelemcau  9866  caucvgre  9867