Theorem ltxrlt 8109

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 8083	. . . . 5 ⊢ < = ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)} ∪ (((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ)))
2	1	breqi 4040	. . . 4 ⊢ (𝐴 < 𝐵 ↔ 𝐴({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)} ∪ (((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ)))𝐵)
3		brun 4085	. . . 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 2259	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ ℝ ↔ 𝐴 ∈ ℝ))
6		breq1 4037	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑥 <ℝ 𝑦 ↔ 𝐴 <ℝ 𝑦))
7	5, 6	3anbi13d 1325	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦) ↔ (𝐴 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝐴 <ℝ 𝑦)))
8		eleq1 2259	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝑦 ∈ ℝ ↔ 𝐵 ∈ ℝ))
9		breq2 4038	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝐴 <ℝ 𝑦 ↔ 𝐴 <ℝ 𝐵))
10	8, 9	3anbi23d 1326	. . . . . 6 ⊢ (𝑦 = 𝐵 → ((𝐴 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝐴 <ℝ 𝑦) ↔ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 <ℝ 𝐵)))
11		eqid 2196	. . . . . 6 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)}
12	7, 10, 11	brabg 4304	. . . . 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 1001	. . . . 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 4085	. . . . 5 ⊢ (𝐴(((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))𝐵 ↔ (𝐴((ℝ ∪ {-∞}) × {+∞})𝐵 ∨ 𝐴({-∞} × ℝ)𝐵))
16		brxp 4695	. . . . . . . . . . 11 ⊢ (𝐴((ℝ ∪ {-∞}) × {+∞})𝐵 ↔ (𝐴 ∈ (ℝ ∪ {-∞}) ∧ 𝐵 ∈ {+∞}))
17	16	simprbi 275	. . . . . . . . . 10 ⊢ (𝐴((ℝ ∪ {-∞}) × {+∞})𝐵 → 𝐵 ∈ {+∞})
18		elsni 3641	. . . . . . . . . 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 8091	. . . . . . . . 9 ⊢ (𝐵 ∈ ℝ → 𝐵 ≠ +∞)
22	21	neneqd 2388	. . . . . . . 8 ⊢ (𝐵 ∈ ℝ → ¬ 𝐵 = +∞)
23		pm2.24 622	. . . . . . . 8 ⊢ (𝐵 = +∞ → (¬ 𝐵 = +∞ → 𝐴 <ℝ 𝐵))
24	20, 22, 23	syl6ci 1456	. . . . . . 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 4695	. . . . . . . . . . 11 ⊢ (𝐴({-∞} × ℝ)𝐵 ↔ (𝐴 ∈ {-∞} ∧ 𝐵 ∈ ℝ))
27	26	simplbi 274	. . . . . . . . . 10 ⊢ (𝐴({-∞} × ℝ)𝐵 → 𝐴 ∈ {-∞})
28		elsni 3641	. . . . . . . . . 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 8092	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ -∞)
32	31	neneqd 2388	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → ¬ 𝐴 = -∞)
33		pm2.24 622	. . . . . . . 8 ⊢ (𝐴 = -∞ → (¬ 𝐴 = -∞ → 𝐴 <ℝ 𝐵))
34	30, 32, 33	syl6ci 1456	. . . . . . 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 718	. . . . 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 718	. . 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 1019	. . . . . 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 734	. . . 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 1207	. 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 709   ∧ w3a 980   = wceq 1364   ∈ wcel 2167   ∪ cun 3155  {csn 3623   class class class wbr 4034  {copab 4094   × cxp 4662  ℝcr 7895   <_ℝ cltrr 7900  +∞cpnf 8075  -∞cmnf 8076   < clt 8078

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-br 4035  df-opab 4096  df-xp 4670  df-pnf 8080  df-mnf 8081  df-ltxr 8083

This theorem is referenced by:  axltirr  8110  axltwlin  8111  axlttrn  8112  axltadd  8113  axapti  8114  axmulgt0  8115  axsuploc  8116  0lt1  8170  recexre  8622  recexgt0  8624  remulext1  8643  arch  9263  caucvgrelemcau  11162  caucvgre  11163