Theorem ltxrlt 11184

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		brun 5155	. . . . 5 ⊢ (𝐴(((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))𝐵 ↔ (𝐴((ℝ ∪ {-∞}) × {+∞})𝐵 ∨ 𝐴({-∞} × ℝ)𝐵))
2		brxp 5680	. . . . . . 7 ⊢ (𝐴((ℝ ∪ {-∞}) × {+∞})𝐵 ↔ (𝐴 ∈ (ℝ ∪ {-∞}) ∧ 𝐵 ∈ {+∞}))
3		elsni 4602	. . . . . . . 8 ⊢ (𝐵 ∈ {+∞} → 𝐵 = +∞)
4		pnfnre 11155	. . . . . . . . . 10 ⊢ +∞ ∉ ℝ
5	4	neli 3050	. . . . . . . . 9 ⊢ ¬ +∞ ∈ ℝ
6		simpr 486	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐵 ∈ ℝ)
7		eleq1 2826	. . . . . . . . . 10 ⊢ (𝐵 = +∞ → (𝐵 ∈ ℝ ↔ +∞ ∈ ℝ))
8	6, 7	imbitrid 243	. . . . . . . . 9 ⊢ (𝐵 = +∞ → ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → +∞ ∈ ℝ))
9	5, 8	mtoi 198	. . . . . . . 8 ⊢ (𝐵 = +∞ → ¬ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
10	3, 9	syl 17	. . . . . . 7 ⊢ (𝐵 ∈ {+∞} → ¬ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
11	2, 10	simplbiim 506	. . . . . 6 ⊢ (𝐴((ℝ ∪ {-∞}) × {+∞})𝐵 → ¬ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
12		brxp 5680	. . . . . . 7 ⊢ (𝐴({-∞} × ℝ)𝐵 ↔ (𝐴 ∈ {-∞} ∧ 𝐵 ∈ ℝ))
13		elsni 4602	. . . . . . . . 9 ⊢ (𝐴 ∈ {-∞} → 𝐴 = -∞)
14		mnfnre 11157	. . . . . . . . . . 11 ⊢ -∞ ∉ ℝ
15	14	neli 3050	. . . . . . . . . 10 ⊢ ¬ -∞ ∈ ℝ
16		simpl 484	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐴 ∈ ℝ)
17		eleq1 2826	. . . . . . . . . . 11 ⊢ (𝐴 = -∞ → (𝐴 ∈ ℝ ↔ -∞ ∈ ℝ))
18	16, 17	imbitrid 243	. . . . . . . . . 10 ⊢ (𝐴 = -∞ → ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → -∞ ∈ ℝ))
19	15, 18	mtoi 198	. . . . . . . . 9 ⊢ (𝐴 = -∞ → ¬ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
20	13, 19	syl 17	. . . . . . . 8 ⊢ (𝐴 ∈ {-∞} → ¬ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
21	20	adantr 482	. . . . . . 7 ⊢ ((𝐴 ∈ {-∞} ∧ 𝐵 ∈ ℝ) → ¬ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
22	12, 21	sylbi 216	. . . . . 6 ⊢ (𝐴({-∞} × ℝ)𝐵 → ¬ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
23	11, 22	jaoi 856	. . . . 5 ⊢ ((𝐴((ℝ ∪ {-∞}) × {+∞})𝐵 ∨ 𝐴({-∞} × ℝ)𝐵) → ¬ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
24	1, 23	sylbi 216	. . . 4 ⊢ (𝐴(((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))𝐵 → ¬ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
25	24	con2i 139	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ¬ 𝐴(((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))𝐵)
26		df-ltxr 11153	. . . . . . 7 ⊢ < = ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)} ∪ (((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ)))
27	26	equncomi 4114	. . . . . 6 ⊢ < = ((((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ)) ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)})
28	27	breqi 5110	. . . . 5 ⊢ (𝐴 < 𝐵 ↔ 𝐴((((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ)) ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)})𝐵)
29		brun 5155	. . . . 5 ⊢ (𝐴((((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ)) ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)})𝐵 ↔ (𝐴(((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))𝐵 ∨ 𝐴{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)}𝐵))
30		df-or 847	. . . . 5 ⊢ ((𝐴(((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))𝐵 ∨ 𝐴{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)}𝐵) ↔ (¬ 𝐴(((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))𝐵 → 𝐴{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)}𝐵))
31	28, 29, 30	3bitri 297	. . . 4 ⊢ (𝐴 < 𝐵 ↔ (¬ 𝐴(((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))𝐵 → 𝐴{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)}𝐵))
32		biimt 361	. . . 4 ⊢ (¬ 𝐴(((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))𝐵 → (𝐴{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)}𝐵 ↔ (¬ 𝐴(((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))𝐵 → 𝐴{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)}𝐵)))
33	31, 32	bitr4id 290	. . 3 ⊢ (¬ 𝐴(((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))𝐵 → (𝐴 < 𝐵 ↔ 𝐴{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)}𝐵))
34	25, 33	syl 17	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ 𝐴{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)}𝐵))
35		breq12 5109	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 <ℝ 𝑦 ↔ 𝐴 <ℝ 𝐵))
36		df-3an 1090	. . . . 5 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦) ↔ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝑥 <ℝ 𝑦))
37	36	opabbii 5171	. . . 4 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝑥 <ℝ 𝑦)}
38	35, 37	brab2a 5724	. . 3 ⊢ (𝐴{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)}𝐵 ↔ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐴 <ℝ 𝐵))
39	38	baibr 538	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 <ℝ 𝐵 ↔ 𝐴{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)}𝐵))
40	34, 39	bitr4d 282	1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ 𝐴 <ℝ 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 397   ∨ wo 846   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   ∪ cun 3907  {csn 4585   class class class wbr 5104  {copab 5166   × cxp 5630  ℝcr 11009   <_ℝ cltrr 11014  +∞cpnf 11145  -∞cmnf 11146   < clt 11148

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2709  ax-sep 5255  ax-nul 5262  ax-pow 5319  ax-pr 5383  ax-un 7665  ax-resscn 11067

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3064  df-rex 3073  df-rab 3407  df-v 3446  df-sbc 3739  df-csb 3855  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-br 5105  df-opab 5167  df-mpt 5188  df-id 5530  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6446  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-er 8607  df-en 8843  df-dom 8844  df-sdom 8845  df-pnf 11150  df-mnf 11151  df-ltxr 11153

This theorem is referenced by:  axlttri  11185  axlttrn  11186  axltadd  11187  axmulgt0  11188  axsup  11189