Theorem icorempo 38037

Description: Closed-below, open-above intervals of reals. (Contributed by ML, 26-Jul-2020.)

Hypothesis

Ref	Expression
icorempo.1	⊢ 𝐹 = ([,) ↾ (ℝ × ℝ))

Assertion

Ref	Expression
icorempo	⊢ 𝐹 = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)})

Distinct variable group:   𝑥,𝑦,𝑧

Allowed substitution hints:   𝐹(𝑥, 𝑦, 𝑧)

Proof of Theorem icorempo

Step	Hyp	Ref	Expression
1		icorempo.1	. 2 ⊢ 𝐹 = ([,) ↾ (ℝ × ℝ))
2		df-ico 13396	. . . 4 ⊢ [,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)})
3	2	reseq1i 5979	. . 3 ⊢ ([,) ↾ (ℝ × ℝ)) = ((𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}) ↾ (ℝ × ℝ))
4		ressxr 11271	. . . 4 ⊢ ℝ ⊆ ℝ*
5		resmpo 7543	. . . 4 ⊢ ((ℝ ⊆ ℝ* ∧ ℝ ⊆ ℝ*) → ((𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}) ↾ (ℝ × ℝ)) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}))
6	4, 4, 5	mp2an 705	. . 3 ⊢ ((𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}) ↾ (ℝ × ℝ)) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)})
7	3, 6	eqtri 2789	. 2 ⊢ ([,) ↾ (ℝ × ℝ)) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)})
8		nfv 1947	. . . 4 ⊢ Ⅎ𝑧(𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)
9		nfrab1 3439	. . . 4 ⊢ Ⅎ𝑧{𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}
10		nfrab1 3439	. . . 4 ⊢ Ⅎ𝑧{𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}
11		rabid 3440	. . . . . . . 8 ⊢ (𝑧 ∈ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} ↔ (𝑧 ∈ ℝ* ∧ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)))
12		rexr 11273	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ*)
13		nltmnf 13172	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ℝ* → ¬ 𝑥 < -∞)
14	12, 13	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℝ → ¬ 𝑥 < -∞)
15		renemnf 11276	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ℝ → 𝑥 ≠ -∞)
16	15	neneqd 2966	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℝ → ¬ 𝑥 = -∞)
17	14, 16	jca 521	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℝ → (¬ 𝑥 < -∞ ∧ ¬ 𝑥 = -∞))
18		pm4.56 1004	. . . . . . . . . . . . . . 15 ⊢ ((¬ 𝑥 < -∞ ∧ ¬ 𝑥 = -∞) ↔ ¬ (𝑥 < -∞ ∨ 𝑥 = -∞))
19	17, 18	sylib 221	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ → ¬ (𝑥 < -∞ ∨ 𝑥 = -∞))
20		mnfxr 11284	. . . . . . . . . . . . . . 15 ⊢ -∞ ∈ ℝ*
21		xrleloe 13187	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ* ∧ -∞ ∈ ℝ*) → (𝑥 ≤ -∞ ↔ (𝑥 < -∞ ∨ 𝑥 = -∞)))
22	12, 20, 21	sylancl 598	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ → (𝑥 ≤ -∞ ↔ (𝑥 < -∞ ∨ 𝑥 = -∞)))
23	19, 22	mtbird 328	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ → ¬ 𝑥 ≤ -∞)
24		breq2 5118	. . . . . . . . . . . . . 14 ⊢ (𝑧 = -∞ → (𝑥 ≤ 𝑧 ↔ 𝑥 ≤ -∞))
25	24	notbid 321	. . . . . . . . . . . . 13 ⊢ (𝑧 = -∞ → (¬ 𝑥 ≤ 𝑧 ↔ ¬ 𝑥 ≤ -∞))
26	23, 25	syl5ibrcom 250	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ → (𝑧 = -∞ → ¬ 𝑥 ≤ 𝑧))
27	26	con2d 135	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ → (𝑥 ≤ 𝑧 → ¬ 𝑧 = -∞))
28		rexr 11273	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℝ → 𝑦 ∈ ℝ*)
29		pnfnlt 13171	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℝ* → ¬ +∞ < 𝑦)
30		breq1 5117	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = +∞ → (𝑧 < 𝑦 ↔ +∞ < 𝑦))
31	30	notbid 321	. . . . . . . . . . . . . 14 ⊢ (𝑧 = +∞ → (¬ 𝑧 < 𝑦 ↔ ¬ +∞ < 𝑦))
32	29, 31	syl5ibrcom 250	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℝ* → (𝑧 = +∞ → ¬ 𝑧 < 𝑦))
33	32	con2d 135	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℝ* → (𝑧 < 𝑦 → ¬ 𝑧 = +∞))
34	28, 33	syl 18	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℝ → (𝑧 < 𝑦 → ¬ 𝑧 = +∞))
35	27, 34	im2anan9 632	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦) → (¬ 𝑧 = -∞ ∧ ¬ 𝑧 = +∞)))
36	35	anim2d 624	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((𝑧 ∈ ℝ* ∧ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)) → (𝑧 ∈ ℝ* ∧ (¬ 𝑧 = -∞ ∧ ¬ 𝑧 = +∞))))
37		renepnf 11275	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ℝ → 𝑧 ≠ +∞)
38	37	neneqd 2966	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ℝ → ¬ 𝑧 = +∞)
39	38	pm4.71i 569	. . . . . . . . . 10 ⊢ (𝑧 ∈ ℝ ↔ (𝑧 ∈ ℝ ∧ ¬ 𝑧 = +∞))
40		xrnemnf 13160	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ ℝ* ∧ 𝑧 ≠ -∞) ↔ (𝑧 ∈ ℝ ∨ 𝑧 = +∞))
41	40	anbi1i 636	. . . . . . . . . . 11 ⊢ (((𝑧 ∈ ℝ* ∧ 𝑧 ≠ -∞) ∧ ¬ 𝑧 = +∞) ↔ ((𝑧 ∈ ℝ ∨ 𝑧 = +∞) ∧ ¬ 𝑧 = +∞))
42		df-ne 2962	. . . . . . . . . . . . 13 ⊢ (𝑧 ≠ -∞ ↔ ¬ 𝑧 = -∞)
43	42	anbi2i 635	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ ℝ* ∧ 𝑧 ≠ -∞) ↔ (𝑧 ∈ ℝ* ∧ ¬ 𝑧 = -∞))
44	43	anbi1i 636	. . . . . . . . . . 11 ⊢ (((𝑧 ∈ ℝ* ∧ 𝑧 ≠ -∞) ∧ ¬ 𝑧 = +∞) ↔ ((𝑧 ∈ ℝ* ∧ ¬ 𝑧 = -∞) ∧ ¬ 𝑧 = +∞))
45		pm5.61 1016	. . . . . . . . . . 11 ⊢ (((𝑧 ∈ ℝ ∨ 𝑧 = +∞) ∧ ¬ 𝑧 = +∞) ↔ (𝑧 ∈ ℝ ∧ ¬ 𝑧 = +∞))
46	41, 44, 45	3bitr3i 304	. . . . . . . . . 10 ⊢ (((𝑧 ∈ ℝ* ∧ ¬ 𝑧 = -∞) ∧ ¬ 𝑧 = +∞) ↔ (𝑧 ∈ ℝ ∧ ¬ 𝑧 = +∞))
47		anass 474	. . . . . . . . . 10 ⊢ (((𝑧 ∈ ℝ* ∧ ¬ 𝑧 = -∞) ∧ ¬ 𝑧 = +∞) ↔ (𝑧 ∈ ℝ* ∧ (¬ 𝑧 = -∞ ∧ ¬ 𝑧 = +∞)))
48	39, 46, 47	3bitr2ri 303	. . . . . . . . 9 ⊢ ((𝑧 ∈ ℝ* ∧ (¬ 𝑧 = -∞ ∧ ¬ 𝑧 = +∞)) ↔ 𝑧 ∈ ℝ)
49	36, 48	imbitrdi 254	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((𝑧 ∈ ℝ* ∧ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)) → 𝑧 ∈ ℝ))
50	11, 49	biimtrid 245	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑧 ∈ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} → 𝑧 ∈ ℝ))
51	11	simprbi 503	. . . . . . . 8 ⊢ (𝑧 ∈ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} → (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦))
52	51	a1i 11	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑧 ∈ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} → (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)))
53	50, 52	jcad 522	. . . . . 6 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑧 ∈ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} → (𝑧 ∈ ℝ ∧ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦))))
54		rabid 3440	. . . . . 6 ⊢ (𝑧 ∈ {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} ↔ (𝑧 ∈ ℝ ∧ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)))
55	53, 54	imbitrrdi 255	. . . . 5 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑧 ∈ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} → 𝑧 ∈ {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}))
56		rabss2 4034	. . . . . . 7 ⊢ (ℝ ⊆ ℝ* → {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} ⊆ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)})
57	4, 56	ax-mp 5	. . . . . 6 ⊢ {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} ⊆ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}
58	57	sseli 3936	. . . . 5 ⊢ (𝑧 ∈ {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} → 𝑧 ∈ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)})
59	55, 58	impbid1 228	. . . 4 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑧 ∈ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} ↔ 𝑧 ∈ {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}))
60	8, 9, 10, 59	eqrd 3959	. . 3 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} = {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)})
61	60	mpoeq3ia 7501	. 2 ⊢ (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)})
62	1, 7, 61	3eqtri 2793	1 ⊢ 𝐹 = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)})

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2146   ≠ wne 2961  {crab 3419   ⊆ wss 3908   class class class wbr 5114   × cxp 5664   ↾ cres 5668   ∈ cmpo 7425  ℝcr 11117  +∞cpnf 11258  -∞cmnf 11259  ℝ^*cxr 11260   < clt 11261   ≤ cle 11262  [,)cico 13392

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745  ax-cnex 11174  ax-resscn 11175  ax-pre-lttri 11192  ax-pre-lttrn 11193

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-nel 3068  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-po 5574  df-so 5575  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-oprab 7427  df-mpo 7428  df-er 8703  df-en 8953  df-dom 8954  df-sdom 8955  df-pnf 11263  df-mnf 11264  df-xr 11265  df-ltxr 11266  df-le 11267  df-ico 13396

This theorem is used by:  icoreresf  38038  icoreval  38039