Theorem icorempo 37334

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 13390	. . . 4 ⊢ [,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)})
3	2	reseq1i 5996	. . 3 ⊢ ([,) ↾ (ℝ × ℝ)) = ((𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}) ↾ (ℝ × ℝ))
4		ressxr 11303	. . . 4 ⊢ ℝ ⊆ ℝ*
5		resmpo 7553	. . . 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 692	. . 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 2763	. 2 ⊢ ([,) ↾ (ℝ × ℝ)) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)})
8		nfv 1912	. . . 4 ⊢ Ⅎ𝑧(𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)
9		nfrab1 3454	. . . 4 ⊢ Ⅎ𝑧{𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}
10		nfrab1 3454	. . . 4 ⊢ Ⅎ𝑧{𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}
11		rabid 3455	. . . . . . . 8 ⊢ (𝑧 ∈ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} ↔ (𝑧 ∈ ℝ* ∧ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)))
12		rexr 11305	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ*)
13		nltmnf 13169	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ℝ* → ¬ 𝑥 < -∞)
14	12, 13	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℝ → ¬ 𝑥 < -∞)
15		renemnf 11308	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ℝ → 𝑥 ≠ -∞)
16	15	neneqd 2943	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℝ → ¬ 𝑥 = -∞)
17	14, 16	jca 511	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℝ → (¬ 𝑥 < -∞ ∧ ¬ 𝑥 = -∞))
18		pm4.56 990	. . . . . . . . . . . . . . 15 ⊢ ((¬ 𝑥 < -∞ ∧ ¬ 𝑥 = -∞) ↔ ¬ (𝑥 < -∞ ∨ 𝑥 = -∞))
19	17, 18	sylib 218	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ → ¬ (𝑥 < -∞ ∨ 𝑥 = -∞))
20		mnfxr 11316	. . . . . . . . . . . . . . 15 ⊢ -∞ ∈ ℝ*
21		xrleloe 13183	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ* ∧ -∞ ∈ ℝ*) → (𝑥 ≤ -∞ ↔ (𝑥 < -∞ ∨ 𝑥 = -∞)))
22	12, 20, 21	sylancl 586	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ → (𝑥 ≤ -∞ ↔ (𝑥 < -∞ ∨ 𝑥 = -∞)))
23	19, 22	mtbird 325	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ → ¬ 𝑥 ≤ -∞)
24		breq2 5152	. . . . . . . . . . . . . 14 ⊢ (𝑧 = -∞ → (𝑥 ≤ 𝑧 ↔ 𝑥 ≤ -∞))
25	24	notbid 318	. . . . . . . . . . . . 13 ⊢ (𝑧 = -∞ → (¬ 𝑥 ≤ 𝑧 ↔ ¬ 𝑥 ≤ -∞))
26	23, 25	syl5ibrcom 247	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ → (𝑧 = -∞ → ¬ 𝑥 ≤ 𝑧))
27	26	con2d 134	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ → (𝑥 ≤ 𝑧 → ¬ 𝑧 = -∞))
28		rexr 11305	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℝ → 𝑦 ∈ ℝ*)
29		pnfnlt 13168	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℝ* → ¬ +∞ < 𝑦)
30		breq1 5151	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = +∞ → (𝑧 < 𝑦 ↔ +∞ < 𝑦))
31	30	notbid 318	. . . . . . . . . . . . . 14 ⊢ (𝑧 = +∞ → (¬ 𝑧 < 𝑦 ↔ ¬ +∞ < 𝑦))
32	29, 31	syl5ibrcom 247	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℝ* → (𝑧 = +∞ → ¬ 𝑧 < 𝑦))
33	32	con2d 134	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℝ* → (𝑧 < 𝑦 → ¬ 𝑧 = +∞))
34	28, 33	syl 17	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℝ → (𝑧 < 𝑦 → ¬ 𝑧 = +∞))
35	27, 34	im2anan9 620	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦) → (¬ 𝑧 = -∞ ∧ ¬ 𝑧 = +∞)))
36	35	anim2d 612	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((𝑧 ∈ ℝ* ∧ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)) → (𝑧 ∈ ℝ* ∧ (¬ 𝑧 = -∞ ∧ ¬ 𝑧 = +∞))))
37		renepnf 11307	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ℝ → 𝑧 ≠ +∞)
38	37	neneqd 2943	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ℝ → ¬ 𝑧 = +∞)
39	38	pm4.71i 559	. . . . . . . . . 10 ⊢ (𝑧 ∈ ℝ ↔ (𝑧 ∈ ℝ ∧ ¬ 𝑧 = +∞))
40		xrnemnf 13157	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ ℝ* ∧ 𝑧 ≠ -∞) ↔ (𝑧 ∈ ℝ ∨ 𝑧 = +∞))
41	40	anbi1i 624	. . . . . . . . . . 11 ⊢ (((𝑧 ∈ ℝ* ∧ 𝑧 ≠ -∞) ∧ ¬ 𝑧 = +∞) ↔ ((𝑧 ∈ ℝ ∨ 𝑧 = +∞) ∧ ¬ 𝑧 = +∞))
42		df-ne 2939	. . . . . . . . . . . . 13 ⊢ (𝑧 ≠ -∞ ↔ ¬ 𝑧 = -∞)
43	42	anbi2i 623	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ ℝ* ∧ 𝑧 ≠ -∞) ↔ (𝑧 ∈ ℝ* ∧ ¬ 𝑧 = -∞))
44	43	anbi1i 624	. . . . . . . . . . 11 ⊢ (((𝑧 ∈ ℝ* ∧ 𝑧 ≠ -∞) ∧ ¬ 𝑧 = +∞) ↔ ((𝑧 ∈ ℝ* ∧ ¬ 𝑧 = -∞) ∧ ¬ 𝑧 = +∞))
45		pm5.61 1002	. . . . . . . . . . 11 ⊢ (((𝑧 ∈ ℝ ∨ 𝑧 = +∞) ∧ ¬ 𝑧 = +∞) ↔ (𝑧 ∈ ℝ ∧ ¬ 𝑧 = +∞))
46	41, 44, 45	3bitr3i 301	. . . . . . . . . 10 ⊢ (((𝑧 ∈ ℝ* ∧ ¬ 𝑧 = -∞) ∧ ¬ 𝑧 = +∞) ↔ (𝑧 ∈ ℝ ∧ ¬ 𝑧 = +∞))
47		anass 468	. . . . . . . . . 10 ⊢ (((𝑧 ∈ ℝ* ∧ ¬ 𝑧 = -∞) ∧ ¬ 𝑧 = +∞) ↔ (𝑧 ∈ ℝ* ∧ (¬ 𝑧 = -∞ ∧ ¬ 𝑧 = +∞)))
48	39, 46, 47	3bitr2ri 300	. . . . . . . . 9 ⊢ ((𝑧 ∈ ℝ* ∧ (¬ 𝑧 = -∞ ∧ ¬ 𝑧 = +∞)) ↔ 𝑧 ∈ ℝ)
49	36, 48	imbitrdi 251	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((𝑧 ∈ ℝ* ∧ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)) → 𝑧 ∈ ℝ))
50	11, 49	biimtrid 242	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑧 ∈ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} → 𝑧 ∈ ℝ))
51	11	simprbi 496	. . . . . . . 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 512	. . . . . 6 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑧 ∈ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} → (𝑧 ∈ ℝ ∧ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦))))
54		rabid 3455	. . . . . 6 ⊢ (𝑧 ∈ {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} ↔ (𝑧 ∈ ℝ ∧ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)))
55	53, 54	imbitrrdi 252	. . . . 5 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑧 ∈ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} → 𝑧 ∈ {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}))
56		rabss2 4088	. . . . . . 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 3991	. . . . 5 ⊢ (𝑧 ∈ {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} → 𝑧 ∈ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)})
59	55, 58	impbid1 225	. . . 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 4015	. . 3 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} = {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)})
61	60	mpoeq3ia 7511	. 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 2767	1 ⊢ 𝐹 = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)})

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1537   ∈ wcel 2106   ≠ wne 2938  {crab 3433   ⊆ wss 3963   class class class wbr 5148   × cxp 5687   ↾ cres 5691   ∈ cmpo 7433  ℝcr 11152  +∞cpnf 11290  -∞cmnf 11291  ℝ^*cxr 11292   < clt 11293   ≤ cle 11294  [,)cico 13386

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-pre-lttri 11227  ax-pre-lttrn 11228

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-po 5597  df-so 5598  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-oprab 7435  df-mpo 7436  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-ico 13390

This theorem is referenced by:  icoreresf  37335  icoreval  37336