Theorem icorempo 37603

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 13279	. . . 4 ⊢ [,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)})
3	2	reseq1i 5942	. . 3 ⊢ ([,) ↾ (ℝ × ℝ)) = ((𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}) ↾ (ℝ × ℝ))
4		ressxr 11188	. . . 4 ⊢ ℝ ⊆ ℝ*
5		resmpo 7488	. . . 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 693	. . 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 2760	. 2 ⊢ ([,) ↾ (ℝ × ℝ)) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)})
8		nfv 1916	. . . 4 ⊢ Ⅎ𝑧(𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)
9		nfrab1 3421	. . . 4 ⊢ Ⅎ𝑧{𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}
10		nfrab1 3421	. . . 4 ⊢ Ⅎ𝑧{𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}
11		rabid 3422	. . . . . . . 8 ⊢ (𝑧 ∈ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} ↔ (𝑧 ∈ ℝ* ∧ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)))
12		rexr 11190	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ*)
13		nltmnf 13055	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ℝ* → ¬ 𝑥 < -∞)
14	12, 13	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℝ → ¬ 𝑥 < -∞)
15		renemnf 11193	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ℝ → 𝑥 ≠ -∞)
16	15	neneqd 2938	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℝ → ¬ 𝑥 = -∞)
17	14, 16	jca 511	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℝ → (¬ 𝑥 < -∞ ∧ ¬ 𝑥 = -∞))
18		pm4.56 991	. . . . . . . . . . . . . . 15 ⊢ ((¬ 𝑥 < -∞ ∧ ¬ 𝑥 = -∞) ↔ ¬ (𝑥 < -∞ ∨ 𝑥 = -∞))
19	17, 18	sylib 218	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ → ¬ (𝑥 < -∞ ∨ 𝑥 = -∞))
20		mnfxr 11201	. . . . . . . . . . . . . . 15 ⊢ -∞ ∈ ℝ*
21		xrleloe 13070	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ* ∧ -∞ ∈ ℝ*) → (𝑥 ≤ -∞ ↔ (𝑥 < -∞ ∨ 𝑥 = -∞)))
22	12, 20, 21	sylancl 587	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ → (𝑥 ≤ -∞ ↔ (𝑥 < -∞ ∨ 𝑥 = -∞)))
23	19, 22	mtbird 325	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ → ¬ 𝑥 ≤ -∞)
24		breq2 5104	. . . . . . . . . . . . . 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 11190	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℝ → 𝑦 ∈ ℝ*)
29		pnfnlt 13054	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℝ* → ¬ +∞ < 𝑦)
30		breq1 5103	. . . . . . . . . . . . . . 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 621	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦) → (¬ 𝑧 = -∞ ∧ ¬ 𝑧 = +∞)))
36	35	anim2d 613	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((𝑧 ∈ ℝ* ∧ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)) → (𝑧 ∈ ℝ* ∧ (¬ 𝑧 = -∞ ∧ ¬ 𝑧 = +∞))))
37		renepnf 11192	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ℝ → 𝑧 ≠ +∞)
38	37	neneqd 2938	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ℝ → ¬ 𝑧 = +∞)
39	38	pm4.71i 559	. . . . . . . . . 10 ⊢ (𝑧 ∈ ℝ ↔ (𝑧 ∈ ℝ ∧ ¬ 𝑧 = +∞))
40		xrnemnf 13043	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ ℝ* ∧ 𝑧 ≠ -∞) ↔ (𝑧 ∈ ℝ ∨ 𝑧 = +∞))
41	40	anbi1i 625	. . . . . . . . . . 11 ⊢ (((𝑧 ∈ ℝ* ∧ 𝑧 ≠ -∞) ∧ ¬ 𝑧 = +∞) ↔ ((𝑧 ∈ ℝ ∨ 𝑧 = +∞) ∧ ¬ 𝑧 = +∞))
42		df-ne 2934	. . . . . . . . . . . . 13 ⊢ (𝑧 ≠ -∞ ↔ ¬ 𝑧 = -∞)
43	42	anbi2i 624	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ ℝ* ∧ 𝑧 ≠ -∞) ↔ (𝑧 ∈ ℝ* ∧ ¬ 𝑧 = -∞))
44	43	anbi1i 625	. . . . . . . . . . 11 ⊢ (((𝑧 ∈ ℝ* ∧ 𝑧 ≠ -∞) ∧ ¬ 𝑧 = +∞) ↔ ((𝑧 ∈ ℝ* ∧ ¬ 𝑧 = -∞) ∧ ¬ 𝑧 = +∞))
45		pm5.61 1003	. . . . . . . . . . 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 497	. . . . . . . 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 3422	. . . . . 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 4031	. . . . . . 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 3931	. . . . 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 3955	. . 3 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} = {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)})
61	60	mpoeq3ia 7446	. 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 2764	1 ⊢ 𝐹 = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)})

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  {crab 3401   ⊆ wss 3903   class class class wbr 5100   × cxp 5630   ↾ cres 5634   ∈ cmpo 7370  ℝcr 11037  +∞cpnf 11175  -∞cmnf 11176  ℝ^*cxr 11177   < clt 11178   ≤ cle 11179  [,)cico 13275

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-cnex 11094  ax-resscn 11095  ax-pre-lttri 11112  ax-pre-lttrn 11113

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-po 5540  df-so 5541  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 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-oprab 7372  df-mpo 7373  df-er 8645  df-en 8896  df-dom 8897  df-sdom 8898  df-pnf 11180  df-mnf 11181  df-xr 11182  df-ltxr 11183  df-le 11184  df-ico 13279

This theorem is referenced by:  icoreresf  37604  icoreval  37605