Users' Mathboxes Mathbox for ML < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  icorempt2 Structured version   Visualization version   GIF version

Theorem icorempt2 32175
Description: Closed-below, open-above intervals of reals. (Contributed by ML, 26-Jul-2020.)
Hypothesis
Ref Expression
icorempt2.1 𝐹 = ([,) ↾ (ℝ × ℝ))
Assertion
Ref Expression
icorempt2 𝐹 = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)})
Distinct variable group:   𝑥,𝑦,𝑧
Allowed substitution hints:   𝐹(𝑥,𝑦,𝑧)

Proof of Theorem icorempt2
StepHypRef Expression
1 icorempt2.1 . 2 𝐹 = ([,) ↾ (ℝ × ℝ))
2 df-ico 12005 . . . 4 [,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)})
32reseq1i 5297 . . 3 ([,) ↾ (ℝ × ℝ)) = ((𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)}) ↾ (ℝ × ℝ))
4 ressxr 9936 . . . 4 ℝ ⊆ ℝ*
5 resmpt2 6631 . . . 4 ((ℝ ⊆ ℝ* ∧ ℝ ⊆ ℝ*) → ((𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)}) ↾ (ℝ × ℝ)) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)}))
64, 4, 5mp2an 703 . . 3 ((𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)}) ↾ (ℝ × ℝ)) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)})
73, 6eqtri 2628 . 2 ([,) ↾ (ℝ × ℝ)) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)})
8 nfv 1829 . . . 4 𝑧(𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)
9 nfrab1 3095 . . . 4 𝑧{𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)}
10 nfrab1 3095 . . . 4 𝑧{𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)}
11 rabid 3091 . . . . . . . 8 (𝑧 ∈ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)} ↔ (𝑧 ∈ ℝ* ∧ (𝑥𝑧𝑧 < 𝑦)))
12 rexr 9938 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ ℝ → 𝑥 ∈ ℝ*)
13 nltmnf 11797 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ ℝ* → ¬ 𝑥 < -∞)
1412, 13syl 17 . . . . . . . . . . . . . . . 16 (𝑥 ∈ ℝ → ¬ 𝑥 < -∞)
15 renemnf 9941 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ ℝ → 𝑥 ≠ -∞)
1615neneqd 2783 . . . . . . . . . . . . . . . 16 (𝑥 ∈ ℝ → ¬ 𝑥 = -∞)
1714, 16jca 552 . . . . . . . . . . . . . . 15 (𝑥 ∈ ℝ → (¬ 𝑥 < -∞ ∧ ¬ 𝑥 = -∞))
18 pm4.56 514 . . . . . . . . . . . . . . 15 ((¬ 𝑥 < -∞ ∧ ¬ 𝑥 = -∞) ↔ ¬ (𝑥 < -∞ ∨ 𝑥 = -∞))
1917, 18sylib 206 . . . . . . . . . . . . . 14 (𝑥 ∈ ℝ → ¬ (𝑥 < -∞ ∨ 𝑥 = -∞))
20 mnfxr 11780 . . . . . . . . . . . . . . 15 -∞ ∈ ℝ*
21 xrleloe 11809 . . . . . . . . . . . . . . 15 ((𝑥 ∈ ℝ* ∧ -∞ ∈ ℝ*) → (𝑥 ≤ -∞ ↔ (𝑥 < -∞ ∨ 𝑥 = -∞)))
2212, 20, 21sylancl 692 . . . . . . . . . . . . . 14 (𝑥 ∈ ℝ → (𝑥 ≤ -∞ ↔ (𝑥 < -∞ ∨ 𝑥 = -∞)))
2319, 22mtbird 313 . . . . . . . . . . . . 13 (𝑥 ∈ ℝ → ¬ 𝑥 ≤ -∞)
24 breq2 4578 . . . . . . . . . . . . . 14 (𝑧 = -∞ → (𝑥𝑧𝑥 ≤ -∞))
2524notbid 306 . . . . . . . . . . . . 13 (𝑧 = -∞ → (¬ 𝑥𝑧 ↔ ¬ 𝑥 ≤ -∞))
2623, 25syl5ibrcom 235 . . . . . . . . . . . 12 (𝑥 ∈ ℝ → (𝑧 = -∞ → ¬ 𝑥𝑧))
2726con2d 127 . . . . . . . . . . 11 (𝑥 ∈ ℝ → (𝑥𝑧 → ¬ 𝑧 = -∞))
28 rexr 9938 . . . . . . . . . . . 12 (𝑦 ∈ ℝ → 𝑦 ∈ ℝ*)
29 pnfnlt 11796 . . . . . . . . . . . . . 14 (𝑦 ∈ ℝ* → ¬ +∞ < 𝑦)
30 breq1 4577 . . . . . . . . . . . . . . 15 (𝑧 = +∞ → (𝑧 < 𝑦 ↔ +∞ < 𝑦))
3130notbid 306 . . . . . . . . . . . . . 14 (𝑧 = +∞ → (¬ 𝑧 < 𝑦 ↔ ¬ +∞ < 𝑦))
3229, 31syl5ibrcom 235 . . . . . . . . . . . . 13 (𝑦 ∈ ℝ* → (𝑧 = +∞ → ¬ 𝑧 < 𝑦))
3332con2d 127 . . . . . . . . . . . 12 (𝑦 ∈ ℝ* → (𝑧 < 𝑦 → ¬ 𝑧 = +∞))
3428, 33syl 17 . . . . . . . . . . 11 (𝑦 ∈ ℝ → (𝑧 < 𝑦 → ¬ 𝑧 = +∞))
3527, 34im2anan9 875 . . . . . . . . . 10 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((𝑥𝑧𝑧 < 𝑦) → (¬ 𝑧 = -∞ ∧ ¬ 𝑧 = +∞)))
3635anim2d 586 . . . . . . . . 9 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((𝑧 ∈ ℝ* ∧ (𝑥𝑧𝑧 < 𝑦)) → (𝑧 ∈ ℝ* ∧ (¬ 𝑧 = -∞ ∧ ¬ 𝑧 = +∞))))
37 renepnf 9940 . . . . . . . . . . . 12 (𝑧 ∈ ℝ → 𝑧 ≠ +∞)
3837neneqd 2783 . . . . . . . . . . 11 (𝑧 ∈ ℝ → ¬ 𝑧 = +∞)
3938pm4.71i 661 . . . . . . . . . 10 (𝑧 ∈ ℝ ↔ (𝑧 ∈ ℝ ∧ ¬ 𝑧 = +∞))
40 xrnemnf 11785 . . . . . . . . . . . 12 ((𝑧 ∈ ℝ*𝑧 ≠ -∞) ↔ (𝑧 ∈ ℝ ∨ 𝑧 = +∞))
4140anbi1i 726 . . . . . . . . . . 11 (((𝑧 ∈ ℝ*𝑧 ≠ -∞) ∧ ¬ 𝑧 = +∞) ↔ ((𝑧 ∈ ℝ ∨ 𝑧 = +∞) ∧ ¬ 𝑧 = +∞))
42 df-ne 2778 . . . . . . . . . . . . 13 (𝑧 ≠ -∞ ↔ ¬ 𝑧 = -∞)
4342anbi2i 725 . . . . . . . . . . . 12 ((𝑧 ∈ ℝ*𝑧 ≠ -∞) ↔ (𝑧 ∈ ℝ* ∧ ¬ 𝑧 = -∞))
4443anbi1i 726 . . . . . . . . . . 11 (((𝑧 ∈ ℝ*𝑧 ≠ -∞) ∧ ¬ 𝑧 = +∞) ↔ ((𝑧 ∈ ℝ* ∧ ¬ 𝑧 = -∞) ∧ ¬ 𝑧 = +∞))
45 pm5.61 744 . . . . . . . . . . 11 (((𝑧 ∈ ℝ ∨ 𝑧 = +∞) ∧ ¬ 𝑧 = +∞) ↔ (𝑧 ∈ ℝ ∧ ¬ 𝑧 = +∞))
4641, 44, 453bitr3i 288 . . . . . . . . . 10 (((𝑧 ∈ ℝ* ∧ ¬ 𝑧 = -∞) ∧ ¬ 𝑧 = +∞) ↔ (𝑧 ∈ ℝ ∧ ¬ 𝑧 = +∞))
47 anass 678 . . . . . . . . . 10 (((𝑧 ∈ ℝ* ∧ ¬ 𝑧 = -∞) ∧ ¬ 𝑧 = +∞) ↔ (𝑧 ∈ ℝ* ∧ (¬ 𝑧 = -∞ ∧ ¬ 𝑧 = +∞)))
4839, 46, 473bitr2ri 287 . . . . . . . . 9 ((𝑧 ∈ ℝ* ∧ (¬ 𝑧 = -∞ ∧ ¬ 𝑧 = +∞)) ↔ 𝑧 ∈ ℝ)
4936, 48syl6ib 239 . . . . . . . 8 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((𝑧 ∈ ℝ* ∧ (𝑥𝑧𝑧 < 𝑦)) → 𝑧 ∈ ℝ))
5011, 49syl5bi 230 . . . . . . 7 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑧 ∈ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)} → 𝑧 ∈ ℝ))
5111simprbi 478 . . . . . . . 8 (𝑧 ∈ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)} → (𝑥𝑧𝑧 < 𝑦))
5251a1i 11 . . . . . . 7 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑧 ∈ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)} → (𝑥𝑧𝑧 < 𝑦)))
5350, 52jcad 553 . . . . . 6 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑧 ∈ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)} → (𝑧 ∈ ℝ ∧ (𝑥𝑧𝑧 < 𝑦))))
54 rabid 3091 . . . . . 6 (𝑧 ∈ {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)} ↔ (𝑧 ∈ ℝ ∧ (𝑥𝑧𝑧 < 𝑦)))
5553, 54syl6ibr 240 . . . . 5 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑧 ∈ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)} → 𝑧 ∈ {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)}))
56 rabss2 3644 . . . . . . 7 (ℝ ⊆ ℝ* → {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)} ⊆ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)})
574, 56ax-mp 5 . . . . . 6 {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)} ⊆ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)}
5857sseli 3560 . . . . 5 (𝑧 ∈ {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)} → 𝑧 ∈ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)})
5955, 58impbid1 213 . . . 4 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑧 ∈ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)} ↔ 𝑧 ∈ {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)}))
608, 9, 10, 59eqrd 3582 . . 3 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)} = {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)})
6160mpt2eq3ia 6593 . 2 (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)}) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)})
621, 7, 613eqtri 2632 1 𝐹 = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wo 381  wa 382   = wceq 1474  wcel 1976  wne 2776  {crab 2896  wss 3536   class class class wbr 4574   × cxp 5023  cres 5027  cmpt2 6526  cr 9788  +∞cpnf 9924  -∞cmnf 9925  *cxr 9926   < clt 9927  cle 9928  [,)cico 12001
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586  ax-sep 4700  ax-nul 4709  ax-pow 4761  ax-pr 4825  ax-un 6821  ax-cnex 9845  ax-resscn 9846  ax-pre-lttri 9863  ax-pre-lttrn 9864
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ne 2778  df-nel 2779  df-ral 2897  df-rex 2898  df-rab 2901  df-v 3171  df-sbc 3399  df-csb 3496  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-nul 3871  df-if 4033  df-pw 4106  df-sn 4122  df-pr 4124  df-op 4128  df-uni 4364  df-br 4575  df-opab 4635  df-mpt 4636  df-id 4940  df-po 4946  df-so 4947  df-xp 5031  df-rel 5032  df-cnv 5033  df-co 5034  df-dm 5035  df-rn 5036  df-res 5037  df-ima 5038  df-iota 5751  df-fun 5789  df-fn 5790  df-f 5791  df-f1 5792  df-fo 5793  df-f1o 5794  df-fv 5795  df-oprab 6528  df-mpt2 6529  df-er 7603  df-en 7816  df-dom 7817  df-sdom 7818  df-pnf 9929  df-mnf 9930  df-xr 9931  df-ltxr 9932  df-le 9933  df-ico 12005
This theorem is referenced by:  icoreresf  32176  icoreval  32177
  Copyright terms: Public domain W3C validator