Mathbox for Glauco Siliprandi < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  hoicoto2 Structured version   Visualization version   GIF version

Theorem hoicoto2 41607
 Description: The half-open interval expressed using a composition of a function into (ℝ × ℝ) and using two distinct real-valued functions for the borders. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
Hypotheses
Ref Expression
hoicoto2.i (𝜑𝐼:𝑋⟶(ℝ × ℝ))
hoicoto2.a 𝐴 = (𝑘𝑋 ↦ (1st ‘(𝐼𝑘)))
hoicoto2.b 𝐵 = (𝑘𝑋 ↦ (2nd ‘(𝐼𝑘)))
Assertion
Ref Expression
hoicoto2 (𝜑X𝑘𝑋 (([,) ∘ 𝐼)‘𝑘) = X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘)))
Distinct variable groups:   𝑘,𝑋   𝜑,𝑘
Allowed substitution hints:   𝐴(𝑘)   𝐵(𝑘)   𝐼(𝑘)

Proof of Theorem hoicoto2
StepHypRef Expression
1 hoicoto2.i . . . . 5 (𝜑𝐼:𝑋⟶(ℝ × ℝ))
21adantr 474 . . . 4 ((𝜑𝑘𝑋) → 𝐼:𝑋⟶(ℝ × ℝ))
3 simpr 479 . . . 4 ((𝜑𝑘𝑋) → 𝑘𝑋)
42, 3fvovco 40184 . . 3 ((𝜑𝑘𝑋) → (([,) ∘ 𝐼)‘𝑘) = ((1st ‘(𝐼𝑘))[,)(2nd ‘(𝐼𝑘))))
51ffvelrnda 6613 . . . . . . . 8 ((𝜑𝑘𝑋) → (𝐼𝑘) ∈ (ℝ × ℝ))
6 xp1st 7465 . . . . . . . 8 ((𝐼𝑘) ∈ (ℝ × ℝ) → (1st ‘(𝐼𝑘)) ∈ ℝ)
75, 6syl 17 . . . . . . 7 ((𝜑𝑘𝑋) → (1st ‘(𝐼𝑘)) ∈ ℝ)
87elexd 3431 . . . . . 6 ((𝜑𝑘𝑋) → (1st ‘(𝐼𝑘)) ∈ V)
9 hoicoto2.a . . . . . . 7 𝐴 = (𝑘𝑋 ↦ (1st ‘(𝐼𝑘)))
109fvmpt2 6543 . . . . . 6 ((𝑘𝑋 ∧ (1st ‘(𝐼𝑘)) ∈ V) → (𝐴𝑘) = (1st ‘(𝐼𝑘)))
113, 8, 10syl2anc 579 . . . . 5 ((𝜑𝑘𝑋) → (𝐴𝑘) = (1st ‘(𝐼𝑘)))
1211eqcomd 2831 . . . 4 ((𝜑𝑘𝑋) → (1st ‘(𝐼𝑘)) = (𝐴𝑘))
13 xp2nd 7466 . . . . . . . 8 ((𝐼𝑘) ∈ (ℝ × ℝ) → (2nd ‘(𝐼𝑘)) ∈ ℝ)
145, 13syl 17 . . . . . . 7 ((𝜑𝑘𝑋) → (2nd ‘(𝐼𝑘)) ∈ ℝ)
1514elexd 3431 . . . . . 6 ((𝜑𝑘𝑋) → (2nd ‘(𝐼𝑘)) ∈ V)
16 hoicoto2.b . . . . . . 7 𝐵 = (𝑘𝑋 ↦ (2nd ‘(𝐼𝑘)))
1716fvmpt2 6543 . . . . . 6 ((𝑘𝑋 ∧ (2nd ‘(𝐼𝑘)) ∈ V) → (𝐵𝑘) = (2nd ‘(𝐼𝑘)))
183, 15, 17syl2anc 579 . . . . 5 ((𝜑𝑘𝑋) → (𝐵𝑘) = (2nd ‘(𝐼𝑘)))
1918eqcomd 2831 . . . 4 ((𝜑𝑘𝑋) → (2nd ‘(𝐼𝑘)) = (𝐵𝑘))
2012, 19oveq12d 6928 . . 3 ((𝜑𝑘𝑋) → ((1st ‘(𝐼𝑘))[,)(2nd ‘(𝐼𝑘))) = ((𝐴𝑘)[,)(𝐵𝑘)))
214, 20eqtrd 2861 . 2 ((𝜑𝑘𝑋) → (([,) ∘ 𝐼)‘𝑘) = ((𝐴𝑘)[,)(𝐵𝑘)))
2221ixpeq2dva 8196 1 (𝜑X𝑘𝑋 (([,) ∘ 𝐼)‘𝑘) = X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 386   = wceq 1656   ∈ wcel 2164  Vcvv 3414   ↦ cmpt 4954   × cxp 5344   ∘ ccom 5350  ⟶wf 6123  ‘cfv 6127  (class class class)co 6910  1st c1st 7431  2nd c2nd 7432  Xcixp 8181  ℝcr 10258  [,)cico 12472 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-fv 6135  df-ov 6913  df-1st 7433  df-2nd 7434  df-ixp 8182 This theorem is referenced by:  opnvonmbllem2  41635
 Copyright terms: Public domain W3C validator