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

Theorem hsphoif 41314
Description: 𝐻 is a function (that returns the representation of the right side of a half-open interval intersected with a half-space). Step (b) in Lemma 115B of [Fremlin1] p. 29. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
Hypotheses
Ref Expression
hsphoif.h 𝐻 = (𝑥 ∈ ℝ ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑋) ↦ (𝑗𝑋 ↦ if(𝑗𝑌, (𝑎𝑗), if((𝑎𝑗) ≤ 𝑥, (𝑎𝑗), 𝑥)))))
hsphoif.a (𝜑𝐴 ∈ ℝ)
hsphoif.x (𝜑𝑋𝑉)
hsphoif.b (𝜑𝐵:𝑋⟶ℝ)
Assertion
Ref Expression
hsphoif (𝜑 → ((𝐻𝐴)‘𝐵):𝑋⟶ℝ)
Distinct variable groups:   𝐴,𝑎,𝑗,𝑥   𝐵,𝑎,𝑗   𝑋,𝑎,𝑗,𝑥   𝑌,𝑎,𝑥   𝜑,𝑎,𝑗,𝑥
Allowed substitution hints:   𝐵(𝑥)   𝐻(𝑥,𝑗,𝑎)   𝑉(𝑥,𝑗,𝑎)   𝑌(𝑗)

Proof of Theorem hsphoif
StepHypRef Expression
1 hsphoif.b . . . . 5 (𝜑𝐵:𝑋⟶ℝ)
21ffvelrnda 6523 . . . 4 ((𝜑𝑗𝑋) → (𝐵𝑗) ∈ ℝ)
3 hsphoif.a . . . . . 6 (𝜑𝐴 ∈ ℝ)
43adantr 472 . . . . 5 ((𝜑𝑗𝑋) → 𝐴 ∈ ℝ)
52, 4ifcld 4275 . . . 4 ((𝜑𝑗𝑋) → if((𝐵𝑗) ≤ 𝐴, (𝐵𝑗), 𝐴) ∈ ℝ)
62, 5ifcld 4275 . . 3 ((𝜑𝑗𝑋) → if(𝑗𝑌, (𝐵𝑗), if((𝐵𝑗) ≤ 𝐴, (𝐵𝑗), 𝐴)) ∈ ℝ)
7 eqid 2760 . . 3 (𝑗𝑋 ↦ if(𝑗𝑌, (𝐵𝑗), if((𝐵𝑗) ≤ 𝐴, (𝐵𝑗), 𝐴))) = (𝑗𝑋 ↦ if(𝑗𝑌, (𝐵𝑗), if((𝐵𝑗) ≤ 𝐴, (𝐵𝑗), 𝐴)))
86, 7fmptd 6549 . 2 (𝜑 → (𝑗𝑋 ↦ if(𝑗𝑌, (𝐵𝑗), if((𝐵𝑗) ≤ 𝐴, (𝐵𝑗), 𝐴))):𝑋⟶ℝ)
9 hsphoif.h . . . . . 6 𝐻 = (𝑥 ∈ ℝ ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑋) ↦ (𝑗𝑋 ↦ if(𝑗𝑌, (𝑎𝑗), if((𝑎𝑗) ≤ 𝑥, (𝑎𝑗), 𝑥)))))
109a1i 11 . . . . 5 (𝜑𝐻 = (𝑥 ∈ ℝ ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑋) ↦ (𝑗𝑋 ↦ if(𝑗𝑌, (𝑎𝑗), if((𝑎𝑗) ≤ 𝑥, (𝑎𝑗), 𝑥))))))
11 breq2 4808 . . . . . . . . . 10 (𝑥 = 𝐴 → ((𝑎𝑗) ≤ 𝑥 ↔ (𝑎𝑗) ≤ 𝐴))
12 id 22 . . . . . . . . . 10 (𝑥 = 𝐴𝑥 = 𝐴)
1311, 12ifbieq2d 4255 . . . . . . . . 9 (𝑥 = 𝐴 → if((𝑎𝑗) ≤ 𝑥, (𝑎𝑗), 𝑥) = if((𝑎𝑗) ≤ 𝐴, (𝑎𝑗), 𝐴))
1413ifeq2d 4249 . . . . . . . 8 (𝑥 = 𝐴 → if(𝑗𝑌, (𝑎𝑗), if((𝑎𝑗) ≤ 𝑥, (𝑎𝑗), 𝑥)) = if(𝑗𝑌, (𝑎𝑗), if((𝑎𝑗) ≤ 𝐴, (𝑎𝑗), 𝐴)))
1514mpteq2dv 4897 . . . . . . 7 (𝑥 = 𝐴 → (𝑗𝑋 ↦ if(𝑗𝑌, (𝑎𝑗), if((𝑎𝑗) ≤ 𝑥, (𝑎𝑗), 𝑥))) = (𝑗𝑋 ↦ if(𝑗𝑌, (𝑎𝑗), if((𝑎𝑗) ≤ 𝐴, (𝑎𝑗), 𝐴))))
1615mpteq2dv 4897 . . . . . 6 (𝑥 = 𝐴 → (𝑎 ∈ (ℝ ↑𝑚 𝑋) ↦ (𝑗𝑋 ↦ if(𝑗𝑌, (𝑎𝑗), if((𝑎𝑗) ≤ 𝑥, (𝑎𝑗), 𝑥)))) = (𝑎 ∈ (ℝ ↑𝑚 𝑋) ↦ (𝑗𝑋 ↦ if(𝑗𝑌, (𝑎𝑗), if((𝑎𝑗) ≤ 𝐴, (𝑎𝑗), 𝐴)))))
1716adantl 473 . . . . 5 ((𝜑𝑥 = 𝐴) → (𝑎 ∈ (ℝ ↑𝑚 𝑋) ↦ (𝑗𝑋 ↦ if(𝑗𝑌, (𝑎𝑗), if((𝑎𝑗) ≤ 𝑥, (𝑎𝑗), 𝑥)))) = (𝑎 ∈ (ℝ ↑𝑚 𝑋) ↦ (𝑗𝑋 ↦ if(𝑗𝑌, (𝑎𝑗), if((𝑎𝑗) ≤ 𝐴, (𝑎𝑗), 𝐴)))))
18 ovex 6842 . . . . . . 7 (ℝ ↑𝑚 𝑋) ∈ V
1918mptex 6651 . . . . . 6 (𝑎 ∈ (ℝ ↑𝑚 𝑋) ↦ (𝑗𝑋 ↦ if(𝑗𝑌, (𝑎𝑗), if((𝑎𝑗) ≤ 𝐴, (𝑎𝑗), 𝐴)))) ∈ V
2019a1i 11 . . . . 5 (𝜑 → (𝑎 ∈ (ℝ ↑𝑚 𝑋) ↦ (𝑗𝑋 ↦ if(𝑗𝑌, (𝑎𝑗), if((𝑎𝑗) ≤ 𝐴, (𝑎𝑗), 𝐴)))) ∈ V)
2110, 17, 3, 20fvmptd 6451 . . . 4 (𝜑 → (𝐻𝐴) = (𝑎 ∈ (ℝ ↑𝑚 𝑋) ↦ (𝑗𝑋 ↦ if(𝑗𝑌, (𝑎𝑗), if((𝑎𝑗) ≤ 𝐴, (𝑎𝑗), 𝐴)))))
22 fveq1 6352 . . . . . . 7 (𝑎 = 𝐵 → (𝑎𝑗) = (𝐵𝑗))
2322breq1d 4814 . . . . . . . 8 (𝑎 = 𝐵 → ((𝑎𝑗) ≤ 𝐴 ↔ (𝐵𝑗) ≤ 𝐴))
2423, 22ifbieq1d 4253 . . . . . . 7 (𝑎 = 𝐵 → if((𝑎𝑗) ≤ 𝐴, (𝑎𝑗), 𝐴) = if((𝐵𝑗) ≤ 𝐴, (𝐵𝑗), 𝐴))
2522, 24ifeq12d 4250 . . . . . 6 (𝑎 = 𝐵 → if(𝑗𝑌, (𝑎𝑗), if((𝑎𝑗) ≤ 𝐴, (𝑎𝑗), 𝐴)) = if(𝑗𝑌, (𝐵𝑗), if((𝐵𝑗) ≤ 𝐴, (𝐵𝑗), 𝐴)))
2625mpteq2dv 4897 . . . . 5 (𝑎 = 𝐵 → (𝑗𝑋 ↦ if(𝑗𝑌, (𝑎𝑗), if((𝑎𝑗) ≤ 𝐴, (𝑎𝑗), 𝐴))) = (𝑗𝑋 ↦ if(𝑗𝑌, (𝐵𝑗), if((𝐵𝑗) ≤ 𝐴, (𝐵𝑗), 𝐴))))
2726adantl 473 . . . 4 ((𝜑𝑎 = 𝐵) → (𝑗𝑋 ↦ if(𝑗𝑌, (𝑎𝑗), if((𝑎𝑗) ≤ 𝐴, (𝑎𝑗), 𝐴))) = (𝑗𝑋 ↦ if(𝑗𝑌, (𝐵𝑗), if((𝐵𝑗) ≤ 𝐴, (𝐵𝑗), 𝐴))))
28 reex 10239 . . . . . . . 8 ℝ ∈ V
2928a1i 11 . . . . . . 7 (𝜑 → ℝ ∈ V)
30 hsphoif.x . . . . . . 7 (𝜑𝑋𝑉)
3129, 30jca 555 . . . . . 6 (𝜑 → (ℝ ∈ V ∧ 𝑋𝑉))
32 elmapg 8038 . . . . . 6 ((ℝ ∈ V ∧ 𝑋𝑉) → (𝐵 ∈ (ℝ ↑𝑚 𝑋) ↔ 𝐵:𝑋⟶ℝ))
3331, 32syl 17 . . . . 5 (𝜑 → (𝐵 ∈ (ℝ ↑𝑚 𝑋) ↔ 𝐵:𝑋⟶ℝ))
341, 33mpbird 247 . . . 4 (𝜑𝐵 ∈ (ℝ ↑𝑚 𝑋))
35 mptexg 6649 . . . . 5 (𝑋𝑉 → (𝑗𝑋 ↦ if(𝑗𝑌, (𝐵𝑗), if((𝐵𝑗) ≤ 𝐴, (𝐵𝑗), 𝐴))) ∈ V)
3630, 35syl 17 . . . 4 (𝜑 → (𝑗𝑋 ↦ if(𝑗𝑌, (𝐵𝑗), if((𝐵𝑗) ≤ 𝐴, (𝐵𝑗), 𝐴))) ∈ V)
3721, 27, 34, 36fvmptd 6451 . . 3 (𝜑 → ((𝐻𝐴)‘𝐵) = (𝑗𝑋 ↦ if(𝑗𝑌, (𝐵𝑗), if((𝐵𝑗) ≤ 𝐴, (𝐵𝑗), 𝐴))))
3837feq1d 6191 . 2 (𝜑 → (((𝐻𝐴)‘𝐵):𝑋⟶ℝ ↔ (𝑗𝑋 ↦ if(𝑗𝑌, (𝐵𝑗), if((𝐵𝑗) ≤ 𝐴, (𝐵𝑗), 𝐴))):𝑋⟶ℝ))
398, 38mpbird 247 1 (𝜑 → ((𝐻𝐴)‘𝐵):𝑋⟶ℝ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  Vcvv 3340  ifcif 4230   class class class wbr 4804  cmpt 4881  wf 6045  cfv 6049  (class class class)co 6814  𝑚 cmap 8025  cr 10147  cle 10287
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-cnex 10204  ax-resscn 10205
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-map 8027
This theorem is referenced by:  hsphoidmvle2  41323  hsphoidmvle  41324  sge0hsphoire  41327  hoidmvlelem1  41333  hoidmvlelem2  41334  hoidmvlelem4  41336  hspmbllem1  41364  hspmbllem2  41365
  Copyright terms: Public domain W3C validator