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 46215
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)
21ffvelcdmda 7100 . . . 4 ((𝜑𝑗𝑋) → (𝐵𝑗) ∈ ℝ)
3 hsphoif.a . . . . . 6 (𝜑𝐴 ∈ ℝ)
43adantr 479 . . . . 5 ((𝜑𝑗𝑋) → 𝐴 ∈ ℝ)
52, 4ifcld 4579 . . . 4 ((𝜑𝑗𝑋) → if((𝐵𝑗) ≤ 𝐴, (𝐵𝑗), 𝐴) ∈ ℝ)
62, 5ifcld 4579 . . 3 ((𝜑𝑗𝑋) → if(𝑗𝑌, (𝐵𝑗), if((𝐵𝑗) ≤ 𝐴, (𝐵𝑗), 𝐴)) ∈ ℝ)
7 eqid 2726 . . 3 (𝑗𝑋 ↦ if(𝑗𝑌, (𝐵𝑗), if((𝐵𝑗) ≤ 𝐴, (𝐵𝑗), 𝐴))) = (𝑗𝑋 ↦ if(𝑗𝑌, (𝐵𝑗), if((𝐵𝑗) ≤ 𝐴, (𝐵𝑗), 𝐴)))
86, 7fmptd 7130 . 2 (𝜑 → (𝑗𝑋 ↦ if(𝑗𝑌, (𝐵𝑗), if((𝐵𝑗) ≤ 𝐴, (𝐵𝑗), 𝐴))):𝑋⟶ℝ)
9 hsphoif.h . . . . 5 𝐻 = (𝑥 ∈ ℝ ↦ (𝑎 ∈ (ℝ ↑m 𝑋) ↦ (𝑗𝑋 ↦ if(𝑗𝑌, (𝑎𝑗), if((𝑎𝑗) ≤ 𝑥, (𝑎𝑗), 𝑥)))))
10 breq2 5159 . . . . . . . . 9 (𝑥 = 𝐴 → ((𝑎𝑗) ≤ 𝑥 ↔ (𝑎𝑗) ≤ 𝐴))
11 id 22 . . . . . . . . 9 (𝑥 = 𝐴𝑥 = 𝐴)
1210, 11ifbieq2d 4559 . . . . . . . 8 (𝑥 = 𝐴 → if((𝑎𝑗) ≤ 𝑥, (𝑎𝑗), 𝑥) = if((𝑎𝑗) ≤ 𝐴, (𝑎𝑗), 𝐴))
1312ifeq2d 4553 . . . . . . 7 (𝑥 = 𝐴 → if(𝑗𝑌, (𝑎𝑗), if((𝑎𝑗) ≤ 𝑥, (𝑎𝑗), 𝑥)) = if(𝑗𝑌, (𝑎𝑗), if((𝑎𝑗) ≤ 𝐴, (𝑎𝑗), 𝐴)))
1413mpteq2dv 5257 . . . . . 6 (𝑥 = 𝐴 → (𝑗𝑋 ↦ if(𝑗𝑌, (𝑎𝑗), if((𝑎𝑗) ≤ 𝑥, (𝑎𝑗), 𝑥))) = (𝑗𝑋 ↦ if(𝑗𝑌, (𝑎𝑗), if((𝑎𝑗) ≤ 𝐴, (𝑎𝑗), 𝐴))))
1514mpteq2dv 5257 . . . . 5 (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)))))
16 ovex 7459 . . . . . . 7 (ℝ ↑m 𝑋) ∈ V
1716mptex 7242 . . . . . 6 (𝑎 ∈ (ℝ ↑m 𝑋) ↦ (𝑗𝑋 ↦ if(𝑗𝑌, (𝑎𝑗), if((𝑎𝑗) ≤ 𝐴, (𝑎𝑗), 𝐴)))) ∈ V
1817a1i 11 . . . . 5 (𝜑 → (𝑎 ∈ (ℝ ↑m 𝑋) ↦ (𝑗𝑋 ↦ if(𝑗𝑌, (𝑎𝑗), if((𝑎𝑗) ≤ 𝐴, (𝑎𝑗), 𝐴)))) ∈ V)
199, 15, 3, 18fvmptd3 7034 . . . 4 (𝜑 → (𝐻𝐴) = (𝑎 ∈ (ℝ ↑m 𝑋) ↦ (𝑗𝑋 ↦ if(𝑗𝑌, (𝑎𝑗), if((𝑎𝑗) ≤ 𝐴, (𝑎𝑗), 𝐴)))))
20 fveq1 6902 . . . . . . 7 (𝑎 = 𝐵 → (𝑎𝑗) = (𝐵𝑗))
2120breq1d 5165 . . . . . . . 8 (𝑎 = 𝐵 → ((𝑎𝑗) ≤ 𝐴 ↔ (𝐵𝑗) ≤ 𝐴))
2221, 20ifbieq1d 4557 . . . . . . 7 (𝑎 = 𝐵 → if((𝑎𝑗) ≤ 𝐴, (𝑎𝑗), 𝐴) = if((𝐵𝑗) ≤ 𝐴, (𝐵𝑗), 𝐴))
2320, 22ifeq12d 4554 . . . . . 6 (𝑎 = 𝐵 → if(𝑗𝑌, (𝑎𝑗), if((𝑎𝑗) ≤ 𝐴, (𝑎𝑗), 𝐴)) = if(𝑗𝑌, (𝐵𝑗), if((𝐵𝑗) ≤ 𝐴, (𝐵𝑗), 𝐴)))
2423mpteq2dv 5257 . . . . 5 (𝑎 = 𝐵 → (𝑗𝑋 ↦ if(𝑗𝑌, (𝑎𝑗), if((𝑎𝑗) ≤ 𝐴, (𝑎𝑗), 𝐴))) = (𝑗𝑋 ↦ if(𝑗𝑌, (𝐵𝑗), if((𝐵𝑗) ≤ 𝐴, (𝐵𝑗), 𝐴))))
2524adantl 480 . . . 4 ((𝜑𝑎 = 𝐵) → (𝑗𝑋 ↦ if(𝑗𝑌, (𝑎𝑗), if((𝑎𝑗) ≤ 𝐴, (𝑎𝑗), 𝐴))) = (𝑗𝑋 ↦ if(𝑗𝑌, (𝐵𝑗), if((𝐵𝑗) ≤ 𝐴, (𝐵𝑗), 𝐴))))
26 reex 11251 . . . . . . . 8 ℝ ∈ V
2726a1i 11 . . . . . . 7 (𝜑 → ℝ ∈ V)
28 hsphoif.x . . . . . . 7 (𝜑𝑋𝑉)
2927, 28jca 510 . . . . . 6 (𝜑 → (ℝ ∈ V ∧ 𝑋𝑉))
30 elmapg 8870 . . . . . 6 ((ℝ ∈ V ∧ 𝑋𝑉) → (𝐵 ∈ (ℝ ↑m 𝑋) ↔ 𝐵:𝑋⟶ℝ))
3129, 30syl 17 . . . . 5 (𝜑 → (𝐵 ∈ (ℝ ↑m 𝑋) ↔ 𝐵:𝑋⟶ℝ))
321, 31mpbird 256 . . . 4 (𝜑𝐵 ∈ (ℝ ↑m 𝑋))
33 mptexg 7240 . . . . 5 (𝑋𝑉 → (𝑗𝑋 ↦ if(𝑗𝑌, (𝐵𝑗), if((𝐵𝑗) ≤ 𝐴, (𝐵𝑗), 𝐴))) ∈ V)
3428, 33syl 17 . . . 4 (𝜑 → (𝑗𝑋 ↦ if(𝑗𝑌, (𝐵𝑗), if((𝐵𝑗) ≤ 𝐴, (𝐵𝑗), 𝐴))) ∈ V)
3519, 25, 32, 34fvmptd 7018 . . 3 (𝜑 → ((𝐻𝐴)‘𝐵) = (𝑗𝑋 ↦ if(𝑗𝑌, (𝐵𝑗), if((𝐵𝑗) ≤ 𝐴, (𝐵𝑗), 𝐴))))
3635feq1d 6715 . 2 (𝜑 → (((𝐻𝐴)‘𝐵):𝑋⟶ℝ ↔ (𝑗𝑋 ↦ if(𝑗𝑌, (𝐵𝑗), if((𝐵𝑗) ≤ 𝐴, (𝐵𝑗), 𝐴))):𝑋⟶ℝ))
378, 36mpbird 256 1 (𝜑 → ((𝐻𝐴)‘𝐵):𝑋⟶ℝ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1534  wcel 2099  Vcvv 3462  ifcif 4533   class class class wbr 5155  cmpt 5238  wf 6552  cfv 6556  (class class class)co 7426  m cmap 8857  cr 11159  cle 11301
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5292  ax-sep 5306  ax-nul 5313  ax-pow 5371  ax-pr 5435  ax-un 7748  ax-cnex 11216  ax-resscn 11217
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4916  df-iun 5005  df-br 5156  df-opab 5218  df-mpt 5239  df-id 5582  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6508  df-fun 6558  df-fn 6559  df-f 6560  df-f1 6561  df-fo 6562  df-f1o 6563  df-fv 6564  df-ov 7429  df-oprab 7430  df-mpo 7431  df-map 8859
This theorem is referenced by:  hsphoidmvle2  46224  hsphoidmvle  46225  sge0hsphoire  46228  hoidmvlelem1  46234  hoidmvlelem2  46235  hoidmvlelem4  46237  hspmbllem1  46265  hspmbllem2  46266
  Copyright terms: Public domain W3C validator