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Theorem hsphoif 46532
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 7104 . . . 4 ((𝜑𝑗𝑋) → (𝐵𝑗) ∈ ℝ)
3 hsphoif.a . . . . . 6 (𝜑𝐴 ∈ ℝ)
43adantr 480 . . . . 5 ((𝜑𝑗𝑋) → 𝐴 ∈ ℝ)
52, 4ifcld 4577 . . . 4 ((𝜑𝑗𝑋) → if((𝐵𝑗) ≤ 𝐴, (𝐵𝑗), 𝐴) ∈ ℝ)
62, 5ifcld 4577 . . 3 ((𝜑𝑗𝑋) → if(𝑗𝑌, (𝐵𝑗), if((𝐵𝑗) ≤ 𝐴, (𝐵𝑗), 𝐴)) ∈ ℝ)
7 eqid 2735 . . 3 (𝑗𝑋 ↦ if(𝑗𝑌, (𝐵𝑗), if((𝐵𝑗) ≤ 𝐴, (𝐵𝑗), 𝐴))) = (𝑗𝑋 ↦ if(𝑗𝑌, (𝐵𝑗), if((𝐵𝑗) ≤ 𝐴, (𝐵𝑗), 𝐴)))
86, 7fmptd 7134 . 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 5152 . . . . . . . . 9 (𝑥 = 𝐴 → ((𝑎𝑗) ≤ 𝑥 ↔ (𝑎𝑗) ≤ 𝐴))
11 id 22 . . . . . . . . 9 (𝑥 = 𝐴𝑥 = 𝐴)
1210, 11ifbieq2d 4557 . . . . . . . 8 (𝑥 = 𝐴 → if((𝑎𝑗) ≤ 𝑥, (𝑎𝑗), 𝑥) = if((𝑎𝑗) ≤ 𝐴, (𝑎𝑗), 𝐴))
1312ifeq2d 4551 . . . . . . 7 (𝑥 = 𝐴 → if(𝑗𝑌, (𝑎𝑗), if((𝑎𝑗) ≤ 𝑥, (𝑎𝑗), 𝑥)) = if(𝑗𝑌, (𝑎𝑗), if((𝑎𝑗) ≤ 𝐴, (𝑎𝑗), 𝐴)))
1413mpteq2dv 5250 . . . . . 6 (𝑥 = 𝐴 → (𝑗𝑋 ↦ if(𝑗𝑌, (𝑎𝑗), if((𝑎𝑗) ≤ 𝑥, (𝑎𝑗), 𝑥))) = (𝑗𝑋 ↦ if(𝑗𝑌, (𝑎𝑗), if((𝑎𝑗) ≤ 𝐴, (𝑎𝑗), 𝐴))))
1514mpteq2dv 5250 . . . . 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 7464 . . . . . . 7 (ℝ ↑m 𝑋) ∈ V
1716mptex 7243 . . . . . 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 7039 . . . 4 (𝜑 → (𝐻𝐴) = (𝑎 ∈ (ℝ ↑m 𝑋) ↦ (𝑗𝑋 ↦ if(𝑗𝑌, (𝑎𝑗), if((𝑎𝑗) ≤ 𝐴, (𝑎𝑗), 𝐴)))))
20 fveq1 6906 . . . . . . 7 (𝑎 = 𝐵 → (𝑎𝑗) = (𝐵𝑗))
2120breq1d 5158 . . . . . . . 8 (𝑎 = 𝐵 → ((𝑎𝑗) ≤ 𝐴 ↔ (𝐵𝑗) ≤ 𝐴))
2221, 20ifbieq1d 4555 . . . . . . 7 (𝑎 = 𝐵 → if((𝑎𝑗) ≤ 𝐴, (𝑎𝑗), 𝐴) = if((𝐵𝑗) ≤ 𝐴, (𝐵𝑗), 𝐴))
2320, 22ifeq12d 4552 . . . . . 6 (𝑎 = 𝐵 → if(𝑗𝑌, (𝑎𝑗), if((𝑎𝑗) ≤ 𝐴, (𝑎𝑗), 𝐴)) = if(𝑗𝑌, (𝐵𝑗), if((𝐵𝑗) ≤ 𝐴, (𝐵𝑗), 𝐴)))
2423mpteq2dv 5250 . . . . 5 (𝑎 = 𝐵 → (𝑗𝑋 ↦ if(𝑗𝑌, (𝑎𝑗), if((𝑎𝑗) ≤ 𝐴, (𝑎𝑗), 𝐴))) = (𝑗𝑋 ↦ if(𝑗𝑌, (𝐵𝑗), if((𝐵𝑗) ≤ 𝐴, (𝐵𝑗), 𝐴))))
2524adantl 481 . . . 4 ((𝜑𝑎 = 𝐵) → (𝑗𝑋 ↦ if(𝑗𝑌, (𝑎𝑗), if((𝑎𝑗) ≤ 𝐴, (𝑎𝑗), 𝐴))) = (𝑗𝑋 ↦ if(𝑗𝑌, (𝐵𝑗), if((𝐵𝑗) ≤ 𝐴, (𝐵𝑗), 𝐴))))
26 reex 11244 . . . . . . . 8 ℝ ∈ V
2726a1i 11 . . . . . . 7 (𝜑 → ℝ ∈ V)
28 hsphoif.x . . . . . . 7 (𝜑𝑋𝑉)
2927, 28jca 511 . . . . . 6 (𝜑 → (ℝ ∈ V ∧ 𝑋𝑉))
30 elmapg 8878 . . . . . 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 257 . . . 4 (𝜑𝐵 ∈ (ℝ ↑m 𝑋))
33 mptexg 7241 . . . . 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 7023 . . 3 (𝜑 → ((𝐻𝐴)‘𝐵) = (𝑗𝑋 ↦ if(𝑗𝑌, (𝐵𝑗), if((𝐵𝑗) ≤ 𝐴, (𝐵𝑗), 𝐴))))
3635feq1d 6721 . 2 (𝜑 → (((𝐻𝐴)‘𝐵):𝑋⟶ℝ ↔ (𝑗𝑋 ↦ if(𝑗𝑌, (𝐵𝑗), if((𝐵𝑗) ≤ 𝐴, (𝐵𝑗), 𝐴))):𝑋⟶ℝ))
378, 36mpbird 257 1 (𝜑 → ((𝐻𝐴)‘𝐵):𝑋⟶ℝ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  Vcvv 3478  ifcif 4531   class class class wbr 5148  cmpt 5231  wf 6559  cfv 6563  (class class class)co 7431  m cmap 8865  cr 11152  cle 11294
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8867
This theorem is referenced by:  hsphoidmvle2  46541  hsphoidmvle  46542  sge0hsphoire  46545  hoidmvlelem1  46551  hoidmvlelem2  46552  hoidmvlelem4  46554  hspmbllem1  46582  hspmbllem2  46583
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