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Theorem hsphoival 39469
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
hsphoival.h 𝐻 = (𝑥 ∈ ℝ ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑋) ↦ (𝑗𝑋 ↦ if(𝑗𝑌, (𝑎𝑗), if((𝑎𝑗) ≤ 𝑥, (𝑎𝑗), 𝑥)))))
hsphoival.a (𝜑𝐴 ∈ ℝ)
hsphoival.x (𝜑𝑋𝑉)
hsphoival.b (𝜑𝐵:𝑋⟶ℝ)
hsphoival.k (𝜑𝐾𝑋)
Assertion
Ref Expression
hsphoival (𝜑 → (((𝐻𝐴)‘𝐵)‘𝐾) = if(𝐾𝑌, (𝐵𝐾), if((𝐵𝐾) ≤ 𝐴, (𝐵𝐾), 𝐴)))
Distinct variable groups:   𝐴,𝑎,𝑗,𝑥   𝐵,𝑎,𝑗   𝑗,𝐾   𝑋,𝑎,𝑗,𝑥   𝑌,𝑎,𝑗,𝑥   𝜑,𝑎,𝑗,𝑥
Allowed substitution hints:   𝐵(𝑥)   𝐻(𝑥,𝑗,𝑎)   𝐾(𝑥,𝑎)   𝑉(𝑥,𝑗,𝑎)

Proof of Theorem hsphoival
StepHypRef Expression
1 hsphoival.h . . . . 5 𝐻 = (𝑥 ∈ ℝ ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑋) ↦ (𝑗𝑋 ↦ if(𝑗𝑌, (𝑎𝑗), if((𝑎𝑗) ≤ 𝑥, (𝑎𝑗), 𝑥)))))
21a1i 11 . . . 4 (𝜑𝐻 = (𝑥 ∈ ℝ ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑋) ↦ (𝑗𝑋 ↦ if(𝑗𝑌, (𝑎𝑗), if((𝑎𝑗) ≤ 𝑥, (𝑎𝑗), 𝑥))))))
3 breq2 4587 . . . . . . . . 9 (𝑥 = 𝐴 → ((𝑎𝑗) ≤ 𝑥 ↔ (𝑎𝑗) ≤ 𝐴))
4 id 22 . . . . . . . . 9 (𝑥 = 𝐴𝑥 = 𝐴)
53, 4ifbieq2d 4061 . . . . . . . 8 (𝑥 = 𝐴 → if((𝑎𝑗) ≤ 𝑥, (𝑎𝑗), 𝑥) = if((𝑎𝑗) ≤ 𝐴, (𝑎𝑗), 𝐴))
65ifeq2d 4055 . . . . . . 7 (𝑥 = 𝐴 → if(𝑗𝑌, (𝑎𝑗), if((𝑎𝑗) ≤ 𝑥, (𝑎𝑗), 𝑥)) = if(𝑗𝑌, (𝑎𝑗), if((𝑎𝑗) ≤ 𝐴, (𝑎𝑗), 𝐴)))
76mpteq2dv 4673 . . . . . 6 (𝑥 = 𝐴 → (𝑗𝑋 ↦ if(𝑗𝑌, (𝑎𝑗), if((𝑎𝑗) ≤ 𝑥, (𝑎𝑗), 𝑥))) = (𝑗𝑋 ↦ if(𝑗𝑌, (𝑎𝑗), if((𝑎𝑗) ≤ 𝐴, (𝑎𝑗), 𝐴))))
87mpteq2dv 4673 . . . . 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)))))
98adantl 481 . . . 4 ((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)))))
10 hsphoival.a . . . 4 (𝜑𝐴 ∈ ℝ)
11 ovex 6577 . . . . . 6 (ℝ ↑𝑚 𝑋) ∈ V
1211mptex 6390 . . . . 5 (𝑎 ∈ (ℝ ↑𝑚 𝑋) ↦ (𝑗𝑋 ↦ if(𝑗𝑌, (𝑎𝑗), if((𝑎𝑗) ≤ 𝐴, (𝑎𝑗), 𝐴)))) ∈ V
1312a1i 11 . . . 4 (𝜑 → (𝑎 ∈ (ℝ ↑𝑚 𝑋) ↦ (𝑗𝑋 ↦ if(𝑗𝑌, (𝑎𝑗), if((𝑎𝑗) ≤ 𝐴, (𝑎𝑗), 𝐴)))) ∈ V)
142, 9, 10, 13fvmptd 6197 . . 3 (𝜑 → (𝐻𝐴) = (𝑎 ∈ (ℝ ↑𝑚 𝑋) ↦ (𝑗𝑋 ↦ if(𝑗𝑌, (𝑎𝑗), if((𝑎𝑗) ≤ 𝐴, (𝑎𝑗), 𝐴)))))
15 fveq1 6102 . . . . . 6 (𝑎 = 𝐵 → (𝑎𝑗) = (𝐵𝑗))
1615breq1d 4593 . . . . . . 7 (𝑎 = 𝐵 → ((𝑎𝑗) ≤ 𝐴 ↔ (𝐵𝑗) ≤ 𝐴))
1716, 15ifbieq1d 4059 . . . . . 6 (𝑎 = 𝐵 → if((𝑎𝑗) ≤ 𝐴, (𝑎𝑗), 𝐴) = if((𝐵𝑗) ≤ 𝐴, (𝐵𝑗), 𝐴))
1815, 17ifeq12d 4056 . . . . 5 (𝑎 = 𝐵 → if(𝑗𝑌, (𝑎𝑗), if((𝑎𝑗) ≤ 𝐴, (𝑎𝑗), 𝐴)) = if(𝑗𝑌, (𝐵𝑗), if((𝐵𝑗) ≤ 𝐴, (𝐵𝑗), 𝐴)))
1918mpteq2dv 4673 . . . 4 (𝑎 = 𝐵 → (𝑗𝑋 ↦ if(𝑗𝑌, (𝑎𝑗), if((𝑎𝑗) ≤ 𝐴, (𝑎𝑗), 𝐴))) = (𝑗𝑋 ↦ if(𝑗𝑌, (𝐵𝑗), if((𝐵𝑗) ≤ 𝐴, (𝐵𝑗), 𝐴))))
2019adantl 481 . . 3 ((𝜑𝑎 = 𝐵) → (𝑗𝑋 ↦ if(𝑗𝑌, (𝑎𝑗), if((𝑎𝑗) ≤ 𝐴, (𝑎𝑗), 𝐴))) = (𝑗𝑋 ↦ if(𝑗𝑌, (𝐵𝑗), if((𝐵𝑗) ≤ 𝐴, (𝐵𝑗), 𝐴))))
21 hsphoival.b . . . 4 (𝜑𝐵:𝑋⟶ℝ)
22 reex 9906 . . . . . . 7 ℝ ∈ V
2322a1i 11 . . . . . 6 (𝜑 → ℝ ∈ V)
24 hsphoival.x . . . . . 6 (𝜑𝑋𝑉)
2523, 24jca 553 . . . . 5 (𝜑 → (ℝ ∈ V ∧ 𝑋𝑉))
26 elmapg 7757 . . . . 5 ((ℝ ∈ V ∧ 𝑋𝑉) → (𝐵 ∈ (ℝ ↑𝑚 𝑋) ↔ 𝐵:𝑋⟶ℝ))
2725, 26syl 17 . . . 4 (𝜑 → (𝐵 ∈ (ℝ ↑𝑚 𝑋) ↔ 𝐵:𝑋⟶ℝ))
2821, 27mpbird 246 . . 3 (𝜑𝐵 ∈ (ℝ ↑𝑚 𝑋))
29 mptexg 6389 . . . 4 (𝑋𝑉 → (𝑗𝑋 ↦ if(𝑗𝑌, (𝐵𝑗), if((𝐵𝑗) ≤ 𝐴, (𝐵𝑗), 𝐴))) ∈ V)
3024, 29syl 17 . . 3 (𝜑 → (𝑗𝑋 ↦ if(𝑗𝑌, (𝐵𝑗), if((𝐵𝑗) ≤ 𝐴, (𝐵𝑗), 𝐴))) ∈ V)
3114, 20, 28, 30fvmptd 6197 . 2 (𝜑 → ((𝐻𝐴)‘𝐵) = (𝑗𝑋 ↦ if(𝑗𝑌, (𝐵𝑗), if((𝐵𝑗) ≤ 𝐴, (𝐵𝑗), 𝐴))))
32 eleq1 2676 . . . 4 (𝑗 = 𝐾 → (𝑗𝑌𝐾𝑌))
33 fveq2 6103 . . . 4 (𝑗 = 𝐾 → (𝐵𝑗) = (𝐵𝐾))
3433breq1d 4593 . . . . 5 (𝑗 = 𝐾 → ((𝐵𝑗) ≤ 𝐴 ↔ (𝐵𝐾) ≤ 𝐴))
3534, 33ifbieq1d 4059 . . . 4 (𝑗 = 𝐾 → if((𝐵𝑗) ≤ 𝐴, (𝐵𝑗), 𝐴) = if((𝐵𝐾) ≤ 𝐴, (𝐵𝐾), 𝐴))
3632, 33, 35ifbieq12d 4063 . . 3 (𝑗 = 𝐾 → if(𝑗𝑌, (𝐵𝑗), if((𝐵𝑗) ≤ 𝐴, (𝐵𝑗), 𝐴)) = if(𝐾𝑌, (𝐵𝐾), if((𝐵𝐾) ≤ 𝐴, (𝐵𝐾), 𝐴)))
3736adantl 481 . 2 ((𝜑𝑗 = 𝐾) → if(𝑗𝑌, (𝐵𝑗), if((𝐵𝑗) ≤ 𝐴, (𝐵𝑗), 𝐴)) = if(𝐾𝑌, (𝐵𝐾), if((𝐵𝐾) ≤ 𝐴, (𝐵𝐾), 𝐴)))
38 hsphoival.k . 2 (𝜑𝐾𝑋)
3921, 38ffvelrnd 6268 . . 3 (𝜑 → (𝐵𝐾) ∈ ℝ)
4039, 10ifcld 4081 . . 3 (𝜑 → if((𝐵𝐾) ≤ 𝐴, (𝐵𝐾), 𝐴) ∈ ℝ)
41 ifexg 4107 . . 3 (((𝐵𝐾) ∈ ℝ ∧ if((𝐵𝐾) ≤ 𝐴, (𝐵𝐾), 𝐴) ∈ ℝ) → if(𝐾𝑌, (𝐵𝐾), if((𝐵𝐾) ≤ 𝐴, (𝐵𝐾), 𝐴)) ∈ V)
4239, 40, 41syl2anc 691 . 2 (𝜑 → if(𝐾𝑌, (𝐵𝐾), if((𝐵𝐾) ≤ 𝐴, (𝐵𝐾), 𝐴)) ∈ V)
4331, 37, 38, 42fvmptd 6197 1 (𝜑 → (((𝐻𝐴)‘𝐵)‘𝐾) = if(𝐾𝑌, (𝐵𝐾), if((𝐵𝐾) ≤ 𝐴, (𝐵𝐾), 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383   = wceq 1475  wcel 1977  Vcvv 3173  ifcif 4036   class class class wbr 4583  cmpt 4643  wf 5800  cfv 5804  (class class class)co 6549  𝑚 cmap 7744  cr 9814  cle 9954
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-map 7746
This theorem is referenced by:  hsphoidmvle2  39475  hsphoidmvle  39476  hoidmvlelem2  39486  hspmbllem1  39516
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