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Theorem hsphoif 41585
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 6609 . . . 4 ((𝜑𝑗𝑋) → (𝐵𝑗) ∈ ℝ)
3 hsphoif.a . . . . . 6 (𝜑𝐴 ∈ ℝ)
43adantr 474 . . . . 5 ((𝜑𝑗𝑋) → 𝐴 ∈ ℝ)
52, 4ifcld 4352 . . . 4 ((𝜑𝑗𝑋) → if((𝐵𝑗) ≤ 𝐴, (𝐵𝑗), 𝐴) ∈ ℝ)
62, 5ifcld 4352 . . 3 ((𝜑𝑗𝑋) → if(𝑗𝑌, (𝐵𝑗), if((𝐵𝑗) ≤ 𝐴, (𝐵𝑗), 𝐴)) ∈ ℝ)
7 eqid 2826 . . 3 (𝑗𝑋 ↦ if(𝑗𝑌, (𝐵𝑗), if((𝐵𝑗) ≤ 𝐴, (𝐵𝑗), 𝐴))) = (𝑗𝑋 ↦ if(𝑗𝑌, (𝐵𝑗), if((𝐵𝑗) ≤ 𝐴, (𝐵𝑗), 𝐴)))
86, 7fmptd 6634 . 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 4878 . . . . . . . . 9 (𝑥 = 𝐴 → ((𝑎𝑗) ≤ 𝑥 ↔ (𝑎𝑗) ≤ 𝐴))
11 id 22 . . . . . . . . 9 (𝑥 = 𝐴𝑥 = 𝐴)
1210, 11ifbieq2d 4332 . . . . . . . 8 (𝑥 = 𝐴 → if((𝑎𝑗) ≤ 𝑥, (𝑎𝑗), 𝑥) = if((𝑎𝑗) ≤ 𝐴, (𝑎𝑗), 𝐴))
1312ifeq2d 4326 . . . . . . 7 (𝑥 = 𝐴 → if(𝑗𝑌, (𝑎𝑗), if((𝑎𝑗) ≤ 𝑥, (𝑎𝑗), 𝑥)) = if(𝑗𝑌, (𝑎𝑗), if((𝑎𝑗) ≤ 𝐴, (𝑎𝑗), 𝐴)))
1413mpteq2dv 4969 . . . . . 6 (𝑥 = 𝐴 → (𝑗𝑋 ↦ if(𝑗𝑌, (𝑎𝑗), if((𝑎𝑗) ≤ 𝑥, (𝑎𝑗), 𝑥))) = (𝑗𝑋 ↦ if(𝑗𝑌, (𝑎𝑗), if((𝑎𝑗) ≤ 𝐴, (𝑎𝑗), 𝐴))))
1514mpteq2dv 4969 . . . . 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 6938 . . . . . . 7 (ℝ ↑𝑚 𝑋) ∈ V
1716mptex 6743 . . . . . 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 6551 . . . 4 (𝜑 → (𝐻𝐴) = (𝑎 ∈ (ℝ ↑𝑚 𝑋) ↦ (𝑗𝑋 ↦ if(𝑗𝑌, (𝑎𝑗), if((𝑎𝑗) ≤ 𝐴, (𝑎𝑗), 𝐴)))))
20 fveq1 6433 . . . . . . 7 (𝑎 = 𝐵 → (𝑎𝑗) = (𝐵𝑗))
2120breq1d 4884 . . . . . . . 8 (𝑎 = 𝐵 → ((𝑎𝑗) ≤ 𝐴 ↔ (𝐵𝑗) ≤ 𝐴))
2221, 20ifbieq1d 4330 . . . . . . 7 (𝑎 = 𝐵 → if((𝑎𝑗) ≤ 𝐴, (𝑎𝑗), 𝐴) = if((𝐵𝑗) ≤ 𝐴, (𝐵𝑗), 𝐴))
2320, 22ifeq12d 4327 . . . . . 6 (𝑎 = 𝐵 → if(𝑗𝑌, (𝑎𝑗), if((𝑎𝑗) ≤ 𝐴, (𝑎𝑗), 𝐴)) = if(𝑗𝑌, (𝐵𝑗), if((𝐵𝑗) ≤ 𝐴, (𝐵𝑗), 𝐴)))
2423mpteq2dv 4969 . . . . 5 (𝑎 = 𝐵 → (𝑗𝑋 ↦ if(𝑗𝑌, (𝑎𝑗), if((𝑎𝑗) ≤ 𝐴, (𝑎𝑗), 𝐴))) = (𝑗𝑋 ↦ if(𝑗𝑌, (𝐵𝑗), if((𝐵𝑗) ≤ 𝐴, (𝐵𝑗), 𝐴))))
2524adantl 475 . . . 4 ((𝜑𝑎 = 𝐵) → (𝑗𝑋 ↦ if(𝑗𝑌, (𝑎𝑗), if((𝑎𝑗) ≤ 𝐴, (𝑎𝑗), 𝐴))) = (𝑗𝑋 ↦ if(𝑗𝑌, (𝐵𝑗), if((𝐵𝑗) ≤ 𝐴, (𝐵𝑗), 𝐴))))
26 reex 10344 . . . . . . . 8 ℝ ∈ V
2726a1i 11 . . . . . . 7 (𝜑 → ℝ ∈ V)
28 hsphoif.x . . . . . . 7 (𝜑𝑋𝑉)
2927, 28jca 509 . . . . . 6 (𝜑 → (ℝ ∈ V ∧ 𝑋𝑉))
30 elmapg 8136 . . . . . 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 249 . . . 4 (𝜑𝐵 ∈ (ℝ ↑𝑚 𝑋))
33 mptexg 6741 . . . . 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 6536 . . 3 (𝜑 → ((𝐻𝐴)‘𝐵) = (𝑗𝑋 ↦ if(𝑗𝑌, (𝐵𝑗), if((𝐵𝑗) ≤ 𝐴, (𝐵𝑗), 𝐴))))
3635feq1d 6264 . 2 (𝜑 → (((𝐻𝐴)‘𝐵):𝑋⟶ℝ ↔ (𝑗𝑋 ↦ if(𝑗𝑌, (𝐵𝑗), if((𝐵𝑗) ≤ 𝐴, (𝐵𝑗), 𝐴))):𝑋⟶ℝ))
378, 36mpbird 249 1 (𝜑 → ((𝐻𝐴)‘𝐵):𝑋⟶ℝ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386   = wceq 1658  wcel 2166  Vcvv 3415  ifcif 4307   class class class wbr 4874  cmpt 4953  wf 6120  cfv 6124  (class class class)co 6906  𝑚 cmap 8123  cr 10252  cle 10393
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-rep 4995  ax-sep 5006  ax-nul 5014  ax-pow 5066  ax-pr 5128  ax-un 7210  ax-cnex 10309  ax-resscn 10310
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ne 3001  df-ral 3123  df-rex 3124  df-reu 3125  df-rab 3127  df-v 3417  df-sbc 3664  df-csb 3759  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4660  df-iun 4743  df-br 4875  df-opab 4937  df-mpt 4954  df-id 5251  df-xp 5349  df-rel 5350  df-cnv 5351  df-co 5352  df-dm 5353  df-rn 5354  df-res 5355  df-ima 5356  df-iota 6087  df-fun 6126  df-fn 6127  df-f 6128  df-f1 6129  df-fo 6130  df-f1o 6131  df-fv 6132  df-ov 6909  df-oprab 6910  df-mpt2 6911  df-map 8125
This theorem is referenced by:  hsphoidmvle2  41594  hsphoidmvle  41595  sge0hsphoire  41598  hoidmvlelem1  41604  hoidmvlelem2  41605  hoidmvlelem4  41607  hspmbllem1  41635  hspmbllem2  41636
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