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Theorem hsphoival 42729
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 . . . 4 𝐻 = (𝑥 ∈ ℝ ↦ (𝑎 ∈ (ℝ ↑m 𝑋) ↦ (𝑗𝑋 ↦ if(𝑗𝑌, (𝑎𝑗), if((𝑎𝑗) ≤ 𝑥, (𝑎𝑗), 𝑥)))))
2 breq2 5066 . . . . . . . 8 (𝑥 = 𝐴 → ((𝑎𝑗) ≤ 𝑥 ↔ (𝑎𝑗) ≤ 𝐴))
3 id 22 . . . . . . . 8 (𝑥 = 𝐴𝑥 = 𝐴)
42, 3ifbieq2d 4494 . . . . . . 7 (𝑥 = 𝐴 → if((𝑎𝑗) ≤ 𝑥, (𝑎𝑗), 𝑥) = if((𝑎𝑗) ≤ 𝐴, (𝑎𝑗), 𝐴))
54ifeq2d 4488 . . . . . 6 (𝑥 = 𝐴 → if(𝑗𝑌, (𝑎𝑗), if((𝑎𝑗) ≤ 𝑥, (𝑎𝑗), 𝑥)) = if(𝑗𝑌, (𝑎𝑗), if((𝑎𝑗) ≤ 𝐴, (𝑎𝑗), 𝐴)))
65mpteq2dv 5158 . . . . 5 (𝑥 = 𝐴 → (𝑗𝑋 ↦ if(𝑗𝑌, (𝑎𝑗), if((𝑎𝑗) ≤ 𝑥, (𝑎𝑗), 𝑥))) = (𝑗𝑋 ↦ if(𝑗𝑌, (𝑎𝑗), if((𝑎𝑗) ≤ 𝐴, (𝑎𝑗), 𝐴))))
76mpteq2dv 5158 . . . 4 (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)))))
8 hsphoival.a . . . 4 (𝜑𝐴 ∈ ℝ)
9 ovex 7184 . . . . . 6 (ℝ ↑m 𝑋) ∈ V
109mptex 6984 . . . . 5 (𝑎 ∈ (ℝ ↑m 𝑋) ↦ (𝑗𝑋 ↦ if(𝑗𝑌, (𝑎𝑗), if((𝑎𝑗) ≤ 𝐴, (𝑎𝑗), 𝐴)))) ∈ V
1110a1i 11 . . . 4 (𝜑 → (𝑎 ∈ (ℝ ↑m 𝑋) ↦ (𝑗𝑋 ↦ if(𝑗𝑌, (𝑎𝑗), if((𝑎𝑗) ≤ 𝐴, (𝑎𝑗), 𝐴)))) ∈ V)
121, 7, 8, 11fvmptd3 6786 . . 3 (𝜑 → (𝐻𝐴) = (𝑎 ∈ (ℝ ↑m 𝑋) ↦ (𝑗𝑋 ↦ if(𝑗𝑌, (𝑎𝑗), if((𝑎𝑗) ≤ 𝐴, (𝑎𝑗), 𝐴)))))
13 fveq1 6665 . . . . . 6 (𝑎 = 𝐵 → (𝑎𝑗) = (𝐵𝑗))
1413breq1d 5072 . . . . . . 7 (𝑎 = 𝐵 → ((𝑎𝑗) ≤ 𝐴 ↔ (𝐵𝑗) ≤ 𝐴))
1514, 13ifbieq1d 4492 . . . . . 6 (𝑎 = 𝐵 → if((𝑎𝑗) ≤ 𝐴, (𝑎𝑗), 𝐴) = if((𝐵𝑗) ≤ 𝐴, (𝐵𝑗), 𝐴))
1613, 15ifeq12d 4489 . . . . 5 (𝑎 = 𝐵 → if(𝑗𝑌, (𝑎𝑗), if((𝑎𝑗) ≤ 𝐴, (𝑎𝑗), 𝐴)) = if(𝑗𝑌, (𝐵𝑗), if((𝐵𝑗) ≤ 𝐴, (𝐵𝑗), 𝐴)))
1716mpteq2dv 5158 . . . 4 (𝑎 = 𝐵 → (𝑗𝑋 ↦ if(𝑗𝑌, (𝑎𝑗), if((𝑎𝑗) ≤ 𝐴, (𝑎𝑗), 𝐴))) = (𝑗𝑋 ↦ if(𝑗𝑌, (𝐵𝑗), if((𝐵𝑗) ≤ 𝐴, (𝐵𝑗), 𝐴))))
1817adantl 482 . . 3 ((𝜑𝑎 = 𝐵) → (𝑗𝑋 ↦ if(𝑗𝑌, (𝑎𝑗), if((𝑎𝑗) ≤ 𝐴, (𝑎𝑗), 𝐴))) = (𝑗𝑋 ↦ if(𝑗𝑌, (𝐵𝑗), if((𝐵𝑗) ≤ 𝐴, (𝐵𝑗), 𝐴))))
19 hsphoival.b . . . 4 (𝜑𝐵:𝑋⟶ℝ)
20 reex 10620 . . . . . . 7 ℝ ∈ V
2120a1i 11 . . . . . 6 (𝜑 → ℝ ∈ V)
22 hsphoival.x . . . . . 6 (𝜑𝑋𝑉)
2321, 22jca 512 . . . . 5 (𝜑 → (ℝ ∈ V ∧ 𝑋𝑉))
24 elmapg 8412 . . . . 5 ((ℝ ∈ V ∧ 𝑋𝑉) → (𝐵 ∈ (ℝ ↑m 𝑋) ↔ 𝐵:𝑋⟶ℝ))
2523, 24syl 17 . . . 4 (𝜑 → (𝐵 ∈ (ℝ ↑m 𝑋) ↔ 𝐵:𝑋⟶ℝ))
2619, 25mpbird 258 . . 3 (𝜑𝐵 ∈ (ℝ ↑m 𝑋))
27 mptexg 6982 . . . 4 (𝑋𝑉 → (𝑗𝑋 ↦ if(𝑗𝑌, (𝐵𝑗), if((𝐵𝑗) ≤ 𝐴, (𝐵𝑗), 𝐴))) ∈ V)
2822, 27syl 17 . . 3 (𝜑 → (𝑗𝑋 ↦ if(𝑗𝑌, (𝐵𝑗), if((𝐵𝑗) ≤ 𝐴, (𝐵𝑗), 𝐴))) ∈ V)
2912, 18, 26, 28fvmptd 6770 . 2 (𝜑 → ((𝐻𝐴)‘𝐵) = (𝑗𝑋 ↦ if(𝑗𝑌, (𝐵𝑗), if((𝐵𝑗) ≤ 𝐴, (𝐵𝑗), 𝐴))))
30 eleq1 2904 . . . 4 (𝑗 = 𝐾 → (𝑗𝑌𝐾𝑌))
31 fveq2 6666 . . . 4 (𝑗 = 𝐾 → (𝐵𝑗) = (𝐵𝐾))
3231breq1d 5072 . . . . 5 (𝑗 = 𝐾 → ((𝐵𝑗) ≤ 𝐴 ↔ (𝐵𝐾) ≤ 𝐴))
3332, 31ifbieq1d 4492 . . . 4 (𝑗 = 𝐾 → if((𝐵𝑗) ≤ 𝐴, (𝐵𝑗), 𝐴) = if((𝐵𝐾) ≤ 𝐴, (𝐵𝐾), 𝐴))
3430, 31, 33ifbieq12d 4496 . . 3 (𝑗 = 𝐾 → if(𝑗𝑌, (𝐵𝑗), if((𝐵𝑗) ≤ 𝐴, (𝐵𝑗), 𝐴)) = if(𝐾𝑌, (𝐵𝐾), if((𝐵𝐾) ≤ 𝐴, (𝐵𝐾), 𝐴)))
3534adantl 482 . 2 ((𝜑𝑗 = 𝐾) → if(𝑗𝑌, (𝐵𝑗), if((𝐵𝑗) ≤ 𝐴, (𝐵𝑗), 𝐴)) = if(𝐾𝑌, (𝐵𝐾), if((𝐵𝐾) ≤ 𝐴, (𝐵𝐾), 𝐴)))
36 hsphoival.k . 2 (𝜑𝐾𝑋)
3719, 36ffvelrnd 6847 . . 3 (𝜑 → (𝐵𝐾) ∈ ℝ)
3837, 8ifcld 4514 . . 3 (𝜑 → if((𝐵𝐾) ≤ 𝐴, (𝐵𝐾), 𝐴) ∈ ℝ)
39 ifexg 4516 . . 3 (((𝐵𝐾) ∈ ℝ ∧ if((𝐵𝐾) ≤ 𝐴, (𝐵𝐾), 𝐴) ∈ ℝ) → if(𝐾𝑌, (𝐵𝐾), if((𝐵𝐾) ≤ 𝐴, (𝐵𝐾), 𝐴)) ∈ V)
4037, 38, 39syl2anc 584 . 2 (𝜑 → if(𝐾𝑌, (𝐵𝐾), if((𝐵𝐾) ≤ 𝐴, (𝐵𝐾), 𝐴)) ∈ V)
4129, 35, 36, 40fvmptd 6770 1 (𝜑 → (((𝐻𝐴)‘𝐵)‘𝐾) = if(𝐾𝑌, (𝐵𝐾), if((𝐵𝐾) ≤ 𝐴, (𝐵𝐾), 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1530  wcel 2107  Vcvv 3499  ifcif 4469   class class class wbr 5062  cmpt 5142  wf 6347  cfv 6351  (class class class)co 7151  m cmap 8399  cr 10528  cle 10668
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454  ax-cnex 10585  ax-resscn 10586
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-reu 3149  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-ov 7154  df-oprab 7155  df-mpo 7156  df-map 8401
This theorem is referenced by:  hsphoidmvle2  42735  hsphoidmvle  42736  hoidmvlelem2  42746  hspmbllem1  42776
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