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Theorem hoidifhspval3 45013
Description: 𝐷 is a function that returns the representation of the left side of the difference of a half-open interval and a half-space. Used in Lemma 115F of [Fremlin1] p. 31 . (Contributed by Glauco Siliprandi, 24-Dec-2020.)
Hypotheses
Ref Expression
hoidifhspval3.d 𝐷 = (π‘₯ ∈ ℝ ↦ (π‘Ž ∈ (ℝ ↑m 𝑋) ↦ (π‘˜ ∈ 𝑋 ↦ if(π‘˜ = 𝐾, if(π‘₯ ≀ (π‘Žβ€˜π‘˜), (π‘Žβ€˜π‘˜), π‘₯), (π‘Žβ€˜π‘˜)))))
hoidifhspval3.y (πœ‘ β†’ π‘Œ ∈ ℝ)
hoidifhspval3.x (πœ‘ β†’ 𝑋 ∈ 𝑉)
hoidifhspval3.a (πœ‘ β†’ 𝐴:π‘‹βŸΆβ„)
hoidifhspval3.j (πœ‘ β†’ 𝐽 ∈ 𝑋)
Assertion
Ref Expression
hoidifhspval3 (πœ‘ β†’ (((π·β€˜π‘Œ)β€˜π΄)β€˜π½) = if(𝐽 = 𝐾, if(π‘Œ ≀ (π΄β€˜π½), (π΄β€˜π½), π‘Œ), (π΄β€˜π½)))
Distinct variable groups:   𝐴,π‘Ž,π‘˜   π‘˜,𝐽   𝐾,π‘Ž,π‘˜,π‘₯   𝑋,π‘Ž,π‘˜,π‘₯   π‘Œ,π‘Ž,π‘˜,π‘₯   πœ‘,π‘Ž,π‘˜,π‘₯
Allowed substitution hints:   𝐴(π‘₯)   𝐷(π‘₯,π‘˜,π‘Ž)   𝐽(π‘₯,π‘Ž)   𝑉(π‘₯,π‘˜,π‘Ž)

Proof of Theorem hoidifhspval3
StepHypRef Expression
1 hoidifhspval3.d . . 3 𝐷 = (π‘₯ ∈ ℝ ↦ (π‘Ž ∈ (ℝ ↑m 𝑋) ↦ (π‘˜ ∈ 𝑋 ↦ if(π‘˜ = 𝐾, if(π‘₯ ≀ (π‘Žβ€˜π‘˜), (π‘Žβ€˜π‘˜), π‘₯), (π‘Žβ€˜π‘˜)))))
2 hoidifhspval3.y . . 3 (πœ‘ β†’ π‘Œ ∈ ℝ)
3 hoidifhspval3.x . . 3 (πœ‘ β†’ 𝑋 ∈ 𝑉)
4 hoidifhspval3.a . . 3 (πœ‘ β†’ 𝐴:π‘‹βŸΆβ„)
51, 2, 3, 4hoidifhspval2 45009 . 2 (πœ‘ β†’ ((π·β€˜π‘Œ)β€˜π΄) = (π‘˜ ∈ 𝑋 ↦ if(π‘˜ = 𝐾, if(π‘Œ ≀ (π΄β€˜π‘˜), (π΄β€˜π‘˜), π‘Œ), (π΄β€˜π‘˜))))
6 eqeq1 2735 . . . 4 (π‘˜ = 𝐽 β†’ (π‘˜ = 𝐾 ↔ 𝐽 = 𝐾))
7 fveq2 6862 . . . . . 6 (π‘˜ = 𝐽 β†’ (π΄β€˜π‘˜) = (π΄β€˜π½))
87breq2d 5137 . . . . 5 (π‘˜ = 𝐽 β†’ (π‘Œ ≀ (π΄β€˜π‘˜) ↔ π‘Œ ≀ (π΄β€˜π½)))
98, 7ifbieq1d 4530 . . . 4 (π‘˜ = 𝐽 β†’ if(π‘Œ ≀ (π΄β€˜π‘˜), (π΄β€˜π‘˜), π‘Œ) = if(π‘Œ ≀ (π΄β€˜π½), (π΄β€˜π½), π‘Œ))
106, 9, 7ifbieq12d 4534 . . 3 (π‘˜ = 𝐽 β†’ if(π‘˜ = 𝐾, if(π‘Œ ≀ (π΄β€˜π‘˜), (π΄β€˜π‘˜), π‘Œ), (π΄β€˜π‘˜)) = if(𝐽 = 𝐾, if(π‘Œ ≀ (π΄β€˜π½), (π΄β€˜π½), π‘Œ), (π΄β€˜π½)))
1110adantl 482 . 2 ((πœ‘ ∧ π‘˜ = 𝐽) β†’ if(π‘˜ = 𝐾, if(π‘Œ ≀ (π΄β€˜π‘˜), (π΄β€˜π‘˜), π‘Œ), (π΄β€˜π‘˜)) = if(𝐽 = 𝐾, if(π‘Œ ≀ (π΄β€˜π½), (π΄β€˜π½), π‘Œ), (π΄β€˜π½)))
12 hoidifhspval3.j . 2 (πœ‘ β†’ 𝐽 ∈ 𝑋)
13 fvexd 6877 . . . 4 (πœ‘ β†’ (π΄β€˜π½) ∈ V)
142elexd 3479 . . . 4 (πœ‘ β†’ π‘Œ ∈ V)
1513, 14ifcld 4552 . . 3 (πœ‘ β†’ if(π‘Œ ≀ (π΄β€˜π½), (π΄β€˜π½), π‘Œ) ∈ V)
1615, 13ifcld 4552 . 2 (πœ‘ β†’ if(𝐽 = 𝐾, if(π‘Œ ≀ (π΄β€˜π½), (π΄β€˜π½), π‘Œ), (π΄β€˜π½)) ∈ V)
175, 11, 12, 16fvmptd 6975 1 (πœ‘ β†’ (((π·β€˜π‘Œ)β€˜π΄)β€˜π½) = if(𝐽 = 𝐾, if(π‘Œ ≀ (π΄β€˜π½), (π΄β€˜π½), π‘Œ), (π΄β€˜π½)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1541   ∈ wcel 2106  Vcvv 3459  ifcif 4506   class class class wbr 5125   ↦ cmpt 5208  βŸΆwf 6512  β€˜cfv 6516  (class class class)co 7377   ↑m cmap 8787  β„cr 11074   ≀ cle 11214
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5262  ax-sep 5276  ax-nul 5283  ax-pow 5340  ax-pr 5404  ax-un 7692  ax-cnex 11131  ax-resscn 11132
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3365  df-rab 3419  df-v 3461  df-sbc 3758  df-csb 3874  df-dif 3931  df-un 3933  df-in 3935  df-ss 3945  df-nul 4303  df-if 4507  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4886  df-iun 4976  df-br 5126  df-opab 5188  df-mpt 5209  df-id 5551  df-xp 5659  df-rel 5660  df-cnv 5661  df-co 5662  df-dm 5663  df-rn 5664  df-res 5665  df-ima 5666  df-iota 6468  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-ov 7380  df-oprab 7381  df-mpo 7382  df-map 8789
This theorem is referenced by:  hoidifhspdmvle  45014  hspmbllem1  45020  hspmbllem2  45021
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