Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  hsphoival Structured version   Visualization version   GIF version

Theorem hsphoival 45593
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 = (π‘₯ ∈ ℝ ↦ (π‘Ž ∈ (ℝ ↑m 𝑋) ↦ (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ π‘Œ, (π‘Žβ€˜π‘—), if((π‘Žβ€˜π‘—) ≀ π‘₯, (π‘Žβ€˜π‘—), π‘₯)))))
hsphoival.a (πœ‘ β†’ 𝐴 ∈ ℝ)
hsphoival.x (πœ‘ β†’ 𝑋 ∈ 𝑉)
hsphoival.b (πœ‘ β†’ 𝐡:π‘‹βŸΆβ„)
hsphoival.k (πœ‘ β†’ 𝐾 ∈ 𝑋)
Assertion
Ref Expression
hsphoival (πœ‘ β†’ (((π»β€˜π΄)β€˜π΅)β€˜πΎ) = if(𝐾 ∈ π‘Œ, (π΅β€˜πΎ), if((π΅β€˜πΎ) ≀ 𝐴, (π΅β€˜πΎ), 𝐴)))
Distinct variable groups:   𝐴,π‘Ž,𝑗,π‘₯   𝐡,π‘Ž,𝑗   𝑗,𝐾   𝑋,π‘Ž,𝑗,π‘₯   π‘Œ,π‘Ž,𝑗,π‘₯   πœ‘,π‘Ž,𝑗,π‘₯
Allowed substitution hints:   𝐡(π‘₯)   𝐻(π‘₯,𝑗,π‘Ž)   𝐾(π‘₯,π‘Ž)   𝑉(π‘₯,𝑗,π‘Ž)

Proof of Theorem hsphoival
StepHypRef Expression
1 hsphoival.h . . . 4 𝐻 = (π‘₯ ∈ ℝ ↦ (π‘Ž ∈ (ℝ ↑m 𝑋) ↦ (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ π‘Œ, (π‘Žβ€˜π‘—), if((π‘Žβ€˜π‘—) ≀ π‘₯, (π‘Žβ€˜π‘—), π‘₯)))))
2 breq2 5151 . . . . . . . 8 (π‘₯ = 𝐴 β†’ ((π‘Žβ€˜π‘—) ≀ π‘₯ ↔ (π‘Žβ€˜π‘—) ≀ 𝐴))
3 id 22 . . . . . . . 8 (π‘₯ = 𝐴 β†’ π‘₯ = 𝐴)
42, 3ifbieq2d 4553 . . . . . . 7 (π‘₯ = 𝐴 β†’ if((π‘Žβ€˜π‘—) ≀ π‘₯, (π‘Žβ€˜π‘—), π‘₯) = if((π‘Žβ€˜π‘—) ≀ 𝐴, (π‘Žβ€˜π‘—), 𝐴))
54ifeq2d 4547 . . . . . 6 (π‘₯ = 𝐴 β†’ if(𝑗 ∈ π‘Œ, (π‘Žβ€˜π‘—), if((π‘Žβ€˜π‘—) ≀ π‘₯, (π‘Žβ€˜π‘—), π‘₯)) = if(𝑗 ∈ π‘Œ, (π‘Žβ€˜π‘—), if((π‘Žβ€˜π‘—) ≀ 𝐴, (π‘Žβ€˜π‘—), 𝐴)))
65mpteq2dv 5249 . . . . 5 (π‘₯ = 𝐴 β†’ (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ π‘Œ, (π‘Žβ€˜π‘—), if((π‘Žβ€˜π‘—) ≀ π‘₯, (π‘Žβ€˜π‘—), π‘₯))) = (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ π‘Œ, (π‘Žβ€˜π‘—), if((π‘Žβ€˜π‘—) ≀ 𝐴, (π‘Žβ€˜π‘—), 𝐴))))
76mpteq2dv 5249 . . . 4 (π‘₯ = 𝐴 β†’ (π‘Ž ∈ (ℝ ↑m 𝑋) ↦ (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ π‘Œ, (π‘Žβ€˜π‘—), if((π‘Žβ€˜π‘—) ≀ π‘₯, (π‘Žβ€˜π‘—), π‘₯)))) = (π‘Ž ∈ (ℝ ↑m 𝑋) ↦ (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ π‘Œ, (π‘Žβ€˜π‘—), if((π‘Žβ€˜π‘—) ≀ 𝐴, (π‘Žβ€˜π‘—), 𝐴)))))
8 hsphoival.a . . . 4 (πœ‘ β†’ 𝐴 ∈ ℝ)
9 ovex 7444 . . . . . 6 (ℝ ↑m 𝑋) ∈ V
109mptex 7226 . . . . 5 (π‘Ž ∈ (ℝ ↑m 𝑋) ↦ (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ π‘Œ, (π‘Žβ€˜π‘—), if((π‘Žβ€˜π‘—) ≀ 𝐴, (π‘Žβ€˜π‘—), 𝐴)))) ∈ V
1110a1i 11 . . . 4 (πœ‘ β†’ (π‘Ž ∈ (ℝ ↑m 𝑋) ↦ (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ π‘Œ, (π‘Žβ€˜π‘—), if((π‘Žβ€˜π‘—) ≀ 𝐴, (π‘Žβ€˜π‘—), 𝐴)))) ∈ V)
121, 7, 8, 11fvmptd3 7020 . . 3 (πœ‘ β†’ (π»β€˜π΄) = (π‘Ž ∈ (ℝ ↑m 𝑋) ↦ (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ π‘Œ, (π‘Žβ€˜π‘—), if((π‘Žβ€˜π‘—) ≀ 𝐴, (π‘Žβ€˜π‘—), 𝐴)))))
13 fveq1 6889 . . . . . 6 (π‘Ž = 𝐡 β†’ (π‘Žβ€˜π‘—) = (π΅β€˜π‘—))
1413breq1d 5157 . . . . . . 7 (π‘Ž = 𝐡 β†’ ((π‘Žβ€˜π‘—) ≀ 𝐴 ↔ (π΅β€˜π‘—) ≀ 𝐴))
1514, 13ifbieq1d 4551 . . . . . 6 (π‘Ž = 𝐡 β†’ if((π‘Žβ€˜π‘—) ≀ 𝐴, (π‘Žβ€˜π‘—), 𝐴) = if((π΅β€˜π‘—) ≀ 𝐴, (π΅β€˜π‘—), 𝐴))
1613, 15ifeq12d 4548 . . . . 5 (π‘Ž = 𝐡 β†’ if(𝑗 ∈ π‘Œ, (π‘Žβ€˜π‘—), if((π‘Žβ€˜π‘—) ≀ 𝐴, (π‘Žβ€˜π‘—), 𝐴)) = if(𝑗 ∈ π‘Œ, (π΅β€˜π‘—), if((π΅β€˜π‘—) ≀ 𝐴, (π΅β€˜π‘—), 𝐴)))
1716mpteq2dv 5249 . . . 4 (π‘Ž = 𝐡 β†’ (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ π‘Œ, (π‘Žβ€˜π‘—), if((π‘Žβ€˜π‘—) ≀ 𝐴, (π‘Žβ€˜π‘—), 𝐴))) = (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ π‘Œ, (π΅β€˜π‘—), if((π΅β€˜π‘—) ≀ 𝐴, (π΅β€˜π‘—), 𝐴))))
1817adantl 480 . . 3 ((πœ‘ ∧ π‘Ž = 𝐡) β†’ (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ π‘Œ, (π‘Žβ€˜π‘—), if((π‘Žβ€˜π‘—) ≀ 𝐴, (π‘Žβ€˜π‘—), 𝐴))) = (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ π‘Œ, (π΅β€˜π‘—), if((π΅β€˜π‘—) ≀ 𝐴, (π΅β€˜π‘—), 𝐴))))
19 hsphoival.b . . . 4 (πœ‘ β†’ 𝐡:π‘‹βŸΆβ„)
20 reex 11203 . . . . . . 7 ℝ ∈ V
2120a1i 11 . . . . . 6 (πœ‘ β†’ ℝ ∈ V)
22 hsphoival.x . . . . . 6 (πœ‘ β†’ 𝑋 ∈ 𝑉)
2321, 22jca 510 . . . . 5 (πœ‘ β†’ (ℝ ∈ V ∧ 𝑋 ∈ 𝑉))
24 elmapg 8835 . . . . 5 ((ℝ ∈ V ∧ 𝑋 ∈ 𝑉) β†’ (𝐡 ∈ (ℝ ↑m 𝑋) ↔ 𝐡:π‘‹βŸΆβ„))
2523, 24syl 17 . . . 4 (πœ‘ β†’ (𝐡 ∈ (ℝ ↑m 𝑋) ↔ 𝐡:π‘‹βŸΆβ„))
2619, 25mpbird 256 . . 3 (πœ‘ β†’ 𝐡 ∈ (ℝ ↑m 𝑋))
27 mptexg 7224 . . . 4 (𝑋 ∈ 𝑉 β†’ (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ π‘Œ, (π΅β€˜π‘—), if((π΅β€˜π‘—) ≀ 𝐴, (π΅β€˜π‘—), 𝐴))) ∈ V)
2822, 27syl 17 . . 3 (πœ‘ β†’ (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ π‘Œ, (π΅β€˜π‘—), if((π΅β€˜π‘—) ≀ 𝐴, (π΅β€˜π‘—), 𝐴))) ∈ V)
2912, 18, 26, 28fvmptd 7004 . 2 (πœ‘ β†’ ((π»β€˜π΄)β€˜π΅) = (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ π‘Œ, (π΅β€˜π‘—), if((π΅β€˜π‘—) ≀ 𝐴, (π΅β€˜π‘—), 𝐴))))
30 eleq1 2819 . . . 4 (𝑗 = 𝐾 β†’ (𝑗 ∈ π‘Œ ↔ 𝐾 ∈ π‘Œ))
31 fveq2 6890 . . . 4 (𝑗 = 𝐾 β†’ (π΅β€˜π‘—) = (π΅β€˜πΎ))
3231breq1d 5157 . . . . 5 (𝑗 = 𝐾 β†’ ((π΅β€˜π‘—) ≀ 𝐴 ↔ (π΅β€˜πΎ) ≀ 𝐴))
3332, 31ifbieq1d 4551 . . . 4 (𝑗 = 𝐾 β†’ if((π΅β€˜π‘—) ≀ 𝐴, (π΅β€˜π‘—), 𝐴) = if((π΅β€˜πΎ) ≀ 𝐴, (π΅β€˜πΎ), 𝐴))
3430, 31, 33ifbieq12d 4555 . . 3 (𝑗 = 𝐾 β†’ if(𝑗 ∈ π‘Œ, (π΅β€˜π‘—), if((π΅β€˜π‘—) ≀ 𝐴, (π΅β€˜π‘—), 𝐴)) = if(𝐾 ∈ π‘Œ, (π΅β€˜πΎ), if((π΅β€˜πΎ) ≀ 𝐴, (π΅β€˜πΎ), 𝐴)))
3534adantl 480 . 2 ((πœ‘ ∧ 𝑗 = 𝐾) β†’ if(𝑗 ∈ π‘Œ, (π΅β€˜π‘—), if((π΅β€˜π‘—) ≀ 𝐴, (π΅β€˜π‘—), 𝐴)) = if(𝐾 ∈ π‘Œ, (π΅β€˜πΎ), if((π΅β€˜πΎ) ≀ 𝐴, (π΅β€˜πΎ), 𝐴)))
36 hsphoival.k . 2 (πœ‘ β†’ 𝐾 ∈ 𝑋)
3719, 36ffvelcdmd 7086 . . 3 (πœ‘ β†’ (π΅β€˜πΎ) ∈ ℝ)
3837, 8ifcld 4573 . . 3 (πœ‘ β†’ if((π΅β€˜πΎ) ≀ 𝐴, (π΅β€˜πΎ), 𝐴) ∈ ℝ)
3937, 38ifexd 4575 . 2 (πœ‘ β†’ if(𝐾 ∈ π‘Œ, (π΅β€˜πΎ), if((π΅β€˜πΎ) ≀ 𝐴, (π΅β€˜πΎ), 𝐴)) ∈ V)
4029, 35, 36, 39fvmptd 7004 1 (πœ‘ β†’ (((π»β€˜π΄)β€˜π΅)β€˜πΎ) = if(𝐾 ∈ π‘Œ, (π΅β€˜πΎ), if((π΅β€˜πΎ) ≀ 𝐴, (π΅β€˜πΎ), 𝐴)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   = wceq 1539   ∈ wcel 2104  Vcvv 3472  ifcif 4527   class class class wbr 5147   ↦ cmpt 5230  βŸΆwf 6538  β€˜cfv 6542  (class class class)co 7411   ↑m cmap 8822  β„cr 11111   ≀ cle 11253
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-cnex 11168  ax-resscn 11169
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7414  df-oprab 7415  df-mpo 7416  df-map 8824
This theorem is referenced by:  hsphoidmvle2  45599  hsphoidmvle  45600  hoidmvlelem2  45610  hspmbllem1  45640
  Copyright terms: Public domain W3C validator