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Theorem hsphoival 45594
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 5152 . . . . . . . 8 (π‘₯ = 𝐴 β†’ ((π‘Žβ€˜π‘—) ≀ π‘₯ ↔ (π‘Žβ€˜π‘—) ≀ 𝐴))
3 id 22 . . . . . . . 8 (π‘₯ = 𝐴 β†’ π‘₯ = 𝐴)
42, 3ifbieq2d 4554 . . . . . . 7 (π‘₯ = 𝐴 β†’ if((π‘Žβ€˜π‘—) ≀ π‘₯, (π‘Žβ€˜π‘—), π‘₯) = if((π‘Žβ€˜π‘—) ≀ 𝐴, (π‘Žβ€˜π‘—), 𝐴))
54ifeq2d 4548 . . . . . 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 5250 . . . . 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 5250 . . . 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 7445 . . . . . 6 (ℝ ↑m 𝑋) ∈ V
109mptex 7227 . . . . 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 7021 . . 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 6890 . . . . . 6 (π‘Ž = 𝐡 β†’ (π‘Žβ€˜π‘—) = (π΅β€˜π‘—))
1413breq1d 5158 . . . . . . 7 (π‘Ž = 𝐡 β†’ ((π‘Žβ€˜π‘—) ≀ 𝐴 ↔ (π΅β€˜π‘—) ≀ 𝐴))
1514, 13ifbieq1d 4552 . . . . . 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 4549 . . . . 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 5250 . . . 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 481 . . 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 11205 . . . . . . 7 ℝ ∈ V
2120a1i 11 . . . . . 6 (πœ‘ β†’ ℝ ∈ V)
22 hsphoival.x . . . . . 6 (πœ‘ β†’ 𝑋 ∈ 𝑉)
2321, 22jca 511 . . . . 5 (πœ‘ β†’ (ℝ ∈ V ∧ 𝑋 ∈ 𝑉))
24 elmapg 8837 . . . . 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 257 . . 3 (πœ‘ β†’ 𝐡 ∈ (ℝ ↑m 𝑋))
27 mptexg 7225 . . . 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 7005 . 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 2820 . . . 4 (𝑗 = 𝐾 β†’ (𝑗 ∈ π‘Œ ↔ 𝐾 ∈ π‘Œ))
31 fveq2 6891 . . . 4 (𝑗 = 𝐾 β†’ (π΅β€˜π‘—) = (π΅β€˜πΎ))
3231breq1d 5158 . . . . 5 (𝑗 = 𝐾 β†’ ((π΅β€˜π‘—) ≀ 𝐴 ↔ (π΅β€˜πΎ) ≀ 𝐴))
3332, 31ifbieq1d 4552 . . . 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 4556 . . 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 481 . 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 7087 . . 3 (πœ‘ β†’ (π΅β€˜πΎ) ∈ ℝ)
3837, 8ifcld 4574 . . 3 (πœ‘ β†’ if((π΅β€˜πΎ) ≀ 𝐴, (π΅β€˜πΎ), 𝐴) ∈ ℝ)
3937, 38ifexd 4576 . 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 7005 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 395   = wceq 1540   ∈ wcel 2105  Vcvv 3473  ifcif 4528   class class class wbr 5148   ↦ cmpt 5231  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7412   ↑m cmap 8824  β„cr 11113   ≀ cle 11254
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-cnex 11170  ax-resscn 11171
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  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 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417  df-map 8826
This theorem is referenced by:  hsphoidmvle2  45600  hsphoidmvle  45601  hoidmvlelem2  45611  hspmbllem1  45641
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