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Theorem hsphoival 45295
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 5153 . . . . . . . 8 (π‘₯ = 𝐴 β†’ ((π‘Žβ€˜π‘—) ≀ π‘₯ ↔ (π‘Žβ€˜π‘—) ≀ 𝐴))
3 id 22 . . . . . . . 8 (π‘₯ = 𝐴 β†’ π‘₯ = 𝐴)
42, 3ifbieq2d 4555 . . . . . . 7 (π‘₯ = 𝐴 β†’ if((π‘Žβ€˜π‘—) ≀ π‘₯, (π‘Žβ€˜π‘—), π‘₯) = if((π‘Žβ€˜π‘—) ≀ 𝐴, (π‘Žβ€˜π‘—), 𝐴))
54ifeq2d 4549 . . . . . 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 5251 . . . . 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 5251 . . . 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 7442 . . . . . 6 (ℝ ↑m 𝑋) ∈ V
109mptex 7225 . . . . 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 7022 . . 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 6891 . . . . . 6 (π‘Ž = 𝐡 β†’ (π‘Žβ€˜π‘—) = (π΅β€˜π‘—))
1413breq1d 5159 . . . . . . 7 (π‘Ž = 𝐡 β†’ ((π‘Žβ€˜π‘—) ≀ 𝐴 ↔ (π΅β€˜π‘—) ≀ 𝐴))
1514, 13ifbieq1d 4553 . . . . . 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 4550 . . . . 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 5251 . . . 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 483 . . 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 11201 . . . . . . 7 ℝ ∈ V
2120a1i 11 . . . . . 6 (πœ‘ β†’ ℝ ∈ V)
22 hsphoival.x . . . . . 6 (πœ‘ β†’ 𝑋 ∈ 𝑉)
2321, 22jca 513 . . . . 5 (πœ‘ β†’ (ℝ ∈ V ∧ 𝑋 ∈ 𝑉))
24 elmapg 8833 . . . . 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 7223 . . . 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 7006 . 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 2822 . . . 4 (𝑗 = 𝐾 β†’ (𝑗 ∈ π‘Œ ↔ 𝐾 ∈ π‘Œ))
31 fveq2 6892 . . . 4 (𝑗 = 𝐾 β†’ (π΅β€˜π‘—) = (π΅β€˜πΎ))
3231breq1d 5159 . . . . 5 (𝑗 = 𝐾 β†’ ((π΅β€˜π‘—) ≀ 𝐴 ↔ (π΅β€˜πΎ) ≀ 𝐴))
3332, 31ifbieq1d 4553 . . . 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 4557 . . 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 483 . 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 7088 . . 3 (πœ‘ β†’ (π΅β€˜πΎ) ∈ ℝ)
3837, 8ifcld 4575 . . 3 (πœ‘ β†’ if((π΅β€˜πΎ) ≀ 𝐴, (π΅β€˜πΎ), 𝐴) ∈ ℝ)
3937, 38ifexd 4577 . 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 7006 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 397   = wceq 1542   ∈ wcel 2107  Vcvv 3475  ifcif 4529   class class class wbr 5149   ↦ cmpt 5232  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7409   ↑m cmap 8820  β„cr 11109   ≀ cle 11249
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-map 8822
This theorem is referenced by:  hsphoidmvle2  45301  hsphoidmvle  45302  hoidmvlelem2  45312  hspmbllem1  45342
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