Metamath Proof Explorer

Theorem hsphoival

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)

Ref Expression
Hypotheses hsphoival.h ⊢ 𝐻 = ( 𝑥 ∈ ℝ ↦ ( 𝑎 ∈ ( ℝ ↑m 𝑋 ) ↦ ( 𝑗 ∈ 𝑋 ↦ if ( 𝑗 ∈ 𝑌 , ( 𝑎 ‘ 𝑗 ) , if ( ( 𝑎 ‘ 𝑗 ) ≤ 𝑥 , ( 𝑎 ‘ 𝑗 ) , 𝑥 ) ) ) ) )
hsphoival.a ⊢ ( 𝜑 → 𝐴 ∈ ℝ )
hsphoival.x ⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
hsphoival.b ⊢ ( 𝜑 → 𝐵 : 𝑋 ⟶ ℝ )
hsphoival.k ⊢ ( 𝜑 → 𝐾 ∈ 𝑋 )
Assertion hsphoival ( 𝜑 → ( ( ( 𝐻 ‘ 𝐴 ) ‘ 𝐵 ) ‘ 𝐾 ) = if ( 𝐾 ∈ 𝑌 , ( 𝐵 ‘ 𝐾 ) , if ( ( 𝐵 ‘ 𝐾 ) ≤ 𝐴 , ( 𝐵 ‘ 𝐾 ) , 𝐴 ) ) )

Proof

Step Hyp Ref Expression
1 hsphoival.h ⊢ 𝐻 = ( 𝑥 ∈ ℝ ↦ ( 𝑎 ∈ ( ℝ ↑m 𝑋 ) ↦ ( 𝑗 ∈ 𝑋 ↦ if ( 𝑗 ∈ 𝑌 , ( 𝑎 ‘ 𝑗 ) , if ( ( 𝑎 ‘ 𝑗 ) ≤ 𝑥 , ( 𝑎 ‘ 𝑗 ) , 𝑥 ) ) ) ) )
2 hsphoival.a ⊢ ( 𝜑 → 𝐴 ∈ ℝ )
3 hsphoival.x ⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
4 hsphoival.b ⊢ ( 𝜑 → 𝐵 : 𝑋 ⟶ ℝ )
5 hsphoival.k ⊢ ( 𝜑 → 𝐾 ∈ 𝑋 )
6 breq2 ⊢ ( 𝑥 = 𝐴 → ( ( 𝑎 ‘ 𝑗 ) ≤ 𝑥 ↔ ( 𝑎 ‘ 𝑗 ) ≤ 𝐴 ) )
7 id ⊢ ( 𝑥 = 𝐴 → 𝑥 = 𝐴 )
8 6 7 ifbieq2d ⊢ ( 𝑥 = 𝐴 → if ( ( 𝑎 ‘ 𝑗 ) ≤ 𝑥 , ( 𝑎 ‘ 𝑗 ) , 𝑥 ) = if ( ( 𝑎 ‘ 𝑗 ) ≤ 𝐴 , ( 𝑎 ‘ 𝑗 ) , 𝐴 ) )
9 8 ifeq2d ⊢ ( 𝑥 = 𝐴 → if ( 𝑗 ∈ 𝑌 , ( 𝑎 ‘ 𝑗 ) , if ( ( 𝑎 ‘ 𝑗 ) ≤ 𝑥 , ( 𝑎 ‘ 𝑗 ) , 𝑥 ) ) = if ( 𝑗 ∈ 𝑌 , ( 𝑎 ‘ 𝑗 ) , if ( ( 𝑎 ‘ 𝑗 ) ≤ 𝐴 , ( 𝑎 ‘ 𝑗 ) , 𝐴 ) ) )
10 9 mpteq2dv ⊢ ( 𝑥 = 𝐴 → ( 𝑗 ∈ 𝑋 ↦ if ( 𝑗 ∈ 𝑌 , ( 𝑎 ‘ 𝑗 ) , if ( ( 𝑎 ‘ 𝑗 ) ≤ 𝑥 , ( 𝑎 ‘ 𝑗 ) , 𝑥 ) ) ) = ( 𝑗 ∈ 𝑋 ↦ if ( 𝑗 ∈ 𝑌 , ( 𝑎 ‘ 𝑗 ) , if ( ( 𝑎 ‘ 𝑗 ) ≤ 𝐴 , ( 𝑎 ‘ 𝑗 ) , 𝐴 ) ) ) )
11 10 mpteq2dv ⊢ ( 𝑥 = 𝐴 → ( 𝑎 ∈ ( ℝ ↑m 𝑋 ) ↦ ( 𝑗 ∈ 𝑋 ↦ if ( 𝑗 ∈ 𝑌 , ( 𝑎 ‘ 𝑗 ) , if ( ( 𝑎 ‘ 𝑗 ) ≤ 𝑥 , ( 𝑎 ‘ 𝑗 ) , 𝑥 ) ) ) ) = ( 𝑎 ∈ ( ℝ ↑m 𝑋 ) ↦ ( 𝑗 ∈ 𝑋 ↦ if ( 𝑗 ∈ 𝑌 , ( 𝑎 ‘ 𝑗 ) , if ( ( 𝑎 ‘ 𝑗 ) ≤ 𝐴 , ( 𝑎 ‘ 𝑗 ) , 𝐴 ) ) ) ) )
12 ovex ⊢ ( ℝ ↑m 𝑋 ) ∈ V
13 12 mptex ⊢ ( 𝑎 ∈ ( ℝ ↑m 𝑋 ) ↦ ( 𝑗 ∈ 𝑋 ↦ if ( 𝑗 ∈ 𝑌 , ( 𝑎 ‘ 𝑗 ) , if ( ( 𝑎 ‘ 𝑗 ) ≤ 𝐴 , ( 𝑎 ‘ 𝑗 ) , 𝐴 ) ) ) ) ∈ V
14 13 a1i ⊢ ( 𝜑 → ( 𝑎 ∈ ( ℝ ↑m 𝑋 ) ↦ ( 𝑗 ∈ 𝑋 ↦ if ( 𝑗 ∈ 𝑌 , ( 𝑎 ‘ 𝑗 ) , if ( ( 𝑎 ‘ 𝑗 ) ≤ 𝐴 , ( 𝑎 ‘ 𝑗 ) , 𝐴 ) ) ) ) ∈ V )
15 1 11 2 14 fvmptd3 ⊢ ( 𝜑 → ( 𝐻 ‘ 𝐴 ) = ( 𝑎 ∈ ( ℝ ↑m 𝑋 ) ↦ ( 𝑗 ∈ 𝑋 ↦ if ( 𝑗 ∈ 𝑌 , ( 𝑎 ‘ 𝑗 ) , if ( ( 𝑎 ‘ 𝑗 ) ≤ 𝐴 , ( 𝑎 ‘ 𝑗 ) , 𝐴 ) ) ) ) )
16 fveq1 ⊢ ( 𝑎 = 𝐵 → ( 𝑎 ‘ 𝑗 ) = ( 𝐵 ‘ 𝑗 ) )
17 16 breq1d ⊢ ( 𝑎 = 𝐵 → ( ( 𝑎 ‘ 𝑗 ) ≤ 𝐴 ↔ ( 𝐵 ‘ 𝑗 ) ≤ 𝐴 ) )
18 17 16 ifbieq1d ⊢ ( 𝑎 = 𝐵 → if ( ( 𝑎 ‘ 𝑗 ) ≤ 𝐴 , ( 𝑎 ‘ 𝑗 ) , 𝐴 ) = if ( ( 𝐵 ‘ 𝑗 ) ≤ 𝐴 , ( 𝐵 ‘ 𝑗 ) , 𝐴 ) )
19 16 18 ifeq12d ⊢ ( 𝑎 = 𝐵 → if ( 𝑗 ∈ 𝑌 , ( 𝑎 ‘ 𝑗 ) , if ( ( 𝑎 ‘ 𝑗 ) ≤ 𝐴 , ( 𝑎 ‘ 𝑗 ) , 𝐴 ) ) = if ( 𝑗 ∈ 𝑌 , ( 𝐵 ‘ 𝑗 ) , if ( ( 𝐵 ‘ 𝑗 ) ≤ 𝐴 , ( 𝐵 ‘ 𝑗 ) , 𝐴 ) ) )
20 19 mpteq2dv ⊢ ( 𝑎 = 𝐵 → ( 𝑗 ∈ 𝑋 ↦ if ( 𝑗 ∈ 𝑌 , ( 𝑎 ‘ 𝑗 ) , if ( ( 𝑎 ‘ 𝑗 ) ≤ 𝐴 , ( 𝑎 ‘ 𝑗 ) , 𝐴 ) ) ) = ( 𝑗 ∈ 𝑋 ↦ if ( 𝑗 ∈ 𝑌 , ( 𝐵 ‘ 𝑗 ) , if ( ( 𝐵 ‘ 𝑗 ) ≤ 𝐴 , ( 𝐵 ‘ 𝑗 ) , 𝐴 ) ) ) )
21 20 adantl ⊢ ( ( 𝜑 ∧ 𝑎 = 𝐵 ) → ( 𝑗 ∈ 𝑋 ↦ if ( 𝑗 ∈ 𝑌 , ( 𝑎 ‘ 𝑗 ) , if ( ( 𝑎 ‘ 𝑗 ) ≤ 𝐴 , ( 𝑎 ‘ 𝑗 ) , 𝐴 ) ) ) = ( 𝑗 ∈ 𝑋 ↦ if ( 𝑗 ∈ 𝑌 , ( 𝐵 ‘ 𝑗 ) , if ( ( 𝐵 ‘ 𝑗 ) ≤ 𝐴 , ( 𝐵 ‘ 𝑗 ) , 𝐴 ) ) ) )
22 reex ⊢ ℝ ∈ V
23 22 a1i ⊢ ( 𝜑 → ℝ ∈ V )
24 23 3 jca ⊢ ( 𝜑 → ( ℝ ∈ V ∧ 𝑋 ∈ 𝑉 ) )
25 elmapg ⊢ ( ( ℝ ∈ V ∧ 𝑋 ∈ 𝑉 ) → ( 𝐵 ∈ ( ℝ ↑m 𝑋 ) ↔ 𝐵 : 𝑋 ⟶ ℝ ) )
26 24 25 syl ⊢ ( 𝜑 → ( 𝐵 ∈ ( ℝ ↑m 𝑋 ) ↔ 𝐵 : 𝑋 ⟶ ℝ ) )
27 4 26 mpbird ⊢ ( 𝜑 → 𝐵 ∈ ( ℝ ↑m 𝑋 ) )
28 mptexg ⊢ ( 𝑋 ∈ 𝑉 → ( 𝑗 ∈ 𝑋 ↦ if ( 𝑗 ∈ 𝑌 , ( 𝐵 ‘ 𝑗 ) , if ( ( 𝐵 ‘ 𝑗 ) ≤ 𝐴 , ( 𝐵 ‘ 𝑗 ) , 𝐴 ) ) ) ∈ V )
29 3 28 syl ⊢ ( 𝜑 → ( 𝑗 ∈ 𝑋 ↦ if ( 𝑗 ∈ 𝑌 , ( 𝐵 ‘ 𝑗 ) , if ( ( 𝐵 ‘ 𝑗 ) ≤ 𝐴 , ( 𝐵 ‘ 𝑗 ) , 𝐴 ) ) ) ∈ V )
30 15 21 27 29 fvmptd ⊢ ( 𝜑 → ( ( 𝐻 ‘ 𝐴 ) ‘ 𝐵 ) = ( 𝑗 ∈ 𝑋 ↦ if ( 𝑗 ∈ 𝑌 , ( 𝐵 ‘ 𝑗 ) , if ( ( 𝐵 ‘ 𝑗 ) ≤ 𝐴 , ( 𝐵 ‘ 𝑗 ) , 𝐴 ) ) ) )
31 eleq1 ⊢ ( 𝑗 = 𝐾 → ( 𝑗 ∈ 𝑌 ↔ 𝐾 ∈ 𝑌 ) )
32 fveq2 ⊢ ( 𝑗 = 𝐾 → ( 𝐵 ‘ 𝑗 ) = ( 𝐵 ‘ 𝐾 ) )
33 32 breq1d ⊢ ( 𝑗 = 𝐾 → ( ( 𝐵 ‘ 𝑗 ) ≤ 𝐴 ↔ ( 𝐵 ‘ 𝐾 ) ≤ 𝐴 ) )
34 33 32 ifbieq1d ⊢ ( 𝑗 = 𝐾 → if ( ( 𝐵 ‘ 𝑗 ) ≤ 𝐴 , ( 𝐵 ‘ 𝑗 ) , 𝐴 ) = if ( ( 𝐵 ‘ 𝐾 ) ≤ 𝐴 , ( 𝐵 ‘ 𝐾 ) , 𝐴 ) )
35 31 32 34 ifbieq12d ⊢ ( 𝑗 = 𝐾 → if ( 𝑗 ∈ 𝑌 , ( 𝐵 ‘ 𝑗 ) , if ( ( 𝐵 ‘ 𝑗 ) ≤ 𝐴 , ( 𝐵 ‘ 𝑗 ) , 𝐴 ) ) = if ( 𝐾 ∈ 𝑌 , ( 𝐵 ‘ 𝐾 ) , if ( ( 𝐵 ‘ 𝐾 ) ≤ 𝐴 , ( 𝐵 ‘ 𝐾 ) , 𝐴 ) ) )
36 35 adantl ⊢ ( ( 𝜑 ∧ 𝑗 = 𝐾 ) → if ( 𝑗 ∈ 𝑌 , ( 𝐵 ‘ 𝑗 ) , if ( ( 𝐵 ‘ 𝑗 ) ≤ 𝐴 , ( 𝐵 ‘ 𝑗 ) , 𝐴 ) ) = if ( 𝐾 ∈ 𝑌 , ( 𝐵 ‘ 𝐾 ) , if ( ( 𝐵 ‘ 𝐾 ) ≤ 𝐴 , ( 𝐵 ‘ 𝐾 ) , 𝐴 ) ) )
37 4 5 ffvelrnd ⊢ ( 𝜑 → ( 𝐵 ‘ 𝐾 ) ∈ ℝ )
38 37 2 ifcld ⊢ ( 𝜑 → if ( ( 𝐵 ‘ 𝐾 ) ≤ 𝐴 , ( 𝐵 ‘ 𝐾 ) , 𝐴 ) ∈ ℝ )
39 37 38 ifexd ⊢ ( 𝜑 → if ( 𝐾 ∈ 𝑌 , ( 𝐵 ‘ 𝐾 ) , if ( ( 𝐵 ‘ 𝐾 ) ≤ 𝐴 , ( 𝐵 ‘ 𝐾 ) , 𝐴 ) ) ∈ V )
40 30 36 5 39 fvmptd ⊢ ( 𝜑 → ( ( ( 𝐻 ‘ 𝐴 ) ‘ 𝐵 ) ‘ 𝐾 ) = if ( 𝐾 ∈ 𝑌 , ( 𝐵 ‘ 𝐾 ) , if ( ( 𝐵 ‘ 𝐾 ) ≤ 𝐴 , ( 𝐵 ‘ 𝐾 ) , 𝐴 ) ) )