Step |
Hyp |
Ref |
Expression |
1 |
|
hsphoif.h |
⊢ 𝐻 = ( 𝑥 ∈ ℝ ↦ ( 𝑎 ∈ ( ℝ ↑m 𝑋 ) ↦ ( 𝑗 ∈ 𝑋 ↦ if ( 𝑗 ∈ 𝑌 , ( 𝑎 ‘ 𝑗 ) , if ( ( 𝑎 ‘ 𝑗 ) ≤ 𝑥 , ( 𝑎 ‘ 𝑗 ) , 𝑥 ) ) ) ) ) |
2 |
|
hsphoif.a |
⊢ ( 𝜑 → 𝐴 ∈ ℝ ) |
3 |
|
hsphoif.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝑉 ) |
4 |
|
hsphoif.b |
⊢ ( 𝜑 → 𝐵 : 𝑋 ⟶ ℝ ) |
5 |
4
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑋 ) → ( 𝐵 ‘ 𝑗 ) ∈ ℝ ) |
6 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑋 ) → 𝐴 ∈ ℝ ) |
7 |
5 6
|
ifcld |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑋 ) → if ( ( 𝐵 ‘ 𝑗 ) ≤ 𝐴 , ( 𝐵 ‘ 𝑗 ) , 𝐴 ) ∈ ℝ ) |
8 |
5 7
|
ifcld |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑋 ) → if ( 𝑗 ∈ 𝑌 , ( 𝐵 ‘ 𝑗 ) , if ( ( 𝐵 ‘ 𝑗 ) ≤ 𝐴 , ( 𝐵 ‘ 𝑗 ) , 𝐴 ) ) ∈ ℝ ) |
9 |
|
eqid |
⊢ ( 𝑗 ∈ 𝑋 ↦ if ( 𝑗 ∈ 𝑌 , ( 𝐵 ‘ 𝑗 ) , if ( ( 𝐵 ‘ 𝑗 ) ≤ 𝐴 , ( 𝐵 ‘ 𝑗 ) , 𝐴 ) ) ) = ( 𝑗 ∈ 𝑋 ↦ if ( 𝑗 ∈ 𝑌 , ( 𝐵 ‘ 𝑗 ) , if ( ( 𝐵 ‘ 𝑗 ) ≤ 𝐴 , ( 𝐵 ‘ 𝑗 ) , 𝐴 ) ) ) |
10 |
8 9
|
fmptd |
⊢ ( 𝜑 → ( 𝑗 ∈ 𝑋 ↦ if ( 𝑗 ∈ 𝑌 , ( 𝐵 ‘ 𝑗 ) , if ( ( 𝐵 ‘ 𝑗 ) ≤ 𝐴 , ( 𝐵 ‘ 𝑗 ) , 𝐴 ) ) ) : 𝑋 ⟶ ℝ ) |
11 |
|
breq2 |
⊢ ( 𝑥 = 𝐴 → ( ( 𝑎 ‘ 𝑗 ) ≤ 𝑥 ↔ ( 𝑎 ‘ 𝑗 ) ≤ 𝐴 ) ) |
12 |
|
id |
⊢ ( 𝑥 = 𝐴 → 𝑥 = 𝐴 ) |
13 |
11 12
|
ifbieq2d |
⊢ ( 𝑥 = 𝐴 → if ( ( 𝑎 ‘ 𝑗 ) ≤ 𝑥 , ( 𝑎 ‘ 𝑗 ) , 𝑥 ) = if ( ( 𝑎 ‘ 𝑗 ) ≤ 𝐴 , ( 𝑎 ‘ 𝑗 ) , 𝐴 ) ) |
14 |
13
|
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 ) ) ) |
15 |
14
|
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 ) ) ) ) |
16 |
15
|
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 ) ) ) ) ) |
17 |
|
ovex |
⊢ ( ℝ ↑m 𝑋 ) ∈ V |
18 |
17
|
mptex |
⊢ ( 𝑎 ∈ ( ℝ ↑m 𝑋 ) ↦ ( 𝑗 ∈ 𝑋 ↦ if ( 𝑗 ∈ 𝑌 , ( 𝑎 ‘ 𝑗 ) , if ( ( 𝑎 ‘ 𝑗 ) ≤ 𝐴 , ( 𝑎 ‘ 𝑗 ) , 𝐴 ) ) ) ) ∈ V |
19 |
18
|
a1i |
⊢ ( 𝜑 → ( 𝑎 ∈ ( ℝ ↑m 𝑋 ) ↦ ( 𝑗 ∈ 𝑋 ↦ if ( 𝑗 ∈ 𝑌 , ( 𝑎 ‘ 𝑗 ) , if ( ( 𝑎 ‘ 𝑗 ) ≤ 𝐴 , ( 𝑎 ‘ 𝑗 ) , 𝐴 ) ) ) ) ∈ V ) |
20 |
1 16 2 19
|
fvmptd3 |
⊢ ( 𝜑 → ( 𝐻 ‘ 𝐴 ) = ( 𝑎 ∈ ( ℝ ↑m 𝑋 ) ↦ ( 𝑗 ∈ 𝑋 ↦ if ( 𝑗 ∈ 𝑌 , ( 𝑎 ‘ 𝑗 ) , if ( ( 𝑎 ‘ 𝑗 ) ≤ 𝐴 , ( 𝑎 ‘ 𝑗 ) , 𝐴 ) ) ) ) ) |
21 |
|
fveq1 |
⊢ ( 𝑎 = 𝐵 → ( 𝑎 ‘ 𝑗 ) = ( 𝐵 ‘ 𝑗 ) ) |
22 |
21
|
breq1d |
⊢ ( 𝑎 = 𝐵 → ( ( 𝑎 ‘ 𝑗 ) ≤ 𝐴 ↔ ( 𝐵 ‘ 𝑗 ) ≤ 𝐴 ) ) |
23 |
22 21
|
ifbieq1d |
⊢ ( 𝑎 = 𝐵 → if ( ( 𝑎 ‘ 𝑗 ) ≤ 𝐴 , ( 𝑎 ‘ 𝑗 ) , 𝐴 ) = if ( ( 𝐵 ‘ 𝑗 ) ≤ 𝐴 , ( 𝐵 ‘ 𝑗 ) , 𝐴 ) ) |
24 |
21 23
|
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 ) ) ) |
25 |
24
|
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 ) ) ) ) |
26 |
25
|
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 ) ) ) ) |
27 |
|
reex |
⊢ ℝ ∈ V |
28 |
27
|
a1i |
⊢ ( 𝜑 → ℝ ∈ V ) |
29 |
28 3
|
jca |
⊢ ( 𝜑 → ( ℝ ∈ V ∧ 𝑋 ∈ 𝑉 ) ) |
30 |
|
elmapg |
⊢ ( ( ℝ ∈ V ∧ 𝑋 ∈ 𝑉 ) → ( 𝐵 ∈ ( ℝ ↑m 𝑋 ) ↔ 𝐵 : 𝑋 ⟶ ℝ ) ) |
31 |
29 30
|
syl |
⊢ ( 𝜑 → ( 𝐵 ∈ ( ℝ ↑m 𝑋 ) ↔ 𝐵 : 𝑋 ⟶ ℝ ) ) |
32 |
4 31
|
mpbird |
⊢ ( 𝜑 → 𝐵 ∈ ( ℝ ↑m 𝑋 ) ) |
33 |
|
mptexg |
⊢ ( 𝑋 ∈ 𝑉 → ( 𝑗 ∈ 𝑋 ↦ if ( 𝑗 ∈ 𝑌 , ( 𝐵 ‘ 𝑗 ) , if ( ( 𝐵 ‘ 𝑗 ) ≤ 𝐴 , ( 𝐵 ‘ 𝑗 ) , 𝐴 ) ) ) ∈ V ) |
34 |
3 33
|
syl |
⊢ ( 𝜑 → ( 𝑗 ∈ 𝑋 ↦ if ( 𝑗 ∈ 𝑌 , ( 𝐵 ‘ 𝑗 ) , if ( ( 𝐵 ‘ 𝑗 ) ≤ 𝐴 , ( 𝐵 ‘ 𝑗 ) , 𝐴 ) ) ) ∈ V ) |
35 |
20 26 32 34
|
fvmptd |
⊢ ( 𝜑 → ( ( 𝐻 ‘ 𝐴 ) ‘ 𝐵 ) = ( 𝑗 ∈ 𝑋 ↦ if ( 𝑗 ∈ 𝑌 , ( 𝐵 ‘ 𝑗 ) , if ( ( 𝐵 ‘ 𝑗 ) ≤ 𝐴 , ( 𝐵 ‘ 𝑗 ) , 𝐴 ) ) ) ) |
36 |
35
|
feq1d |
⊢ ( 𝜑 → ( ( ( 𝐻 ‘ 𝐴 ) ‘ 𝐵 ) : 𝑋 ⟶ ℝ ↔ ( 𝑗 ∈ 𝑋 ↦ if ( 𝑗 ∈ 𝑌 , ( 𝐵 ‘ 𝑗 ) , if ( ( 𝐵 ‘ 𝑗 ) ≤ 𝐴 , ( 𝐵 ‘ 𝑗 ) , 𝐴 ) ) ) : 𝑋 ⟶ ℝ ) ) |
37 |
10 36
|
mpbird |
⊢ ( 𝜑 → ( ( 𝐻 ‘ 𝐴 ) ‘ 𝐵 ) : 𝑋 ⟶ ℝ ) |