Step |
Hyp |
Ref |
Expression |
1 |
|
smfadd.x |
⊢ Ⅎ 𝑥 𝜑 |
2 |
|
smfadd.s |
⊢ ( 𝜑 → 𝑆 ∈ SAlg ) |
3 |
|
smfadd.a |
⊢ ( 𝜑 → 𝐴 ∈ 𝑉 ) |
4 |
|
smfadd.b |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ ℝ ) |
5 |
|
smfadd.d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐶 ) → 𝐷 ∈ ℝ ) |
6 |
|
smfadd.m |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∈ ( SMblFn ‘ 𝑆 ) ) |
7 |
|
smfadd.n |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐶 ↦ 𝐷 ) ∈ ( SMblFn ‘ 𝑆 ) ) |
8 |
|
nfv |
⊢ Ⅎ 𝑎 𝜑 |
9 |
|
elinel1 |
⊢ ( 𝑥 ∈ ( 𝐴 ∩ 𝐶 ) → 𝑥 ∈ 𝐴 ) |
10 |
9
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐴 ∩ 𝐶 ) ) → 𝑥 ∈ 𝐴 ) |
11 |
1 10
|
ssdf |
⊢ ( 𝜑 → ( 𝐴 ∩ 𝐶 ) ⊆ 𝐴 ) |
12 |
|
eqid |
⊢ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) |
13 |
1 12 4
|
dmmptdf |
⊢ ( 𝜑 → dom ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = 𝐴 ) |
14 |
13
|
eqcomd |
⊢ ( 𝜑 → 𝐴 = dom ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) |
15 |
|
eqid |
⊢ dom ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = dom ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) |
16 |
2 6 15
|
smfdmss |
⊢ ( 𝜑 → dom ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ⊆ ∪ 𝑆 ) |
17 |
14 16
|
eqsstrd |
⊢ ( 𝜑 → 𝐴 ⊆ ∪ 𝑆 ) |
18 |
11 17
|
sstrd |
⊢ ( 𝜑 → ( 𝐴 ∩ 𝐶 ) ⊆ ∪ 𝑆 ) |
19 |
10 4
|
syldan |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐴 ∩ 𝐶 ) ) → 𝐵 ∈ ℝ ) |
20 |
|
elinel2 |
⊢ ( 𝑥 ∈ ( 𝐴 ∩ 𝐶 ) → 𝑥 ∈ 𝐶 ) |
21 |
20
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐴 ∩ 𝐶 ) ) → 𝑥 ∈ 𝐶 ) |
22 |
21 5
|
syldan |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐴 ∩ 𝐶 ) ) → 𝐷 ∈ ℝ ) |
23 |
19 22
|
readdcld |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐴 ∩ 𝐶 ) ) → ( 𝐵 + 𝐷 ) ∈ ℝ ) |
24 |
|
eqid |
⊢ ( 𝑥 ∈ ( 𝐴 ∩ 𝐶 ) ↦ ( 𝐵 + 𝐷 ) ) = ( 𝑥 ∈ ( 𝐴 ∩ 𝐶 ) ↦ ( 𝐵 + 𝐷 ) ) |
25 |
1 23 24
|
fmptdf |
⊢ ( 𝜑 → ( 𝑥 ∈ ( 𝐴 ∩ 𝐶 ) ↦ ( 𝐵 + 𝐷 ) ) : ( 𝐴 ∩ 𝐶 ) ⟶ ℝ ) |
26 |
25
|
fvmptelrn |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐴 ∩ 𝐶 ) ) → ( 𝐵 + 𝐷 ) ∈ ℝ ) |
27 |
|
nfv |
⊢ Ⅎ 𝑥 𝑎 ∈ ℝ |
28 |
1 27
|
nfan |
⊢ Ⅎ 𝑥 ( 𝜑 ∧ 𝑎 ∈ ℝ ) |
29 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ℝ ) → 𝑆 ∈ SAlg ) |
30 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ℝ ) → 𝐴 ∈ 𝑉 ) |
31 |
4
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ ℝ ) ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ ℝ ) |
32 |
5
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ ℝ ) ∧ 𝑥 ∈ 𝐶 ) → 𝐷 ∈ ℝ ) |
33 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ℝ ) → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∈ ( SMblFn ‘ 𝑆 ) ) |
34 |
7
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ℝ ) → ( 𝑥 ∈ 𝐶 ↦ 𝐷 ) ∈ ( SMblFn ‘ 𝑆 ) ) |
35 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ℝ ) → 𝑎 ∈ ℝ ) |
36 |
|
oveq2 |
⊢ ( 𝑟 = 𝑞 → ( 𝑝 + 𝑟 ) = ( 𝑝 + 𝑞 ) ) |
37 |
36
|
breq1d |
⊢ ( 𝑟 = 𝑞 → ( ( 𝑝 + 𝑟 ) < 𝑎 ↔ ( 𝑝 + 𝑞 ) < 𝑎 ) ) |
38 |
37
|
cbvrabv |
⊢ { 𝑟 ∈ ℚ ∣ ( 𝑝 + 𝑟 ) < 𝑎 } = { 𝑞 ∈ ℚ ∣ ( 𝑝 + 𝑞 ) < 𝑎 } |
39 |
38
|
mpteq2i |
⊢ ( 𝑝 ∈ ℚ ↦ { 𝑟 ∈ ℚ ∣ ( 𝑝 + 𝑟 ) < 𝑎 } ) = ( 𝑝 ∈ ℚ ↦ { 𝑞 ∈ ℚ ∣ ( 𝑝 + 𝑞 ) < 𝑎 } ) |
40 |
28 29 30 31 32 33 34 35 39
|
smfaddlem2 |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ℝ ) → { 𝑥 ∈ ( 𝐴 ∩ 𝐶 ) ∣ ( 𝐵 + 𝐷 ) < 𝑎 } ∈ ( 𝑆 ↾t ( 𝐴 ∩ 𝐶 ) ) ) |
41 |
1 8 2 18 26 40
|
issmfdmpt |
⊢ ( 𝜑 → ( 𝑥 ∈ ( 𝐴 ∩ 𝐶 ) ↦ ( 𝐵 + 𝐷 ) ) ∈ ( SMblFn ‘ 𝑆 ) ) |