Step |
Hyp |
Ref |
Expression |
1 |
|
smfaddlem2.x |
⊢ Ⅎ 𝑥 𝜑 |
2 |
|
smfaddlem2.s |
⊢ ( 𝜑 → 𝑆 ∈ SAlg ) |
3 |
|
smfaddlem2.a |
⊢ ( 𝜑 → 𝐴 ∈ 𝑉 ) |
4 |
|
smfaddlem2.b |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ ℝ ) |
5 |
|
smfaddlem2.d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐶 ) → 𝐷 ∈ ℝ ) |
6 |
|
smfaddlem2.m |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∈ ( SMblFn ‘ 𝑆 ) ) |
7 |
|
smfaddlem2.7 |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐶 ↦ 𝐷 ) ∈ ( SMblFn ‘ 𝑆 ) ) |
8 |
|
smfaddlem2.r |
⊢ ( 𝜑 → 𝑅 ∈ ℝ ) |
9 |
|
smfaddlem2.k |
⊢ 𝐾 = ( 𝑝 ∈ ℚ ↦ { 𝑞 ∈ ℚ ∣ ( 𝑝 + 𝑞 ) < 𝑅 } ) |
10 |
1 4 5 8 9
|
smfaddlem1 |
⊢ ( 𝜑 → { 𝑥 ∈ ( 𝐴 ∩ 𝐶 ) ∣ ( 𝐵 + 𝐷 ) < 𝑅 } = ∪ 𝑝 ∈ ℚ ∪ 𝑞 ∈ ( 𝐾 ‘ 𝑝 ) { 𝑥 ∈ ( 𝐴 ∩ 𝐶 ) ∣ ( 𝐵 < 𝑝 ∧ 𝐷 < 𝑞 ) } ) |
11 |
|
elinel1 |
⊢ ( 𝑥 ∈ ( 𝐴 ∩ 𝐶 ) → 𝑥 ∈ 𝐴 ) |
12 |
11
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐴 ∩ 𝐶 ) ) → 𝑥 ∈ 𝐴 ) |
13 |
1 12
|
ssdf |
⊢ ( 𝜑 → ( 𝐴 ∩ 𝐶 ) ⊆ 𝐴 ) |
14 |
3 13
|
ssexd |
⊢ ( 𝜑 → ( 𝐴 ∩ 𝐶 ) ∈ V ) |
15 |
|
eqid |
⊢ ( 𝑆 ↾t ( 𝐴 ∩ 𝐶 ) ) = ( 𝑆 ↾t ( 𝐴 ∩ 𝐶 ) ) |
16 |
2 14 15
|
subsalsal |
⊢ ( 𝜑 → ( 𝑆 ↾t ( 𝐴 ∩ 𝐶 ) ) ∈ SAlg ) |
17 |
|
qct |
⊢ ℚ ≼ ω |
18 |
17
|
a1i |
⊢ ( 𝜑 → ℚ ≼ ω ) |
19 |
16
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑝 ∈ ℚ ) → ( 𝑆 ↾t ( 𝐴 ∩ 𝐶 ) ) ∈ SAlg ) |
20 |
|
qex |
⊢ ℚ ∈ V |
21 |
20
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑝 ∈ ℚ ) → ℚ ∈ V ) |
22 |
9
|
a1i |
⊢ ( 𝜑 → 𝐾 = ( 𝑝 ∈ ℚ ↦ { 𝑞 ∈ ℚ ∣ ( 𝑝 + 𝑞 ) < 𝑅 } ) ) |
23 |
20
|
rabex |
⊢ { 𝑞 ∈ ℚ ∣ ( 𝑝 + 𝑞 ) < 𝑅 } ∈ V |
24 |
23
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑝 ∈ ℚ ) → { 𝑞 ∈ ℚ ∣ ( 𝑝 + 𝑞 ) < 𝑅 } ∈ V ) |
25 |
22 24
|
fvmpt2d |
⊢ ( ( 𝜑 ∧ 𝑝 ∈ ℚ ) → ( 𝐾 ‘ 𝑝 ) = { 𝑞 ∈ ℚ ∣ ( 𝑝 + 𝑞 ) < 𝑅 } ) |
26 |
|
ssrab2 |
⊢ { 𝑞 ∈ ℚ ∣ ( 𝑝 + 𝑞 ) < 𝑅 } ⊆ ℚ |
27 |
25 26
|
eqsstrdi |
⊢ ( ( 𝜑 ∧ 𝑝 ∈ ℚ ) → ( 𝐾 ‘ 𝑝 ) ⊆ ℚ ) |
28 |
|
ssdomg |
⊢ ( ℚ ∈ V → ( ( 𝐾 ‘ 𝑝 ) ⊆ ℚ → ( 𝐾 ‘ 𝑝 ) ≼ ℚ ) ) |
29 |
21 27 28
|
sylc |
⊢ ( ( 𝜑 ∧ 𝑝 ∈ ℚ ) → ( 𝐾 ‘ 𝑝 ) ≼ ℚ ) |
30 |
17
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑝 ∈ ℚ ) → ℚ ≼ ω ) |
31 |
|
domtr |
⊢ ( ( ( 𝐾 ‘ 𝑝 ) ≼ ℚ ∧ ℚ ≼ ω ) → ( 𝐾 ‘ 𝑝 ) ≼ ω ) |
32 |
29 30 31
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑝 ∈ ℚ ) → ( 𝐾 ‘ 𝑝 ) ≼ ω ) |
33 |
|
inrab |
⊢ ( { 𝑥 ∈ ( 𝐴 ∩ 𝐶 ) ∣ 𝐵 < 𝑝 } ∩ { 𝑥 ∈ ( 𝐴 ∩ 𝐶 ) ∣ 𝐷 < 𝑞 } ) = { 𝑥 ∈ ( 𝐴 ∩ 𝐶 ) ∣ ( 𝐵 < 𝑝 ∧ 𝐷 < 𝑞 ) } |
34 |
16
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑝 ∈ ℚ ) ∧ 𝑞 ∈ ( 𝐾 ‘ 𝑝 ) ) → ( 𝑆 ↾t ( 𝐴 ∩ 𝐶 ) ) ∈ SAlg ) |
35 |
|
nfv |
⊢ Ⅎ 𝑥 𝑝 ∈ ℚ |
36 |
1 35
|
nfan |
⊢ Ⅎ 𝑥 ( 𝜑 ∧ 𝑝 ∈ ℚ ) |
37 |
|
nfv |
⊢ Ⅎ 𝑥 𝑞 ∈ ( 𝐾 ‘ 𝑝 ) |
38 |
36 37
|
nfan |
⊢ Ⅎ 𝑥 ( ( 𝜑 ∧ 𝑝 ∈ ℚ ) ∧ 𝑞 ∈ ( 𝐾 ‘ 𝑝 ) ) |
39 |
2
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑝 ∈ ℚ ) ∧ 𝑞 ∈ ( 𝐾 ‘ 𝑝 ) ) → 𝑆 ∈ SAlg ) |
40 |
12 4
|
syldan |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐴 ∩ 𝐶 ) ) → 𝐵 ∈ ℝ ) |
41 |
40
|
ad4ant14 |
⊢ ( ( ( ( 𝜑 ∧ 𝑝 ∈ ℚ ) ∧ 𝑞 ∈ ( 𝐾 ‘ 𝑝 ) ) ∧ 𝑥 ∈ ( 𝐴 ∩ 𝐶 ) ) → 𝐵 ∈ ℝ ) |
42 |
2 6 13
|
sssmfmpt |
⊢ ( 𝜑 → ( 𝑥 ∈ ( 𝐴 ∩ 𝐶 ) ↦ 𝐵 ) ∈ ( SMblFn ‘ 𝑆 ) ) |
43 |
42
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑝 ∈ ℚ ) ∧ 𝑞 ∈ ( 𝐾 ‘ 𝑝 ) ) → ( 𝑥 ∈ ( 𝐴 ∩ 𝐶 ) ↦ 𝐵 ) ∈ ( SMblFn ‘ 𝑆 ) ) |
44 |
|
qre |
⊢ ( 𝑝 ∈ ℚ → 𝑝 ∈ ℝ ) |
45 |
44
|
ad2antlr |
⊢ ( ( ( 𝜑 ∧ 𝑝 ∈ ℚ ) ∧ 𝑞 ∈ ( 𝐾 ‘ 𝑝 ) ) → 𝑝 ∈ ℝ ) |
46 |
38 39 41 43 45
|
smfpimltmpt |
⊢ ( ( ( 𝜑 ∧ 𝑝 ∈ ℚ ) ∧ 𝑞 ∈ ( 𝐾 ‘ 𝑝 ) ) → { 𝑥 ∈ ( 𝐴 ∩ 𝐶 ) ∣ 𝐵 < 𝑝 } ∈ ( 𝑆 ↾t ( 𝐴 ∩ 𝐶 ) ) ) |
47 |
|
elinel2 |
⊢ ( 𝑥 ∈ ( 𝐴 ∩ 𝐶 ) → 𝑥 ∈ 𝐶 ) |
48 |
47
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐴 ∩ 𝐶 ) ) → 𝑥 ∈ 𝐶 ) |
49 |
48 5
|
syldan |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐴 ∩ 𝐶 ) ) → 𝐷 ∈ ℝ ) |
50 |
49
|
ad4ant14 |
⊢ ( ( ( ( 𝜑 ∧ 𝑝 ∈ ℚ ) ∧ 𝑞 ∈ ( 𝐾 ‘ 𝑝 ) ) ∧ 𝑥 ∈ ( 𝐴 ∩ 𝐶 ) ) → 𝐷 ∈ ℝ ) |
51 |
1 48
|
ssdf |
⊢ ( 𝜑 → ( 𝐴 ∩ 𝐶 ) ⊆ 𝐶 ) |
52 |
2 7 51
|
sssmfmpt |
⊢ ( 𝜑 → ( 𝑥 ∈ ( 𝐴 ∩ 𝐶 ) ↦ 𝐷 ) ∈ ( SMblFn ‘ 𝑆 ) ) |
53 |
52
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑝 ∈ ℚ ) ∧ 𝑞 ∈ ( 𝐾 ‘ 𝑝 ) ) → ( 𝑥 ∈ ( 𝐴 ∩ 𝐶 ) ↦ 𝐷 ) ∈ ( SMblFn ‘ 𝑆 ) ) |
54 |
44
|
ssriv |
⊢ ℚ ⊆ ℝ |
55 |
27
|
sselda |
⊢ ( ( ( 𝜑 ∧ 𝑝 ∈ ℚ ) ∧ 𝑞 ∈ ( 𝐾 ‘ 𝑝 ) ) → 𝑞 ∈ ℚ ) |
56 |
54 55
|
sselid |
⊢ ( ( ( 𝜑 ∧ 𝑝 ∈ ℚ ) ∧ 𝑞 ∈ ( 𝐾 ‘ 𝑝 ) ) → 𝑞 ∈ ℝ ) |
57 |
38 39 50 53 56
|
smfpimltmpt |
⊢ ( ( ( 𝜑 ∧ 𝑝 ∈ ℚ ) ∧ 𝑞 ∈ ( 𝐾 ‘ 𝑝 ) ) → { 𝑥 ∈ ( 𝐴 ∩ 𝐶 ) ∣ 𝐷 < 𝑞 } ∈ ( 𝑆 ↾t ( 𝐴 ∩ 𝐶 ) ) ) |
58 |
34 46 57
|
salincld |
⊢ ( ( ( 𝜑 ∧ 𝑝 ∈ ℚ ) ∧ 𝑞 ∈ ( 𝐾 ‘ 𝑝 ) ) → ( { 𝑥 ∈ ( 𝐴 ∩ 𝐶 ) ∣ 𝐵 < 𝑝 } ∩ { 𝑥 ∈ ( 𝐴 ∩ 𝐶 ) ∣ 𝐷 < 𝑞 } ) ∈ ( 𝑆 ↾t ( 𝐴 ∩ 𝐶 ) ) ) |
59 |
33 58
|
eqeltrrid |
⊢ ( ( ( 𝜑 ∧ 𝑝 ∈ ℚ ) ∧ 𝑞 ∈ ( 𝐾 ‘ 𝑝 ) ) → { 𝑥 ∈ ( 𝐴 ∩ 𝐶 ) ∣ ( 𝐵 < 𝑝 ∧ 𝐷 < 𝑞 ) } ∈ ( 𝑆 ↾t ( 𝐴 ∩ 𝐶 ) ) ) |
60 |
19 32 59
|
saliuncl |
⊢ ( ( 𝜑 ∧ 𝑝 ∈ ℚ ) → ∪ 𝑞 ∈ ( 𝐾 ‘ 𝑝 ) { 𝑥 ∈ ( 𝐴 ∩ 𝐶 ) ∣ ( 𝐵 < 𝑝 ∧ 𝐷 < 𝑞 ) } ∈ ( 𝑆 ↾t ( 𝐴 ∩ 𝐶 ) ) ) |
61 |
16 18 60
|
saliuncl |
⊢ ( 𝜑 → ∪ 𝑝 ∈ ℚ ∪ 𝑞 ∈ ( 𝐾 ‘ 𝑝 ) { 𝑥 ∈ ( 𝐴 ∩ 𝐶 ) ∣ ( 𝐵 < 𝑝 ∧ 𝐷 < 𝑞 ) } ∈ ( 𝑆 ↾t ( 𝐴 ∩ 𝐶 ) ) ) |
62 |
10 61
|
eqeltrd |
⊢ ( 𝜑 → { 𝑥 ∈ ( 𝐴 ∩ 𝐶 ) ∣ ( 𝐵 + 𝐷 ) < 𝑅 } ∈ ( 𝑆 ↾t ( 𝐴 ∩ 𝐶 ) ) ) |