Proof of Theorem smfaddlem2
| Step | Hyp | Ref
| Expression |
| 1 | | smfaddlem2.x |
. . 3
⊢
Ⅎ𝑥𝜑 |
| 2 | | smfaddlem2.b |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) |
| 3 | | smfaddlem2.d |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐷 ∈ ℝ) |
| 4 | | smfaddlem2.r |
. . 3
⊢ (𝜑 → 𝑅 ∈ ℝ) |
| 5 | | smfaddlem2.k |
. . 3
⊢ 𝐾 = (𝑝 ∈ ℚ ↦ {𝑞 ∈ ℚ ∣ (𝑝 + 𝑞) < 𝑅}) |
| 6 | 1, 2, 3, 4, 5 | smfaddlem1 46778 |
. 2
⊢ (𝜑 → {𝑥 ∈ (𝐴 ∩ 𝐶) ∣ (𝐵 + 𝐷) < 𝑅} = ∪
𝑝 ∈ ℚ ∪ 𝑞 ∈ (𝐾‘𝑝){𝑥 ∈ (𝐴 ∩ 𝐶) ∣ (𝐵 < 𝑝 ∧ 𝐷 < 𝑞)}) |
| 7 | | smfaddlem2.s |
. . . 4
⊢ (𝜑 → 𝑆 ∈ SAlg) |
| 8 | | smfaddlem2.a |
. . . . 5
⊢ (𝜑 → 𝐴 ∈ 𝑉) |
| 9 | | elinel1 4201 |
. . . . . . 7
⊢ (𝑥 ∈ (𝐴 ∩ 𝐶) → 𝑥 ∈ 𝐴) |
| 10 | 9 | adantl 481 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐶)) → 𝑥 ∈ 𝐴) |
| 11 | 1, 10 | ssdf 45080 |
. . . . 5
⊢ (𝜑 → (𝐴 ∩ 𝐶) ⊆ 𝐴) |
| 12 | 8, 11 | ssexd 5324 |
. . . 4
⊢ (𝜑 → (𝐴 ∩ 𝐶) ∈ V) |
| 13 | | eqid 2737 |
. . . 4
⊢ (𝑆 ↾t (𝐴 ∩ 𝐶)) = (𝑆 ↾t (𝐴 ∩ 𝐶)) |
| 14 | 7, 12, 13 | subsalsal 46374 |
. . 3
⊢ (𝜑 → (𝑆 ↾t (𝐴 ∩ 𝐶)) ∈ SAlg) |
| 15 | | qct 45373 |
. . . 4
⊢ ℚ
≼ ω |
| 16 | 15 | a1i 11 |
. . 3
⊢ (𝜑 → ℚ ≼
ω) |
| 17 | 14 | adantr 480 |
. . . 4
⊢ ((𝜑 ∧ 𝑝 ∈ ℚ) → (𝑆 ↾t (𝐴 ∩ 𝐶)) ∈ SAlg) |
| 18 | | qex 13003 |
. . . . . . 7
⊢ ℚ
∈ V |
| 19 | 18 | a1i 11 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑝 ∈ ℚ) → ℚ ∈
V) |
| 20 | 5 | a1i 11 |
. . . . . . . 8
⊢ (𝜑 → 𝐾 = (𝑝 ∈ ℚ ↦ {𝑞 ∈ ℚ ∣ (𝑝 + 𝑞) < 𝑅})) |
| 21 | 18 | rabex 5339 |
. . . . . . . . 9
⊢ {𝑞 ∈ ℚ ∣ (𝑝 + 𝑞) < 𝑅} ∈ V |
| 22 | 21 | a1i 11 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑝 ∈ ℚ) → {𝑞 ∈ ℚ ∣ (𝑝 + 𝑞) < 𝑅} ∈ V) |
| 23 | 20, 22 | fvmpt2d 7029 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑝 ∈ ℚ) → (𝐾‘𝑝) = {𝑞 ∈ ℚ ∣ (𝑝 + 𝑞) < 𝑅}) |
| 24 | | ssrab2 4080 |
. . . . . . 7
⊢ {𝑞 ∈ ℚ ∣ (𝑝 + 𝑞) < 𝑅} ⊆ ℚ |
| 25 | 23, 24 | eqsstrdi 4028 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑝 ∈ ℚ) → (𝐾‘𝑝) ⊆ ℚ) |
| 26 | | ssdomg 9040 |
. . . . . 6
⊢ (ℚ
∈ V → ((𝐾‘𝑝) ⊆ ℚ → (𝐾‘𝑝) ≼ ℚ)) |
| 27 | 19, 25, 26 | sylc 65 |
. . . . 5
⊢ ((𝜑 ∧ 𝑝 ∈ ℚ) → (𝐾‘𝑝) ≼ ℚ) |
| 28 | 15 | a1i 11 |
. . . . 5
⊢ ((𝜑 ∧ 𝑝 ∈ ℚ) → ℚ ≼
ω) |
| 29 | | domtr 9047 |
. . . . 5
⊢ (((𝐾‘𝑝) ≼ ℚ ∧ ℚ ≼
ω) → (𝐾‘𝑝) ≼ ω) |
| 30 | 27, 28, 29 | syl2anc 584 |
. . . 4
⊢ ((𝜑 ∧ 𝑝 ∈ ℚ) → (𝐾‘𝑝) ≼ ω) |
| 31 | | inrab 4316 |
. . . . 5
⊢ ({𝑥 ∈ (𝐴 ∩ 𝐶) ∣ 𝐵 < 𝑝} ∩ {𝑥 ∈ (𝐴 ∩ 𝐶) ∣ 𝐷 < 𝑞}) = {𝑥 ∈ (𝐴 ∩ 𝐶) ∣ (𝐵 < 𝑝 ∧ 𝐷 < 𝑞)} |
| 32 | 14 | ad2antrr 726 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ (𝐾‘𝑝)) → (𝑆 ↾t (𝐴 ∩ 𝐶)) ∈ SAlg) |
| 33 | | nfv 1914 |
. . . . . . . . 9
⊢
Ⅎ𝑥 𝑝 ∈ ℚ |
| 34 | 1, 33 | nfan 1899 |
. . . . . . . 8
⊢
Ⅎ𝑥(𝜑 ∧ 𝑝 ∈ ℚ) |
| 35 | | nfv 1914 |
. . . . . . . 8
⊢
Ⅎ𝑥 𝑞 ∈ (𝐾‘𝑝) |
| 36 | 34, 35 | nfan 1899 |
. . . . . . 7
⊢
Ⅎ𝑥((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ (𝐾‘𝑝)) |
| 37 | 7 | ad2antrr 726 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ (𝐾‘𝑝)) → 𝑆 ∈ SAlg) |
| 38 | 10, 2 | syldan 591 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐶)) → 𝐵 ∈ ℝ) |
| 39 | 38 | ad4ant14 752 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ (𝐾‘𝑝)) ∧ 𝑥 ∈ (𝐴 ∩ 𝐶)) → 𝐵 ∈ ℝ) |
| 40 | | smfaddlem2.m |
. . . . . . . . 9
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆)) |
| 41 | 7, 40, 11 | sssmfmpt 46765 |
. . . . . . . 8
⊢ (𝜑 → (𝑥 ∈ (𝐴 ∩ 𝐶) ↦ 𝐵) ∈ (SMblFn‘𝑆)) |
| 42 | 41 | ad2antrr 726 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ (𝐾‘𝑝)) → (𝑥 ∈ (𝐴 ∩ 𝐶) ↦ 𝐵) ∈ (SMblFn‘𝑆)) |
| 43 | | qre 12995 |
. . . . . . . 8
⊢ (𝑝 ∈ ℚ → 𝑝 ∈
ℝ) |
| 44 | 43 | ad2antlr 727 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ (𝐾‘𝑝)) → 𝑝 ∈ ℝ) |
| 45 | 36, 37, 39, 42, 44 | smfpimltmpt 46761 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ (𝐾‘𝑝)) → {𝑥 ∈ (𝐴 ∩ 𝐶) ∣ 𝐵 < 𝑝} ∈ (𝑆 ↾t (𝐴 ∩ 𝐶))) |
| 46 | | elinel2 4202 |
. . . . . . . . . 10
⊢ (𝑥 ∈ (𝐴 ∩ 𝐶) → 𝑥 ∈ 𝐶) |
| 47 | 46 | adantl 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐶)) → 𝑥 ∈ 𝐶) |
| 48 | 47, 3 | syldan 591 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐶)) → 𝐷 ∈ ℝ) |
| 49 | 48 | ad4ant14 752 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ (𝐾‘𝑝)) ∧ 𝑥 ∈ (𝐴 ∩ 𝐶)) → 𝐷 ∈ ℝ) |
| 50 | | smfaddlem2.7 |
. . . . . . . . 9
⊢ (𝜑 → (𝑥 ∈ 𝐶 ↦ 𝐷) ∈ (SMblFn‘𝑆)) |
| 51 | 1, 47 | ssdf 45080 |
. . . . . . . . 9
⊢ (𝜑 → (𝐴 ∩ 𝐶) ⊆ 𝐶) |
| 52 | 7, 50, 51 | sssmfmpt 46765 |
. . . . . . . 8
⊢ (𝜑 → (𝑥 ∈ (𝐴 ∩ 𝐶) ↦ 𝐷) ∈ (SMblFn‘𝑆)) |
| 53 | 52 | ad2antrr 726 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ (𝐾‘𝑝)) → (𝑥 ∈ (𝐴 ∩ 𝐶) ↦ 𝐷) ∈ (SMblFn‘𝑆)) |
| 54 | 43 | ssriv 3987 |
. . . . . . . 8
⊢ ℚ
⊆ ℝ |
| 55 | 25 | sselda 3983 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ (𝐾‘𝑝)) → 𝑞 ∈ ℚ) |
| 56 | 54, 55 | sselid 3981 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ (𝐾‘𝑝)) → 𝑞 ∈ ℝ) |
| 57 | 36, 37, 49, 53, 56 | smfpimltmpt 46761 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ (𝐾‘𝑝)) → {𝑥 ∈ (𝐴 ∩ 𝐶) ∣ 𝐷 < 𝑞} ∈ (𝑆 ↾t (𝐴 ∩ 𝐶))) |
| 58 | 32, 45, 57 | salincld 46367 |
. . . . 5
⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ (𝐾‘𝑝)) → ({𝑥 ∈ (𝐴 ∩ 𝐶) ∣ 𝐵 < 𝑝} ∩ {𝑥 ∈ (𝐴 ∩ 𝐶) ∣ 𝐷 < 𝑞}) ∈ (𝑆 ↾t (𝐴 ∩ 𝐶))) |
| 59 | 31, 58 | eqeltrrid 2846 |
. . . 4
⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ (𝐾‘𝑝)) → {𝑥 ∈ (𝐴 ∩ 𝐶) ∣ (𝐵 < 𝑝 ∧ 𝐷 < 𝑞)} ∈ (𝑆 ↾t (𝐴 ∩ 𝐶))) |
| 60 | 17, 30, 59 | saliuncl 46338 |
. . 3
⊢ ((𝜑 ∧ 𝑝 ∈ ℚ) → ∪ 𝑞 ∈ (𝐾‘𝑝){𝑥 ∈ (𝐴 ∩ 𝐶) ∣ (𝐵 < 𝑝 ∧ 𝐷 < 𝑞)} ∈ (𝑆 ↾t (𝐴 ∩ 𝐶))) |
| 61 | 14, 16, 60 | saliuncl 46338 |
. 2
⊢ (𝜑 → ∪ 𝑝 ∈ ℚ ∪ 𝑞 ∈ (𝐾‘𝑝){𝑥 ∈ (𝐴 ∩ 𝐶) ∣ (𝐵 < 𝑝 ∧ 𝐷 < 𝑞)} ∈ (𝑆 ↾t (𝐴 ∩ 𝐶))) |
| 62 | 6, 61 | eqeltrd 2841 |
1
⊢ (𝜑 → {𝑥 ∈ (𝐴 ∩ 𝐶) ∣ (𝐵 + 𝐷) < 𝑅} ∈ (𝑆 ↾t (𝐴 ∩ 𝐶))) |