Step | Hyp | Ref
| Expression |
1 | | prn0 10126 |
. . . . 5
⊢ (𝐵 ∈ P →
𝐵 ≠
∅) |
2 | | n0 4160 |
. . . . 5
⊢ (𝐵 ≠ ∅ ↔
∃𝑦 𝑦 ∈ 𝐵) |
3 | 1, 2 | sylib 210 |
. . . 4
⊢ (𝐵 ∈ P →
∃𝑦 𝑦 ∈ 𝐵) |
4 | 3 | adantl 475 |
. . 3
⊢ ((𝐴 ∈ P ∧
𝐵 ∈ P)
→ ∃𝑦 𝑦 ∈ 𝐵) |
5 | | addclpr 10155 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ P ∧
𝐵 ∈ P)
→ (𝐴
+P 𝐵) ∈ P) |
6 | 5 | adantr 474 |
. . . . . . . . . . 11
⊢ (((𝐴 ∈ P ∧
𝐵 ∈ P)
∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝐴 +P 𝐵) ∈
P) |
7 | | df-plp 10120 |
. . . . . . . . . . . . 13
⊢
+P = (𝑤 ∈ P, 𝑣 ∈ P ↦ {𝑥 ∣ ∃𝑦 ∈ 𝑤 ∃𝑧 ∈ 𝑣 𝑥 = (𝑦 +Q 𝑧)}) |
8 | | addclnq 10082 |
. . . . . . . . . . . . 13
⊢ ((𝑦 ∈ Q ∧
𝑧 ∈ Q)
→ (𝑦
+Q 𝑧) ∈ Q) |
9 | 7, 8 | genpprecl 10138 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ P ∧
𝐵 ∈ P)
→ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑥 +Q 𝑦) ∈ (𝐴 +P 𝐵))) |
10 | 9 | imp 397 |
. . . . . . . . . . 11
⊢ (((𝐴 ∈ P ∧
𝐵 ∈ P)
∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝑥 +Q 𝑦) ∈ (𝐴 +P 𝐵)) |
11 | | elprnq 10128 |
. . . . . . . . . . . . 13
⊢ (((𝐴 +P
𝐵) ∈ P
∧ (𝑥
+Q 𝑦) ∈ (𝐴 +P 𝐵)) → (𝑥 +Q 𝑦) ∈
Q) |
12 | | addnqf 10085 |
. . . . . . . . . . . . . . 15
⊢
+Q :(Q ×
Q)⟶Q |
13 | 12 | fdmi 6288 |
. . . . . . . . . . . . . 14
⊢ dom
+Q = (Q ×
Q) |
14 | | 0nnq 10061 |
. . . . . . . . . . . . . 14
⊢ ¬
∅ ∈ Q |
15 | 13, 14 | ndmovrcl 7080 |
. . . . . . . . . . . . 13
⊢ ((𝑥 +Q
𝑦) ∈ Q
→ (𝑥 ∈
Q ∧ 𝑦
∈ Q)) |
16 | | ltaddnq 10111 |
. . . . . . . . . . . . 13
⊢ ((𝑥 ∈ Q ∧
𝑦 ∈ Q)
→ 𝑥
<Q (𝑥 +Q 𝑦)) |
17 | 11, 15, 16 | 3syl 18 |
. . . . . . . . . . . 12
⊢ (((𝐴 +P
𝐵) ∈ P
∧ (𝑥
+Q 𝑦) ∈ (𝐴 +P 𝐵)) → 𝑥 <Q (𝑥 +Q
𝑦)) |
18 | | prcdnq 10130 |
. . . . . . . . . . . 12
⊢ (((𝐴 +P
𝐵) ∈ P
∧ (𝑥
+Q 𝑦) ∈ (𝐴 +P 𝐵)) → (𝑥 <Q (𝑥 +Q
𝑦) → 𝑥 ∈ (𝐴 +P 𝐵))) |
19 | 17, 18 | mpd 15 |
. . . . . . . . . . 11
⊢ (((𝐴 +P
𝐵) ∈ P
∧ (𝑥
+Q 𝑦) ∈ (𝐴 +P 𝐵)) → 𝑥 ∈ (𝐴 +P 𝐵)) |
20 | 6, 10, 19 | syl2anc 581 |
. . . . . . . . . 10
⊢ (((𝐴 ∈ P ∧
𝐵 ∈ P)
∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ (𝐴 +P 𝐵)) |
21 | 20 | exp32 413 |
. . . . . . . . 9
⊢ ((𝐴 ∈ P ∧
𝐵 ∈ P)
→ (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 → 𝑥 ∈ (𝐴 +P 𝐵)))) |
22 | 21 | com23 86 |
. . . . . . . 8
⊢ ((𝐴 ∈ P ∧
𝐵 ∈ P)
→ (𝑦 ∈ 𝐵 → (𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐴 +P 𝐵)))) |
23 | 22 | alrimdv 2030 |
. . . . . . 7
⊢ ((𝐴 ∈ P ∧
𝐵 ∈ P)
→ (𝑦 ∈ 𝐵 → ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐴 +P 𝐵)))) |
24 | | dfss2 3815 |
. . . . . . 7
⊢ (𝐴 ⊆ (𝐴 +P 𝐵) ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐴 +P 𝐵))) |
25 | 23, 24 | syl6ibr 244 |
. . . . . 6
⊢ ((𝐴 ∈ P ∧
𝐵 ∈ P)
→ (𝑦 ∈ 𝐵 → 𝐴 ⊆ (𝐴 +P 𝐵))) |
26 | | vex 3417 |
. . . . . . . . 9
⊢ 𝑦 ∈ V |
27 | 26 | prlem934 10170 |
. . . . . . . 8
⊢ (𝐴 ∈ P →
∃𝑥 ∈ 𝐴 ¬ (𝑥 +Q 𝑦) ∈ 𝐴) |
28 | 27 | adantr 474 |
. . . . . . 7
⊢ ((𝐴 ∈ P ∧
𝐵 ∈ P)
→ ∃𝑥 ∈
𝐴 ¬ (𝑥 +Q 𝑦) ∈ 𝐴) |
29 | | eleq2 2895 |
. . . . . . . . . . . . 13
⊢ (𝐴 = (𝐴 +P 𝐵) → ((𝑥 +Q 𝑦) ∈ 𝐴 ↔ (𝑥 +Q 𝑦) ∈ (𝐴 +P 𝐵))) |
30 | 29 | biimprcd 242 |
. . . . . . . . . . . 12
⊢ ((𝑥 +Q
𝑦) ∈ (𝐴 +P
𝐵) → (𝐴 = (𝐴 +P 𝐵) → (𝑥 +Q 𝑦) ∈ 𝐴)) |
31 | 30 | con3d 150 |
. . . . . . . . . . 11
⊢ ((𝑥 +Q
𝑦) ∈ (𝐴 +P
𝐵) → (¬ (𝑥 +Q
𝑦) ∈ 𝐴 → ¬ 𝐴 = (𝐴 +P 𝐵))) |
32 | 9, 31 | syl6 35 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ P ∧
𝐵 ∈ P)
→ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (¬ (𝑥 +Q 𝑦) ∈ 𝐴 → ¬ 𝐴 = (𝐴 +P 𝐵)))) |
33 | 32 | expd 406 |
. . . . . . . . 9
⊢ ((𝐴 ∈ P ∧
𝐵 ∈ P)
→ (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 → (¬ (𝑥 +Q 𝑦) ∈ 𝐴 → ¬ 𝐴 = (𝐴 +P 𝐵))))) |
34 | 33 | com34 91 |
. . . . . . . 8
⊢ ((𝐴 ∈ P ∧
𝐵 ∈ P)
→ (𝑥 ∈ 𝐴 → (¬ (𝑥 +Q
𝑦) ∈ 𝐴 → (𝑦 ∈ 𝐵 → ¬ 𝐴 = (𝐴 +P 𝐵))))) |
35 | 34 | rexlimdv 3239 |
. . . . . . 7
⊢ ((𝐴 ∈ P ∧
𝐵 ∈ P)
→ (∃𝑥 ∈
𝐴 ¬ (𝑥 +Q 𝑦) ∈ 𝐴 → (𝑦 ∈ 𝐵 → ¬ 𝐴 = (𝐴 +P 𝐵)))) |
36 | 28, 35 | mpd 15 |
. . . . . 6
⊢ ((𝐴 ∈ P ∧
𝐵 ∈ P)
→ (𝑦 ∈ 𝐵 → ¬ 𝐴 = (𝐴 +P 𝐵))) |
37 | 25, 36 | jcad 510 |
. . . . 5
⊢ ((𝐴 ∈ P ∧
𝐵 ∈ P)
→ (𝑦 ∈ 𝐵 → (𝐴 ⊆ (𝐴 +P 𝐵) ∧ ¬ 𝐴 = (𝐴 +P 𝐵)))) |
38 | | dfpss2 3918 |
. . . . 5
⊢ (𝐴 ⊊ (𝐴 +P 𝐵) ↔ (𝐴 ⊆ (𝐴 +P 𝐵) ∧ ¬ 𝐴 = (𝐴 +P 𝐵))) |
39 | 37, 38 | syl6ibr 244 |
. . . 4
⊢ ((𝐴 ∈ P ∧
𝐵 ∈ P)
→ (𝑦 ∈ 𝐵 → 𝐴 ⊊ (𝐴 +P 𝐵))) |
40 | 39 | exlimdv 2034 |
. . 3
⊢ ((𝐴 ∈ P ∧
𝐵 ∈ P)
→ (∃𝑦 𝑦 ∈ 𝐵 → 𝐴 ⊊ (𝐴 +P 𝐵))) |
41 | 4, 40 | mpd 15 |
. 2
⊢ ((𝐴 ∈ P ∧
𝐵 ∈ P)
→ 𝐴 ⊊ (𝐴 +P
𝐵)) |
42 | | ltprord 10167 |
. . 3
⊢ ((𝐴 ∈ P ∧
(𝐴
+P 𝐵) ∈ P) → (𝐴<P
(𝐴
+P 𝐵) ↔ 𝐴 ⊊ (𝐴 +P 𝐵))) |
43 | 5, 42 | syldan 587 |
. 2
⊢ ((𝐴 ∈ P ∧
𝐵 ∈ P)
→ (𝐴<P (𝐴 +P
𝐵) ↔ 𝐴 ⊊ (𝐴 +P 𝐵))) |
44 | 41, 43 | mpbird 249 |
1
⊢ ((𝐴 ∈ P ∧
𝐵 ∈ P)
→ 𝐴<P (𝐴 +P
𝐵)) |