Step | Hyp | Ref
| Expression |
1 | | prn0 10745 |
. . . . 5
⊢ (𝐵 ∈ P →
𝐵 ≠
∅) |
2 | | n0 4280 |
. . . . 5
⊢ (𝐵 ≠ ∅ ↔
∃𝑦 𝑦 ∈ 𝐵) |
3 | 1, 2 | sylib 217 |
. . . 4
⊢ (𝐵 ∈ P →
∃𝑦 𝑦 ∈ 𝐵) |
4 | 3 | adantl 482 |
. . 3
⊢ ((𝐴 ∈ P ∧
𝐵 ∈ P)
→ ∃𝑦 𝑦 ∈ 𝐵) |
5 | | addclpr 10774 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ P ∧
𝐵 ∈ P)
→ (𝐴
+P 𝐵) ∈ P) |
6 | | df-plp 10739 |
. . . . . . . . . . . . 13
⊢
+P = (𝑤 ∈ P, 𝑣 ∈ P ↦ {𝑥 ∣ ∃𝑦 ∈ 𝑤 ∃𝑧 ∈ 𝑣 𝑥 = (𝑦 +Q 𝑧)}) |
7 | | addclnq 10701 |
. . . . . . . . . . . . 13
⊢ ((𝑦 ∈ Q ∧
𝑧 ∈ Q)
→ (𝑦
+Q 𝑧) ∈ Q) |
8 | 6, 7 | genpprecl 10757 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ P ∧
𝐵 ∈ P)
→ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑥 +Q 𝑦) ∈ (𝐴 +P 𝐵))) |
9 | 8 | imp 407 |
. . . . . . . . . . 11
⊢ (((𝐴 ∈ P ∧
𝐵 ∈ P)
∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝑥 +Q 𝑦) ∈ (𝐴 +P 𝐵)) |
10 | | elprnq 10747 |
. . . . . . . . . . . . 13
⊢ (((𝐴 +P
𝐵) ∈ P
∧ (𝑥
+Q 𝑦) ∈ (𝐴 +P 𝐵)) → (𝑥 +Q 𝑦) ∈
Q) |
11 | | addnqf 10704 |
. . . . . . . . . . . . . . 15
⊢
+Q :(Q ×
Q)⟶Q |
12 | 11 | fdmi 6612 |
. . . . . . . . . . . . . 14
⊢ dom
+Q = (Q ×
Q) |
13 | | 0nnq 10680 |
. . . . . . . . . . . . . 14
⊢ ¬
∅ ∈ Q |
14 | 12, 13 | ndmovrcl 7458 |
. . . . . . . . . . . . 13
⊢ ((𝑥 +Q
𝑦) ∈ Q
→ (𝑥 ∈
Q ∧ 𝑦
∈ Q)) |
15 | | ltaddnq 10730 |
. . . . . . . . . . . . 13
⊢ ((𝑥 ∈ Q ∧
𝑦 ∈ Q)
→ 𝑥
<Q (𝑥 +Q 𝑦)) |
16 | 10, 14, 15 | 3syl 18 |
. . . . . . . . . . . 12
⊢ (((𝐴 +P
𝐵) ∈ P
∧ (𝑥
+Q 𝑦) ∈ (𝐴 +P 𝐵)) → 𝑥 <Q (𝑥 +Q
𝑦)) |
17 | | prcdnq 10749 |
. . . . . . . . . . . 12
⊢ (((𝐴 +P
𝐵) ∈ P
∧ (𝑥
+Q 𝑦) ∈ (𝐴 +P 𝐵)) → (𝑥 <Q (𝑥 +Q
𝑦) → 𝑥 ∈ (𝐴 +P 𝐵))) |
18 | 16, 17 | mpd 15 |
. . . . . . . . . . 11
⊢ (((𝐴 +P
𝐵) ∈ P
∧ (𝑥
+Q 𝑦) ∈ (𝐴 +P 𝐵)) → 𝑥 ∈ (𝐴 +P 𝐵)) |
19 | 5, 9, 18 | syl2an2r 682 |
. . . . . . . . . 10
⊢ (((𝐴 ∈ P ∧
𝐵 ∈ P)
∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ (𝐴 +P 𝐵)) |
20 | 19 | exp32 421 |
. . . . . . . . 9
⊢ ((𝐴 ∈ P ∧
𝐵 ∈ P)
→ (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 → 𝑥 ∈ (𝐴 +P 𝐵)))) |
21 | 20 | com23 86 |
. . . . . . . 8
⊢ ((𝐴 ∈ P ∧
𝐵 ∈ P)
→ (𝑦 ∈ 𝐵 → (𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐴 +P 𝐵)))) |
22 | 21 | alrimdv 1932 |
. . . . . . 7
⊢ ((𝐴 ∈ P ∧
𝐵 ∈ P)
→ (𝑦 ∈ 𝐵 → ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐴 +P 𝐵)))) |
23 | | dfss2 3907 |
. . . . . . 7
⊢ (𝐴 ⊆ (𝐴 +P 𝐵) ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐴 +P 𝐵))) |
24 | 22, 23 | syl6ibr 251 |
. . . . . 6
⊢ ((𝐴 ∈ P ∧
𝐵 ∈ P)
→ (𝑦 ∈ 𝐵 → 𝐴 ⊆ (𝐴 +P 𝐵))) |
25 | | vex 3436 |
. . . . . . . . 9
⊢ 𝑦 ∈ V |
26 | 25 | prlem934 10789 |
. . . . . . . 8
⊢ (𝐴 ∈ P →
∃𝑥 ∈ 𝐴 ¬ (𝑥 +Q 𝑦) ∈ 𝐴) |
27 | 26 | adantr 481 |
. . . . . . 7
⊢ ((𝐴 ∈ P ∧
𝐵 ∈ P)
→ ∃𝑥 ∈
𝐴 ¬ (𝑥 +Q 𝑦) ∈ 𝐴) |
28 | | eleq2 2827 |
. . . . . . . . . . . . 13
⊢ (𝐴 = (𝐴 +P 𝐵) → ((𝑥 +Q 𝑦) ∈ 𝐴 ↔ (𝑥 +Q 𝑦) ∈ (𝐴 +P 𝐵))) |
29 | 28 | biimprcd 249 |
. . . . . . . . . . . 12
⊢ ((𝑥 +Q
𝑦) ∈ (𝐴 +P
𝐵) → (𝐴 = (𝐴 +P 𝐵) → (𝑥 +Q 𝑦) ∈ 𝐴)) |
30 | 29 | con3d 152 |
. . . . . . . . . . 11
⊢ ((𝑥 +Q
𝑦) ∈ (𝐴 +P
𝐵) → (¬ (𝑥 +Q
𝑦) ∈ 𝐴 → ¬ 𝐴 = (𝐴 +P 𝐵))) |
31 | 8, 30 | syl6 35 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ P ∧
𝐵 ∈ P)
→ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (¬ (𝑥 +Q 𝑦) ∈ 𝐴 → ¬ 𝐴 = (𝐴 +P 𝐵)))) |
32 | 31 | expd 416 |
. . . . . . . . 9
⊢ ((𝐴 ∈ P ∧
𝐵 ∈ P)
→ (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 → (¬ (𝑥 +Q 𝑦) ∈ 𝐴 → ¬ 𝐴 = (𝐴 +P 𝐵))))) |
33 | 32 | com34 91 |
. . . . . . . 8
⊢ ((𝐴 ∈ P ∧
𝐵 ∈ P)
→ (𝑥 ∈ 𝐴 → (¬ (𝑥 +Q
𝑦) ∈ 𝐴 → (𝑦 ∈ 𝐵 → ¬ 𝐴 = (𝐴 +P 𝐵))))) |
34 | 33 | rexlimdv 3212 |
. . . . . . 7
⊢ ((𝐴 ∈ P ∧
𝐵 ∈ P)
→ (∃𝑥 ∈
𝐴 ¬ (𝑥 +Q 𝑦) ∈ 𝐴 → (𝑦 ∈ 𝐵 → ¬ 𝐴 = (𝐴 +P 𝐵)))) |
35 | 27, 34 | mpd 15 |
. . . . . 6
⊢ ((𝐴 ∈ P ∧
𝐵 ∈ P)
→ (𝑦 ∈ 𝐵 → ¬ 𝐴 = (𝐴 +P 𝐵))) |
36 | 24, 35 | jcad 513 |
. . . . 5
⊢ ((𝐴 ∈ P ∧
𝐵 ∈ P)
→ (𝑦 ∈ 𝐵 → (𝐴 ⊆ (𝐴 +P 𝐵) ∧ ¬ 𝐴 = (𝐴 +P 𝐵)))) |
37 | | dfpss2 4020 |
. . . . 5
⊢ (𝐴 ⊊ (𝐴 +P 𝐵) ↔ (𝐴 ⊆ (𝐴 +P 𝐵) ∧ ¬ 𝐴 = (𝐴 +P 𝐵))) |
38 | 36, 37 | syl6ibr 251 |
. . . 4
⊢ ((𝐴 ∈ P ∧
𝐵 ∈ P)
→ (𝑦 ∈ 𝐵 → 𝐴 ⊊ (𝐴 +P 𝐵))) |
39 | 38 | exlimdv 1936 |
. . 3
⊢ ((𝐴 ∈ P ∧
𝐵 ∈ P)
→ (∃𝑦 𝑦 ∈ 𝐵 → 𝐴 ⊊ (𝐴 +P 𝐵))) |
40 | 4, 39 | mpd 15 |
. 2
⊢ ((𝐴 ∈ P ∧
𝐵 ∈ P)
→ 𝐴 ⊊ (𝐴 +P
𝐵)) |
41 | | ltprord 10786 |
. . 3
⊢ ((𝐴 ∈ P ∧
(𝐴
+P 𝐵) ∈ P) → (𝐴<P
(𝐴
+P 𝐵) ↔ 𝐴 ⊊ (𝐴 +P 𝐵))) |
42 | 5, 41 | syldan 591 |
. 2
⊢ ((𝐴 ∈ P ∧
𝐵 ∈ P)
→ (𝐴<P (𝐴 +P
𝐵) ↔ 𝐴 ⊊ (𝐴 +P 𝐵))) |
43 | 40, 42 | mpbird 256 |
1
⊢ ((𝐴 ∈ P ∧
𝐵 ∈ P)
→ 𝐴<P (𝐴 +P
𝐵)) |