| Step | Hyp | Ref
| Expression |
| 1 | | prn0 11029 |
. . . . 5
⊢ (𝐵 ∈ P →
𝐵 ≠
∅) |
| 2 | | n0 4353 |
. . . . 5
⊢ (𝐵 ≠ ∅ ↔
∃𝑦 𝑦 ∈ 𝐵) |
| 3 | 1, 2 | sylib 218 |
. . . 4
⊢ (𝐵 ∈ P →
∃𝑦 𝑦 ∈ 𝐵) |
| 4 | 3 | adantl 481 |
. . 3
⊢ ((𝐴 ∈ P ∧
𝐵 ∈ P)
→ ∃𝑦 𝑦 ∈ 𝐵) |
| 5 | | addclpr 11058 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ P ∧
𝐵 ∈ P)
→ (𝐴
+P 𝐵) ∈ P) |
| 6 | | df-plp 11023 |
. . . . . . . . . . . . 13
⊢
+P = (𝑤 ∈ P, 𝑣 ∈ P ↦ {𝑥 ∣ ∃𝑦 ∈ 𝑤 ∃𝑧 ∈ 𝑣 𝑥 = (𝑦 +Q 𝑧)}) |
| 7 | | addclnq 10985 |
. . . . . . . . . . . . 13
⊢ ((𝑦 ∈ Q ∧
𝑧 ∈ Q)
→ (𝑦
+Q 𝑧) ∈ Q) |
| 8 | 6, 7 | genpprecl 11041 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ P ∧
𝐵 ∈ P)
→ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑥 +Q 𝑦) ∈ (𝐴 +P 𝐵))) |
| 9 | 8 | imp 406 |
. . . . . . . . . . 11
⊢ (((𝐴 ∈ P ∧
𝐵 ∈ P)
∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝑥 +Q 𝑦) ∈ (𝐴 +P 𝐵)) |
| 10 | | elprnq 11031 |
. . . . . . . . . . . . 13
⊢ (((𝐴 +P
𝐵) ∈ P
∧ (𝑥
+Q 𝑦) ∈ (𝐴 +P 𝐵)) → (𝑥 +Q 𝑦) ∈
Q) |
| 11 | | addnqf 10988 |
. . . . . . . . . . . . . . 15
⊢
+Q :(Q ×
Q)⟶Q |
| 12 | 11 | fdmi 6747 |
. . . . . . . . . . . . . 14
⊢ dom
+Q = (Q ×
Q) |
| 13 | | 0nnq 10964 |
. . . . . . . . . . . . . 14
⊢ ¬
∅ ∈ Q |
| 14 | 12, 13 | ndmovrcl 7619 |
. . . . . . . . . . . . 13
⊢ ((𝑥 +Q
𝑦) ∈ Q
→ (𝑥 ∈
Q ∧ 𝑦
∈ Q)) |
| 15 | | ltaddnq 11014 |
. . . . . . . . . . . . 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 11033 |
. . . . . . . . . . . 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 685 |
. . . . . . . . . 10
⊢ (((𝐴 ∈ P ∧
𝐵 ∈ P)
∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ (𝐴 +P 𝐵)) |
| 20 | 19 | exp32 420 |
. . . . . . . . 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 1929 |
. . . . . . 7
⊢ ((𝐴 ∈ P ∧
𝐵 ∈ P)
→ (𝑦 ∈ 𝐵 → ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐴 +P 𝐵)))) |
| 23 | | df-ss 3968 |
. . . . . . 7
⊢ (𝐴 ⊆ (𝐴 +P 𝐵) ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐴 +P 𝐵))) |
| 24 | 22, 23 | imbitrrdi 252 |
. . . . . 6
⊢ ((𝐴 ∈ P ∧
𝐵 ∈ P)
→ (𝑦 ∈ 𝐵 → 𝐴 ⊆ (𝐴 +P 𝐵))) |
| 25 | | vex 3484 |
. . . . . . . . 9
⊢ 𝑦 ∈ V |
| 26 | 25 | prlem934 11073 |
. . . . . . . 8
⊢ (𝐴 ∈ P →
∃𝑥 ∈ 𝐴 ¬ (𝑥 +Q 𝑦) ∈ 𝐴) |
| 27 | 26 | adantr 480 |
. . . . . . 7
⊢ ((𝐴 ∈ P ∧
𝐵 ∈ P)
→ ∃𝑥 ∈
𝐴 ¬ (𝑥 +Q 𝑦) ∈ 𝐴) |
| 28 | | eleq2 2830 |
. . . . . . . . . . . . 13
⊢ (𝐴 = (𝐴 +P 𝐵) → ((𝑥 +Q 𝑦) ∈ 𝐴 ↔ (𝑥 +Q 𝑦) ∈ (𝐴 +P 𝐵))) |
| 29 | 28 | biimprcd 250 |
. . . . . . . . . . . 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 415 |
. . . . . . . . 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 3153 |
. . . . . . 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 512 |
. . . . 5
⊢ ((𝐴 ∈ P ∧
𝐵 ∈ P)
→ (𝑦 ∈ 𝐵 → (𝐴 ⊆ (𝐴 +P 𝐵) ∧ ¬ 𝐴 = (𝐴 +P 𝐵)))) |
| 37 | | dfpss2 4088 |
. . . . 5
⊢ (𝐴 ⊊ (𝐴 +P 𝐵) ↔ (𝐴 ⊆ (𝐴 +P 𝐵) ∧ ¬ 𝐴 = (𝐴 +P 𝐵))) |
| 38 | 36, 37 | imbitrrdi 252 |
. . . 4
⊢ ((𝐴 ∈ P ∧
𝐵 ∈ P)
→ (𝑦 ∈ 𝐵 → 𝐴 ⊊ (𝐴 +P 𝐵))) |
| 39 | 38 | exlimdv 1933 |
. . 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 11070 |
. . 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 257 |
1
⊢ ((𝐴 ∈ P ∧
𝐵 ∈ P)
→ 𝐴<P (𝐴 +P
𝐵)) |