Proof of Theorem pclunN
Step | Hyp | Ref
| Expression |
1 | | simp1 1061 |
. . 3
⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → 𝐾 ∈ 𝑉) |
2 | | pclun.a |
. . . 4
⊢ 𝐴 = (Atoms‘𝐾) |
3 | | pclun.p |
. . . 4
⊢ + =
(+𝑃‘𝐾) |
4 | 2, 3 | paddunssN 35094 |
. . 3
⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (𝑋 ∪ 𝑌) ⊆ (𝑋 + 𝑌)) |
5 | 2, 3 | paddssat 35100 |
. . 3
⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (𝑋 + 𝑌) ⊆ 𝐴) |
6 | | pclun.c |
. . . 4
⊢ 𝑈 = (PCl‘𝐾) |
7 | 2, 6 | pclssN 35180 |
. . 3
⊢ ((𝐾 ∈ 𝑉 ∧ (𝑋 ∪ 𝑌) ⊆ (𝑋 + 𝑌) ∧ (𝑋 + 𝑌) ⊆ 𝐴) → (𝑈‘(𝑋 ∪ 𝑌)) ⊆ (𝑈‘(𝑋 + 𝑌))) |
8 | 1, 4, 5, 7 | syl3anc 1326 |
. 2
⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (𝑈‘(𝑋 ∪ 𝑌)) ⊆ (𝑈‘(𝑋 + 𝑌))) |
9 | | unss 3787 |
. . . . . . . . 9
⊢ ((𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ↔ (𝑋 ∪ 𝑌) ⊆ 𝐴) |
10 | 9 | biimpi 206 |
. . . . . . . 8
⊢ ((𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (𝑋 ∪ 𝑌) ⊆ 𝐴) |
11 | 10 | 3adant1 1079 |
. . . . . . 7
⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (𝑋 ∪ 𝑌) ⊆ 𝐴) |
12 | 2, 6 | pclssidN 35181 |
. . . . . . 7
⊢ ((𝐾 ∈ 𝑉 ∧ (𝑋 ∪ 𝑌) ⊆ 𝐴) → (𝑋 ∪ 𝑌) ⊆ (𝑈‘(𝑋 ∪ 𝑌))) |
13 | 1, 11, 12 | syl2anc 693 |
. . . . . 6
⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (𝑋 ∪ 𝑌) ⊆ (𝑈‘(𝑋 ∪ 𝑌))) |
14 | | unss 3787 |
. . . . . 6
⊢ ((𝑋 ⊆ (𝑈‘(𝑋 ∪ 𝑌)) ∧ 𝑌 ⊆ (𝑈‘(𝑋 ∪ 𝑌))) ↔ (𝑋 ∪ 𝑌) ⊆ (𝑈‘(𝑋 ∪ 𝑌))) |
15 | 13, 14 | sylibr 224 |
. . . . 5
⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (𝑋 ⊆ (𝑈‘(𝑋 ∪ 𝑌)) ∧ 𝑌 ⊆ (𝑈‘(𝑋 ∪ 𝑌)))) |
16 | | simp2 1062 |
. . . . . 6
⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → 𝑋 ⊆ 𝐴) |
17 | | simp3 1063 |
. . . . . 6
⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → 𝑌 ⊆ 𝐴) |
18 | | eqid 2622 |
. . . . . . . 8
⊢
(PSubSp‘𝐾) =
(PSubSp‘𝐾) |
19 | 2, 18, 6 | pclclN 35177 |
. . . . . . 7
⊢ ((𝐾 ∈ 𝑉 ∧ (𝑋 ∪ 𝑌) ⊆ 𝐴) → (𝑈‘(𝑋 ∪ 𝑌)) ∈ (PSubSp‘𝐾)) |
20 | 1, 11, 19 | syl2anc 693 |
. . . . . 6
⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (𝑈‘(𝑋 ∪ 𝑌)) ∈ (PSubSp‘𝐾)) |
21 | 2, 18, 3 | paddss 35131 |
. . . . . 6
⊢ ((𝐾 ∈ 𝑉 ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ (𝑈‘(𝑋 ∪ 𝑌)) ∈ (PSubSp‘𝐾))) → ((𝑋 ⊆ (𝑈‘(𝑋 ∪ 𝑌)) ∧ 𝑌 ⊆ (𝑈‘(𝑋 ∪ 𝑌))) ↔ (𝑋 + 𝑌) ⊆ (𝑈‘(𝑋 ∪ 𝑌)))) |
22 | 1, 16, 17, 20, 21 | syl13anc 1328 |
. . . . 5
⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → ((𝑋 ⊆ (𝑈‘(𝑋 ∪ 𝑌)) ∧ 𝑌 ⊆ (𝑈‘(𝑋 ∪ 𝑌))) ↔ (𝑋 + 𝑌) ⊆ (𝑈‘(𝑋 ∪ 𝑌)))) |
23 | 15, 22 | mpbid 222 |
. . . 4
⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (𝑋 + 𝑌) ⊆ (𝑈‘(𝑋 ∪ 𝑌))) |
24 | 2, 18 | psubssat 35040 |
. . . . 5
⊢ ((𝐾 ∈ 𝑉 ∧ (𝑈‘(𝑋 ∪ 𝑌)) ∈ (PSubSp‘𝐾)) → (𝑈‘(𝑋 ∪ 𝑌)) ⊆ 𝐴) |
25 | 1, 20, 24 | syl2anc 693 |
. . . 4
⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (𝑈‘(𝑋 ∪ 𝑌)) ⊆ 𝐴) |
26 | 2, 6 | pclssN 35180 |
. . . 4
⊢ ((𝐾 ∈ 𝑉 ∧ (𝑋 + 𝑌) ⊆ (𝑈‘(𝑋 ∪ 𝑌)) ∧ (𝑈‘(𝑋 ∪ 𝑌)) ⊆ 𝐴) → (𝑈‘(𝑋 + 𝑌)) ⊆ (𝑈‘(𝑈‘(𝑋 ∪ 𝑌)))) |
27 | 1, 23, 25, 26 | syl3anc 1326 |
. . 3
⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (𝑈‘(𝑋 + 𝑌)) ⊆ (𝑈‘(𝑈‘(𝑋 ∪ 𝑌)))) |
28 | 18, 6 | pclidN 35182 |
. . . 4
⊢ ((𝐾 ∈ 𝑉 ∧ (𝑈‘(𝑋 ∪ 𝑌)) ∈ (PSubSp‘𝐾)) → (𝑈‘(𝑈‘(𝑋 ∪ 𝑌))) = (𝑈‘(𝑋 ∪ 𝑌))) |
29 | 1, 20, 28 | syl2anc 693 |
. . 3
⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (𝑈‘(𝑈‘(𝑋 ∪ 𝑌))) = (𝑈‘(𝑋 ∪ 𝑌))) |
30 | 27, 29 | sseqtrd 3641 |
. 2
⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (𝑈‘(𝑋 + 𝑌)) ⊆ (𝑈‘(𝑋 ∪ 𝑌))) |
31 | 8, 30 | eqssd 3620 |
1
⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (𝑈‘(𝑋 ∪ 𝑌)) = (𝑈‘(𝑋 + 𝑌))) |