Step | Hyp | Ref
| Expression |
1 | | polval2.o |
. . 3
⊢ ⊥ =
(oc‘𝐾) |
2 | | polval2.a |
. . 3
⊢ 𝐴 = (Atoms‘𝐾) |
3 | | polval2.m |
. . 3
⊢ 𝑀 = (pmap‘𝐾) |
4 | | polval2.p |
. . 3
⊢ 𝑃 =
(⊥𝑃‘𝐾) |
5 | 1, 2, 3, 4 | polvalN 34209 |
. 2
⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → (𝑃‘𝑋) = (𝐴 ∩ ∩
𝑝 ∈ 𝑋 (𝑀‘( ⊥ ‘𝑝)))) |
6 | | hlop 33667 |
. . . . . 6
⊢ (𝐾 ∈ HL → 𝐾 ∈ OP) |
7 | 6 | ad2antrr 758 |
. . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ 𝑝 ∈ 𝑋) → 𝐾 ∈ OP) |
8 | | ssel2 3563 |
. . . . . . 7
⊢ ((𝑋 ⊆ 𝐴 ∧ 𝑝 ∈ 𝑋) → 𝑝 ∈ 𝐴) |
9 | 8 | adantll 746 |
. . . . . 6
⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ 𝑝 ∈ 𝑋) → 𝑝 ∈ 𝐴) |
10 | | eqid 2610 |
. . . . . . 7
⊢
(Base‘𝐾) =
(Base‘𝐾) |
11 | 10, 2 | atbase 33594 |
. . . . . 6
⊢ (𝑝 ∈ 𝐴 → 𝑝 ∈ (Base‘𝐾)) |
12 | 9, 11 | syl 17 |
. . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ 𝑝 ∈ 𝑋) → 𝑝 ∈ (Base‘𝐾)) |
13 | 10, 1 | opoccl 33499 |
. . . . 5
⊢ ((𝐾 ∈ OP ∧ 𝑝 ∈ (Base‘𝐾)) → ( ⊥ ‘𝑝) ∈ (Base‘𝐾)) |
14 | 7, 12, 13 | syl2anc 691 |
. . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ 𝑝 ∈ 𝑋) → ( ⊥ ‘𝑝) ∈ (Base‘𝐾)) |
15 | 14 | ralrimiva 2949 |
. . 3
⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ∀𝑝 ∈ 𝑋 ( ⊥ ‘𝑝) ∈ (Base‘𝐾)) |
16 | | eqid 2610 |
. . . 4
⊢
(glb‘𝐾) =
(glb‘𝐾) |
17 | 10, 16, 2, 3 | pmapglb2xN 34076 |
. . 3
⊢ ((𝐾 ∈ HL ∧ ∀𝑝 ∈ 𝑋 ( ⊥ ‘𝑝) ∈ (Base‘𝐾)) → (𝑀‘((glb‘𝐾)‘{𝑥 ∣ ∃𝑝 ∈ 𝑋 𝑥 = ( ⊥ ‘𝑝)})) = (𝐴 ∩ ∩
𝑝 ∈ 𝑋 (𝑀‘( ⊥ ‘𝑝)))) |
18 | 15, 17 | syldan 486 |
. 2
⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → (𝑀‘((glb‘𝐾)‘{𝑥 ∣ ∃𝑝 ∈ 𝑋 𝑥 = ( ⊥ ‘𝑝)})) = (𝐴 ∩ ∩
𝑝 ∈ 𝑋 (𝑀‘( ⊥ ‘𝑝)))) |
19 | | polval2.u |
. . . . . 6
⊢ 𝑈 = (lub‘𝐾) |
20 | 10, 19, 16, 1 | glbconxN 33682 |
. . . . 5
⊢ ((𝐾 ∈ HL ∧ ∀𝑝 ∈ 𝑋 ( ⊥ ‘𝑝) ∈ (Base‘𝐾)) → ((glb‘𝐾)‘{𝑥 ∣ ∃𝑝 ∈ 𝑋 𝑥 = ( ⊥ ‘𝑝)}) = ( ⊥ ‘(𝑈‘{𝑥 ∣ ∃𝑝 ∈ 𝑋 𝑥 = ( ⊥ ‘( ⊥
‘𝑝))}))) |
21 | 15, 20 | syldan 486 |
. . . 4
⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ((glb‘𝐾)‘{𝑥 ∣ ∃𝑝 ∈ 𝑋 𝑥 = ( ⊥ ‘𝑝)}) = ( ⊥ ‘(𝑈‘{𝑥 ∣ ∃𝑝 ∈ 𝑋 𝑥 = ( ⊥ ‘( ⊥
‘𝑝))}))) |
22 | 10, 1 | opococ 33500 |
. . . . . . . . . . 11
⊢ ((𝐾 ∈ OP ∧ 𝑝 ∈ (Base‘𝐾)) → ( ⊥ ‘( ⊥
‘𝑝)) = 𝑝) |
23 | 7, 12, 22 | syl2anc 691 |
. . . . . . . . . 10
⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ 𝑝 ∈ 𝑋) → ( ⊥ ‘( ⊥
‘𝑝)) = 𝑝) |
24 | 23 | eqeq2d 2620 |
. . . . . . . . 9
⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ 𝑝 ∈ 𝑋) → (𝑥 = ( ⊥ ‘( ⊥
‘𝑝)) ↔ 𝑥 = 𝑝)) |
25 | 24 | rexbidva 3031 |
. . . . . . . 8
⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → (∃𝑝 ∈ 𝑋 𝑥 = ( ⊥ ‘( ⊥
‘𝑝)) ↔
∃𝑝 ∈ 𝑋 𝑥 = 𝑝)) |
26 | 25 | abbidv 2728 |
. . . . . . 7
⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → {𝑥 ∣ ∃𝑝 ∈ 𝑋 𝑥 = ( ⊥ ‘( ⊥
‘𝑝))} = {𝑥 ∣ ∃𝑝 ∈ 𝑋 𝑥 = 𝑝}) |
27 | | df-rex 2902 |
. . . . . . . . . 10
⊢
(∃𝑝 ∈
𝑋 𝑥 = 𝑝 ↔ ∃𝑝(𝑝 ∈ 𝑋 ∧ 𝑥 = 𝑝)) |
28 | | equcom 1932 |
. . . . . . . . . . . . 13
⊢ (𝑥 = 𝑝 ↔ 𝑝 = 𝑥) |
29 | 28 | anbi2i 726 |
. . . . . . . . . . . 12
⊢ ((𝑝 ∈ 𝑋 ∧ 𝑥 = 𝑝) ↔ (𝑝 ∈ 𝑋 ∧ 𝑝 = 𝑥)) |
30 | | ancom 465 |
. . . . . . . . . . . 12
⊢ ((𝑝 ∈ 𝑋 ∧ 𝑝 = 𝑥) ↔ (𝑝 = 𝑥 ∧ 𝑝 ∈ 𝑋)) |
31 | 29, 30 | bitri 263 |
. . . . . . . . . . 11
⊢ ((𝑝 ∈ 𝑋 ∧ 𝑥 = 𝑝) ↔ (𝑝 = 𝑥 ∧ 𝑝 ∈ 𝑋)) |
32 | 31 | exbii 1764 |
. . . . . . . . . 10
⊢
(∃𝑝(𝑝 ∈ 𝑋 ∧ 𝑥 = 𝑝) ↔ ∃𝑝(𝑝 = 𝑥 ∧ 𝑝 ∈ 𝑋)) |
33 | | vex 3176 |
. . . . . . . . . . 11
⊢ 𝑥 ∈ V |
34 | | eleq1 2676 |
. . . . . . . . . . 11
⊢ (𝑝 = 𝑥 → (𝑝 ∈ 𝑋 ↔ 𝑥 ∈ 𝑋)) |
35 | 33, 34 | ceqsexv 3215 |
. . . . . . . . . 10
⊢
(∃𝑝(𝑝 = 𝑥 ∧ 𝑝 ∈ 𝑋) ↔ 𝑥 ∈ 𝑋) |
36 | 27, 32, 35 | 3bitri 285 |
. . . . . . . . 9
⊢
(∃𝑝 ∈
𝑋 𝑥 = 𝑝 ↔ 𝑥 ∈ 𝑋) |
37 | 36 | abbii 2726 |
. . . . . . . 8
⊢ {𝑥 ∣ ∃𝑝 ∈ 𝑋 𝑥 = 𝑝} = {𝑥 ∣ 𝑥 ∈ 𝑋} |
38 | | abid2 2732 |
. . . . . . . 8
⊢ {𝑥 ∣ 𝑥 ∈ 𝑋} = 𝑋 |
39 | 37, 38 | eqtri 2632 |
. . . . . . 7
⊢ {𝑥 ∣ ∃𝑝 ∈ 𝑋 𝑥 = 𝑝} = 𝑋 |
40 | 26, 39 | syl6eq 2660 |
. . . . . 6
⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → {𝑥 ∣ ∃𝑝 ∈ 𝑋 𝑥 = ( ⊥ ‘( ⊥
‘𝑝))} = 𝑋) |
41 | 40 | fveq2d 6107 |
. . . . 5
⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → (𝑈‘{𝑥 ∣ ∃𝑝 ∈ 𝑋 𝑥 = ( ⊥ ‘( ⊥
‘𝑝))}) = (𝑈‘𝑋)) |
42 | 41 | fveq2d 6107 |
. . . 4
⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ( ⊥ ‘(𝑈‘{𝑥 ∣ ∃𝑝 ∈ 𝑋 𝑥 = ( ⊥ ‘( ⊥
‘𝑝))})) = ( ⊥
‘(𝑈‘𝑋))) |
43 | 21, 42 | eqtrd 2644 |
. . 3
⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ((glb‘𝐾)‘{𝑥 ∣ ∃𝑝 ∈ 𝑋 𝑥 = ( ⊥ ‘𝑝)}) = ( ⊥ ‘(𝑈‘𝑋))) |
44 | 43 | fveq2d 6107 |
. 2
⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → (𝑀‘((glb‘𝐾)‘{𝑥 ∣ ∃𝑝 ∈ 𝑋 𝑥 = ( ⊥ ‘𝑝)})) = (𝑀‘( ⊥ ‘(𝑈‘𝑋)))) |
45 | 5, 18, 44 | 3eqtr2d 2650 |
1
⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → (𝑃‘𝑋) = (𝑀‘( ⊥ ‘(𝑈‘𝑋)))) |