| 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 39907 |
. 2
⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → (𝑃‘𝑋) = (𝐴 ∩ ∩
𝑝 ∈ 𝑋 (𝑀‘( ⊥ ‘𝑝)))) |
| 6 | | hlop 39363 |
. . . . . 6
⊢ (𝐾 ∈ HL → 𝐾 ∈ OP) |
| 7 | 6 | ad2antrr 726 |
. . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ 𝑝 ∈ 𝑋) → 𝐾 ∈ OP) |
| 8 | | ssel2 3978 |
. . . . . . 7
⊢ ((𝑋 ⊆ 𝐴 ∧ 𝑝 ∈ 𝑋) → 𝑝 ∈ 𝐴) |
| 9 | 8 | adantll 714 |
. . . . . 6
⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ 𝑝 ∈ 𝑋) → 𝑝 ∈ 𝐴) |
| 10 | | eqid 2737 |
. . . . . . 7
⊢
(Base‘𝐾) =
(Base‘𝐾) |
| 11 | 10, 2 | atbase 39290 |
. . . . . 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 39195 |
. . . . 5
⊢ ((𝐾 ∈ OP ∧ 𝑝 ∈ (Base‘𝐾)) → ( ⊥ ‘𝑝) ∈ (Base‘𝐾)) |
| 14 | 7, 12, 13 | syl2anc 584 |
. . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ 𝑝 ∈ 𝑋) → ( ⊥ ‘𝑝) ∈ (Base‘𝐾)) |
| 15 | 14 | ralrimiva 3146 |
. . 3
⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ∀𝑝 ∈ 𝑋 ( ⊥ ‘𝑝) ∈ (Base‘𝐾)) |
| 16 | | eqid 2737 |
. . . 4
⊢
(glb‘𝐾) =
(glb‘𝐾) |
| 17 | 10, 16, 2, 3 | pmapglb2xN 39774 |
. . 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 591 |
. 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 39380 |
. . . . 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 591 |
. . . 4
⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ((glb‘𝐾)‘{𝑥 ∣ ∃𝑝 ∈ 𝑋 𝑥 = ( ⊥ ‘𝑝)}) = ( ⊥ ‘(𝑈‘{𝑥 ∣ ∃𝑝 ∈ 𝑋 𝑥 = ( ⊥ ‘( ⊥
‘𝑝))}))) |
| 22 | 10, 1 | opococ 39196 |
. . . . . . . . . . 11
⊢ ((𝐾 ∈ OP ∧ 𝑝 ∈ (Base‘𝐾)) → ( ⊥ ‘( ⊥
‘𝑝)) = 𝑝) |
| 23 | 7, 12, 22 | syl2anc 584 |
. . . . . . . . . 10
⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ 𝑝 ∈ 𝑋) → ( ⊥ ‘( ⊥
‘𝑝)) = 𝑝) |
| 24 | 23 | eqeq2d 2748 |
. . . . . . . . 9
⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ 𝑝 ∈ 𝑋) → (𝑥 = ( ⊥ ‘( ⊥
‘𝑝)) ↔ 𝑥 = 𝑝)) |
| 25 | 24 | rexbidva 3177 |
. . . . . . . 8
⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → (∃𝑝 ∈ 𝑋 𝑥 = ( ⊥ ‘( ⊥
‘𝑝)) ↔
∃𝑝 ∈ 𝑋 𝑥 = 𝑝)) |
| 26 | 25 | abbidv 2808 |
. . . . . . 7
⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → {𝑥 ∣ ∃𝑝 ∈ 𝑋 𝑥 = ( ⊥ ‘( ⊥
‘𝑝))} = {𝑥 ∣ ∃𝑝 ∈ 𝑋 𝑥 = 𝑝}) |
| 27 | | df-rex 3071 |
. . . . . . . . . 10
⊢
(∃𝑝 ∈
𝑋 𝑥 = 𝑝 ↔ ∃𝑝(𝑝 ∈ 𝑋 ∧ 𝑥 = 𝑝)) |
| 28 | | equcom 2017 |
. . . . . . . . . . . 12
⊢ (𝑥 = 𝑝 ↔ 𝑝 = 𝑥) |
| 29 | 28 | anbi1ci 626 |
. . . . . . . . . . 11
⊢ ((𝑝 ∈ 𝑋 ∧ 𝑥 = 𝑝) ↔ (𝑝 = 𝑥 ∧ 𝑝 ∈ 𝑋)) |
| 30 | 29 | exbii 1848 |
. . . . . . . . . 10
⊢
(∃𝑝(𝑝 ∈ 𝑋 ∧ 𝑥 = 𝑝) ↔ ∃𝑝(𝑝 = 𝑥 ∧ 𝑝 ∈ 𝑋)) |
| 31 | | eleq1w 2824 |
. . . . . . . . . . 11
⊢ (𝑝 = 𝑥 → (𝑝 ∈ 𝑋 ↔ 𝑥 ∈ 𝑋)) |
| 32 | 31 | equsexvw 2004 |
. . . . . . . . . 10
⊢
(∃𝑝(𝑝 = 𝑥 ∧ 𝑝 ∈ 𝑋) ↔ 𝑥 ∈ 𝑋) |
| 33 | 27, 30, 32 | 3bitri 297 |
. . . . . . . . 9
⊢
(∃𝑝 ∈
𝑋 𝑥 = 𝑝 ↔ 𝑥 ∈ 𝑋) |
| 34 | 33 | abbii 2809 |
. . . . . . . 8
⊢ {𝑥 ∣ ∃𝑝 ∈ 𝑋 𝑥 = 𝑝} = {𝑥 ∣ 𝑥 ∈ 𝑋} |
| 35 | | abid2 2879 |
. . . . . . . 8
⊢ {𝑥 ∣ 𝑥 ∈ 𝑋} = 𝑋 |
| 36 | 34, 35 | eqtri 2765 |
. . . . . . 7
⊢ {𝑥 ∣ ∃𝑝 ∈ 𝑋 𝑥 = 𝑝} = 𝑋 |
| 37 | 26, 36 | eqtrdi 2793 |
. . . . . 6
⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → {𝑥 ∣ ∃𝑝 ∈ 𝑋 𝑥 = ( ⊥ ‘( ⊥
‘𝑝))} = 𝑋) |
| 38 | 37 | fveq2d 6910 |
. . . . 5
⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → (𝑈‘{𝑥 ∣ ∃𝑝 ∈ 𝑋 𝑥 = ( ⊥ ‘( ⊥
‘𝑝))}) = (𝑈‘𝑋)) |
| 39 | 38 | fveq2d 6910 |
. . . 4
⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ( ⊥ ‘(𝑈‘{𝑥 ∣ ∃𝑝 ∈ 𝑋 𝑥 = ( ⊥ ‘( ⊥
‘𝑝))})) = ( ⊥
‘(𝑈‘𝑋))) |
| 40 | 21, 39 | eqtrd 2777 |
. . 3
⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ((glb‘𝐾)‘{𝑥 ∣ ∃𝑝 ∈ 𝑋 𝑥 = ( ⊥ ‘𝑝)}) = ( ⊥ ‘(𝑈‘𝑋))) |
| 41 | 40 | fveq2d 6910 |
. 2
⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → (𝑀‘((glb‘𝐾)‘{𝑥 ∣ ∃𝑝 ∈ 𝑋 𝑥 = ( ⊥ ‘𝑝)})) = (𝑀‘( ⊥ ‘(𝑈‘𝑋)))) |
| 42 | 5, 18, 41 | 3eqtr2d 2783 |
1
⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → (𝑃‘𝑋) = (𝑀‘( ⊥ ‘(𝑈‘𝑋)))) |