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Mirrors > Home > MPE Home > Th. List > Mathboxes > 2polvalN | Structured version Visualization version GIF version |
Description: Value of double polarity. (Contributed by NM, 25-Jan-2012.) (New usage is discouraged.) |
Ref | Expression |
---|---|
2polval.u | ⊢ 𝑈 = (lub‘𝐾) |
2polval.a | ⊢ 𝐴 = (Atoms‘𝐾) |
2polval.m | ⊢ 𝑀 = (pmap‘𝐾) |
2polval.p | ⊢ ⊥ = (⊥𝑃‘𝐾) |
Ref | Expression |
---|---|
2polvalN | ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ( ⊥ ‘( ⊥ ‘𝑋)) = (𝑀‘(𝑈‘𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2polval.u | . . . 4 ⊢ 𝑈 = (lub‘𝐾) | |
2 | eqid 2821 | . . . 4 ⊢ (oc‘𝐾) = (oc‘𝐾) | |
3 | 2polval.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
4 | 2polval.m | . . . 4 ⊢ 𝑀 = (pmap‘𝐾) | |
5 | 2polval.p | . . . 4 ⊢ ⊥ = (⊥𝑃‘𝐾) | |
6 | 1, 2, 3, 4, 5 | polval2N 37036 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ( ⊥ ‘𝑋) = (𝑀‘((oc‘𝐾)‘(𝑈‘𝑋)))) |
7 | 6 | fveq2d 6668 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ( ⊥ ‘( ⊥ ‘𝑋)) = ( ⊥ ‘(𝑀‘((oc‘𝐾)‘(𝑈‘𝑋))))) |
8 | hlop 36492 | . . . . 5 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP) | |
9 | 8 | adantr 483 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → 𝐾 ∈ OP) |
10 | hlclat 36488 | . . . . 5 ⊢ (𝐾 ∈ HL → 𝐾 ∈ CLat) | |
11 | eqid 2821 | . . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
12 | 11, 3 | atssbase 36420 | . . . . . 6 ⊢ 𝐴 ⊆ (Base‘𝐾) |
13 | sstr 3974 | . . . . . 6 ⊢ ((𝑋 ⊆ 𝐴 ∧ 𝐴 ⊆ (Base‘𝐾)) → 𝑋 ⊆ (Base‘𝐾)) | |
14 | 12, 13 | mpan2 689 | . . . . 5 ⊢ (𝑋 ⊆ 𝐴 → 𝑋 ⊆ (Base‘𝐾)) |
15 | 11, 1 | clatlubcl 17716 | . . . . 5 ⊢ ((𝐾 ∈ CLat ∧ 𝑋 ⊆ (Base‘𝐾)) → (𝑈‘𝑋) ∈ (Base‘𝐾)) |
16 | 10, 14, 15 | syl2an 597 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → (𝑈‘𝑋) ∈ (Base‘𝐾)) |
17 | 11, 2 | opoccl 36324 | . . . 4 ⊢ ((𝐾 ∈ OP ∧ (𝑈‘𝑋) ∈ (Base‘𝐾)) → ((oc‘𝐾)‘(𝑈‘𝑋)) ∈ (Base‘𝐾)) |
18 | 9, 16, 17 | syl2anc 586 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ((oc‘𝐾)‘(𝑈‘𝑋)) ∈ (Base‘𝐾)) |
19 | 11, 2, 4, 5 | polpmapN 37042 | . . 3 ⊢ ((𝐾 ∈ HL ∧ ((oc‘𝐾)‘(𝑈‘𝑋)) ∈ (Base‘𝐾)) → ( ⊥ ‘(𝑀‘((oc‘𝐾)‘(𝑈‘𝑋)))) = (𝑀‘((oc‘𝐾)‘((oc‘𝐾)‘(𝑈‘𝑋))))) |
20 | 18, 19 | syldan 593 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ( ⊥ ‘(𝑀‘((oc‘𝐾)‘(𝑈‘𝑋)))) = (𝑀‘((oc‘𝐾)‘((oc‘𝐾)‘(𝑈‘𝑋))))) |
21 | 11, 2 | opococ 36325 | . . . 4 ⊢ ((𝐾 ∈ OP ∧ (𝑈‘𝑋) ∈ (Base‘𝐾)) → ((oc‘𝐾)‘((oc‘𝐾)‘(𝑈‘𝑋))) = (𝑈‘𝑋)) |
22 | 9, 16, 21 | syl2anc 586 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ((oc‘𝐾)‘((oc‘𝐾)‘(𝑈‘𝑋))) = (𝑈‘𝑋)) |
23 | 22 | fveq2d 6668 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → (𝑀‘((oc‘𝐾)‘((oc‘𝐾)‘(𝑈‘𝑋)))) = (𝑀‘(𝑈‘𝑋))) |
24 | 7, 20, 23 | 3eqtrd 2860 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ( ⊥ ‘( ⊥ ‘𝑋)) = (𝑀‘(𝑈‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ⊆ wss 3935 ‘cfv 6349 Basecbs 16477 occoc 16567 lubclub 17546 CLatccla 17711 OPcops 36302 Atomscatm 36393 HLchlt 36480 pmapcpmap 36627 ⊥𝑃cpolN 37032 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-riotaBAD 36083 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-iin 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-undef 7933 df-proset 17532 df-poset 17550 df-plt 17562 df-lub 17578 df-glb 17579 df-join 17580 df-meet 17581 df-p0 17643 df-p1 17644 df-lat 17650 df-clat 17712 df-oposet 36306 df-ol 36308 df-oml 36309 df-covers 36396 df-ats 36397 df-atl 36428 df-cvlat 36452 df-hlat 36481 df-pmap 36634 df-polarityN 37033 |
This theorem is referenced by: 2polssN 37045 3polN 37046 sspmaplubN 37055 2pmaplubN 37056 paddunN 37057 pnonsingN 37063 pmapidclN 37072 poml4N 37083 |
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