Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > 2pmaplubN | Structured version Visualization version GIF version |
Description: Double projective map of an LUB. (Contributed by NM, 6-Mar-2012.) (New usage is discouraged.) |
Ref | Expression |
---|---|
sspmaplub.u | ⊢ 𝑈 = (lub‘𝐾) |
sspmaplub.a | ⊢ 𝐴 = (Atoms‘𝐾) |
sspmaplub.m | ⊢ 𝑀 = (pmap‘𝐾) |
Ref | Expression |
---|---|
2pmaplubN | ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴) → (𝑀‘(𝑈‘(𝑀‘(𝑈‘𝑆)))) = (𝑀‘(𝑈‘𝑆))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sspmaplub.u | . . . . . . 7 ⊢ 𝑈 = (lub‘𝐾) | |
2 | sspmaplub.a | . . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾) | |
3 | sspmaplub.m | . . . . . . 7 ⊢ 𝑀 = (pmap‘𝐾) | |
4 | eqid 2821 | . . . . . . 7 ⊢ (⊥𝑃‘𝐾) = (⊥𝑃‘𝐾) | |
5 | 1, 2, 3, 4 | 2polvalN 37044 | . . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴) → ((⊥𝑃‘𝐾)‘((⊥𝑃‘𝐾)‘𝑆)) = (𝑀‘(𝑈‘𝑆))) |
6 | 5 | fveq2d 6668 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴) → ((⊥𝑃‘𝐾)‘((⊥𝑃‘𝐾)‘((⊥𝑃‘𝐾)‘𝑆))) = ((⊥𝑃‘𝐾)‘(𝑀‘(𝑈‘𝑆)))) |
7 | 6 | fveq2d 6668 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴) → ((⊥𝑃‘𝐾)‘((⊥𝑃‘𝐾)‘((⊥𝑃‘𝐾)‘((⊥𝑃‘𝐾)‘𝑆)))) = ((⊥𝑃‘𝐾)‘((⊥𝑃‘𝐾)‘(𝑀‘(𝑈‘𝑆))))) |
8 | 2, 4 | polssatN 37038 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴) → ((⊥𝑃‘𝐾)‘𝑆) ⊆ 𝐴) |
9 | 2, 4 | 3polN 37046 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ ((⊥𝑃‘𝐾)‘𝑆) ⊆ 𝐴) → ((⊥𝑃‘𝐾)‘((⊥𝑃‘𝐾)‘((⊥𝑃‘𝐾)‘((⊥𝑃‘𝐾)‘𝑆)))) = ((⊥𝑃‘𝐾)‘((⊥𝑃‘𝐾)‘𝑆))) |
10 | 8, 9 | syldan 593 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴) → ((⊥𝑃‘𝐾)‘((⊥𝑃‘𝐾)‘((⊥𝑃‘𝐾)‘((⊥𝑃‘𝐾)‘𝑆)))) = ((⊥𝑃‘𝐾)‘((⊥𝑃‘𝐾)‘𝑆))) |
11 | 7, 10 | eqtr3d 2858 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴) → ((⊥𝑃‘𝐾)‘((⊥𝑃‘𝐾)‘(𝑀‘(𝑈‘𝑆)))) = ((⊥𝑃‘𝐾)‘((⊥𝑃‘𝐾)‘𝑆))) |
12 | hlclat 36488 | . . . . . 6 ⊢ (𝐾 ∈ HL → 𝐾 ∈ CLat) | |
13 | eqid 2821 | . . . . . . . 8 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
14 | 13, 2 | atssbase 36420 | . . . . . . 7 ⊢ 𝐴 ⊆ (Base‘𝐾) |
15 | sstr 3974 | . . . . . . 7 ⊢ ((𝑆 ⊆ 𝐴 ∧ 𝐴 ⊆ (Base‘𝐾)) → 𝑆 ⊆ (Base‘𝐾)) | |
16 | 14, 15 | mpan2 689 | . . . . . 6 ⊢ (𝑆 ⊆ 𝐴 → 𝑆 ⊆ (Base‘𝐾)) |
17 | 13, 1 | clatlubcl 17716 | . . . . . 6 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ (Base‘𝐾)) → (𝑈‘𝑆) ∈ (Base‘𝐾)) |
18 | 12, 16, 17 | syl2an 597 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴) → (𝑈‘𝑆) ∈ (Base‘𝐾)) |
19 | 13, 2, 3 | pmapssat 36889 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑈‘𝑆) ∈ (Base‘𝐾)) → (𝑀‘(𝑈‘𝑆)) ⊆ 𝐴) |
20 | 18, 19 | syldan 593 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴) → (𝑀‘(𝑈‘𝑆)) ⊆ 𝐴) |
21 | 1, 2, 3, 4 | 2polvalN 37044 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑀‘(𝑈‘𝑆)) ⊆ 𝐴) → ((⊥𝑃‘𝐾)‘((⊥𝑃‘𝐾)‘(𝑀‘(𝑈‘𝑆)))) = (𝑀‘(𝑈‘(𝑀‘(𝑈‘𝑆))))) |
22 | 20, 21 | syldan 593 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴) → ((⊥𝑃‘𝐾)‘((⊥𝑃‘𝐾)‘(𝑀‘(𝑈‘𝑆)))) = (𝑀‘(𝑈‘(𝑀‘(𝑈‘𝑆))))) |
23 | 11, 22 | eqtr3d 2858 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴) → ((⊥𝑃‘𝐾)‘((⊥𝑃‘𝐾)‘𝑆)) = (𝑀‘(𝑈‘(𝑀‘(𝑈‘𝑆))))) |
24 | 23, 5 | eqtr3d 2858 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴) → (𝑀‘(𝑈‘(𝑀‘(𝑈‘𝑆)))) = (𝑀‘(𝑈‘𝑆))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ⊆ wss 3935 ‘cfv 6349 Basecbs 16477 lubclub 17546 CLatccla 17711 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-psubsp 36633 df-pmap 36634 df-polarityN 37033 |
This theorem is referenced by: paddunN 37057 |
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