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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 2728 | . . . . . . 7 ⊢ (⊥𝑃‘𝐾) = (⊥𝑃‘𝐾) | |
5 | 1, 2, 3, 4 | 2polvalN 39387 | . . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴) → ((⊥𝑃‘𝐾)‘((⊥𝑃‘𝐾)‘𝑆)) = (𝑀‘(𝑈‘𝑆))) |
6 | 5 | fveq2d 6901 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴) → ((⊥𝑃‘𝐾)‘((⊥𝑃‘𝐾)‘((⊥𝑃‘𝐾)‘𝑆))) = ((⊥𝑃‘𝐾)‘(𝑀‘(𝑈‘𝑆)))) |
7 | 6 | fveq2d 6901 | . . . 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 39381 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴) → ((⊥𝑃‘𝐾)‘𝑆) ⊆ 𝐴) |
9 | 2, 4 | 3polN 39389 | . . . . 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 590 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴) → ((⊥𝑃‘𝐾)‘((⊥𝑃‘𝐾)‘((⊥𝑃‘𝐾)‘((⊥𝑃‘𝐾)‘𝑆)))) = ((⊥𝑃‘𝐾)‘((⊥𝑃‘𝐾)‘𝑆))) |
11 | 7, 10 | eqtr3d 2770 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴) → ((⊥𝑃‘𝐾)‘((⊥𝑃‘𝐾)‘(𝑀‘(𝑈‘𝑆)))) = ((⊥𝑃‘𝐾)‘((⊥𝑃‘𝐾)‘𝑆))) |
12 | hlclat 38830 | . . . . . 6 ⊢ (𝐾 ∈ HL → 𝐾 ∈ CLat) | |
13 | eqid 2728 | . . . . . . . 8 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
14 | 13, 2 | atssbase 38762 | . . . . . . 7 ⊢ 𝐴 ⊆ (Base‘𝐾) |
15 | sstr 3988 | . . . . . . 7 ⊢ ((𝑆 ⊆ 𝐴 ∧ 𝐴 ⊆ (Base‘𝐾)) → 𝑆 ⊆ (Base‘𝐾)) | |
16 | 14, 15 | mpan2 690 | . . . . . 6 ⊢ (𝑆 ⊆ 𝐴 → 𝑆 ⊆ (Base‘𝐾)) |
17 | 13, 1 | clatlubcl 18495 | . . . . . 6 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ (Base‘𝐾)) → (𝑈‘𝑆) ∈ (Base‘𝐾)) |
18 | 12, 16, 17 | syl2an 595 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴) → (𝑈‘𝑆) ∈ (Base‘𝐾)) |
19 | 13, 2, 3 | pmapssat 39232 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑈‘𝑆) ∈ (Base‘𝐾)) → (𝑀‘(𝑈‘𝑆)) ⊆ 𝐴) |
20 | 18, 19 | syldan 590 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴) → (𝑀‘(𝑈‘𝑆)) ⊆ 𝐴) |
21 | 1, 2, 3, 4 | 2polvalN 39387 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑀‘(𝑈‘𝑆)) ⊆ 𝐴) → ((⊥𝑃‘𝐾)‘((⊥𝑃‘𝐾)‘(𝑀‘(𝑈‘𝑆)))) = (𝑀‘(𝑈‘(𝑀‘(𝑈‘𝑆))))) |
22 | 20, 21 | syldan 590 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴) → ((⊥𝑃‘𝐾)‘((⊥𝑃‘𝐾)‘(𝑀‘(𝑈‘𝑆)))) = (𝑀‘(𝑈‘(𝑀‘(𝑈‘𝑆))))) |
23 | 11, 22 | eqtr3d 2770 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴) → ((⊥𝑃‘𝐾)‘((⊥𝑃‘𝐾)‘𝑆)) = (𝑀‘(𝑈‘(𝑀‘(𝑈‘𝑆))))) |
24 | 23, 5 | eqtr3d 2770 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴) → (𝑀‘(𝑈‘(𝑀‘(𝑈‘𝑆)))) = (𝑀‘(𝑈‘𝑆))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ⊆ wss 3947 ‘cfv 6548 Basecbs 17180 lubclub 18301 CLatccla 18490 Atomscatm 38735 HLchlt 38822 pmapcpmap 38970 ⊥𝑃cpolN 39375 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-proset 18287 df-poset 18305 df-plt 18322 df-lub 18338 df-glb 18339 df-join 18340 df-meet 18341 df-p0 18417 df-p1 18418 df-lat 18424 df-clat 18491 df-oposet 38648 df-ol 38650 df-oml 38651 df-covers 38738 df-ats 38739 df-atl 38770 df-cvlat 38794 df-hlat 38823 df-psubsp 38976 df-pmap 38977 df-polarityN 39376 |
This theorem is referenced by: paddunN 39400 |
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