Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
||
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 2739 | . . . . . . 7 ⊢ (⊥𝑃‘𝐾) = (⊥𝑃‘𝐾) | |
5 | 1, 2, 3, 4 | 2polvalN 37701 | . . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴) → ((⊥𝑃‘𝐾)‘((⊥𝑃‘𝐾)‘𝑆)) = (𝑀‘(𝑈‘𝑆))) |
6 | 5 | fveq2d 6742 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴) → ((⊥𝑃‘𝐾)‘((⊥𝑃‘𝐾)‘((⊥𝑃‘𝐾)‘𝑆))) = ((⊥𝑃‘𝐾)‘(𝑀‘(𝑈‘𝑆)))) |
7 | 6 | fveq2d 6742 | . . . 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 37695 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴) → ((⊥𝑃‘𝐾)‘𝑆) ⊆ 𝐴) |
9 | 2, 4 | 3polN 37703 | . . . . 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 594 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴) → ((⊥𝑃‘𝐾)‘((⊥𝑃‘𝐾)‘((⊥𝑃‘𝐾)‘((⊥𝑃‘𝐾)‘𝑆)))) = ((⊥𝑃‘𝐾)‘((⊥𝑃‘𝐾)‘𝑆))) |
11 | 7, 10 | eqtr3d 2781 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴) → ((⊥𝑃‘𝐾)‘((⊥𝑃‘𝐾)‘(𝑀‘(𝑈‘𝑆)))) = ((⊥𝑃‘𝐾)‘((⊥𝑃‘𝐾)‘𝑆))) |
12 | hlclat 37145 | . . . . . 6 ⊢ (𝐾 ∈ HL → 𝐾 ∈ CLat) | |
13 | eqid 2739 | . . . . . . . 8 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
14 | 13, 2 | atssbase 37077 | . . . . . . 7 ⊢ 𝐴 ⊆ (Base‘𝐾) |
15 | sstr 3925 | . . . . . . 7 ⊢ ((𝑆 ⊆ 𝐴 ∧ 𝐴 ⊆ (Base‘𝐾)) → 𝑆 ⊆ (Base‘𝐾)) | |
16 | 14, 15 | mpan2 691 | . . . . . 6 ⊢ (𝑆 ⊆ 𝐴 → 𝑆 ⊆ (Base‘𝐾)) |
17 | 13, 1 | clatlubcl 18041 | . . . . . 6 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ (Base‘𝐾)) → (𝑈‘𝑆) ∈ (Base‘𝐾)) |
18 | 12, 16, 17 | syl2an 599 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴) → (𝑈‘𝑆) ∈ (Base‘𝐾)) |
19 | 13, 2, 3 | pmapssat 37546 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑈‘𝑆) ∈ (Base‘𝐾)) → (𝑀‘(𝑈‘𝑆)) ⊆ 𝐴) |
20 | 18, 19 | syldan 594 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴) → (𝑀‘(𝑈‘𝑆)) ⊆ 𝐴) |
21 | 1, 2, 3, 4 | 2polvalN 37701 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑀‘(𝑈‘𝑆)) ⊆ 𝐴) → ((⊥𝑃‘𝐾)‘((⊥𝑃‘𝐾)‘(𝑀‘(𝑈‘𝑆)))) = (𝑀‘(𝑈‘(𝑀‘(𝑈‘𝑆))))) |
22 | 20, 21 | syldan 594 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴) → ((⊥𝑃‘𝐾)‘((⊥𝑃‘𝐾)‘(𝑀‘(𝑈‘𝑆)))) = (𝑀‘(𝑈‘(𝑀‘(𝑈‘𝑆))))) |
23 | 11, 22 | eqtr3d 2781 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴) → ((⊥𝑃‘𝐾)‘((⊥𝑃‘𝐾)‘𝑆)) = (𝑀‘(𝑈‘(𝑀‘(𝑈‘𝑆))))) |
24 | 23, 5 | eqtr3d 2781 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴) → (𝑀‘(𝑈‘(𝑀‘(𝑈‘𝑆)))) = (𝑀‘(𝑈‘𝑆))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ⊆ wss 3883 ‘cfv 6400 Basecbs 16792 lubclub 17848 CLatccla 18036 Atomscatm 37050 HLchlt 37137 pmapcpmap 37284 ⊥𝑃cpolN 37689 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2710 ax-rep 5195 ax-sep 5208 ax-nul 5215 ax-pow 5274 ax-pr 5338 ax-un 7544 ax-riotaBAD 36740 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2818 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-reu 3071 df-rmo 3072 df-rab 3073 df-v 3425 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4254 df-if 4456 df-pw 4531 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4836 df-iun 4922 df-iin 4923 df-br 5070 df-opab 5132 df-mpt 5152 df-id 5471 df-xp 5574 df-rel 5575 df-cnv 5576 df-co 5577 df-dm 5578 df-rn 5579 df-res 5580 df-ima 5581 df-iota 6358 df-fun 6402 df-fn 6403 df-f 6404 df-f1 6405 df-fo 6406 df-f1o 6407 df-fv 6408 df-riota 7191 df-ov 7237 df-oprab 7238 df-undef 8038 df-proset 17834 df-poset 17852 df-plt 17868 df-lub 17884 df-glb 17885 df-join 17886 df-meet 17887 df-p0 17963 df-p1 17964 df-lat 17970 df-clat 18037 df-oposet 36963 df-ol 36965 df-oml 36966 df-covers 37053 df-ats 37054 df-atl 37085 df-cvlat 37109 df-hlat 37138 df-psubsp 37290 df-pmap 37291 df-polarityN 37690 |
This theorem is referenced by: paddunN 37714 |
Copyright terms: Public domain | W3C validator |