![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > sspmaplubN | Structured version Visualization version GIF version |
Description: A set of atoms is a subset of the projective map of its 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 |
---|---|
sspmaplubN | ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴) → 𝑆 ⊆ (𝑀‘(𝑈‘𝑆))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sspmaplub.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
2 | eqid 2733 | . . 3 ⊢ (⊥𝑃‘𝐾) = (⊥𝑃‘𝐾) | |
3 | 1, 2 | 2polssN 38724 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴) → 𝑆 ⊆ ((⊥𝑃‘𝐾)‘((⊥𝑃‘𝐾)‘𝑆))) |
4 | sspmaplub.u | . . 3 ⊢ 𝑈 = (lub‘𝐾) | |
5 | sspmaplub.m | . . 3 ⊢ 𝑀 = (pmap‘𝐾) | |
6 | 4, 1, 5, 2 | 2polvalN 38723 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴) → ((⊥𝑃‘𝐾)‘((⊥𝑃‘𝐾)‘𝑆)) = (𝑀‘(𝑈‘𝑆))) |
7 | 3, 6 | sseqtrd 4021 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴) → 𝑆 ⊆ (𝑀‘(𝑈‘𝑆))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ⊆ wss 3947 ‘cfv 6540 lubclub 18258 Atomscatm 38071 HLchlt 38158 pmapcpmap 38306 ⊥𝑃cpolN 38711 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7360 df-ov 7407 df-oprab 7408 df-proset 18244 df-poset 18262 df-plt 18279 df-lub 18295 df-glb 18296 df-join 18297 df-meet 18298 df-p0 18374 df-p1 18375 df-lat 18381 df-clat 18448 df-oposet 37984 df-ol 37986 df-oml 37987 df-covers 38074 df-ats 38075 df-atl 38106 df-cvlat 38130 df-hlat 38159 df-pmap 38313 df-polarityN 38712 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |