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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 2727 | . . . 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 39368 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ( ⊥ ‘𝑋) = (𝑀‘((oc‘𝐾)‘(𝑈‘𝑋)))) |
7 | 6 | fveq2d 6895 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ( ⊥ ‘( ⊥ ‘𝑋)) = ( ⊥ ‘(𝑀‘((oc‘𝐾)‘(𝑈‘𝑋))))) |
8 | hlop 38823 | . . . . 5 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP) | |
9 | 8 | adantr 480 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → 𝐾 ∈ OP) |
10 | hlclat 38819 | . . . . 5 ⊢ (𝐾 ∈ HL → 𝐾 ∈ CLat) | |
11 | eqid 2727 | . . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
12 | 11, 3 | atssbase 38751 | . . . . . 6 ⊢ 𝐴 ⊆ (Base‘𝐾) |
13 | sstr 3986 | . . . . . 6 ⊢ ((𝑋 ⊆ 𝐴 ∧ 𝐴 ⊆ (Base‘𝐾)) → 𝑋 ⊆ (Base‘𝐾)) | |
14 | 12, 13 | mpan2 690 | . . . . 5 ⊢ (𝑋 ⊆ 𝐴 → 𝑋 ⊆ (Base‘𝐾)) |
15 | 11, 1 | clatlubcl 18488 | . . . . 5 ⊢ ((𝐾 ∈ CLat ∧ 𝑋 ⊆ (Base‘𝐾)) → (𝑈‘𝑋) ∈ (Base‘𝐾)) |
16 | 10, 14, 15 | syl2an 595 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → (𝑈‘𝑋) ∈ (Base‘𝐾)) |
17 | 11, 2 | opoccl 38655 | . . . 4 ⊢ ((𝐾 ∈ OP ∧ (𝑈‘𝑋) ∈ (Base‘𝐾)) → ((oc‘𝐾)‘(𝑈‘𝑋)) ∈ (Base‘𝐾)) |
18 | 9, 16, 17 | syl2anc 583 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ((oc‘𝐾)‘(𝑈‘𝑋)) ∈ (Base‘𝐾)) |
19 | 11, 2, 4, 5 | polpmapN 39374 | . . 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 590 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ( ⊥ ‘(𝑀‘((oc‘𝐾)‘(𝑈‘𝑋)))) = (𝑀‘((oc‘𝐾)‘((oc‘𝐾)‘(𝑈‘𝑋))))) |
21 | 11, 2 | opococ 38656 | . . . 4 ⊢ ((𝐾 ∈ OP ∧ (𝑈‘𝑋) ∈ (Base‘𝐾)) → ((oc‘𝐾)‘((oc‘𝐾)‘(𝑈‘𝑋))) = (𝑈‘𝑋)) |
22 | 9, 16, 21 | syl2anc 583 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ((oc‘𝐾)‘((oc‘𝐾)‘(𝑈‘𝑋))) = (𝑈‘𝑋)) |
23 | 22 | fveq2d 6895 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → (𝑀‘((oc‘𝐾)‘((oc‘𝐾)‘(𝑈‘𝑋)))) = (𝑀‘(𝑈‘𝑋))) |
24 | 7, 20, 23 | 3eqtrd 2771 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ( ⊥ ‘( ⊥ ‘𝑋)) = (𝑀‘(𝑈‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ⊆ wss 3944 ‘cfv 6542 Basecbs 17173 occoc 17234 lubclub 18294 CLatccla 18483 OPcops 38633 Atomscatm 38724 HLchlt 38811 pmapcpmap 38959 ⊥𝑃cpolN 39364 |
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 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 |
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 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-iin 4994 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-proset 18280 df-poset 18298 df-plt 18315 df-lub 18331 df-glb 18332 df-join 18333 df-meet 18334 df-p0 18410 df-p1 18411 df-lat 18417 df-clat 18484 df-oposet 38637 df-ol 38639 df-oml 38640 df-covers 38727 df-ats 38728 df-atl 38759 df-cvlat 38783 df-hlat 38812 df-pmap 38966 df-polarityN 39365 |
This theorem is referenced by: 2polssN 39377 3polN 39378 sspmaplubN 39387 2pmaplubN 39388 paddunN 39389 pnonsingN 39395 pmapidclN 39404 poml4N 39415 |
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