polsubclN - Mathbox for Norm Megill

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Theorem polsubclN 39946

Description: A polarity is a closed projective subspace. (Contributed by NM, 24-Jan-2012.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
polsubcl.a	⊢ 𝐴 = (Atoms‘𝐾)
polsubcl.p	⊢ ⊥ = (⊥_𝑃‘𝐾)
polsubcl.c	⊢ 𝐶 = (PSubCl‘𝐾)

**Assertion**
Ref	Expression
polsubclN	⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ( ⊥ ‘𝑋) ∈ 𝐶)

**Proof of Theorem polsubclN**
Step	Hyp	Ref	Expression
1		eqid 2729	. . 3 ⊢ (lub‘𝐾) = (lub‘𝐾)
2		eqid 2729	. . 3 ⊢ (oc‘𝐾) = (oc‘𝐾)
3		polsubcl.a	. . 3 ⊢ 𝐴 = (Atoms‘𝐾)
4		eqid 2729	. . 3 ⊢ (pmap‘𝐾) = (pmap‘𝐾)
5		polsubcl.p	. . 3 ⊢ ⊥ = (⊥_𝑃‘𝐾)
6	1, 2, 3, 4, 5	polval2N 39900	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ( ⊥ ‘𝑋) = ((pmap‘𝐾)‘((oc‘𝐾)‘((lub‘𝐾)‘𝑋))))
7		hlop 39355	. . . . 5 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP)
8	7	adantr 480	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → 𝐾 ∈ OP)
9		hlclat 39351	. . . . 5 ⊢ (𝐾 ∈ HL → 𝐾 ∈ CLat)
10		eqid 2729	. . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾)
11	10, 3	atssbase 39283	. . . . . 6 ⊢ 𝐴 ⊆ (Base‘𝐾)
12		sstr 3955	. . . . . 6 ⊢ ((𝑋 ⊆ 𝐴 ∧ 𝐴 ⊆ (Base‘𝐾)) → 𝑋 ⊆ (Base‘𝐾))
13	11, 12	mpan2 691	. . . . 5 ⊢ (𝑋 ⊆ 𝐴 → 𝑋 ⊆ (Base‘𝐾))
14	10, 1	clatlubcl 18462	. . . . 5 ⊢ ((𝐾 ∈ CLat ∧ 𝑋 ⊆ (Base‘𝐾)) → ((lub‘𝐾)‘𝑋) ∈ (Base‘𝐾))
15	9, 13, 14	syl2an 596	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ((lub‘𝐾)‘𝑋) ∈ (Base‘𝐾))
16	10, 2	opoccl 39187	. . . 4 ⊢ ((𝐾 ∈ OP ∧ ((lub‘𝐾)‘𝑋) ∈ (Base‘𝐾)) → ((oc‘𝐾)‘((lub‘𝐾)‘𝑋)) ∈ (Base‘𝐾))
17	8, 15, 16	syl2anc 584	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ((oc‘𝐾)‘((lub‘𝐾)‘𝑋)) ∈ (Base‘𝐾))
18		polsubcl.c	. . . 4 ⊢ 𝐶 = (PSubCl‘𝐾)
19	10, 4, 18	pmapsubclN 39940	. . 3 ⊢ ((𝐾 ∈ HL ∧ ((oc‘𝐾)‘((lub‘𝐾)‘𝑋)) ∈ (Base‘𝐾)) → ((pmap‘𝐾)‘((oc‘𝐾)‘((lub‘𝐾)‘𝑋))) ∈ 𝐶)
20	17, 19	syldan 591	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ((pmap‘𝐾)‘((oc‘𝐾)‘((lub‘𝐾)‘𝑋))) ∈ 𝐶)
21	6, 20	eqeltrd 2828	1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ( ⊥ ‘𝑋) ∈ 𝐶)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ⊆ wss 3914 ‘cfv 6511 Basecbs 17179 occoc 17228 lubclub 18270 CLatccla 18457 OPcops 39165 Atomscatm 39256 HLchlt 39343 pmapcpmap 39491 ⊥_𝑃cpolN 39896 PSubClcpscN 39928

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-iun 4957 df-iin 4958 df-br 5108 df-opab 5170 df-mpt 5189 df-id 5533 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-riota 7344 df-ov 7390 df-oprab 7391 df-proset 18255 df-poset 18274 df-plt 18289 df-lub 18305 df-glb 18306 df-join 18307 df-meet 18308 df-p0 18384 df-p1 18385 df-lat 18391 df-clat 18458 df-oposet 39169 df-ol 39171 df-oml 39172 df-covers 39259 df-ats 39260 df-atl 39291 df-cvlat 39315 df-hlat 39344 df-pmap 39498 df-polarityN 39897 df-psubclN 39929

This theorem is referenced by: osumcllem9N 39958 pexmidN 39963

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