polsubclN - Mathbox for Norm Megill

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Theorem polsubclN 40398

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 2737	. . 3 ⊢ (lub‘𝐾) = (lub‘𝐾)
2		eqid 2737	. . 3 ⊢ (oc‘𝐾) = (oc‘𝐾)
3		polsubcl.a	. . 3 ⊢ 𝐴 = (Atoms‘𝐾)
4		eqid 2737	. . 3 ⊢ (pmap‘𝐾) = (pmap‘𝐾)
5		polsubcl.p	. . 3 ⊢ ⊥ = (⊥_𝑃‘𝐾)
6	1, 2, 3, 4, 5	polval2N 40352	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ( ⊥ ‘𝑋) = ((pmap‘𝐾)‘((oc‘𝐾)‘((lub‘𝐾)‘𝑋))))
7		hlop 39808	. . . . 5 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP)
8	7	adantr 480	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → 𝐾 ∈ OP)
9		hlclat 39804	. . . . 5 ⊢ (𝐾 ∈ HL → 𝐾 ∈ CLat)
10		eqid 2737	. . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾)
11	10, 3	atssbase 39736	. . . . . 6 ⊢ 𝐴 ⊆ (Base‘𝐾)
12		sstr 3931	. . . . . 6 ⊢ ((𝑋 ⊆ 𝐴 ∧ 𝐴 ⊆ (Base‘𝐾)) → 𝑋 ⊆ (Base‘𝐾))
13	11, 12	mpan2 692	. . . . 5 ⊢ (𝑋 ⊆ 𝐴 → 𝑋 ⊆ (Base‘𝐾))
14	10, 1	clatlubcl 18469	. . . . 5 ⊢ ((𝐾 ∈ CLat ∧ 𝑋 ⊆ (Base‘𝐾)) → ((lub‘𝐾)‘𝑋) ∈ (Base‘𝐾))
15	9, 13, 14	syl2an 597	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ((lub‘𝐾)‘𝑋) ∈ (Base‘𝐾))
16	10, 2	opoccl 39640	. . . 4 ⊢ ((𝐾 ∈ OP ∧ ((lub‘𝐾)‘𝑋) ∈ (Base‘𝐾)) → ((oc‘𝐾)‘((lub‘𝐾)‘𝑋)) ∈ (Base‘𝐾))
17	8, 15, 16	syl2anc 585	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ((oc‘𝐾)‘((lub‘𝐾)‘𝑋)) ∈ (Base‘𝐾))
18		polsubcl.c	. . . 4 ⊢ 𝐶 = (PSubCl‘𝐾)
19	10, 4, 18	pmapsubclN 40392	. . 3 ⊢ ((𝐾 ∈ HL ∧ ((oc‘𝐾)‘((lub‘𝐾)‘𝑋)) ∈ (Base‘𝐾)) → ((pmap‘𝐾)‘((oc‘𝐾)‘((lub‘𝐾)‘𝑋))) ∈ 𝐶)
20	17, 19	syldan 592	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ((pmap‘𝐾)‘((oc‘𝐾)‘((lub‘𝐾)‘𝑋))) ∈ 𝐶)
21	6, 20	eqeltrd 2837	1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ( ⊥ ‘𝑋) ∈ 𝐶)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ⊆ wss 3890 ‘cfv 6499 Basecbs 17179 occoc 17228 lubclub 18275 CLatccla 18464 OPcops 39618 Atomscatm 39709 HLchlt 39796 pmapcpmap 39943 ⊥_𝑃cpolN 40348 PSubClcpscN 40380

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5213 ax-sep 5232 ax-nul 5242 ax-pow 5308 ax-pr 5376 ax-un 7689

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-iun 4936 df-iin 4937 df-br 5087 df-opab 5149 df-mpt 5168 df-id 5526 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6455 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-riota 7324 df-ov 7370 df-oprab 7371 df-proset 18260 df-poset 18279 df-plt 18294 df-lub 18310 df-glb 18311 df-join 18312 df-meet 18313 df-p0 18389 df-p1 18390 df-lat 18398 df-clat 18465 df-oposet 39622 df-ol 39624 df-oml 39625 df-covers 39712 df-ats 39713 df-atl 39744 df-cvlat 39768 df-hlat 39797 df-pmap 39950 df-polarityN 40349 df-psubclN 40381

This theorem is referenced by: osumcllem9N 40410 pexmidN 40415

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