1psubclN - Mathbox for Norm Megill

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Theorem 1psubclN 41267

Description: The set of all atoms is a closed projective subspace. (Contributed by NM, 25-Jan-2012.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
1psubcl.a	⊢ 𝐴 = (Atoms‘𝐾)
1psubcl.c	⊢ 𝐶 = (PSubCl‘𝐾)

**Assertion**
Ref	Expression
1psubclN	⊢ (𝐾 ∈ HL → 𝐴 ∈ 𝐶)

**Proof of Theorem 1psubclN**
Step	Hyp	Ref	Expression
1		ssidd 3936	. 2 ⊢ (𝐾 ∈ HL → 𝐴 ⊆ 𝐴)
2		1psubcl.a	. . . . 5 ⊢ 𝐴 = (Atoms‘𝐾)
3		eqid 2763	. . . . 5 ⊢ (⊥_𝑃‘𝐾) = (⊥_𝑃‘𝐾)
4	2, 3	pol1N 41233	. . . 4 ⊢ (𝐾 ∈ HL → ((⊥_𝑃‘𝐾)‘𝐴) = ∅)
5	4	fveq2d 6920	. . 3 ⊢ (𝐾 ∈ HL → ((⊥_𝑃‘𝐾)‘((⊥_𝑃‘𝐾)‘𝐴)) = ((⊥_𝑃‘𝐾)‘∅))
6	2, 3	pol0N 41232	. . 3 ⊢ (𝐾 ∈ HL → ((⊥_𝑃‘𝐾)‘∅) = 𝐴)
7	5, 6	eqtrd 2798	. 2 ⊢ (𝐾 ∈ HL → ((⊥_𝑃‘𝐾)‘((⊥_𝑃‘𝐾)‘𝐴)) = 𝐴)
8		1psubcl.c	. . 3 ⊢ 𝐶 = (PSubCl‘𝐾)
9	2, 3, 8	ispsubclN 41260	. 2 ⊢ (𝐾 ∈ HL → (𝐴 ∈ 𝐶 ↔ (𝐴 ⊆ 𝐴 ∧ ((⊥_𝑃‘𝐾)‘((⊥_𝑃‘𝐾)‘𝐴)) = 𝐴)))
10	1, 7, 9	mpbir2and 726	1 ⊢ (𝐾 ∈ HL → 𝐴 ∈ 𝐶)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 = wceq 1570 ∈ wcel 2147 ⊆ wss 3900 ∅c0 4281 ‘cfv 6566 Atomscatm 40586 HLchlt 40673 ⊥_𝑃cpolN 41225 PSubClcpscN 41257

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5238 ax-sep 5255 ax-nul 5266 ax-pow 5334 ax-pr 5398 ax-un 7783

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4841 df-uni 4873 df-iun 4958 df-iin 4959 df-br 5109 df-opab 5174 df-mpt 5193 df-id 5553 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-res 5670 df-ima 5671 df-iota 6520 df-fun 6568 df-fn 6569 df-f 6570 df-f1 6571 df-fo 6572 df-f1o 6573 df-fv 6574 df-riota 7408 df-ov 7454 df-oprab 7455 df-proset 18518 df-poset 18537 df-plt 18552 df-lub 18568 df-glb 18569 df-join 18570 df-meet 18571 df-p0 18647 df-p1 18648 df-lat 18656 df-clat 18723 df-oposet 40499 df-ol 40501 df-oml 40502 df-covers 40589 df-ats 40590 df-atl 40621 df-cvlat 40645 df-hlat 40674 df-pmap 40827 df-polarityN 41226 df-psubclN 41258

This theorem is used by: (None)

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