1psubclN - Mathbox for Norm Megill

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Theorem 1psubclN 41001

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 3954	. 2 ⊢ (𝐾 ∈ HL → 𝐴 ⊆ 𝐴)
2		1psubcl.a	. . . . 5 ⊢ 𝐴 = (Atoms‘𝐾)
3		eqid 2761	. . . . 5 ⊢ (⊥_𝑃‘𝐾) = (⊥_𝑃‘𝐾)
4	2, 3	pol1N 40967	. . . 4 ⊢ (𝐾 ∈ HL → ((⊥_𝑃‘𝐾)‘𝐴) = ∅)
5	4	fveq2d 6889	. . 3 ⊢ (𝐾 ∈ HL → ((⊥_𝑃‘𝐾)‘((⊥_𝑃‘𝐾)‘𝐴)) = ((⊥_𝑃‘𝐾)‘∅))
6	2, 3	pol0N 40966	. . 3 ⊢ (𝐾 ∈ HL → ((⊥_𝑃‘𝐾)‘∅) = 𝐴)
7	5, 6	eqtrd 2796	. 2 ⊢ (𝐾 ∈ HL → ((⊥_𝑃‘𝐾)‘((⊥_𝑃‘𝐾)‘𝐴)) = 𝐴)
8		1psubcl.c	. . 3 ⊢ 𝐶 = (PSubCl‘𝐾)
9	2, 3, 8	ispsubclN 40994	. 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 2145 ⊆ wss 3899 ∅c0 4279 ‘cfv 6538 Atomscatm 40320 HLchlt 40407 ⊥_𝑃cpolN 40959 PSubClcpscN 40991

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 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7751

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 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-id 5546 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-riota 7377 df-ov 7423 df-oprab 7424 df-proset 18468 df-poset 18487 df-plt 18502 df-lub 18518 df-glb 18519 df-join 18520 df-meet 18521 df-p0 18597 df-p1 18598 df-lat 18606 df-clat 18673 df-oposet 40233 df-ol 40235 df-oml 40236 df-covers 40323 df-ats 40324 df-atl 40355 df-cvlat 40379 df-hlat 40408 df-pmap 40561 df-polarityN 40960 df-psubclN 40992

This theorem is used by: (None)

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