Theorem pclun2N

Description: The projective subspace closure of the union of two subspaces equals their projective sum. (Contributed by NM, 12-Sep-2013) (New usage is discouraged.)

	Ref	Expression
Hypotheses	pclun2.s	⊢ 𝑆 = ( PSubSp ‘ 𝐾 )
	pclun2.p	⊢ + = ( +𝑃 ‘ 𝐾 )
	pclun2.c	⊢ 𝑈 = ( PCl ‘ 𝐾 )
Assertion	pclun2N	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆 ) → ( 𝑈 ‘ ( 𝑋 ∪ 𝑌 ) ) = ( 𝑋 + 𝑌 ) )

Proof

Step	Hyp	Ref	Expression
1		pclun2.s	⊢ 𝑆 = ( PSubSp ‘ 𝐾 )
2		pclun2.p	⊢ + = ( +𝑃 ‘ 𝐾 )
3		pclun2.c	⊢ 𝑈 = ( PCl ‘ 𝐾 )
4		simp1	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆 ) → 𝐾 ∈ HL )
5		eqid	⊢ ( Atoms ‘ 𝐾 ) = ( Atoms ‘ 𝐾 )
6	5 1	psubssat	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ) → 𝑋 ⊆ ( Atoms ‘ 𝐾 ) )
7	6	3adant3	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆 ) → 𝑋 ⊆ ( Atoms ‘ 𝐾 ) )
8	5 1	psubssat	⊢ ( ( 𝐾 ∈ HL ∧ 𝑌 ∈ 𝑆 ) → 𝑌 ⊆ ( Atoms ‘ 𝐾 ) )
9	8	3adant2	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆 ) → 𝑌 ⊆ ( Atoms ‘ 𝐾 ) )
10	5 2 3	pclunN	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ ( Atoms ‘ 𝐾 ) ∧ 𝑌 ⊆ ( Atoms ‘ 𝐾 ) ) → ( 𝑈 ‘ ( 𝑋 ∪ 𝑌 ) ) = ( 𝑈 ‘ ( 𝑋 + 𝑌 ) ) )
11	4 7 9 10	syl3anc	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆 ) → ( 𝑈 ‘ ( 𝑋 ∪ 𝑌 ) ) = ( 𝑈 ‘ ( 𝑋 + 𝑌 ) ) )
12	1 2	paddclN	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆 ) → ( 𝑋 + 𝑌 ) ∈ 𝑆 )
13	1 3	pclidN	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 + 𝑌 ) ∈ 𝑆 ) → ( 𝑈 ‘ ( 𝑋 + 𝑌 ) ) = ( 𝑋 + 𝑌 ) )
14	4 12 13	syl2anc	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆 ) → ( 𝑈 ‘ ( 𝑋 + 𝑌 ) ) = ( 𝑋 + 𝑌 ) )
15	11 14	eqtrd	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆 ) → ( 𝑈 ‘ ( 𝑋 ∪ 𝑌 ) ) = ( 𝑋 + 𝑌 ) )