Theorem sspadd1

Description: A projective subspace sum is a superset of its first summand. ( ssun1 analog.) (Contributed by NM, 3-Jan-2012)

	Ref	Expression
Hypotheses	padd0.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
	padd0.p	⊢ + = ( +𝑃 ‘ 𝐾 )
Assertion	sspadd1	⊢ ( ( 𝐾 ∈ 𝐵 ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) → 𝑋 ⊆ ( 𝑋 + 𝑌 ) )

Proof

Step	Hyp	Ref	Expression
1		padd0.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
2		padd0.p	⊢ + = ( +𝑃 ‘ 𝐾 )
3		ssun1	⊢ 𝑋 ⊆ ( 𝑋 ∪ 𝑌 )
4		ssun1	⊢ ( 𝑋 ∪ 𝑌 ) ⊆ ( ( 𝑋 ∪ 𝑌 ) ∪ { 𝑝 ∈ 𝐴 ∣ ∃ 𝑞 ∈ 𝑋 ∃ 𝑟 ∈ 𝑌 𝑝 ( le ‘ 𝐾 ) ( 𝑞 ( join ‘ 𝐾 ) 𝑟 ) } )
5	3 4	sstri	⊢ 𝑋 ⊆ ( ( 𝑋 ∪ 𝑌 ) ∪ { 𝑝 ∈ 𝐴 ∣ ∃ 𝑞 ∈ 𝑋 ∃ 𝑟 ∈ 𝑌 𝑝 ( le ‘ 𝐾 ) ( 𝑞 ( join ‘ 𝐾 ) 𝑟 ) } )
6		eqid	⊢ ( le ‘ 𝐾 ) = ( le ‘ 𝐾 )
7		eqid	⊢ ( join ‘ 𝐾 ) = ( join ‘ 𝐾 )
8	6 7 1 2	paddval	⊢ ( ( 𝐾 ∈ 𝐵 ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) → ( 𝑋 + 𝑌 ) = ( ( 𝑋 ∪ 𝑌 ) ∪ { 𝑝 ∈ 𝐴 ∣ ∃ 𝑞 ∈ 𝑋 ∃ 𝑟 ∈ 𝑌 𝑝 ( le ‘ 𝐾 ) ( 𝑞 ( join ‘ 𝐾 ) 𝑟 ) } ) )
9	5 8	sseqtrrid	⊢ ( ( 𝐾 ∈ 𝐵 ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) → 𝑋 ⊆ ( 𝑋 + 𝑌 ) )