Theorem dihsumssj

Description: The subspace sum of two isomorphisms of lattice elements is less than the isomorphism of their lattice join. (Contributed by NM, 23-Sep-2014)

	Ref	Expression
Hypotheses	dihsumssj.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	dihsumssj.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	dihsumssj.j	⊢ ∨ = ( join ‘ 𝐾 )
	dihsumssj.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
	dihsumssj.p	⊢ ⊕ = ( LSSum ‘ 𝑈 )
	dihsumssj.i	⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
	dihsumssj.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
	dihsumssj.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
	dihsumssj.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
Assertion	dihsumssj	⊢ ( 𝜑 → ( ( 𝐼 ‘ 𝑋 ) ⊕ ( 𝐼 ‘ 𝑌 ) ) ⊆ ( 𝐼 ‘ ( 𝑋 ∨ 𝑌 ) ) )

Proof

Step	Hyp	Ref	Expression
1		dihsumssj.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		dihsumssj.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
3		dihsumssj.j	⊢ ∨ = ( join ‘ 𝐾 )
4		dihsumssj.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
5		dihsumssj.p	⊢ ⊕ = ( LSSum ‘ 𝑈 )
6		dihsumssj.i	⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
7		dihsumssj.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
8		dihsumssj.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
9		dihsumssj.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
10		eqid	⊢ ( Base ‘ 𝑈 ) = ( Base ‘ 𝑈 )
11		eqid	⊢ ( ( joinH ‘ 𝐾 ) ‘ 𝑊 ) = ( ( joinH ‘ 𝐾 ) ‘ 𝑊 )
12	1 2 6 4 10	dihss	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ) → ( 𝐼 ‘ 𝑋 ) ⊆ ( Base ‘ 𝑈 ) )
13	7 8 12	syl2anc	⊢ ( 𝜑 → ( 𝐼 ‘ 𝑋 ) ⊆ ( Base ‘ 𝑈 ) )
14	1 2 6 4 10	dihss	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑌 ∈ 𝐵 ) → ( 𝐼 ‘ 𝑌 ) ⊆ ( Base ‘ 𝑈 ) )
15	7 9 14	syl2anc	⊢ ( 𝜑 → ( 𝐼 ‘ 𝑌 ) ⊆ ( Base ‘ 𝑈 ) )
16	2 4 10 5 11 7 13 15	djhsumss	⊢ ( 𝜑 → ( ( 𝐼 ‘ 𝑋 ) ⊕ ( 𝐼 ‘ 𝑌 ) ) ⊆ ( ( 𝐼 ‘ 𝑋 ) ( ( joinH ‘ 𝐾 ) ‘ 𝑊 ) ( 𝐼 ‘ 𝑌 ) ) )
17	1 3 2 6 11	djhlj	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝐼 ‘ ( 𝑋 ∨ 𝑌 ) ) = ( ( 𝐼 ‘ 𝑋 ) ( ( joinH ‘ 𝐾 ) ‘ 𝑊 ) ( 𝐼 ‘ 𝑌 ) ) )
18	7 8 9 17	syl12anc	⊢ ( 𝜑 → ( 𝐼 ‘ ( 𝑋 ∨ 𝑌 ) ) = ( ( 𝐼 ‘ 𝑋 ) ( ( joinH ‘ 𝐾 ) ‘ 𝑊 ) ( 𝐼 ‘ 𝑌 ) ) )
19	16 18	sseqtrrd	⊢ ( 𝜑 → ( ( 𝐼 ‘ 𝑋 ) ⊕ ( 𝐼 ‘ 𝑌 ) ) ⊆ ( 𝐼 ‘ ( 𝑋 ∨ 𝑌 ) ) )