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Metamath Proof Explorer

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 ) ) )