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Metamath Proof Explorer

Theorem paddvaln0N

Description: Projective subspace sum operation value for nonempty sets. (Contributed by NM, 27-Jan-2012) (New usage is discouraged.)

Ref Expression
Hypotheses paddfval.l ⊢ ≤ = ( le ‘ 𝐾 )
paddfval.j ⊢ ∨ = ( join ‘ 𝐾 )
paddfval.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
paddfval.p ⊢ + = ( +𝑃 ‘ 𝐾 )
Assertion paddvaln0N ( ( ( 𝐾 ∈ Lat ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) ∧ ( 𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅ ) ) → ( 𝑋 + 𝑌 ) = { 𝑝 ∈ 𝐴 ∣ ∃ 𝑞 ∈ 𝑋 ∃ 𝑟 ∈ 𝑌 𝑝 ≤ ( 𝑞 ∨ 𝑟 ) } )

Proof

Step Hyp Ref Expression
1 paddfval.l ⊢ ≤ = ( le ‘ 𝐾 )
2 paddfval.j ⊢ ∨ = ( join ‘ 𝐾 )
3 paddfval.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
4 paddfval.p ⊢ + = ( +𝑃 ‘ 𝐾 )
5 1 2 3 4 elpaddn0 ⊢ ( ( ( 𝐾 ∈ Lat ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) ∧ ( 𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅ ) ) → ( 𝑠 ∈ ( 𝑋 + 𝑌 ) ↔ ( 𝑠 ∈ 𝐴 ∧ ∃ 𝑞 ∈ 𝑋 ∃ 𝑟 ∈ 𝑌 𝑠 ≤ ( 𝑞 ∨ 𝑟 ) ) ) )
6 breq1 ⊢ ( 𝑝 = 𝑠 → ( 𝑝 ≤ ( 𝑞 ∨ 𝑟 ) ↔ 𝑠 ≤ ( 𝑞 ∨ 𝑟 ) ) )
7 6 2rexbidv ⊢ ( 𝑝 = 𝑠 → ( ∃ 𝑞 ∈ 𝑋 ∃ 𝑟 ∈ 𝑌 𝑝 ≤ ( 𝑞 ∨ 𝑟 ) ↔ ∃ 𝑞 ∈ 𝑋 ∃ 𝑟 ∈ 𝑌 𝑠 ≤ ( 𝑞 ∨ 𝑟 ) ) )
8 7 elrab ⊢ ( 𝑠 ∈ { 𝑝 ∈ 𝐴 ∣ ∃ 𝑞 ∈ 𝑋 ∃ 𝑟 ∈ 𝑌 𝑝 ≤ ( 𝑞 ∨ 𝑟 ) } ↔ ( 𝑠 ∈ 𝐴 ∧ ∃ 𝑞 ∈ 𝑋 ∃ 𝑟 ∈ 𝑌 𝑠 ≤ ( 𝑞 ∨ 𝑟 ) ) )
9 5 8 bitr4di ⊢ ( ( ( 𝐾 ∈ Lat ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) ∧ ( 𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅ ) ) → ( 𝑠 ∈ ( 𝑋 + 𝑌 ) ↔ 𝑠 ∈ { 𝑝 ∈ 𝐴 ∣ ∃ 𝑞 ∈ 𝑋 ∃ 𝑟 ∈ 𝑌 𝑝 ≤ ( 𝑞 ∨ 𝑟 ) } ) )
10 9 eqrdv ⊢ ( ( ( 𝐾 ∈ Lat ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) ∧ ( 𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅ ) ) → ( 𝑋 + 𝑌 ) = { 𝑝 ∈ 𝐴 ∣ ∃ 𝑞 ∈ 𝑋 ∃ 𝑟 ∈ 𝑌 𝑝 ≤ ( 𝑞 ∨ 𝑟 ) } )