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 ) } )