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Metamath Proof Explorer

Theorem elpadd

Description: Member of a projective subspace sum. (Contributed by NM, 29-Dec-2011)

Ref Expression
Hypotheses paddfval.l ⊢ ≤ = ( le ‘ 𝐾 )
paddfval.j ⊢ ∨ = ( join ‘ 𝐾 )
paddfval.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
paddfval.p ⊢ + = ( +𝑃 ‘ 𝐾 )
Assertion elpadd ( ( 𝐾 ∈ 𝐵 ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) → ( 𝑆 ∈ ( 𝑋 + 𝑌 ) ↔ ( ( 𝑆 ∈ 𝑋 ∨ 𝑆 ∈ 𝑌 ) ∨ ( 𝑆 ∈ 𝐴 ∧ ∃ 𝑞 ∈ 𝑋 ∃ 𝑟 ∈ 𝑌 𝑆 ≤ ( 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 paddval ⊢ ( ( 𝐾 ∈ 𝐵 ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) → ( 𝑋 + 𝑌 ) = ( ( 𝑋 ∪ 𝑌 ) ∪ { 𝑝 ∈ 𝐴 ∣ ∃ 𝑞 ∈ 𝑋 ∃ 𝑟 ∈ 𝑌 𝑝 ≤ ( 𝑞 ∨ 𝑟 ) } ) )
6 5 eleq2d ⊢ ( ( 𝐾 ∈ 𝐵 ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) → ( 𝑆 ∈ ( 𝑋 + 𝑌 ) ↔ 𝑆 ∈ ( ( 𝑋 ∪ 𝑌 ) ∪ { 𝑝 ∈ 𝐴 ∣ ∃ 𝑞 ∈ 𝑋 ∃ 𝑟 ∈ 𝑌 𝑝 ≤ ( 𝑞 ∨ 𝑟 ) } ) ) )
7 elun ⊢ ( 𝑆 ∈ ( ( 𝑋 ∪ 𝑌 ) ∪ { 𝑝 ∈ 𝐴 ∣ ∃ 𝑞 ∈ 𝑋 ∃ 𝑟 ∈ 𝑌 𝑝 ≤ ( 𝑞 ∨ 𝑟 ) } ) ↔ ( 𝑆 ∈ ( 𝑋 ∪ 𝑌 ) ∨ 𝑆 ∈ { 𝑝 ∈ 𝐴 ∣ ∃ 𝑞 ∈ 𝑋 ∃ 𝑟 ∈ 𝑌 𝑝 ≤ ( 𝑞 ∨ 𝑟 ) } ) )
8 elun ⊢ ( 𝑆 ∈ ( 𝑋 ∪ 𝑌 ) ↔ ( 𝑆 ∈ 𝑋 ∨ 𝑆 ∈ 𝑌 ) )
9 breq1 ⊢ ( 𝑝 = 𝑆 → ( 𝑝 ≤ ( 𝑞 ∨ 𝑟 ) ↔ 𝑆 ≤ ( 𝑞 ∨ 𝑟 ) ) )
10 9 2rexbidv ⊢ ( 𝑝 = 𝑆 → ( ∃ 𝑞 ∈ 𝑋 ∃ 𝑟 ∈ 𝑌 𝑝 ≤ ( 𝑞 ∨ 𝑟 ) ↔ ∃ 𝑞 ∈ 𝑋 ∃ 𝑟 ∈ 𝑌 𝑆 ≤ ( 𝑞 ∨ 𝑟 ) ) )
11 10 elrab ⊢ ( 𝑆 ∈ { 𝑝 ∈ 𝐴 ∣ ∃ 𝑞 ∈ 𝑋 ∃ 𝑟 ∈ 𝑌 𝑝 ≤ ( 𝑞 ∨ 𝑟 ) } ↔ ( 𝑆 ∈ 𝐴 ∧ ∃ 𝑞 ∈ 𝑋 ∃ 𝑟 ∈ 𝑌 𝑆 ≤ ( 𝑞 ∨ 𝑟 ) ) )
12 8 11 orbi12i ⊢ ( ( 𝑆 ∈ ( 𝑋 ∪ 𝑌 ) ∨ 𝑆 ∈ { 𝑝 ∈ 𝐴 ∣ ∃ 𝑞 ∈ 𝑋 ∃ 𝑟 ∈ 𝑌 𝑝 ≤ ( 𝑞 ∨ 𝑟 ) } ) ↔ ( ( 𝑆 ∈ 𝑋 ∨ 𝑆 ∈ 𝑌 ) ∨ ( 𝑆 ∈ 𝐴 ∧ ∃ 𝑞 ∈ 𝑋 ∃ 𝑟 ∈ 𝑌 𝑆 ≤ ( 𝑞 ∨ 𝑟 ) ) ) )
13 7 12 bitri ⊢ ( 𝑆 ∈ ( ( 𝑋 ∪ 𝑌 ) ∪ { 𝑝 ∈ 𝐴 ∣ ∃ 𝑞 ∈ 𝑋 ∃ 𝑟 ∈ 𝑌 𝑝 ≤ ( 𝑞 ∨ 𝑟 ) } ) ↔ ( ( 𝑆 ∈ 𝑋 ∨ 𝑆 ∈ 𝑌 ) ∨ ( 𝑆 ∈ 𝐴 ∧ ∃ 𝑞 ∈ 𝑋 ∃ 𝑟 ∈ 𝑌 𝑆 ≤ ( 𝑞 ∨ 𝑟 ) ) ) )
14 6 13 bitrdi ⊢ ( ( 𝐾 ∈ 𝐵 ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) → ( 𝑆 ∈ ( 𝑋 + 𝑌 ) ↔ ( ( 𝑆 ∈ 𝑋 ∨ 𝑆 ∈ 𝑌 ) ∨ ( 𝑆 ∈ 𝐴 ∧ ∃ 𝑞 ∈ 𝑋 ∃ 𝑟 ∈ 𝑌 𝑆 ≤ ( 𝑞 ∨ 𝑟 ) ) ) ) )