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Metamath Proof Explorer

Theorem cvrat42

Description: Commuted version of cvrat4 . (Contributed by NM, 28-Jan-2012)

Ref Expression
Hypotheses cvrat4.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
cvrat4.l ⊢ ≤ = ( le ‘ 𝐾 )
cvrat4.j ⊢ ∨ = ( join ‘ 𝐾 )
cvrat4.z ⊢ 0 = ( 0. ‘ 𝐾 )
cvrat4.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
Assertion cvrat42 ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → ( ( 𝑋 ≠ 0 ∧ 𝑃 ≤ ( 𝑋 ∨ 𝑄 ) ) → ∃ 𝑟 ∈ 𝐴 ( 𝑟 ≤ 𝑋 ∧ 𝑃 ≤ ( 𝑟 ∨ 𝑄 ) ) ) )

Proof

Step Hyp Ref Expression
1 cvrat4.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 cvrat4.l ⊢ ≤ = ( le ‘ 𝐾 )
3 cvrat4.j ⊢ ∨ = ( join ‘ 𝐾 )
4 cvrat4.z ⊢ 0 = ( 0. ‘ 𝐾 )
5 cvrat4.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
6 1 2 3 4 5 cvrat4 ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → ( ( 𝑋 ≠ 0 ∧ 𝑃 ≤ ( 𝑋 ∨ 𝑄 ) ) → ∃ 𝑟 ∈ 𝐴 ( 𝑟 ≤ 𝑋 ∧ 𝑃 ≤ ( 𝑄 ∨ 𝑟 ) ) ) )
7 hllat ⊢ ( 𝐾 ∈ HL → 𝐾 ∈ Lat )
8 7 ad2antrr ⊢ ( ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) ∧ 𝑟 ∈ 𝐴 ) → 𝐾 ∈ Lat )
9 simplr3 ⊢ ( ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) ∧ 𝑟 ∈ 𝐴 ) → 𝑄 ∈ 𝐴 )
10 1 5 atbase ⊢ ( 𝑄 ∈ 𝐴 → 𝑄 ∈ 𝐵 )
11 9 10 syl ⊢ ( ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) ∧ 𝑟 ∈ 𝐴 ) → 𝑄 ∈ 𝐵 )
12 1 5 atbase ⊢ ( 𝑟 ∈ 𝐴 → 𝑟 ∈ 𝐵 )
13 12 adantl ⊢ ( ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) ∧ 𝑟 ∈ 𝐴 ) → 𝑟 ∈ 𝐵 )
14 1 3 latjcom ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑄 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵 ) → ( 𝑄 ∨ 𝑟 ) = ( 𝑟 ∨ 𝑄 ) )
15 8 11 13 14 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) ∧ 𝑟 ∈ 𝐴 ) → ( 𝑄 ∨ 𝑟 ) = ( 𝑟 ∨ 𝑄 ) )
16 15 breq2d ⊢ ( ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) ∧ 𝑟 ∈ 𝐴 ) → ( 𝑃 ≤ ( 𝑄 ∨ 𝑟 ) ↔ 𝑃 ≤ ( 𝑟 ∨ 𝑄 ) ) )
17 16 anbi2d ⊢ ( ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) ∧ 𝑟 ∈ 𝐴 ) → ( ( 𝑟 ≤ 𝑋 ∧ 𝑃 ≤ ( 𝑄 ∨ 𝑟 ) ) ↔ ( 𝑟 ≤ 𝑋 ∧ 𝑃 ≤ ( 𝑟 ∨ 𝑄 ) ) ) )
18 17 rexbidva ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → ( ∃ 𝑟 ∈ 𝐴 ( 𝑟 ≤ 𝑋 ∧ 𝑃 ≤ ( 𝑄 ∨ 𝑟 ) ) ↔ ∃ 𝑟 ∈ 𝐴 ( 𝑟 ≤ 𝑋 ∧ 𝑃 ≤ ( 𝑟 ∨ 𝑄 ) ) ) )
19 6 18 sylibd ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → ( ( 𝑋 ≠ 0 ∧ 𝑃 ≤ ( 𝑋 ∨ 𝑄 ) ) → ∃ 𝑟 ∈ 𝐴 ( 𝑟 ≤ 𝑋 ∧ 𝑃 ≤ ( 𝑟 ∨ 𝑄 ) ) ) )