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Metamath Proof Explorer

Theorem cdlemg5

Description: TODO: Is there a simpler more direct proof, that could be placed earlier e.g. near lhpexle ? TODO: The .\/ hypothesis is unused. FIX COMMENT. (Contributed by NM, 26-Apr-2013)

Ref Expression
Hypotheses cdlemg5.l ⊢ ≤ = ( le ‘ 𝐾 )
cdlemg5.j ⊢ ∨ = ( join ‘ 𝐾 )
cdlemg5.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
cdlemg5.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
Assertion cdlemg5 ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ∃ 𝑞 ∈ 𝐴 ( 𝑃 ≠ 𝑞 ∧ ¬ 𝑞 ≤ 𝑊 ) )

Proof

Step Hyp Ref Expression
1 cdlemg5.l ⊢ ≤ = ( le ‘ 𝐾 )
2 cdlemg5.j ⊢ ∨ = ( join ‘ 𝐾 )
3 cdlemg5.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
4 cdlemg5.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
5 1 3 4 lhpexle ⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ∃ 𝑟 ∈ 𝐴 𝑟 ≤ 𝑊 )
6 5 adantr ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ∃ 𝑟 ∈ 𝐴 𝑟 ≤ 𝑊 )
7 simpll ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) ∧ ( 𝑟 ∈ 𝐴 ∧ 𝑟 ≤ 𝑊 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
8 simpr ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) ∧ ( 𝑟 ∈ 𝐴 ∧ 𝑟 ≤ 𝑊 ) ) → ( 𝑟 ∈ 𝐴 ∧ 𝑟 ≤ 𝑊 ) )
9 simplr ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) ∧ ( 𝑟 ∈ 𝐴 ∧ 𝑟 ≤ 𝑊 ) ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) )
10 1 2 3 4 cdlemf1 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑟 ∈ 𝐴 ∧ 𝑟 ≤ 𝑊 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ∃ 𝑞 ∈ 𝐴 ( 𝑃 ≠ 𝑞 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝑟 ≤ ( 𝑃 ∨ 𝑞 ) ) )
11 7 8 9 10 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) ∧ ( 𝑟 ∈ 𝐴 ∧ 𝑟 ≤ 𝑊 ) ) → ∃ 𝑞 ∈ 𝐴 ( 𝑃 ≠ 𝑞 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝑟 ≤ ( 𝑃 ∨ 𝑞 ) ) )
12 3simpa ⊢ ( ( 𝑃 ≠ 𝑞 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝑟 ≤ ( 𝑃 ∨ 𝑞 ) ) → ( 𝑃 ≠ 𝑞 ∧ ¬ 𝑞 ≤ 𝑊 ) )
13 12 reximi ⊢ ( ∃ 𝑞 ∈ 𝐴 ( 𝑃 ≠ 𝑞 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝑟 ≤ ( 𝑃 ∨ 𝑞 ) ) → ∃ 𝑞 ∈ 𝐴 ( 𝑃 ≠ 𝑞 ∧ ¬ 𝑞 ≤ 𝑊 ) )
14 11 13 syl ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) ∧ ( 𝑟 ∈ 𝐴 ∧ 𝑟 ≤ 𝑊 ) ) → ∃ 𝑞 ∈ 𝐴 ( 𝑃 ≠ 𝑞 ∧ ¬ 𝑞 ≤ 𝑊 ) )
15 6 14 rexlimddv ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ∃ 𝑞 ∈ 𝐴 ( 𝑃 ≠ 𝑞 ∧ ¬ 𝑞 ≤ 𝑊 ) )