Metamath Proof Explorer


Theorem cdleme25b

Description: Transform cdleme24 . TODO get rid of $d's on U , N (Contributed by NM, 1-Jan-2013)

Ref Expression
Hypotheses cdleme24.b 𝐵 = ( Base ‘ 𝐾 )
cdleme24.l = ( le ‘ 𝐾 )
cdleme24.j = ( join ‘ 𝐾 )
cdleme24.m = ( meet ‘ 𝐾 )
cdleme24.a 𝐴 = ( Atoms ‘ 𝐾 )
cdleme24.h 𝐻 = ( LHyp ‘ 𝐾 )
cdleme24.u 𝑈 = ( ( 𝑃 𝑄 ) 𝑊 )
cdleme24.f 𝐹 = ( ( 𝑠 𝑈 ) ( 𝑄 ( ( 𝑃 𝑠 ) 𝑊 ) ) )
cdleme24.n 𝑁 = ( ( 𝑃 𝑄 ) ( 𝐹 ( ( 𝑅 𝑠 ) 𝑊 ) ) )
Assertion cdleme25b ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ ( 𝑃𝑄𝑅 ( 𝑃 𝑄 ) ) ) → ∃ 𝑢𝐵𝑠𝐴 ( ( ¬ 𝑠 𝑊 ∧ ¬ 𝑠 ( 𝑃 𝑄 ) ) → 𝑢 = 𝑁 ) )

Proof

Step Hyp Ref Expression
1 cdleme24.b 𝐵 = ( Base ‘ 𝐾 )
2 cdleme24.l = ( le ‘ 𝐾 )
3 cdleme24.j = ( join ‘ 𝐾 )
4 cdleme24.m = ( meet ‘ 𝐾 )
5 cdleme24.a 𝐴 = ( Atoms ‘ 𝐾 )
6 cdleme24.h 𝐻 = ( LHyp ‘ 𝐾 )
7 cdleme24.u 𝑈 = ( ( 𝑃 𝑄 ) 𝑊 )
8 cdleme24.f 𝐹 = ( ( 𝑠 𝑈 ) ( 𝑄 ( ( 𝑃 𝑠 ) 𝑊 ) ) )
9 cdleme24.n 𝑁 = ( ( 𝑃 𝑄 ) ( 𝐹 ( ( 𝑅 𝑠 ) 𝑊 ) ) )
10 1 2 3 4 5 6 7 8 9 cdleme25a ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ ( 𝑃𝑄𝑅 ( 𝑃 𝑄 ) ) ) → ∃ 𝑠𝐴 ( ( ¬ 𝑠 𝑊 ∧ ¬ 𝑠 ( 𝑃 𝑄 ) ) ∧ 𝑁𝐵 ) )
11 eqid ( ( 𝑡 𝑈 ) ( 𝑄 ( ( 𝑃 𝑡 ) 𝑊 ) ) ) = ( ( 𝑡 𝑈 ) ( 𝑄 ( ( 𝑃 𝑡 ) 𝑊 ) ) )
12 eqid ( ( 𝑃 𝑄 ) ( ( ( 𝑡 𝑈 ) ( 𝑄 ( ( 𝑃 𝑡 ) 𝑊 ) ) ) ( ( 𝑅 𝑡 ) 𝑊 ) ) ) = ( ( 𝑃 𝑄 ) ( ( ( 𝑡 𝑈 ) ( 𝑄 ( ( 𝑃 𝑡 ) 𝑊 ) ) ) ( ( 𝑅 𝑡 ) 𝑊 ) ) )
13 1 2 3 4 5 6 7 8 9 11 12 cdleme24 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ ( 𝑃𝑄𝑅 ( 𝑃 𝑄 ) ) ) → ∀ 𝑠𝐴𝑡𝐴 ( ( ( ¬ 𝑠 𝑊 ∧ ¬ 𝑠 ( 𝑃 𝑄 ) ) ∧ ( ¬ 𝑡 𝑊 ∧ ¬ 𝑡 ( 𝑃 𝑄 ) ) ) → 𝑁 = ( ( 𝑃 𝑄 ) ( ( ( 𝑡 𝑈 ) ( 𝑄 ( ( 𝑃 𝑡 ) 𝑊 ) ) ) ( ( 𝑅 𝑡 ) 𝑊 ) ) ) ) )
14 breq1 ( 𝑠 = 𝑡 → ( 𝑠 𝑊𝑡 𝑊 ) )
15 14 notbid ( 𝑠 = 𝑡 → ( ¬ 𝑠 𝑊 ↔ ¬ 𝑡 𝑊 ) )
16 breq1 ( 𝑠 = 𝑡 → ( 𝑠 ( 𝑃 𝑄 ) ↔ 𝑡 ( 𝑃 𝑄 ) ) )
17 16 notbid ( 𝑠 = 𝑡 → ( ¬ 𝑠 ( 𝑃 𝑄 ) ↔ ¬ 𝑡 ( 𝑃 𝑄 ) ) )
18 15 17 anbi12d ( 𝑠 = 𝑡 → ( ( ¬ 𝑠 𝑊 ∧ ¬ 𝑠 ( 𝑃 𝑄 ) ) ↔ ( ¬ 𝑡 𝑊 ∧ ¬ 𝑡 ( 𝑃 𝑄 ) ) ) )
19 oveq1 ( 𝑠 = 𝑡 → ( 𝑠 𝑈 ) = ( 𝑡 𝑈 ) )
20 oveq2 ( 𝑠 = 𝑡 → ( 𝑃 𝑠 ) = ( 𝑃 𝑡 ) )
21 20 oveq1d ( 𝑠 = 𝑡 → ( ( 𝑃 𝑠 ) 𝑊 ) = ( ( 𝑃 𝑡 ) 𝑊 ) )
22 21 oveq2d ( 𝑠 = 𝑡 → ( 𝑄 ( ( 𝑃 𝑠 ) 𝑊 ) ) = ( 𝑄 ( ( 𝑃 𝑡 ) 𝑊 ) ) )
23 19 22 oveq12d ( 𝑠 = 𝑡 → ( ( 𝑠 𝑈 ) ( 𝑄 ( ( 𝑃 𝑠 ) 𝑊 ) ) ) = ( ( 𝑡 𝑈 ) ( 𝑄 ( ( 𝑃 𝑡 ) 𝑊 ) ) ) )
24 8 23 syl5eq ( 𝑠 = 𝑡𝐹 = ( ( 𝑡 𝑈 ) ( 𝑄 ( ( 𝑃 𝑡 ) 𝑊 ) ) ) )
25 oveq2 ( 𝑠 = 𝑡 → ( 𝑅 𝑠 ) = ( 𝑅 𝑡 ) )
26 25 oveq1d ( 𝑠 = 𝑡 → ( ( 𝑅 𝑠 ) 𝑊 ) = ( ( 𝑅 𝑡 ) 𝑊 ) )
27 24 26 oveq12d ( 𝑠 = 𝑡 → ( 𝐹 ( ( 𝑅 𝑠 ) 𝑊 ) ) = ( ( ( 𝑡 𝑈 ) ( 𝑄 ( ( 𝑃 𝑡 ) 𝑊 ) ) ) ( ( 𝑅 𝑡 ) 𝑊 ) ) )
28 27 oveq2d ( 𝑠 = 𝑡 → ( ( 𝑃 𝑄 ) ( 𝐹 ( ( 𝑅 𝑠 ) 𝑊 ) ) ) = ( ( 𝑃 𝑄 ) ( ( ( 𝑡 𝑈 ) ( 𝑄 ( ( 𝑃 𝑡 ) 𝑊 ) ) ) ( ( 𝑅 𝑡 ) 𝑊 ) ) ) )
29 9 28 syl5eq ( 𝑠 = 𝑡𝑁 = ( ( 𝑃 𝑄 ) ( ( ( 𝑡 𝑈 ) ( 𝑄 ( ( 𝑃 𝑡 ) 𝑊 ) ) ) ( ( 𝑅 𝑡 ) 𝑊 ) ) ) )
30 18 29 reusv3 ( ∃ 𝑠𝐴 ( ( ¬ 𝑠 𝑊 ∧ ¬ 𝑠 ( 𝑃 𝑄 ) ) ∧ 𝑁𝐵 ) → ( ∀ 𝑠𝐴𝑡𝐴 ( ( ( ¬ 𝑠 𝑊 ∧ ¬ 𝑠 ( 𝑃 𝑄 ) ) ∧ ( ¬ 𝑡 𝑊 ∧ ¬ 𝑡 ( 𝑃 𝑄 ) ) ) → 𝑁 = ( ( 𝑃 𝑄 ) ( ( ( 𝑡 𝑈 ) ( 𝑄 ( ( 𝑃 𝑡 ) 𝑊 ) ) ) ( ( 𝑅 𝑡 ) 𝑊 ) ) ) ) ↔ ∃ 𝑢𝐵𝑠𝐴 ( ( ¬ 𝑠 𝑊 ∧ ¬ 𝑠 ( 𝑃 𝑄 ) ) → 𝑢 = 𝑁 ) ) )
31 30 biimpd ( ∃ 𝑠𝐴 ( ( ¬ 𝑠 𝑊 ∧ ¬ 𝑠 ( 𝑃 𝑄 ) ) ∧ 𝑁𝐵 ) → ( ∀ 𝑠𝐴𝑡𝐴 ( ( ( ¬ 𝑠 𝑊 ∧ ¬ 𝑠 ( 𝑃 𝑄 ) ) ∧ ( ¬ 𝑡 𝑊 ∧ ¬ 𝑡 ( 𝑃 𝑄 ) ) ) → 𝑁 = ( ( 𝑃 𝑄 ) ( ( ( 𝑡 𝑈 ) ( 𝑄 ( ( 𝑃 𝑡 ) 𝑊 ) ) ) ( ( 𝑅 𝑡 ) 𝑊 ) ) ) ) → ∃ 𝑢𝐵𝑠𝐴 ( ( ¬ 𝑠 𝑊 ∧ ¬ 𝑠 ( 𝑃 𝑄 ) ) → 𝑢 = 𝑁 ) ) )
32 10 13 31 sylc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ ( 𝑃𝑄𝑅 ( 𝑃 𝑄 ) ) ) → ∃ 𝑢𝐵𝑠𝐴 ( ( ¬ 𝑠 𝑊 ∧ ¬ 𝑠 ( 𝑃 𝑄 ) ) → 𝑢 = 𝑁 ) )