Metamath Proof Explorer


Theorem cdlemefr29clN

Description: Show closure of the unique element in cdleme29c . TODO fix comment. TODO Not needed? (Contributed by NM, 29-Mar-2013) (New usage is discouraged.)

Ref Expression
Hypotheses cdlemefr27.b 𝐵 = ( Base ‘ 𝐾 )
cdlemefr27.l = ( le ‘ 𝐾 )
cdlemefr27.j = ( join ‘ 𝐾 )
cdlemefr27.m = ( meet ‘ 𝐾 )
cdlemefr27.a 𝐴 = ( Atoms ‘ 𝐾 )
cdlemefr27.h 𝐻 = ( LHyp ‘ 𝐾 )
cdlemefr27.u 𝑈 = ( ( 𝑃 𝑄 ) 𝑊 )
cdlemefr27.c 𝐶 = ( ( 𝑠 𝑈 ) ( 𝑄 ( ( 𝑃 𝑠 ) 𝑊 ) ) )
cdlemefr27.n 𝑁 = if ( 𝑠 ( 𝑃 𝑄 ) , 𝐼 , 𝐶 )
cdlemefr29cl.o 𝑂 = ( 𝑧𝐵𝑠𝐴 ( ( ¬ 𝑠 𝑊 ∧ ( 𝑠 ( 𝑅 𝑊 ) ) = 𝑅 ) → 𝑧 = ( 𝑁 ( 𝑅 𝑊 ) ) ) )
Assertion cdlemefr29clN ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ¬ 𝑅 ( 𝑃 𝑄 ) ) → 𝑂𝐵 )

Proof

Step Hyp Ref Expression
1 cdlemefr27.b 𝐵 = ( Base ‘ 𝐾 )
2 cdlemefr27.l = ( le ‘ 𝐾 )
3 cdlemefr27.j = ( join ‘ 𝐾 )
4 cdlemefr27.m = ( meet ‘ 𝐾 )
5 cdlemefr27.a 𝐴 = ( Atoms ‘ 𝐾 )
6 cdlemefr27.h 𝐻 = ( LHyp ‘ 𝐾 )
7 cdlemefr27.u 𝑈 = ( ( 𝑃 𝑄 ) 𝑊 )
8 cdlemefr27.c 𝐶 = ( ( 𝑠 𝑈 ) ( 𝑄 ( ( 𝑃 𝑠 ) 𝑊 ) ) )
9 cdlemefr27.n 𝑁 = if ( 𝑠 ( 𝑃 𝑄 ) , 𝐼 , 𝐶 )
10 cdlemefr29cl.o 𝑂 = ( 𝑧𝐵𝑠𝐴 ( ( ¬ 𝑠 𝑊 ∧ ( 𝑠 ( 𝑅 𝑊 ) ) = 𝑅 ) → 𝑧 = ( 𝑁 ( 𝑅 𝑊 ) ) ) )
11 breq1 ( 𝑠 = 𝑅 → ( 𝑠 ( 𝑃 𝑄 ) ↔ 𝑅 ( 𝑃 𝑄 ) ) )
12 11 notbid ( 𝑠 = 𝑅 → ( ¬ 𝑠 ( 𝑃 𝑄 ) ↔ ¬ 𝑅 ( 𝑃 𝑄 ) ) )
13 simp11 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ 𝑃𝑄 ∧ ( 𝑠𝐴 ∧ ( ¬ 𝑠 𝑊 ∧ ¬ 𝑠 ( 𝑃 𝑄 ) ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
14 simp12l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ 𝑃𝑄 ∧ ( 𝑠𝐴 ∧ ( ¬ 𝑠 𝑊 ∧ ¬ 𝑠 ( 𝑃 𝑄 ) ) ) ) → 𝑃𝐴 )
15 simp13l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ 𝑃𝑄 ∧ ( 𝑠𝐴 ∧ ( ¬ 𝑠 𝑊 ∧ ¬ 𝑠 ( 𝑃 𝑄 ) ) ) ) → 𝑄𝐴 )
16 simp3l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ 𝑃𝑄 ∧ ( 𝑠𝐴 ∧ ( ¬ 𝑠 𝑊 ∧ ¬ 𝑠 ( 𝑃 𝑄 ) ) ) ) → 𝑠𝐴 )
17 simp3rr ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ 𝑃𝑄 ∧ ( 𝑠𝐴 ∧ ( ¬ 𝑠 𝑊 ∧ ¬ 𝑠 ( 𝑃 𝑄 ) ) ) ) → ¬ 𝑠 ( 𝑃 𝑄 ) )
18 simp2 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ 𝑃𝑄 ∧ ( 𝑠𝐴 ∧ ( ¬ 𝑠 𝑊 ∧ ¬ 𝑠 ( 𝑃 𝑄 ) ) ) ) → 𝑃𝑄 )
19 1 2 3 4 5 6 7 8 9 cdlemefr27cl ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑃𝐴𝑄𝐴 ) ∧ ( 𝑠𝐴 ∧ ¬ 𝑠 ( 𝑃 𝑄 ) ∧ 𝑃𝑄 ) ) → 𝑁𝐵 )
20 13 14 15 16 17 18 19 syl33anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ 𝑃𝑄 ∧ ( 𝑠𝐴 ∧ ( ¬ 𝑠 𝑊 ∧ ¬ 𝑠 ( 𝑃 𝑄 ) ) ) ) → 𝑁𝐵 )
21 1 2 3 4 5 6 7 8 9 cdlemefr32snb ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ¬ 𝑅 ( 𝑃 𝑄 ) ) → 𝑅 / 𝑠 𝑁𝐵 )
22 1 2 3 4 5 6 12 20 21 10 cdlemefrs29clN ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ¬ 𝑅 ( 𝑃 𝑄 ) ) → 𝑂𝐵 )