Metamath Proof Explorer


Theorem cdlemeg47b

Description: TODO: FIX COMMENT. (Contributed by NM, 1-Apr-2013)

Ref Expression
Hypotheses cdlemef47.b 𝐵 = ( Base ‘ 𝐾 )
cdlemef47.l = ( le ‘ 𝐾 )
cdlemef47.j = ( join ‘ 𝐾 )
cdlemef47.m = ( meet ‘ 𝐾 )
cdlemef47.a 𝐴 = ( Atoms ‘ 𝐾 )
cdlemef47.h 𝐻 = ( LHyp ‘ 𝐾 )
cdlemef47.v 𝑉 = ( ( 𝑄 𝑃 ) 𝑊 )
cdlemef47.n 𝑁 = ( ( 𝑣 𝑉 ) ( 𝑃 ( ( 𝑄 𝑣 ) 𝑊 ) ) )
cdlemefs47.o 𝑂 = ( ( 𝑄 𝑃 ) ( 𝑁 ( ( 𝑢 𝑣 ) 𝑊 ) ) )
cdlemef47.g 𝐺 = ( 𝑎𝐵 ↦ if ( ( 𝑄𝑃 ∧ ¬ 𝑎 𝑊 ) , ( 𝑐𝐵𝑢𝐴 ( ( ¬ 𝑢 𝑊 ∧ ( 𝑢 ( 𝑎 𝑊 ) ) = 𝑎 ) → 𝑐 = ( if ( 𝑢 ( 𝑄 𝑃 ) , ( 𝑏𝐵𝑣𝐴 ( ( ¬ 𝑣 𝑊 ∧ ¬ 𝑣 ( 𝑄 𝑃 ) ) → 𝑏 = 𝑂 ) ) , 𝑢 / 𝑣 𝑁 ) ( 𝑎 𝑊 ) ) ) ) , 𝑎 ) )
Assertion cdlemeg47b ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) → ( 𝐺𝑆 ) = 𝑆 / 𝑣 𝑁 )

Proof

Step Hyp Ref Expression
1 cdlemef47.b 𝐵 = ( Base ‘ 𝐾 )
2 cdlemef47.l = ( le ‘ 𝐾 )
3 cdlemef47.j = ( join ‘ 𝐾 )
4 cdlemef47.m = ( meet ‘ 𝐾 )
5 cdlemef47.a 𝐴 = ( Atoms ‘ 𝐾 )
6 cdlemef47.h 𝐻 = ( LHyp ‘ 𝐾 )
7 cdlemef47.v 𝑉 = ( ( 𝑄 𝑃 ) 𝑊 )
8 cdlemef47.n 𝑁 = ( ( 𝑣 𝑉 ) ( 𝑃 ( ( 𝑄 𝑣 ) 𝑊 ) ) )
9 cdlemefs47.o 𝑂 = ( ( 𝑄 𝑃 ) ( 𝑁 ( ( 𝑢 𝑣 ) 𝑊 ) ) )
10 cdlemef47.g 𝐺 = ( 𝑎𝐵 ↦ if ( ( 𝑄𝑃 ∧ ¬ 𝑎 𝑊 ) , ( 𝑐𝐵𝑢𝐴 ( ( ¬ 𝑢 𝑊 ∧ ( 𝑢 ( 𝑎 𝑊 ) ) = 𝑎 ) → 𝑐 = ( if ( 𝑢 ( 𝑄 𝑃 ) , ( 𝑏𝐵𝑣𝐴 ( ( ¬ 𝑣 𝑊 ∧ ¬ 𝑣 ( 𝑄 𝑃 ) ) → 𝑏 = 𝑂 ) ) , 𝑢 / 𝑣 𝑁 ) ( 𝑎 𝑊 ) ) ) ) , 𝑎 ) )
11 3 5 cdleme46f2g2 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) → ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) ∧ ( 𝑄𝑃 ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ¬ 𝑆 ( 𝑄 𝑃 ) ) )
12 1 2 3 4 5 6 7 8 10 cdlemefr45 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) ∧ ( 𝑄𝑃 ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ¬ 𝑆 ( 𝑄 𝑃 ) ) → ( 𝐺𝑆 ) = 𝑆 / 𝑣 𝑁 )
13 11 12 syl ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) → ( 𝐺𝑆 ) = 𝑆 / 𝑣 𝑁 )