Metamath Proof Explorer


Theorem cdlemeg46fvaw

Description: Show that ( FR ) is an atom not under W when R is an atom not under W . (Contributed by NM, 1-Apr-2013)

Ref Expression
Hypotheses cdlemef46g.b 𝐵 = ( Base ‘ 𝐾 )
cdlemef46g.l = ( le ‘ 𝐾 )
cdlemef46g.j = ( join ‘ 𝐾 )
cdlemef46g.m = ( meet ‘ 𝐾 )
cdlemef46g.a 𝐴 = ( Atoms ‘ 𝐾 )
cdlemef46g.h 𝐻 = ( LHyp ‘ 𝐾 )
cdlemef46g.u 𝑈 = ( ( 𝑃 𝑄 ) 𝑊 )
cdlemef46g.d 𝐷 = ( ( 𝑡 𝑈 ) ( 𝑄 ( ( 𝑃 𝑡 ) 𝑊 ) ) )
cdlemefs46g.e 𝐸 = ( ( 𝑃 𝑄 ) ( 𝐷 ( ( 𝑠 𝑡 ) 𝑊 ) ) )
cdlemef46g.f 𝐹 = ( 𝑥𝐵 ↦ if ( ( 𝑃𝑄 ∧ ¬ 𝑥 𝑊 ) , ( 𝑧𝐵𝑠𝐴 ( ( ¬ 𝑠 𝑊 ∧ ( 𝑠 ( 𝑥 𝑊 ) ) = 𝑥 ) → 𝑧 = ( if ( 𝑠 ( 𝑃 𝑄 ) , ( 𝑦𝐵𝑡𝐴 ( ( ¬ 𝑡 𝑊 ∧ ¬ 𝑡 ( 𝑃 𝑄 ) ) → 𝑦 = 𝐸 ) ) , 𝑠 / 𝑡 𝐷 ) ( 𝑥 𝑊 ) ) ) ) , 𝑥 ) )
cdlemef46.v 𝑉 = ( ( 𝑄 𝑃 ) 𝑊 )
cdlemef46.n 𝑁 = ( ( 𝑣 𝑉 ) ( 𝑃 ( ( 𝑄 𝑣 ) 𝑊 ) ) )
cdlemefs46.o 𝑂 = ( ( 𝑄 𝑃 ) ( 𝑁 ( ( 𝑢 𝑣 ) 𝑊 ) ) )
cdlemef46.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 cdlemeg46fvaw ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ 𝑃𝑄 ) → ( ( 𝐺𝑅 ) ∈ 𝐴 ∧ ¬ ( 𝐺𝑅 ) 𝑊 ) )

Proof

Step Hyp Ref Expression
1 cdlemef46g.b 𝐵 = ( Base ‘ 𝐾 )
2 cdlemef46g.l = ( le ‘ 𝐾 )
3 cdlemef46g.j = ( join ‘ 𝐾 )
4 cdlemef46g.m = ( meet ‘ 𝐾 )
5 cdlemef46g.a 𝐴 = ( Atoms ‘ 𝐾 )
6 cdlemef46g.h 𝐻 = ( LHyp ‘ 𝐾 )
7 cdlemef46g.u 𝑈 = ( ( 𝑃 𝑄 ) 𝑊 )
8 cdlemef46g.d 𝐷 = ( ( 𝑡 𝑈 ) ( 𝑄 ( ( 𝑃 𝑡 ) 𝑊 ) ) )
9 cdlemefs46g.e 𝐸 = ( ( 𝑃 𝑄 ) ( 𝐷 ( ( 𝑠 𝑡 ) 𝑊 ) ) )
10 cdlemef46g.f 𝐹 = ( 𝑥𝐵 ↦ if ( ( 𝑃𝑄 ∧ ¬ 𝑥 𝑊 ) , ( 𝑧𝐵𝑠𝐴 ( ( ¬ 𝑠 𝑊 ∧ ( 𝑠 ( 𝑥 𝑊 ) ) = 𝑥 ) → 𝑧 = ( if ( 𝑠 ( 𝑃 𝑄 ) , ( 𝑦𝐵𝑡𝐴 ( ( ¬ 𝑡 𝑊 ∧ ¬ 𝑡 ( 𝑃 𝑄 ) ) → 𝑦 = 𝐸 ) ) , 𝑠 / 𝑡 𝐷 ) ( 𝑥 𝑊 ) ) ) ) , 𝑥 ) )
11 cdlemef46.v 𝑉 = ( ( 𝑄 𝑃 ) 𝑊 )
12 cdlemef46.n 𝑁 = ( ( 𝑣 𝑉 ) ( 𝑃 ( ( 𝑄 𝑣 ) 𝑊 ) ) )
13 cdlemefs46.o 𝑂 = ( ( 𝑄 𝑃 ) ( 𝑁 ( ( 𝑢 𝑣 ) 𝑊 ) ) )
14 cdlemef46.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 ) )
15 simp11 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ 𝑃𝑄 ) → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
16 simp13 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ 𝑃𝑄 ) → ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) )
17 simp12 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ 𝑃𝑄 ) → ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) )
18 simp2 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ 𝑃𝑄 ) → ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) )
19 1 2 3 4 5 6 11 12 13 14 cdleme46fvaw ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) → ( ( 𝐺𝑅 ) ∈ 𝐴 ∧ ¬ ( 𝐺𝑅 ) 𝑊 ) )
20 15 16 17 18 19 syl31anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ 𝑃𝑄 ) → ( ( 𝐺𝑅 ) ∈ 𝐴 ∧ ¬ ( 𝐺𝑅 ) 𝑊 ) )