Metamath Proof Explorer


Theorem cdleme32sn2awN

Description: Show that [_ R / s ]_ N is an atom not under W when -. R .<_ ( P .\/ Q ) . (Contributed by NM, 6-Mar-2013) (New usage is discouraged.)

Ref Expression
Hypotheses cdleme32.b 𝐵 = ( Base ‘ 𝐾 )
cdleme32.l = ( le ‘ 𝐾 )
cdleme32.j = ( join ‘ 𝐾 )
cdleme32.m = ( meet ‘ 𝐾 )
cdleme32.a 𝐴 = ( Atoms ‘ 𝐾 )
cdleme32.h 𝐻 = ( LHyp ‘ 𝐾 )
cdleme32.u 𝑈 = ( ( 𝑃 𝑄 ) 𝑊 )
cdleme32.c 𝐶 = ( ( 𝑠 𝑈 ) ( 𝑄 ( ( 𝑃 𝑠 ) 𝑊 ) ) )
cdleme32.d 𝐷 = ( ( 𝑡 𝑈 ) ( 𝑄 ( ( 𝑃 𝑡 ) 𝑊 ) ) )
cdleme32.e 𝐸 = ( ( 𝑃 𝑄 ) ( 𝐷 ( ( 𝑠 𝑡 ) 𝑊 ) ) )
cdleme32.i 𝐼 = ( 𝑦𝐵𝑡𝐴 ( ( ¬ 𝑡 𝑊 ∧ ¬ 𝑡 ( 𝑃 𝑄 ) ) → 𝑦 = 𝐸 ) )
cdleme32.n 𝑁 = if ( 𝑠 ( 𝑃 𝑄 ) , 𝐼 , 𝐶 )
Assertion cdleme32sn2awN ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ¬ 𝑅 ( 𝑃 𝑄 ) ) → ( 𝑅 / 𝑠 𝑁𝐴 ∧ ¬ 𝑅 / 𝑠 𝑁 𝑊 ) )

Proof

Step Hyp Ref Expression
1 cdleme32.b 𝐵 = ( Base ‘ 𝐾 )
2 cdleme32.l = ( le ‘ 𝐾 )
3 cdleme32.j = ( join ‘ 𝐾 )
4 cdleme32.m = ( meet ‘ 𝐾 )
5 cdleme32.a 𝐴 = ( Atoms ‘ 𝐾 )
6 cdleme32.h 𝐻 = ( LHyp ‘ 𝐾 )
7 cdleme32.u 𝑈 = ( ( 𝑃 𝑄 ) 𝑊 )
8 cdleme32.c 𝐶 = ( ( 𝑠 𝑈 ) ( 𝑄 ( ( 𝑃 𝑠 ) 𝑊 ) ) )
9 cdleme32.d 𝐷 = ( ( 𝑡 𝑈 ) ( 𝑄 ( ( 𝑃 𝑡 ) 𝑊 ) ) )
10 cdleme32.e 𝐸 = ( ( 𝑃 𝑄 ) ( 𝐷 ( ( 𝑠 𝑡 ) 𝑊 ) ) )
11 cdleme32.i 𝐼 = ( 𝑦𝐵𝑡𝐴 ( ( ¬ 𝑡 𝑊 ∧ ¬ 𝑡 ( 𝑃 𝑄 ) ) → 𝑦 = 𝐸 ) )
12 cdleme32.n 𝑁 = if ( 𝑠 ( 𝑃 𝑄 ) , 𝐼 , 𝐶 )
13 1 2 3 4 5 6 7 8 12 cdlemefr32sn2aw ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ¬ 𝑅 ( 𝑃 𝑄 ) ) → ( 𝑅 / 𝑠 𝑁𝐴 ∧ ¬ 𝑅 / 𝑠 𝑁 𝑊 ) )