Metamath Proof Explorer


Theorem cdleme42g

Description: Part of proof of Lemma E in Crawley p. 113. (Contributed by NM, 8-Mar-2013)

Ref Expression
Hypotheses cdleme41.b 𝐵 = ( Base ‘ 𝐾 )
cdleme41.l = ( le ‘ 𝐾 )
cdleme41.j = ( join ‘ 𝐾 )
cdleme41.m = ( meet ‘ 𝐾 )
cdleme41.a 𝐴 = ( Atoms ‘ 𝐾 )
cdleme41.h 𝐻 = ( LHyp ‘ 𝐾 )
cdleme41.u 𝑈 = ( ( 𝑃 𝑄 ) 𝑊 )
cdleme41.d 𝐷 = ( ( 𝑠 𝑈 ) ( 𝑄 ( ( 𝑃 𝑠 ) 𝑊 ) ) )
cdleme41.e 𝐸 = ( ( 𝑡 𝑈 ) ( 𝑄 ( ( 𝑃 𝑡 ) 𝑊 ) ) )
cdleme41.g 𝐺 = ( ( 𝑃 𝑄 ) ( 𝐸 ( ( 𝑠 𝑡 ) 𝑊 ) ) )
cdleme41.i 𝐼 = ( 𝑦𝐵𝑡𝐴 ( ( ¬ 𝑡 𝑊 ∧ ¬ 𝑡 ( 𝑃 𝑄 ) ) → 𝑦 = 𝐺 ) )
cdleme41.n 𝑁 = if ( 𝑠 ( 𝑃 𝑄 ) , 𝐼 , 𝐷 )
cdleme41.o 𝑂 = ( 𝑧𝐵𝑠𝐴 ( ( ¬ 𝑠 𝑊 ∧ ( 𝑠 ( 𝑥 𝑊 ) ) = 𝑥 ) → 𝑧 = ( 𝑁 ( 𝑥 𝑊 ) ) ) )
cdleme41.f 𝐹 = ( 𝑥𝐵 ↦ if ( ( 𝑃𝑄 ∧ ¬ 𝑥 𝑊 ) , 𝑂 , 𝑥 ) )
cdleme34e.v 𝑉 = ( ( 𝑅 𝑆 ) 𝑊 )
Assertion cdleme42g ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ 𝑃𝑄 ) → ( 𝐹 ‘ ( 𝑅 𝑆 ) ) = ( ( 𝐹𝑅 ) 𝑉 ) )

Proof

Step Hyp Ref Expression
1 cdleme41.b 𝐵 = ( Base ‘ 𝐾 )
2 cdleme41.l = ( le ‘ 𝐾 )
3 cdleme41.j = ( join ‘ 𝐾 )
4 cdleme41.m = ( meet ‘ 𝐾 )
5 cdleme41.a 𝐴 = ( Atoms ‘ 𝐾 )
6 cdleme41.h 𝐻 = ( LHyp ‘ 𝐾 )
7 cdleme41.u 𝑈 = ( ( 𝑃 𝑄 ) 𝑊 )
8 cdleme41.d 𝐷 = ( ( 𝑠 𝑈 ) ( 𝑄 ( ( 𝑃 𝑠 ) 𝑊 ) ) )
9 cdleme41.e 𝐸 = ( ( 𝑡 𝑈 ) ( 𝑄 ( ( 𝑃 𝑡 ) 𝑊 ) ) )
10 cdleme41.g 𝐺 = ( ( 𝑃 𝑄 ) ( 𝐸 ( ( 𝑠 𝑡 ) 𝑊 ) ) )
11 cdleme41.i 𝐼 = ( 𝑦𝐵𝑡𝐴 ( ( ¬ 𝑡 𝑊 ∧ ¬ 𝑡 ( 𝑃 𝑄 ) ) → 𝑦 = 𝐺 ) )
12 cdleme41.n 𝑁 = if ( 𝑠 ( 𝑃 𝑄 ) , 𝐼 , 𝐷 )
13 cdleme41.o 𝑂 = ( 𝑧𝐵𝑠𝐴 ( ( ¬ 𝑠 𝑊 ∧ ( 𝑠 ( 𝑥 𝑊 ) ) = 𝑥 ) → 𝑧 = ( 𝑁 ( 𝑥 𝑊 ) ) ) )
14 cdleme41.f 𝐹 = ( 𝑥𝐵 ↦ if ( ( 𝑃𝑄 ∧ ¬ 𝑥 𝑊 ) , 𝑂 , 𝑥 ) )
15 cdleme34e.v 𝑉 = ( ( 𝑅 𝑆 ) 𝑊 )
16 simp11 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ 𝑃𝑄 ) → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
17 simp2l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ 𝑃𝑄 ) → ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) )
18 simp2r ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ 𝑃𝑄 ) → ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) )
19 1 2 3 4 5 6 15 cdleme42a ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) → ( 𝑅 𝑆 ) = ( 𝑅 𝑉 ) )
20 16 17 18 19 syl3anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ 𝑃𝑄 ) → ( 𝑅 𝑆 ) = ( 𝑅 𝑉 ) )
21 20 fveq2d ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ 𝑃𝑄 ) → ( 𝐹 ‘ ( 𝑅 𝑆 ) ) = ( 𝐹 ‘ ( 𝑅 𝑉 ) ) )
22 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 cdleme42f ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ 𝑃𝑄 ) → ( 𝐹 ‘ ( 𝑅 𝑉 ) ) = ( ( 𝐹𝑅 ) 𝑉 ) )
23 21 22 eqtrd ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ 𝑃𝑄 ) → ( 𝐹 ‘ ( 𝑅 𝑆 ) ) = ( ( 𝐹𝑅 ) 𝑉 ) )