Metamath Proof Explorer


Theorem cdleme32a

Description: Part of proof of Lemma D in Crawley p. 113. (Contributed by NM, 19-Feb-2013)

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 )
cdleme32.o 𝑂 = ( 𝑧𝐵𝑠𝐴 ( ( ¬ 𝑠 𝑊 ∧ ( 𝑠 ( 𝑥 𝑊 ) ) = 𝑥 ) → 𝑧 = ( 𝑁 ( 𝑥 𝑊 ) ) ) )
cdleme32.f 𝐹 = ( 𝑥𝐵 ↦ if ( ( 𝑃𝑄 ∧ ¬ 𝑥 𝑊 ) , 𝑂 , 𝑥 ) )
Assertion cdleme32a ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑋𝐵 ∧ ( 𝑃𝑄 ∧ ¬ 𝑋 𝑊 ) ) ∧ ( ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ ( 𝑠 ( 𝑋 𝑊 ) ) = 𝑋 ) ) → ( 𝐹𝑋 ) = ( 𝑁 ( 𝑋 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 cdleme32.o 𝑂 = ( 𝑧𝐵𝑠𝐴 ( ( ¬ 𝑠 𝑊 ∧ ( 𝑠 ( 𝑥 𝑊 ) ) = 𝑥 ) → 𝑧 = ( 𝑁 ( 𝑥 𝑊 ) ) ) )
14 cdleme32.f 𝐹 = ( 𝑥𝐵 ↦ if ( ( 𝑃𝑄 ∧ ¬ 𝑥 𝑊 ) , 𝑂 , 𝑥 ) )
15 1 fvexi 𝐵 ∈ V
16 anass ( ( ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ ( 𝑠 ( 𝑋 𝑊 ) ) = 𝑋 ) ↔ ( 𝑠𝐴 ∧ ( ¬ 𝑠 𝑊 ∧ ( 𝑠 ( 𝑋 𝑊 ) ) = 𝑋 ) ) )
17 eqid ( 𝑧𝐵𝑠𝐴 ( ( ¬ 𝑠 𝑊 ∧ ( 𝑠 ( 𝑋 𝑊 ) ) = 𝑋 ) → 𝑧 = ( 𝑁 ( 𝑋 𝑊 ) ) ) ) = ( 𝑧𝐵𝑠𝐴 ( ( ¬ 𝑠 𝑊 ∧ ( 𝑠 ( 𝑋 𝑊 ) ) = 𝑋 ) → 𝑧 = ( 𝑁 ( 𝑋 𝑊 ) ) ) )
18 13 14 17 cdleme31fv1 ( ( 𝑋𝐵 ∧ ( 𝑃𝑄 ∧ ¬ 𝑋 𝑊 ) ) → ( 𝐹𝑋 ) = ( 𝑧𝐵𝑠𝐴 ( ( ¬ 𝑠 𝑊 ∧ ( 𝑠 ( 𝑋 𝑊 ) ) = 𝑋 ) → 𝑧 = ( 𝑁 ( 𝑋 𝑊 ) ) ) ) )
19 18 adantl ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑋𝐵 ∧ ( 𝑃𝑄 ∧ ¬ 𝑋 𝑊 ) ) ) → ( 𝐹𝑋 ) = ( 𝑧𝐵𝑠𝐴 ( ( ¬ 𝑠 𝑊 ∧ ( 𝑠 ( 𝑋 𝑊 ) ) = 𝑋 ) → 𝑧 = ( 𝑁 ( 𝑋 𝑊 ) ) ) ) )
20 1 2 3 4 5 6 7 8 9 10 11 12 13 14 cdleme32fvcl ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ 𝑋𝐵 ) → ( 𝐹𝑋 ) ∈ 𝐵 )
21 20 adantrr ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑋𝐵 ∧ ( 𝑃𝑄 ∧ ¬ 𝑋 𝑊 ) ) ) → ( 𝐹𝑋 ) ∈ 𝐵 )
22 19 21 riotasvd ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑋𝐵 ∧ ( 𝑃𝑄 ∧ ¬ 𝑋 𝑊 ) ) ) ∧ 𝐵 ∈ V ) → ( ( 𝑠𝐴 ∧ ( ¬ 𝑠 𝑊 ∧ ( 𝑠 ( 𝑋 𝑊 ) ) = 𝑋 ) ) → ( 𝐹𝑋 ) = ( 𝑁 ( 𝑋 𝑊 ) ) ) )
23 16 22 syl5bi ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑋𝐵 ∧ ( 𝑃𝑄 ∧ ¬ 𝑋 𝑊 ) ) ) ∧ 𝐵 ∈ V ) → ( ( ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ ( 𝑠 ( 𝑋 𝑊 ) ) = 𝑋 ) → ( 𝐹𝑋 ) = ( 𝑁 ( 𝑋 𝑊 ) ) ) )
24 15 23 mpan2 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑋𝐵 ∧ ( 𝑃𝑄 ∧ ¬ 𝑋 𝑊 ) ) ) → ( ( ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ ( 𝑠 ( 𝑋 𝑊 ) ) = 𝑋 ) → ( 𝐹𝑋 ) = ( 𝑁 ( 𝑋 𝑊 ) ) ) )
25 24 3impia ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑋𝐵 ∧ ( 𝑃𝑄 ∧ ¬ 𝑋 𝑊 ) ) ∧ ( ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ ( 𝑠 ( 𝑋 𝑊 ) ) = 𝑋 ) ) → ( 𝐹𝑋 ) = ( 𝑁 ( 𝑋 𝑊 ) ) )