Metamath Proof Explorer


Theorem cdleme32b

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 cdleme32b ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑋𝐵𝑌𝐵 ∧ ( 𝑃𝑄 ∧ ¬ 𝑋 𝑊 ) ) ∧ ( ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ ( 𝑠 ( 𝑋 𝑊 ) ) = 𝑋𝑋 𝑌 ) ) → ( 𝐹𝑌 ) = ( 𝑁 ( 𝑌 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 simp1 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑋𝐵𝑌𝐵 ∧ ( 𝑃𝑄 ∧ ¬ 𝑋 𝑊 ) ) ∧ ( ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ ( 𝑠 ( 𝑋 𝑊 ) ) = 𝑋𝑋 𝑌 ) ) → ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) )
16 simp22 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑋𝐵𝑌𝐵 ∧ ( 𝑃𝑄 ∧ ¬ 𝑋 𝑊 ) ) ∧ ( ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ ( 𝑠 ( 𝑋 𝑊 ) ) = 𝑋𝑋 𝑌 ) ) → 𝑌𝐵 )
17 simp23l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑋𝐵𝑌𝐵 ∧ ( 𝑃𝑄 ∧ ¬ 𝑋 𝑊 ) ) ∧ ( ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ ( 𝑠 ( 𝑋 𝑊 ) ) = 𝑋𝑋 𝑌 ) ) → 𝑃𝑄 )
18 simp23r ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑋𝐵𝑌𝐵 ∧ ( 𝑃𝑄 ∧ ¬ 𝑋 𝑊 ) ) ∧ ( ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ ( 𝑠 ( 𝑋 𝑊 ) ) = 𝑋𝑋 𝑌 ) ) → ¬ 𝑋 𝑊 )
19 simp33 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑋𝐵𝑌𝐵 ∧ ( 𝑃𝑄 ∧ ¬ 𝑋 𝑊 ) ) ∧ ( ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ ( 𝑠 ( 𝑋 𝑊 ) ) = 𝑋𝑋 𝑌 ) ) → 𝑋 𝑌 )
20 simp11l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑋𝐵𝑌𝐵 ∧ ( 𝑃𝑄 ∧ ¬ 𝑋 𝑊 ) ) ∧ ( ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ ( 𝑠 ( 𝑋 𝑊 ) ) = 𝑋𝑋 𝑌 ) ) → 𝐾 ∈ HL )
21 20 hllatd ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑋𝐵𝑌𝐵 ∧ ( 𝑃𝑄 ∧ ¬ 𝑋 𝑊 ) ) ∧ ( ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ ( 𝑠 ( 𝑋 𝑊 ) ) = 𝑋𝑋 𝑌 ) ) → 𝐾 ∈ Lat )
22 simp21 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑋𝐵𝑌𝐵 ∧ ( 𝑃𝑄 ∧ ¬ 𝑋 𝑊 ) ) ∧ ( ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ ( 𝑠 ( 𝑋 𝑊 ) ) = 𝑋𝑋 𝑌 ) ) → 𝑋𝐵 )
23 simp11r ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑋𝐵𝑌𝐵 ∧ ( 𝑃𝑄 ∧ ¬ 𝑋 𝑊 ) ) ∧ ( ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ ( 𝑠 ( 𝑋 𝑊 ) ) = 𝑋𝑋 𝑌 ) ) → 𝑊𝐻 )
24 1 6 lhpbase ( 𝑊𝐻𝑊𝐵 )
25 23 24 syl ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑋𝐵𝑌𝐵 ∧ ( 𝑃𝑄 ∧ ¬ 𝑋 𝑊 ) ) ∧ ( ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ ( 𝑠 ( 𝑋 𝑊 ) ) = 𝑋𝑋 𝑌 ) ) → 𝑊𝐵 )
26 1 2 lattr ( ( 𝐾 ∈ Lat ∧ ( 𝑋𝐵𝑌𝐵𝑊𝐵 ) ) → ( ( 𝑋 𝑌𝑌 𝑊 ) → 𝑋 𝑊 ) )
27 21 22 16 25 26 syl13anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑋𝐵𝑌𝐵 ∧ ( 𝑃𝑄 ∧ ¬ 𝑋 𝑊 ) ) ∧ ( ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ ( 𝑠 ( 𝑋 𝑊 ) ) = 𝑋𝑋 𝑌 ) ) → ( ( 𝑋 𝑌𝑌 𝑊 ) → 𝑋 𝑊 ) )
28 19 27 mpand ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑋𝐵𝑌𝐵 ∧ ( 𝑃𝑄 ∧ ¬ 𝑋 𝑊 ) ) ∧ ( ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ ( 𝑠 ( 𝑋 𝑊 ) ) = 𝑋𝑋 𝑌 ) ) → ( 𝑌 𝑊𝑋 𝑊 ) )
29 18 28 mtod ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑋𝐵𝑌𝐵 ∧ ( 𝑃𝑄 ∧ ¬ 𝑋 𝑊 ) ) ∧ ( ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ ( 𝑠 ( 𝑋 𝑊 ) ) = 𝑋𝑋 𝑌 ) ) → ¬ 𝑌 𝑊 )
30 17 29 jca ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑋𝐵𝑌𝐵 ∧ ( 𝑃𝑄 ∧ ¬ 𝑋 𝑊 ) ) ∧ ( ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ ( 𝑠 ( 𝑋 𝑊 ) ) = 𝑋𝑋 𝑌 ) ) → ( 𝑃𝑄 ∧ ¬ 𝑌 𝑊 ) )
31 simp31 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑋𝐵𝑌𝐵 ∧ ( 𝑃𝑄 ∧ ¬ 𝑋 𝑊 ) ) ∧ ( ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ ( 𝑠 ( 𝑋 𝑊 ) ) = 𝑋𝑋 𝑌 ) ) → ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) )
32 simp11 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑋𝐵𝑌𝐵 ∧ ( 𝑃𝑄 ∧ ¬ 𝑋 𝑊 ) ) ∧ ( ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ ( 𝑠 ( 𝑋 𝑊 ) ) = 𝑋𝑋 𝑌 ) ) → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
33 simp31l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑋𝐵𝑌𝐵 ∧ ( 𝑃𝑄 ∧ ¬ 𝑋 𝑊 ) ) ∧ ( ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ ( 𝑠 ( 𝑋 𝑊 ) ) = 𝑋𝑋 𝑌 ) ) → 𝑠𝐴 )
34 22 18 jca ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑋𝐵𝑌𝐵 ∧ ( 𝑃𝑄 ∧ ¬ 𝑋 𝑊 ) ) ∧ ( ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ ( 𝑠 ( 𝑋 𝑊 ) ) = 𝑋𝑋 𝑌 ) ) → ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) )
35 simp32 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑋𝐵𝑌𝐵 ∧ ( 𝑃𝑄 ∧ ¬ 𝑋 𝑊 ) ) ∧ ( ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ ( 𝑠 ( 𝑋 𝑊 ) ) = 𝑋𝑋 𝑌 ) ) → ( 𝑠 ( 𝑋 𝑊 ) ) = 𝑋 )
36 1 2 3 4 5 6 cdleme30a ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑠𝐴 ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ∧ 𝑌𝐵 ) ∧ ( ( 𝑠 ( 𝑋 𝑊 ) ) = 𝑋𝑋 𝑌 ) ) → ( 𝑠 ( 𝑌 𝑊 ) ) = 𝑌 )
37 32 33 34 16 35 19 36 syl132anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑋𝐵𝑌𝐵 ∧ ( 𝑃𝑄 ∧ ¬ 𝑋 𝑊 ) ) ∧ ( ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ ( 𝑠 ( 𝑋 𝑊 ) ) = 𝑋𝑋 𝑌 ) ) → ( 𝑠 ( 𝑌 𝑊 ) ) = 𝑌 )
38 1 2 3 4 5 6 7 8 9 10 11 12 13 14 cdleme32a ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑌𝐵 ∧ ( 𝑃𝑄 ∧ ¬ 𝑌 𝑊 ) ) ∧ ( ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ ( 𝑠 ( 𝑌 𝑊 ) ) = 𝑌 ) ) → ( 𝐹𝑌 ) = ( 𝑁 ( 𝑌 𝑊 ) ) )
39 15 16 30 31 37 38 syl122anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑋𝐵𝑌𝐵 ∧ ( 𝑃𝑄 ∧ ¬ 𝑋 𝑊 ) ) ∧ ( ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ ( 𝑠 ( 𝑋 𝑊 ) ) = 𝑋𝑋 𝑌 ) ) → ( 𝐹𝑌 ) = ( 𝑁 ( 𝑌 𝑊 ) ) )