Metamath Proof Explorer

Theorem cdleme32d

Description: Part of proof of Lemma D in Crawley p. 113. (Contributed by NM, 20-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 = ( ℩ 𝑦 ∈ 𝐵 ∀ 𝑡 ∈ 𝐴 ( ( ¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑦 = 𝐸 ) )
cdleme32.n ⊢ 𝑁 = if ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) , 𝐼 , 𝐶 )
cdleme32.o ⊢ 𝑂 = ( ℩ 𝑧 ∈ 𝐵 ∀ 𝑠 ∈ 𝐴 ( ( ¬ 𝑠 ≤ 𝑊 ∧ ( 𝑠 ∨ ( 𝑥 ∧ 𝑊 ) ) = 𝑥 ) → 𝑧 = ( 𝑁 ∨ ( 𝑥 ∧ 𝑊 ) ) ) )
cdleme32.f ⊢ 𝐹 = ( 𝑥 ∈ 𝐵 ↦ if ( ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑥 ≤ 𝑊 ) , 𝑂 , 𝑥 ) )
Assertion cdleme32d ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ 𝑋 ≤ 𝑌 ) → ( 𝐹 ‘ 𝑋 ) ≤ ( 𝐹 ‘ 𝑌 ) )

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 = ( ℩ 𝑦 ∈ 𝐵 ∀ 𝑡 ∈ 𝐴 ( ( ¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑦 = 𝐸 ) )
12 cdleme32.n ⊢ 𝑁 = if ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) , 𝐼 , 𝐶 )
13 cdleme32.o ⊢ 𝑂 = ( ℩ 𝑧 ∈ 𝐵 ∀ 𝑠 ∈ 𝐴 ( ( ¬ 𝑠 ≤ 𝑊 ∧ ( 𝑠 ∨ ( 𝑥 ∧ 𝑊 ) ) = 𝑥 ) → 𝑧 = ( 𝑁 ∨ ( 𝑥 ∧ 𝑊 ) ) ) )
14 cdleme32.f ⊢ 𝐹 = ( 𝑥 ∈ 𝐵 ↦ if ( ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑥 ≤ 𝑊 ) , 𝑂 , 𝑥 ) )
15 simp11 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ 𝑋 ≤ 𝑌 ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
16 simp21 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ 𝑋 ≤ 𝑌 ) → 𝑋 ∈ 𝐵 )
17 simp23r ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ 𝑋 ≤ 𝑌 ) → ¬ 𝑋 ≤ 𝑊 )
18 1 2 3 4 5 6 lhpmcvr2 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) → ∃ 𝑠 ∈ 𝐴 ( ¬ 𝑠 ≤ 𝑊 ∧ ( 𝑠 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) )
19 15 16 17 18 syl12anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ 𝑋 ≤ 𝑌 ) → ∃ 𝑠 ∈ 𝐴 ( ¬ 𝑠 ≤ 𝑊 ∧ ( 𝑠 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) )
20 nfv ⊢ Ⅎ 𝑠 ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ 𝑋 ≤ 𝑌 )
21 nfcv ⊢ Ⅎ 𝑠 𝐵
22 nfv ⊢ Ⅎ 𝑠 ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑥 ≤ 𝑊 )
23 nfra1 ⊢ Ⅎ 𝑠 ∀ 𝑠 ∈ 𝐴 ( ( ¬ 𝑠 ≤ 𝑊 ∧ ( 𝑠 ∨ ( 𝑥 ∧ 𝑊 ) ) = 𝑥 ) → 𝑧 = ( 𝑁 ∨ ( 𝑥 ∧ 𝑊 ) ) )
24 23 21 nfriota ⊢ Ⅎ 𝑠 ( ℩ 𝑧 ∈ 𝐵 ∀ 𝑠 ∈ 𝐴 ( ( ¬ 𝑠 ≤ 𝑊 ∧ ( 𝑠 ∨ ( 𝑥 ∧ 𝑊 ) ) = 𝑥 ) → 𝑧 = ( 𝑁 ∨ ( 𝑥 ∧ 𝑊 ) ) ) )
25 13 24 nfcxfr ⊢ Ⅎ 𝑠 𝑂
26 nfcv ⊢ Ⅎ 𝑠 𝑥
27 22 25 26 nfif ⊢ Ⅎ 𝑠 if ( ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑥 ≤ 𝑊 ) , 𝑂 , 𝑥 )
28 21 27 nfmpt ⊢ Ⅎ 𝑠 ( 𝑥 ∈ 𝐵 ↦ if ( ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑥 ≤ 𝑊 ) , 𝑂 , 𝑥 ) )
29 14 28 nfcxfr ⊢ Ⅎ 𝑠 𝐹
30 nfcv ⊢ Ⅎ 𝑠 𝑋
31 29 30 nffv ⊢ Ⅎ 𝑠 ( 𝐹 ‘ 𝑋 )
32 nfcv ⊢ Ⅎ 𝑠 ≤
33 nfcv ⊢ Ⅎ 𝑠 𝑌
34 29 33 nffv ⊢ Ⅎ 𝑠 ( 𝐹 ‘ 𝑌 )
35 31 32 34 nfbr ⊢ Ⅎ 𝑠 ( 𝐹 ‘ 𝑋 ) ≤ ( 𝐹 ‘ 𝑌 )
36 simpl1 ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ 𝑋 ≤ 𝑌 ) ∧ ( 𝑠 ∈ 𝐴 ∧ ( ¬ 𝑠 ≤ 𝑊 ∧ ( 𝑠 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ) → ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) )
37 simpl2 ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ 𝑋 ≤ 𝑌 ) ∧ ( 𝑠 ∈ 𝐴 ∧ ( ¬ 𝑠 ≤ 𝑊 ∧ ( 𝑠 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ) → ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊 ) ) )
38 simprl ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ 𝑋 ≤ 𝑌 ) ∧ ( 𝑠 ∈ 𝐴 ∧ ( ¬ 𝑠 ≤ 𝑊 ∧ ( 𝑠 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ) → 𝑠 ∈ 𝐴 )
39 simprrl ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ 𝑋 ≤ 𝑌 ) ∧ ( 𝑠 ∈ 𝐴 ∧ ( ¬ 𝑠 ≤ 𝑊 ∧ ( 𝑠 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ) → ¬ 𝑠 ≤ 𝑊 )
40 38 39 jca ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ 𝑋 ≤ 𝑌 ) ∧ ( 𝑠 ∈ 𝐴 ∧ ( ¬ 𝑠 ≤ 𝑊 ∧ ( 𝑠 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ) → ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) )
41 simprrr ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ 𝑋 ≤ 𝑌 ) ∧ ( 𝑠 ∈ 𝐴 ∧ ( ¬ 𝑠 ≤ 𝑊 ∧ ( 𝑠 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ) → ( 𝑠 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 )
42 simpl3 ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ 𝑋 ≤ 𝑌 ) ∧ ( 𝑠 ∈ 𝐴 ∧ ( ¬ 𝑠 ≤ 𝑊 ∧ ( 𝑠 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ) → 𝑋 ≤ 𝑌 )
43 1 2 3 4 5 6 7 8 9 10 11 12 13 14 cdleme32c ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ∧ ( 𝑠 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ 𝑋 ≤ 𝑌 ) ) → ( 𝐹 ‘ 𝑋 ) ≤ ( 𝐹 ‘ 𝑌 ) )
44 36 37 40 41 42 43 syl113anc ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ 𝑋 ≤ 𝑌 ) ∧ ( 𝑠 ∈ 𝐴 ∧ ( ¬ 𝑠 ≤ 𝑊 ∧ ( 𝑠 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ) → ( 𝐹 ‘ 𝑋 ) ≤ ( 𝐹 ‘ 𝑌 ) )
45 44 exp32 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ 𝑋 ≤ 𝑌 ) → ( 𝑠 ∈ 𝐴 → ( ( ¬ 𝑠 ≤ 𝑊 ∧ ( 𝑠 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) → ( 𝐹 ‘ 𝑋 ) ≤ ( 𝐹 ‘ 𝑌 ) ) ) )
46 20 35 45 rexlimd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ 𝑋 ≤ 𝑌 ) → ( ∃ 𝑠 ∈ 𝐴 ( ¬ 𝑠 ≤ 𝑊 ∧ ( 𝑠 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) → ( 𝐹 ‘ 𝑋 ) ≤ ( 𝐹 ‘ 𝑌 ) ) )
47 19 46 mpd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ 𝑋 ≤ 𝑌 ) → ( 𝐹 ‘ 𝑋 ) ≤ ( 𝐹 ‘ 𝑌 ) )