Metamath Proof Explorer


Theorem cdleme32f

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 = ( 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 cdleme32f ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑋𝐵𝑌𝐵 ) ∧ ¬ ( 𝑃𝑄 ∧ ¬ 𝑋 𝑊 ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑌 𝑊 ) ) ∧ 𝑋 𝑌 ) → ( 𝐹𝑋 ) ( 𝐹𝑌 ) )

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 simp11 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑋𝐵𝑌𝐵 ) ∧ ¬ ( 𝑃𝑄 ∧ ¬ 𝑋 𝑊 ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑌 𝑊 ) ) ∧ 𝑋 𝑌 ) → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
16 simp21r ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑋𝐵𝑌𝐵 ) ∧ ¬ ( 𝑃𝑄 ∧ ¬ 𝑋 𝑊 ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑌 𝑊 ) ) ∧ 𝑋 𝑌 ) → 𝑌𝐵 )
17 simp23r ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑋𝐵𝑌𝐵 ) ∧ ¬ ( 𝑃𝑄 ∧ ¬ 𝑋 𝑊 ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑌 𝑊 ) ) ∧ 𝑋 𝑌 ) → ¬ 𝑌 𝑊 )
18 1 2 3 4 5 6 lhpmcvr2 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) → ∃ 𝑠𝐴 ( ¬ 𝑠 𝑊 ∧ ( 𝑠 ( 𝑌 𝑊 ) ) = 𝑌 ) )
19 15 16 17 18 syl12anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑋𝐵𝑌𝐵 ) ∧ ¬ ( 𝑃𝑄 ∧ ¬ 𝑋 𝑊 ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑌 𝑊 ) ) ∧ 𝑋 𝑌 ) → ∃ 𝑠𝐴 ( ¬ 𝑠 𝑊 ∧ ( 𝑠 ( 𝑌 𝑊 ) ) = 𝑌 ) )
20 nfv 𝑠 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑋𝐵𝑌𝐵 ) ∧ ¬ ( 𝑃𝑄 ∧ ¬ 𝑋 𝑊 ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑌 𝑊 ) ) ∧ 𝑋 𝑌 )
21 nfcv 𝑠 𝐵
22 nfv 𝑠 ( 𝑃𝑄 ∧ ¬ 𝑥 𝑊 )
23 nfra1 𝑠𝑠𝐴 ( ( ¬ 𝑠 𝑊 ∧ ( 𝑠 ( 𝑥 𝑊 ) ) = 𝑥 ) → 𝑧 = ( 𝑁 ( 𝑥 𝑊 ) ) )
24 23 21 nfriota 𝑠 ( 𝑧𝐵𝑠𝐴 ( ( ¬ 𝑠 𝑊 ∧ ( 𝑠 ( 𝑥 𝑊 ) ) = 𝑥 ) → 𝑧 = ( 𝑁 ( 𝑥 𝑊 ) ) ) )
25 13 24 nfcxfr 𝑠 𝑂
26 nfcv 𝑠 𝑥
27 22 25 26 nfif 𝑠 if ( ( 𝑃𝑄 ∧ ¬ 𝑥 𝑊 ) , 𝑂 , 𝑥 )
28 21 27 nfmpt 𝑠 ( 𝑥𝐵 ↦ if ( ( 𝑃𝑄 ∧ ¬ 𝑥 𝑊 ) , 𝑂 , 𝑥 ) )
29 14 28 nfcxfr 𝑠 𝐹
30 nfcv 𝑠 𝑋
31 29 30 nffv 𝑠 ( 𝐹𝑋 )
32 nfcv 𝑠
33 nfcv 𝑠 𝑌
34 29 33 nffv 𝑠 ( 𝐹𝑌 )
35 31 32 34 nfbr 𝑠 ( 𝐹𝑋 ) ( 𝐹𝑌 )
36 simpl1 ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑋𝐵𝑌𝐵 ) ∧ ¬ ( 𝑃𝑄 ∧ ¬ 𝑋 𝑊 ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑌 𝑊 ) ) ∧ 𝑋 𝑌 ) ∧ ( 𝑠𝐴 ∧ ( ¬ 𝑠 𝑊 ∧ ( 𝑠 ( 𝑌 𝑊 ) ) = 𝑌 ) ) ) → ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) )
37 simpl2 ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑋𝐵𝑌𝐵 ) ∧ ¬ ( 𝑃𝑄 ∧ ¬ 𝑋 𝑊 ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑌 𝑊 ) ) ∧ 𝑋 𝑌 ) ∧ ( 𝑠𝐴 ∧ ( ¬ 𝑠 𝑊 ∧ ( 𝑠 ( 𝑌 𝑊 ) ) = 𝑌 ) ) ) → ( ( 𝑋𝐵𝑌𝐵 ) ∧ ¬ ( 𝑃𝑄 ∧ ¬ 𝑋 𝑊 ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑌 𝑊 ) ) )
38 simprl ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑋𝐵𝑌𝐵 ) ∧ ¬ ( 𝑃𝑄 ∧ ¬ 𝑋 𝑊 ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑌 𝑊 ) ) ∧ 𝑋 𝑌 ) ∧ ( 𝑠𝐴 ∧ ( ¬ 𝑠 𝑊 ∧ ( 𝑠 ( 𝑌 𝑊 ) ) = 𝑌 ) ) ) → 𝑠𝐴 )
39 simprrl ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑋𝐵𝑌𝐵 ) ∧ ¬ ( 𝑃𝑄 ∧ ¬ 𝑋 𝑊 ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑌 𝑊 ) ) ∧ 𝑋 𝑌 ) ∧ ( 𝑠𝐴 ∧ ( ¬ 𝑠 𝑊 ∧ ( 𝑠 ( 𝑌 𝑊 ) ) = 𝑌 ) ) ) → ¬ 𝑠 𝑊 )
40 38 39 jca ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑋𝐵𝑌𝐵 ) ∧ ¬ ( 𝑃𝑄 ∧ ¬ 𝑋 𝑊 ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑌 𝑊 ) ) ∧ 𝑋 𝑌 ) ∧ ( 𝑠𝐴 ∧ ( ¬ 𝑠 𝑊 ∧ ( 𝑠 ( 𝑌 𝑊 ) ) = 𝑌 ) ) ) → ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) )
41 simprrr ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑋𝐵𝑌𝐵 ) ∧ ¬ ( 𝑃𝑄 ∧ ¬ 𝑋 𝑊 ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑌 𝑊 ) ) ∧ 𝑋 𝑌 ) ∧ ( 𝑠𝐴 ∧ ( ¬ 𝑠 𝑊 ∧ ( 𝑠 ( 𝑌 𝑊 ) ) = 𝑌 ) ) ) → ( 𝑠 ( 𝑌 𝑊 ) ) = 𝑌 )
42 simpl3 ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑋𝐵𝑌𝐵 ) ∧ ¬ ( 𝑃𝑄 ∧ ¬ 𝑋 𝑊 ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑌 𝑊 ) ) ∧ 𝑋 𝑌 ) ∧ ( 𝑠𝐴 ∧ ( ¬ 𝑠 𝑊 ∧ ( 𝑠 ( 𝑌 𝑊 ) ) = 𝑌 ) ) ) → 𝑋 𝑌 )
43 1 2 3 4 5 6 7 8 9 10 11 12 13 14 cdleme32e ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑋𝐵𝑌𝐵 ) ∧ ¬ ( 𝑃𝑄 ∧ ¬ 𝑋 𝑊 ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑌 𝑊 ) ) ∧ ( ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ ( 𝑠 ( 𝑌 𝑊 ) ) = 𝑌𝑋 𝑌 ) ) → ( 𝐹𝑋 ) ( 𝐹𝑌 ) )
44 36 37 40 41 42 43 syl113anc ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑋𝐵𝑌𝐵 ) ∧ ¬ ( 𝑃𝑄 ∧ ¬ 𝑋 𝑊 ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑌 𝑊 ) ) ∧ 𝑋 𝑌 ) ∧ ( 𝑠𝐴 ∧ ( ¬ 𝑠 𝑊 ∧ ( 𝑠 ( 𝑌 𝑊 ) ) = 𝑌 ) ) ) → ( 𝐹𝑋 ) ( 𝐹𝑌 ) )
45 44 exp32 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑋𝐵𝑌𝐵 ) ∧ ¬ ( 𝑃𝑄 ∧ ¬ 𝑋 𝑊 ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑌 𝑊 ) ) ∧ 𝑋 𝑌 ) → ( 𝑠𝐴 → ( ( ¬ 𝑠 𝑊 ∧ ( 𝑠 ( 𝑌 𝑊 ) ) = 𝑌 ) → ( 𝐹𝑋 ) ( 𝐹𝑌 ) ) ) )
46 20 35 45 rexlimd ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑋𝐵𝑌𝐵 ) ∧ ¬ ( 𝑃𝑄 ∧ ¬ 𝑋 𝑊 ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑌 𝑊 ) ) ∧ 𝑋 𝑌 ) → ( ∃ 𝑠𝐴 ( ¬ 𝑠 𝑊 ∧ ( 𝑠 ( 𝑌 𝑊 ) ) = 𝑌 ) → ( 𝐹𝑋 ) ( 𝐹𝑌 ) ) )
47 19 46 mpd ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑋𝐵𝑌𝐵 ) ∧ ¬ ( 𝑃𝑄 ∧ ¬ 𝑋 𝑊 ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑌 𝑊 ) ) ∧ 𝑋 𝑌 ) → ( 𝐹𝑋 ) ( 𝐹𝑌 ) )