Metamath Proof Explorer


Theorem cdleme42mgN

Description: Part of proof of Lemma E in Crawley p. 113. TODO: FIX COMMENT . f preserves join: f(r \/ s) = f(r) \/ s, p. 115 10th line from bottom. TODO: Use instead of cdleme42mN ? Combine with cdleme42mN ? (Contributed by NM, 20-Mar-2013) (New usage is discouraged.)

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 ) )
Assertion cdleme42mgN ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ) → ( 𝐹 ‘ ( 𝑅 𝑆 ) ) = ( ( 𝐹𝑅 ) ( 𝐹𝑆 ) ) )

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 simpl1l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ) → 𝐾 ∈ HL )
16 15 hllatd ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ) → 𝐾 ∈ Lat )
17 simprll ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ) → 𝑅𝐴 )
18 1 5 atbase ( 𝑅𝐴𝑅𝐵 )
19 17 18 syl ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ) → 𝑅𝐵 )
20 simprrl ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ) → 𝑆𝐴 )
21 1 5 atbase ( 𝑆𝐴𝑆𝐵 )
22 20 21 syl ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ) → 𝑆𝐵 )
23 16 19 22 3jca ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ) → ( 𝐾 ∈ Lat ∧ 𝑅𝐵𝑆𝐵 ) )
24 1 3 latjcl ( ( 𝐾 ∈ Lat ∧ 𝑅𝐵𝑆𝐵 ) → ( 𝑅 𝑆 ) ∈ 𝐵 )
25 14 cdleme31id ( ( ( 𝑅 𝑆 ) ∈ 𝐵𝑃 = 𝑄 ) → ( 𝐹 ‘ ( 𝑅 𝑆 ) ) = ( 𝑅 𝑆 ) )
26 24 25 sylan ( ( ( 𝐾 ∈ Lat ∧ 𝑅𝐵𝑆𝐵 ) ∧ 𝑃 = 𝑄 ) → ( 𝐹 ‘ ( 𝑅 𝑆 ) ) = ( 𝑅 𝑆 ) )
27 14 cdleme31id ( ( 𝑅𝐵𝑃 = 𝑄 ) → ( 𝐹𝑅 ) = 𝑅 )
28 27 3ad2antl2 ( ( ( 𝐾 ∈ Lat ∧ 𝑅𝐵𝑆𝐵 ) ∧ 𝑃 = 𝑄 ) → ( 𝐹𝑅 ) = 𝑅 )
29 14 cdleme31id ( ( 𝑆𝐵𝑃 = 𝑄 ) → ( 𝐹𝑆 ) = 𝑆 )
30 29 3ad2antl3 ( ( ( 𝐾 ∈ Lat ∧ 𝑅𝐵𝑆𝐵 ) ∧ 𝑃 = 𝑄 ) → ( 𝐹𝑆 ) = 𝑆 )
31 28 30 oveq12d ( ( ( 𝐾 ∈ Lat ∧ 𝑅𝐵𝑆𝐵 ) ∧ 𝑃 = 𝑄 ) → ( ( 𝐹𝑅 ) ( 𝐹𝑆 ) ) = ( 𝑅 𝑆 ) )
32 26 31 eqtr4d ( ( ( 𝐾 ∈ Lat ∧ 𝑅𝐵𝑆𝐵 ) ∧ 𝑃 = 𝑄 ) → ( 𝐹 ‘ ( 𝑅 𝑆 ) ) = ( ( 𝐹𝑅 ) ( 𝐹𝑆 ) ) )
33 23 32 sylan ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ) ∧ 𝑃 = 𝑄 ) → ( 𝐹 ‘ ( 𝑅 𝑆 ) ) = ( ( 𝐹𝑅 ) ( 𝐹𝑆 ) ) )
34 simpll ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ) ∧ 𝑃𝑄 ) → ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) )
35 simpr ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ) ∧ 𝑃𝑄 ) → 𝑃𝑄 )
36 simplrl ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ) ∧ 𝑃𝑄 ) → ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) )
37 simplrr ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ) ∧ 𝑃𝑄 ) → ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) )
38 1 2 3 4 5 6 7 8 9 10 11 12 13 14 cdleme42mN ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ) → ( 𝐹 ‘ ( 𝑅 𝑆 ) ) = ( ( 𝐹𝑅 ) ( 𝐹𝑆 ) ) )
39 34 35 36 37 38 syl13anc ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ) ∧ 𝑃𝑄 ) → ( 𝐹 ‘ ( 𝑅 𝑆 ) ) = ( ( 𝐹𝑅 ) ( 𝐹𝑆 ) ) )
40 33 39 pm2.61dane ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ) → ( 𝐹 ‘ ( 𝑅 𝑆 ) ) = ( ( 𝐹𝑅 ) ( 𝐹𝑆 ) ) )