Metamath Proof Explorer


Theorem cdlemeg49lebilem

Description: Part of proof of Lemma D in Crawley p. 113. TODO: fix comment. (Contributed by NM, 9-Apr-2013)

Ref Expression
Hypotheses cdlemef46g.b 𝐵 = ( Base ‘ 𝐾 )
cdlemef46g.l = ( le ‘ 𝐾 )
cdlemef46g.j = ( join ‘ 𝐾 )
cdlemef46g.m = ( meet ‘ 𝐾 )
cdlemef46g.a 𝐴 = ( Atoms ‘ 𝐾 )
cdlemef46g.h 𝐻 = ( LHyp ‘ 𝐾 )
cdlemef46g.u 𝑈 = ( ( 𝑃 𝑄 ) 𝑊 )
cdlemef46g.d 𝐷 = ( ( 𝑡 𝑈 ) ( 𝑄 ( ( 𝑃 𝑡 ) 𝑊 ) ) )
cdlemefs46g.e 𝐸 = ( ( 𝑃 𝑄 ) ( 𝐷 ( ( 𝑠 𝑡 ) 𝑊 ) ) )
cdlemef46g.f 𝐹 = ( 𝑥𝐵 ↦ if ( ( 𝑃𝑄 ∧ ¬ 𝑥 𝑊 ) , ( 𝑧𝐵𝑠𝐴 ( ( ¬ 𝑠 𝑊 ∧ ( 𝑠 ( 𝑥 𝑊 ) ) = 𝑥 ) → 𝑧 = ( if ( 𝑠 ( 𝑃 𝑄 ) , ( 𝑦𝐵𝑡𝐴 ( ( ¬ 𝑡 𝑊 ∧ ¬ 𝑡 ( 𝑃 𝑄 ) ) → 𝑦 = 𝐸 ) ) , 𝑠 / 𝑡 𝐷 ) ( 𝑥 𝑊 ) ) ) ) , 𝑥 ) )
cdlemef46.v 𝑉 = ( ( 𝑄 𝑃 ) 𝑊 )
cdlemef46.n 𝑁 = ( ( 𝑣 𝑉 ) ( 𝑃 ( ( 𝑄 𝑣 ) 𝑊 ) ) )
cdlemefs46.o 𝑂 = ( ( 𝑄 𝑃 ) ( 𝑁 ( ( 𝑢 𝑣 ) 𝑊 ) ) )
cdlemef46.g 𝐺 = ( 𝑎𝐵 ↦ if ( ( 𝑄𝑃 ∧ ¬ 𝑎 𝑊 ) , ( 𝑐𝐵𝑢𝐴 ( ( ¬ 𝑢 𝑊 ∧ ( 𝑢 ( 𝑎 𝑊 ) ) = 𝑎 ) → 𝑐 = ( if ( 𝑢 ( 𝑄 𝑃 ) , ( 𝑏𝐵𝑣𝐴 ( ( ¬ 𝑣 𝑊 ∧ ¬ 𝑣 ( 𝑄 𝑃 ) ) → 𝑏 = 𝑂 ) ) , 𝑢 / 𝑣 𝑁 ) ( 𝑎 𝑊 ) ) ) ) , 𝑎 ) )
Assertion cdlemeg49lebilem ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑋𝐵𝑌𝐵 ) ) → ( 𝑋 𝑌 ↔ ( 𝐹𝑋 ) ( 𝐹𝑌 ) ) )

Proof

Step Hyp Ref Expression
1 cdlemef46g.b 𝐵 = ( Base ‘ 𝐾 )
2 cdlemef46g.l = ( le ‘ 𝐾 )
3 cdlemef46g.j = ( join ‘ 𝐾 )
4 cdlemef46g.m = ( meet ‘ 𝐾 )
5 cdlemef46g.a 𝐴 = ( Atoms ‘ 𝐾 )
6 cdlemef46g.h 𝐻 = ( LHyp ‘ 𝐾 )
7 cdlemef46g.u 𝑈 = ( ( 𝑃 𝑄 ) 𝑊 )
8 cdlemef46g.d 𝐷 = ( ( 𝑡 𝑈 ) ( 𝑄 ( ( 𝑃 𝑡 ) 𝑊 ) ) )
9 cdlemefs46g.e 𝐸 = ( ( 𝑃 𝑄 ) ( 𝐷 ( ( 𝑠 𝑡 ) 𝑊 ) ) )
10 cdlemef46g.f 𝐹 = ( 𝑥𝐵 ↦ if ( ( 𝑃𝑄 ∧ ¬ 𝑥 𝑊 ) , ( 𝑧𝐵𝑠𝐴 ( ( ¬ 𝑠 𝑊 ∧ ( 𝑠 ( 𝑥 𝑊 ) ) = 𝑥 ) → 𝑧 = ( if ( 𝑠 ( 𝑃 𝑄 ) , ( 𝑦𝐵𝑡𝐴 ( ( ¬ 𝑡 𝑊 ∧ ¬ 𝑡 ( 𝑃 𝑄 ) ) → 𝑦 = 𝐸 ) ) , 𝑠 / 𝑡 𝐷 ) ( 𝑥 𝑊 ) ) ) ) , 𝑥 ) )
11 cdlemef46.v 𝑉 = ( ( 𝑄 𝑃 ) 𝑊 )
12 cdlemef46.n 𝑁 = ( ( 𝑣 𝑉 ) ( 𝑃 ( ( 𝑄 𝑣 ) 𝑊 ) ) )
13 cdlemefs46.o 𝑂 = ( ( 𝑄 𝑃 ) ( 𝑁 ( ( 𝑢 𝑣 ) 𝑊 ) ) )
14 cdlemef46.g 𝐺 = ( 𝑎𝐵 ↦ if ( ( 𝑄𝑃 ∧ ¬ 𝑎 𝑊 ) , ( 𝑐𝐵𝑢𝐴 ( ( ¬ 𝑢 𝑊 ∧ ( 𝑢 ( 𝑎 𝑊 ) ) = 𝑎 ) → 𝑐 = ( if ( 𝑢 ( 𝑄 𝑃 ) , ( 𝑏𝐵𝑣𝐴 ( ( ¬ 𝑣 𝑊 ∧ ¬ 𝑣 ( 𝑄 𝑃 ) ) → 𝑏 = 𝑂 ) ) , 𝑢 / 𝑣 𝑁 ) ( 𝑎 𝑊 ) ) ) ) , 𝑎 ) )
15 vex 𝑠 ∈ V
16 eqid ( ( 𝑠 𝑈 ) ( 𝑄 ( ( 𝑃 𝑠 ) 𝑊 ) ) ) = ( ( 𝑠 𝑈 ) ( 𝑄 ( ( 𝑃 𝑠 ) 𝑊 ) ) )
17 8 16 cdleme31sc ( 𝑠 ∈ V → 𝑠 / 𝑡 𝐷 = ( ( 𝑠 𝑈 ) ( 𝑄 ( ( 𝑃 𝑠 ) 𝑊 ) ) ) )
18 15 17 ax-mp 𝑠 / 𝑡 𝐷 = ( ( 𝑠 𝑈 ) ( 𝑄 ( ( 𝑃 𝑠 ) 𝑊 ) ) )
19 eqid ( 𝑦𝐵𝑡𝐴 ( ( ¬ 𝑡 𝑊 ∧ ¬ 𝑡 ( 𝑃 𝑄 ) ) → 𝑦 = 𝐸 ) ) = ( 𝑦𝐵𝑡𝐴 ( ( ¬ 𝑡 𝑊 ∧ ¬ 𝑡 ( 𝑃 𝑄 ) ) → 𝑦 = 𝐸 ) )
20 eqid if ( 𝑠 ( 𝑃 𝑄 ) , ( 𝑦𝐵𝑡𝐴 ( ( ¬ 𝑡 𝑊 ∧ ¬ 𝑡 ( 𝑃 𝑄 ) ) → 𝑦 = 𝐸 ) ) , 𝑠 / 𝑡 𝐷 ) = if ( 𝑠 ( 𝑃 𝑄 ) , ( 𝑦𝐵𝑡𝐴 ( ( ¬ 𝑡 𝑊 ∧ ¬ 𝑡 ( 𝑃 𝑄 ) ) → 𝑦 = 𝐸 ) ) , 𝑠 / 𝑡 𝐷 )
21 eqid ( 𝑧𝐵𝑠𝐴 ( ( ¬ 𝑠 𝑊 ∧ ( 𝑠 ( 𝑥 𝑊 ) ) = 𝑥 ) → 𝑧 = ( if ( 𝑠 ( 𝑃 𝑄 ) , ( 𝑦𝐵𝑡𝐴 ( ( ¬ 𝑡 𝑊 ∧ ¬ 𝑡 ( 𝑃 𝑄 ) ) → 𝑦 = 𝐸 ) ) , 𝑠 / 𝑡 𝐷 ) ( 𝑥 𝑊 ) ) ) ) = ( 𝑧𝐵𝑠𝐴 ( ( ¬ 𝑠 𝑊 ∧ ( 𝑠 ( 𝑥 𝑊 ) ) = 𝑥 ) → 𝑧 = ( if ( 𝑠 ( 𝑃 𝑄 ) , ( 𝑦𝐵𝑡𝐴 ( ( ¬ 𝑡 𝑊 ∧ ¬ 𝑡 ( 𝑃 𝑄 ) ) → 𝑦 = 𝐸 ) ) , 𝑠 / 𝑡 𝐷 ) ( 𝑥 𝑊 ) ) ) )
22 1 2 3 4 5 6 7 18 8 9 19 20 21 10 cdleme32le ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑋𝐵𝑌𝐵 ) ∧ 𝑋 𝑌 ) → ( 𝐹𝑋 ) ( 𝐹𝑌 ) )
23 22 3expia ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑋𝐵𝑌𝐵 ) ) → ( 𝑋 𝑌 → ( 𝐹𝑋 ) ( 𝐹𝑌 ) ) )
24 simp1 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑋𝐵𝑌𝐵 ) ∧ ( 𝐹𝑋 ) ( 𝐹𝑌 ) ) → ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) )
25 simp2l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑋𝐵𝑌𝐵 ) ∧ ( 𝐹𝑋 ) ( 𝐹𝑌 ) ) → 𝑋𝐵 )
26 biid ( 𝑠 ( 𝑃 𝑄 ) ↔ 𝑠 ( 𝑃 𝑄 ) )
27 26 18 ifbieq2i if ( 𝑠 ( 𝑃 𝑄 ) , ( 𝑦𝐵𝑡𝐴 ( ( ¬ 𝑡 𝑊 ∧ ¬ 𝑡 ( 𝑃 𝑄 ) ) → 𝑦 = 𝐸 ) ) , 𝑠 / 𝑡 𝐷 ) = if ( 𝑠 ( 𝑃 𝑄 ) , ( 𝑦𝐵𝑡𝐴 ( ( ¬ 𝑡 𝑊 ∧ ¬ 𝑡 ( 𝑃 𝑄 ) ) → 𝑦 = 𝐸 ) ) , ( ( 𝑠 𝑈 ) ( 𝑄 ( ( 𝑃 𝑠 ) 𝑊 ) ) ) )
28 1 2 3 4 5 6 7 16 8 9 19 27 21 10 cdleme32fvcl ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ 𝑋𝐵 ) → ( 𝐹𝑋 ) ∈ 𝐵 )
29 24 25 28 syl2anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑋𝐵𝑌𝐵 ) ∧ ( 𝐹𝑋 ) ( 𝐹𝑌 ) ) → ( 𝐹𝑋 ) ∈ 𝐵 )
30 simp2r ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑋𝐵𝑌𝐵 ) ∧ ( 𝐹𝑋 ) ( 𝐹𝑌 ) ) → 𝑌𝐵 )
31 1 2 3 4 5 6 7 16 8 9 19 27 21 10 cdleme32fvcl ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ 𝑌𝐵 ) → ( 𝐹𝑌 ) ∈ 𝐵 )
32 24 30 31 syl2anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑋𝐵𝑌𝐵 ) ∧ ( 𝐹𝑋 ) ( 𝐹𝑌 ) ) → ( 𝐹𝑌 ) ∈ 𝐵 )
33 simp3 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑋𝐵𝑌𝐵 ) ∧ ( 𝐹𝑋 ) ( 𝐹𝑌 ) ) → ( 𝐹𝑋 ) ( 𝐹𝑌 ) )
34 1 2 3 4 5 6 11 12 13 14 cdlemeg49le ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝐹𝑋 ) ∈ 𝐵 ∧ ( 𝐹𝑌 ) ∈ 𝐵 ) ∧ ( 𝐹𝑋 ) ( 𝐹𝑌 ) ) → ( 𝐺 ‘ ( 𝐹𝑋 ) ) ( 𝐺 ‘ ( 𝐹𝑌 ) ) )
35 24 29 32 33 34 syl121anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑋𝐵𝑌𝐵 ) ∧ ( 𝐹𝑋 ) ( 𝐹𝑌 ) ) → ( 𝐺 ‘ ( 𝐹𝑋 ) ) ( 𝐺 ‘ ( 𝐹𝑌 ) ) )
36 1 2 3 4 5 6 7 8 9 10 11 12 13 14 cdleme48gfv ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ 𝑋𝐵 ) → ( 𝐺 ‘ ( 𝐹𝑋 ) ) = 𝑋 )
37 36 adantrr ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑋𝐵𝑌𝐵 ) ) → ( 𝐺 ‘ ( 𝐹𝑋 ) ) = 𝑋 )
38 1 2 3 4 5 6 7 8 9 10 11 12 13 14 cdleme48gfv ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ 𝑌𝐵 ) → ( 𝐺 ‘ ( 𝐹𝑌 ) ) = 𝑌 )
39 38 adantrl ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑋𝐵𝑌𝐵 ) ) → ( 𝐺 ‘ ( 𝐹𝑌 ) ) = 𝑌 )
40 37 39 breq12d ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑋𝐵𝑌𝐵 ) ) → ( ( 𝐺 ‘ ( 𝐹𝑋 ) ) ( 𝐺 ‘ ( 𝐹𝑌 ) ) ↔ 𝑋 𝑌 ) )
41 40 3adant3 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑋𝐵𝑌𝐵 ) ∧ ( 𝐹𝑋 ) ( 𝐹𝑌 ) ) → ( ( 𝐺 ‘ ( 𝐹𝑋 ) ) ( 𝐺 ‘ ( 𝐹𝑌 ) ) ↔ 𝑋 𝑌 ) )
42 35 41 mpbid ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑋𝐵𝑌𝐵 ) ∧ ( 𝐹𝑋 ) ( 𝐹𝑌 ) ) → 𝑋 𝑌 )
43 42 3expia ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑋𝐵𝑌𝐵 ) ) → ( ( 𝐹𝑋 ) ( 𝐹𝑌 ) → 𝑋 𝑌 ) )
44 23 43 impbid ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑋𝐵𝑌𝐵 ) ) → ( 𝑋 𝑌 ↔ ( 𝐹𝑋 ) ( 𝐹𝑌 ) ) )