Metamath Proof Explorer


Theorem 4atexlemunv

Description: Lemma for 4atexlem7 . (Contributed by NM, 21-Nov-2012)

Ref Expression
Hypotheses 4thatlem.ph ( 𝜑 ↔ ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑆𝐴 ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ∧ ( 𝑃 𝑅 ) = ( 𝑄 𝑅 ) ) ∧ ( 𝑇𝐴 ∧ ( 𝑈 𝑇 ) = ( 𝑉 𝑇 ) ) ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) ) )
4thatlem0.l = ( le ‘ 𝐾 )
4thatlem0.j = ( join ‘ 𝐾 )
4thatlem0.m = ( meet ‘ 𝐾 )
4thatlem0.a 𝐴 = ( Atoms ‘ 𝐾 )
4thatlem0.h 𝐻 = ( LHyp ‘ 𝐾 )
4thatlem0.u 𝑈 = ( ( 𝑃 𝑄 ) 𝑊 )
4thatlem0.v 𝑉 = ( ( 𝑃 𝑆 ) 𝑊 )
Assertion 4atexlemunv ( 𝜑𝑈𝑉 )

Proof

Step Hyp Ref Expression
1 4thatlem.ph ( 𝜑 ↔ ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑆𝐴 ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ∧ ( 𝑃 𝑅 ) = ( 𝑄 𝑅 ) ) ∧ ( 𝑇𝐴 ∧ ( 𝑈 𝑇 ) = ( 𝑉 𝑇 ) ) ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) ) )
2 4thatlem0.l = ( le ‘ 𝐾 )
3 4thatlem0.j = ( join ‘ 𝐾 )
4 4thatlem0.m = ( meet ‘ 𝐾 )
5 4thatlem0.a 𝐴 = ( Atoms ‘ 𝐾 )
6 4thatlem0.h 𝐻 = ( LHyp ‘ 𝐾 )
7 4thatlem0.u 𝑈 = ( ( 𝑃 𝑄 ) 𝑊 )
8 4thatlem0.v 𝑉 = ( ( 𝑃 𝑆 ) 𝑊 )
9 1 4atexlemnslpq ( 𝜑 → ¬ 𝑆 ( 𝑃 𝑄 ) )
10 1 4atexlemk ( 𝜑𝐾 ∈ HL )
11 1 4atexlemp ( 𝜑𝑃𝐴 )
12 1 4atexlems ( 𝜑𝑆𝐴 )
13 2 3 5 hlatlej2 ( ( 𝐾 ∈ HL ∧ 𝑃𝐴𝑆𝐴 ) → 𝑆 ( 𝑃 𝑆 ) )
14 10 11 12 13 syl3anc ( 𝜑𝑆 ( 𝑃 𝑆 ) )
15 14 adantr ( ( 𝜑𝑈 = 𝑉 ) → 𝑆 ( 𝑃 𝑆 ) )
16 1 4atexlemkl ( 𝜑𝐾 ∈ Lat )
17 1 3 5 4atexlempsb ( 𝜑 → ( 𝑃 𝑆 ) ∈ ( Base ‘ 𝐾 ) )
18 1 6 4atexlemwb ( 𝜑𝑊 ∈ ( Base ‘ 𝐾 ) )
19 eqid ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
20 19 2 4 latmle1 ( ( 𝐾 ∈ Lat ∧ ( 𝑃 𝑆 ) ∈ ( Base ‘ 𝐾 ) ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) → ( ( 𝑃 𝑆 ) 𝑊 ) ( 𝑃 𝑆 ) )
21 16 17 18 20 syl3anc ( 𝜑 → ( ( 𝑃 𝑆 ) 𝑊 ) ( 𝑃 𝑆 ) )
22 8 21 eqbrtrid ( 𝜑𝑉 ( 𝑃 𝑆 ) )
23 1 4atexlemkc ( 𝜑𝐾 ∈ CvLat )
24 1 2 3 4 5 6 7 8 4atexlemv ( 𝜑𝑉𝐴 )
25 19 2 4 latmle2 ( ( 𝐾 ∈ Lat ∧ ( 𝑃 𝑆 ) ∈ ( Base ‘ 𝐾 ) ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) → ( ( 𝑃 𝑆 ) 𝑊 ) 𝑊 )
26 16 17 18 25 syl3anc ( 𝜑 → ( ( 𝑃 𝑆 ) 𝑊 ) 𝑊 )
27 8 26 eqbrtrid ( 𝜑𝑉 𝑊 )
28 1 4atexlempw ( 𝜑 → ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) )
29 28 simprd ( 𝜑 → ¬ 𝑃 𝑊 )
30 nbrne2 ( ( 𝑉 𝑊 ∧ ¬ 𝑃 𝑊 ) → 𝑉𝑃 )
31 27 29 30 syl2anc ( 𝜑𝑉𝑃 )
32 2 3 5 cvlatexchb1 ( ( 𝐾 ∈ CvLat ∧ ( 𝑉𝐴𝑆𝐴𝑃𝐴 ) ∧ 𝑉𝑃 ) → ( 𝑉 ( 𝑃 𝑆 ) ↔ ( 𝑃 𝑉 ) = ( 𝑃 𝑆 ) ) )
33 23 24 12 11 31 32 syl131anc ( 𝜑 → ( 𝑉 ( 𝑃 𝑆 ) ↔ ( 𝑃 𝑉 ) = ( 𝑃 𝑆 ) ) )
34 22 33 mpbid ( 𝜑 → ( 𝑃 𝑉 ) = ( 𝑃 𝑆 ) )
35 34 adantr ( ( 𝜑𝑈 = 𝑉 ) → ( 𝑃 𝑉 ) = ( 𝑃 𝑆 ) )
36 oveq2 ( 𝑈 = 𝑉 → ( 𝑃 𝑈 ) = ( 𝑃 𝑉 ) )
37 36 eqcomd ( 𝑈 = 𝑉 → ( 𝑃 𝑉 ) = ( 𝑃 𝑈 ) )
38 1 4atexlemq ( 𝜑𝑄𝐴 )
39 19 3 5 hlatjcl ( ( 𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴 ) → ( 𝑃 𝑄 ) ∈ ( Base ‘ 𝐾 ) )
40 10 11 38 39 syl3anc ( 𝜑 → ( 𝑃 𝑄 ) ∈ ( Base ‘ 𝐾 ) )
41 19 2 4 latmle1 ( ( 𝐾 ∈ Lat ∧ ( 𝑃 𝑄 ) ∈ ( Base ‘ 𝐾 ) ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) → ( ( 𝑃 𝑄 ) 𝑊 ) ( 𝑃 𝑄 ) )
42 16 40 18 41 syl3anc ( 𝜑 → ( ( 𝑃 𝑄 ) 𝑊 ) ( 𝑃 𝑄 ) )
43 7 42 eqbrtrid ( 𝜑𝑈 ( 𝑃 𝑄 ) )
44 1 2 3 4 5 6 7 4atexlemu ( 𝜑𝑈𝐴 )
45 19 2 4 latmle2 ( ( 𝐾 ∈ Lat ∧ ( 𝑃 𝑄 ) ∈ ( Base ‘ 𝐾 ) ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) → ( ( 𝑃 𝑄 ) 𝑊 ) 𝑊 )
46 16 40 18 45 syl3anc ( 𝜑 → ( ( 𝑃 𝑄 ) 𝑊 ) 𝑊 )
47 7 46 eqbrtrid ( 𝜑𝑈 𝑊 )
48 nbrne2 ( ( 𝑈 𝑊 ∧ ¬ 𝑃 𝑊 ) → 𝑈𝑃 )
49 47 29 48 syl2anc ( 𝜑𝑈𝑃 )
50 2 3 5 cvlatexchb1 ( ( 𝐾 ∈ CvLat ∧ ( 𝑈𝐴𝑄𝐴𝑃𝐴 ) ∧ 𝑈𝑃 ) → ( 𝑈 ( 𝑃 𝑄 ) ↔ ( 𝑃 𝑈 ) = ( 𝑃 𝑄 ) ) )
51 23 44 38 11 49 50 syl131anc ( 𝜑 → ( 𝑈 ( 𝑃 𝑄 ) ↔ ( 𝑃 𝑈 ) = ( 𝑃 𝑄 ) ) )
52 43 51 mpbid ( 𝜑 → ( 𝑃 𝑈 ) = ( 𝑃 𝑄 ) )
53 37 52 sylan9eqr ( ( 𝜑𝑈 = 𝑉 ) → ( 𝑃 𝑉 ) = ( 𝑃 𝑄 ) )
54 35 53 eqtr3d ( ( 𝜑𝑈 = 𝑉 ) → ( 𝑃 𝑆 ) = ( 𝑃 𝑄 ) )
55 15 54 breqtrd ( ( 𝜑𝑈 = 𝑉 ) → 𝑆 ( 𝑃 𝑄 ) )
56 55 ex ( 𝜑 → ( 𝑈 = 𝑉𝑆 ( 𝑃 𝑄 ) ) )
57 56 necon3bd ( 𝜑 → ( ¬ 𝑆 ( 𝑃 𝑄 ) → 𝑈𝑉 ) )
58 9 57 mpd ( 𝜑𝑈𝑉 )