Metamath Proof Explorer

Theorem dihjat

Description: Isomorphism H of lattice join of two atoms. (Contributed by NM, 29-Sep-2014)

Ref Expression
Hypotheses dihjat.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
dihjat.j ⊢ ∨ = ( join ‘ 𝐾 )
dihjat.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
dihjat.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
dihjat.s ⊢ ⊕ = ( LSSum ‘ 𝑈 )
dihjat.i ⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
dihjat.k ⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
dihjat.p ⊢ ( 𝜑 → 𝑃 ∈ 𝐴 )
dihjat.q ⊢ ( 𝜑 → 𝑄 ∈ 𝐴 )
Assertion dihjat ( 𝜑 → ( 𝐼 ‘ ( 𝑃 ∨ 𝑄 ) ) = ( ( 𝐼 ‘ 𝑃 ) ⊕ ( 𝐼 ‘ 𝑄 ) ) )

Proof

Step Hyp Ref Expression
1 dihjat.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2 dihjat.j ⊢ ∨ = ( join ‘ 𝐾 )
3 dihjat.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
4 dihjat.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
5 dihjat.s ⊢ ⊕ = ( LSSum ‘ 𝑈 )
6 dihjat.i ⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
7 dihjat.k ⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
8 dihjat.p ⊢ ( 𝜑 → 𝑃 ∈ 𝐴 )
9 dihjat.q ⊢ ( 𝜑 → 𝑄 ∈ 𝐴 )
10 eqid ⊢ ( le ‘ 𝐾 ) = ( le ‘ 𝐾 )
11 7 adantr ⊢ ( ( 𝜑 ∧ ( 𝑃 ( le ‘ 𝐾 ) 𝑊 ∧ 𝑄 ( le ‘ 𝐾 ) 𝑊 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
12 8 adantr ⊢ ( ( 𝜑 ∧ ( 𝑃 ( le ‘ 𝐾 ) 𝑊 ∧ 𝑄 ( le ‘ 𝐾 ) 𝑊 ) ) → 𝑃 ∈ 𝐴 )
13 simprl ⊢ ( ( 𝜑 ∧ ( 𝑃 ( le ‘ 𝐾 ) 𝑊 ∧ 𝑄 ( le ‘ 𝐾 ) 𝑊 ) ) → 𝑃 ( le ‘ 𝐾 ) 𝑊 )
14 12 13 jca ⊢ ( ( 𝜑 ∧ ( 𝑃 ( le ‘ 𝐾 ) 𝑊 ∧ 𝑄 ( le ‘ 𝐾 ) 𝑊 ) ) → ( 𝑃 ∈ 𝐴 ∧ 𝑃 ( le ‘ 𝐾 ) 𝑊 ) )
15 9 adantr ⊢ ( ( 𝜑 ∧ ( 𝑃 ( le ‘ 𝐾 ) 𝑊 ∧ 𝑄 ( le ‘ 𝐾 ) 𝑊 ) ) → 𝑄 ∈ 𝐴 )
16 simprr ⊢ ( ( 𝜑 ∧ ( 𝑃 ( le ‘ 𝐾 ) 𝑊 ∧ 𝑄 ( le ‘ 𝐾 ) 𝑊 ) ) → 𝑄 ( le ‘ 𝐾 ) 𝑊 )
17 15 16 jca ⊢ ( ( 𝜑 ∧ ( 𝑃 ( le ‘ 𝐾 ) 𝑊 ∧ 𝑄 ( le ‘ 𝐾 ) 𝑊 ) ) → ( 𝑄 ∈ 𝐴 ∧ 𝑄 ( le ‘ 𝐾 ) 𝑊 ) )
18 10 1 2 3 4 5 6 11 14 17 dihjatb ⊢ ( ( 𝜑 ∧ ( 𝑃 ( le ‘ 𝐾 ) 𝑊 ∧ 𝑄 ( le ‘ 𝐾 ) 𝑊 ) ) → ( 𝐼 ‘ ( 𝑃 ∨ 𝑄 ) ) = ( ( 𝐼 ‘ 𝑃 ) ⊕ ( 𝐼 ‘ 𝑄 ) ) )
19 eqid ⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
20 7 adantr ⊢ ( ( 𝜑 ∧ ( 𝑃 ( le ‘ 𝐾 ) 𝑊 ∧ ¬ 𝑄 ( le ‘ 𝐾 ) 𝑊 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
21 19 3 atbase ⊢ ( 𝑃 ∈ 𝐴 → 𝑃 ∈ ( Base ‘ 𝐾 ) )
22 8 21 syl ⊢ ( 𝜑 → 𝑃 ∈ ( Base ‘ 𝐾 ) )
23 22 adantr ⊢ ( ( 𝜑 ∧ ( 𝑃 ( le ‘ 𝐾 ) 𝑊 ∧ ¬ 𝑄 ( le ‘ 𝐾 ) 𝑊 ) ) → 𝑃 ∈ ( Base ‘ 𝐾 ) )
24 simprl ⊢ ( ( 𝜑 ∧ ( 𝑃 ( le ‘ 𝐾 ) 𝑊 ∧ ¬ 𝑄 ( le ‘ 𝐾 ) 𝑊 ) ) → 𝑃 ( le ‘ 𝐾 ) 𝑊 )
25 23 24 jca ⊢ ( ( 𝜑 ∧ ( 𝑃 ( le ‘ 𝐾 ) 𝑊 ∧ ¬ 𝑄 ( le ‘ 𝐾 ) 𝑊 ) ) → ( 𝑃 ∈ ( Base ‘ 𝐾 ) ∧ 𝑃 ( le ‘ 𝐾 ) 𝑊 ) )
26 9 adantr ⊢ ( ( 𝜑 ∧ ( 𝑃 ( le ‘ 𝐾 ) 𝑊 ∧ ¬ 𝑄 ( le ‘ 𝐾 ) 𝑊 ) ) → 𝑄 ∈ 𝐴 )
27 simprr ⊢ ( ( 𝜑 ∧ ( 𝑃 ( le ‘ 𝐾 ) 𝑊 ∧ ¬ 𝑄 ( le ‘ 𝐾 ) 𝑊 ) ) → ¬ 𝑄 ( le ‘ 𝐾 ) 𝑊 )
28 26 27 jca ⊢ ( ( 𝜑 ∧ ( 𝑃 ( le ‘ 𝐾 ) 𝑊 ∧ ¬ 𝑄 ( le ‘ 𝐾 ) 𝑊 ) ) → ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ( le ‘ 𝐾 ) 𝑊 ) )
29 19 10 1 2 3 4 5 6 20 25 28 dihjatc ⊢ ( ( 𝜑 ∧ ( 𝑃 ( le ‘ 𝐾 ) 𝑊 ∧ ¬ 𝑄 ( le ‘ 𝐾 ) 𝑊 ) ) → ( 𝐼 ‘ ( 𝑃 ∨ 𝑄 ) ) = ( ( 𝐼 ‘ 𝑃 ) ⊕ ( 𝐼 ‘ 𝑄 ) ) )
30 7 adantr ⊢ ( ( 𝜑 ∧ ( ¬ 𝑃 ( le ‘ 𝐾 ) 𝑊 ∧ 𝑄 ( le ‘ 𝐾 ) 𝑊 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
31 19 3 atbase ⊢ ( 𝑄 ∈ 𝐴 → 𝑄 ∈ ( Base ‘ 𝐾 ) )
32 9 31 syl ⊢ ( 𝜑 → 𝑄 ∈ ( Base ‘ 𝐾 ) )
33 32 adantr ⊢ ( ( 𝜑 ∧ ( ¬ 𝑃 ( le ‘ 𝐾 ) 𝑊 ∧ 𝑄 ( le ‘ 𝐾 ) 𝑊 ) ) → 𝑄 ∈ ( Base ‘ 𝐾 ) )
34 simprr ⊢ ( ( 𝜑 ∧ ( ¬ 𝑃 ( le ‘ 𝐾 ) 𝑊 ∧ 𝑄 ( le ‘ 𝐾 ) 𝑊 ) ) → 𝑄 ( le ‘ 𝐾 ) 𝑊 )
35 33 34 jca ⊢ ( ( 𝜑 ∧ ( ¬ 𝑃 ( le ‘ 𝐾 ) 𝑊 ∧ 𝑄 ( le ‘ 𝐾 ) 𝑊 ) ) → ( 𝑄 ∈ ( Base ‘ 𝐾 ) ∧ 𝑄 ( le ‘ 𝐾 ) 𝑊 ) )
36 8 adantr ⊢ ( ( 𝜑 ∧ ( ¬ 𝑃 ( le ‘ 𝐾 ) 𝑊 ∧ 𝑄 ( le ‘ 𝐾 ) 𝑊 ) ) → 𝑃 ∈ 𝐴 )
37 simprl ⊢ ( ( 𝜑 ∧ ( ¬ 𝑃 ( le ‘ 𝐾 ) 𝑊 ∧ 𝑄 ( le ‘ 𝐾 ) 𝑊 ) ) → ¬ 𝑃 ( le ‘ 𝐾 ) 𝑊 )
38 36 37 jca ⊢ ( ( 𝜑 ∧ ( ¬ 𝑃 ( le ‘ 𝐾 ) 𝑊 ∧ 𝑄 ( le ‘ 𝐾 ) 𝑊 ) ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ( le ‘ 𝐾 ) 𝑊 ) )
39 19 10 1 2 3 4 5 6 30 35 38 dihjatc ⊢ ( ( 𝜑 ∧ ( ¬ 𝑃 ( le ‘ 𝐾 ) 𝑊 ∧ 𝑄 ( le ‘ 𝐾 ) 𝑊 ) ) → ( 𝐼 ‘ ( 𝑄 ∨ 𝑃 ) ) = ( ( 𝐼 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ 𝑃 ) ) )
40 7 simpld ⊢ ( 𝜑 → 𝐾 ∈ HL )
41 2 3 hlatjcom ⊢ ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ( 𝑃 ∨ 𝑄 ) = ( 𝑄 ∨ 𝑃 ) )
42 40 8 9 41 syl3anc ⊢ ( 𝜑 → ( 𝑃 ∨ 𝑄 ) = ( 𝑄 ∨ 𝑃 ) )
43 42 fveq2d ⊢ ( 𝜑 → ( 𝐼 ‘ ( 𝑃 ∨ 𝑄 ) ) = ( 𝐼 ‘ ( 𝑄 ∨ 𝑃 ) ) )
44 43 adantr ⊢ ( ( 𝜑 ∧ ( ¬ 𝑃 ( le ‘ 𝐾 ) 𝑊 ∧ 𝑄 ( le ‘ 𝐾 ) 𝑊 ) ) → ( 𝐼 ‘ ( 𝑃 ∨ 𝑄 ) ) = ( 𝐼 ‘ ( 𝑄 ∨ 𝑃 ) ) )
45 1 4 7 dvhlmod ⊢ ( 𝜑 → 𝑈 ∈ LMod )
46 lmodabl ⊢ ( 𝑈 ∈ LMod → 𝑈 ∈ Abel )
47 45 46 syl ⊢ ( 𝜑 → 𝑈 ∈ Abel )
48 eqid ⊢ ( LSubSp ‘ 𝑈 ) = ( LSubSp ‘ 𝑈 )
49 48 lsssssubg ⊢ ( 𝑈 ∈ LMod → ( LSubSp ‘ 𝑈 ) ⊆ ( SubGrp ‘ 𝑈 ) )
50 45 49 syl ⊢ ( 𝜑 → ( LSubSp ‘ 𝑈 ) ⊆ ( SubGrp ‘ 𝑈 ) )
51 19 1 6 4 48 dihlss ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ∈ ( Base ‘ 𝐾 ) ) → ( 𝐼 ‘ 𝑃 ) ∈ ( LSubSp ‘ 𝑈 ) )
52 7 22 51 syl2anc ⊢ ( 𝜑 → ( 𝐼 ‘ 𝑃 ) ∈ ( LSubSp ‘ 𝑈 ) )
53 50 52 sseldd ⊢ ( 𝜑 → ( 𝐼 ‘ 𝑃 ) ∈ ( SubGrp ‘ 𝑈 ) )
54 19 1 6 4 48 dihlss ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑄 ∈ ( Base ‘ 𝐾 ) ) → ( 𝐼 ‘ 𝑄 ) ∈ ( LSubSp ‘ 𝑈 ) )
55 7 32 54 syl2anc ⊢ ( 𝜑 → ( 𝐼 ‘ 𝑄 ) ∈ ( LSubSp ‘ 𝑈 ) )
56 50 55 sseldd ⊢ ( 𝜑 → ( 𝐼 ‘ 𝑄 ) ∈ ( SubGrp ‘ 𝑈 ) )
57 5 lsmcom ⊢ ( ( 𝑈 ∈ Abel ∧ ( 𝐼 ‘ 𝑃 ) ∈ ( SubGrp ‘ 𝑈 ) ∧ ( 𝐼 ‘ 𝑄 ) ∈ ( SubGrp ‘ 𝑈 ) ) → ( ( 𝐼 ‘ 𝑃 ) ⊕ ( 𝐼 ‘ 𝑄 ) ) = ( ( 𝐼 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ 𝑃 ) ) )
58 47 53 56 57 syl3anc ⊢ ( 𝜑 → ( ( 𝐼 ‘ 𝑃 ) ⊕ ( 𝐼 ‘ 𝑄 ) ) = ( ( 𝐼 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ 𝑃 ) ) )
59 58 adantr ⊢ ( ( 𝜑 ∧ ( ¬ 𝑃 ( le ‘ 𝐾 ) 𝑊 ∧ 𝑄 ( le ‘ 𝐾 ) 𝑊 ) ) → ( ( 𝐼 ‘ 𝑃 ) ⊕ ( 𝐼 ‘ 𝑄 ) ) = ( ( 𝐼 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ 𝑃 ) ) )
60 39 44 59 3eqtr4d ⊢ ( ( 𝜑 ∧ ( ¬ 𝑃 ( le ‘ 𝐾 ) 𝑊 ∧ 𝑄 ( le ‘ 𝐾 ) 𝑊 ) ) → ( 𝐼 ‘ ( 𝑃 ∨ 𝑄 ) ) = ( ( 𝐼 ‘ 𝑃 ) ⊕ ( 𝐼 ‘ 𝑄 ) ) )
61 7 adantr ⊢ ( ( 𝜑 ∧ ( ¬ 𝑃 ( le ‘ 𝐾 ) 𝑊 ∧ ¬ 𝑄 ( le ‘ 𝐾 ) 𝑊 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
62 8 adantr ⊢ ( ( 𝜑 ∧ ( ¬ 𝑃 ( le ‘ 𝐾 ) 𝑊 ∧ ¬ 𝑄 ( le ‘ 𝐾 ) 𝑊 ) ) → 𝑃 ∈ 𝐴 )
63 simprl ⊢ ( ( 𝜑 ∧ ( ¬ 𝑃 ( le ‘ 𝐾 ) 𝑊 ∧ ¬ 𝑄 ( le ‘ 𝐾 ) 𝑊 ) ) → ¬ 𝑃 ( le ‘ 𝐾 ) 𝑊 )
64 62 63 jca ⊢ ( ( 𝜑 ∧ ( ¬ 𝑃 ( le ‘ 𝐾 ) 𝑊 ∧ ¬ 𝑄 ( le ‘ 𝐾 ) 𝑊 ) ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ( le ‘ 𝐾 ) 𝑊 ) )
65 9 adantr ⊢ ( ( 𝜑 ∧ ( ¬ 𝑃 ( le ‘ 𝐾 ) 𝑊 ∧ ¬ 𝑄 ( le ‘ 𝐾 ) 𝑊 ) ) → 𝑄 ∈ 𝐴 )
66 simprr ⊢ ( ( 𝜑 ∧ ( ¬ 𝑃 ( le ‘ 𝐾 ) 𝑊 ∧ ¬ 𝑄 ( le ‘ 𝐾 ) 𝑊 ) ) → ¬ 𝑄 ( le ‘ 𝐾 ) 𝑊 )
67 65 66 jca ⊢ ( ( 𝜑 ∧ ( ¬ 𝑃 ( le ‘ 𝐾 ) 𝑊 ∧ ¬ 𝑄 ( le ‘ 𝐾 ) 𝑊 ) ) → ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ( le ‘ 𝐾 ) 𝑊 ) )
68 10 1 2 3 4 5 6 61 64 67 dihjatcc ⊢ ( ( 𝜑 ∧ ( ¬ 𝑃 ( le ‘ 𝐾 ) 𝑊 ∧ ¬ 𝑄 ( le ‘ 𝐾 ) 𝑊 ) ) → ( 𝐼 ‘ ( 𝑃 ∨ 𝑄 ) ) = ( ( 𝐼 ‘ 𝑃 ) ⊕ ( 𝐼 ‘ 𝑄 ) ) )
69 18 29 60 68 4casesdan ⊢ ( 𝜑 → ( 𝐼 ‘ ( 𝑃 ∨ 𝑄 ) ) = ( ( 𝐼 ‘ 𝑃 ) ⊕ ( 𝐼 ‘ 𝑄 ) ) )