Metamath Proof Explorer

Theorem dihjatcclem2

Description: Lemma for isomorphism H of lattice join of two atoms not under the fiducial hyperplane. (Contributed by NM, 26-Sep-2014)

Ref Expression
Hypotheses dihjatcclem.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
dihjatcclem.l ⊢ ≤ = ( le ‘ 𝐾 )
dihjatcclem.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
dihjatcclem.j ⊢ ∨ = ( join ‘ 𝐾 )
dihjatcclem.m ⊢ ∧ = ( meet ‘ 𝐾 )
dihjatcclem.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
dihjatcclem.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
dihjatcclem.s ⊢ ⊕ = ( LSSum ‘ 𝑈 )
dihjatcclem.i ⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
dihjatcclem.v ⊢ 𝑉 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 )
dihjatcclem.k ⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
dihjatcclem.p ⊢ ( 𝜑 → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) )
dihjatcclem.q ⊢ ( 𝜑 → ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) )
dihjatcclem2.c ⊢ ( 𝜑 → ( 𝐼 ‘ 𝑉 ) ⊆ ( ( 𝐼 ‘ 𝑃 ) ⊕ ( 𝐼 ‘ 𝑄 ) ) )
Assertion dihjatcclem2 ( 𝜑 → ( 𝐼 ‘ ( 𝑃 ∨ 𝑄 ) ) = ( ( 𝐼 ‘ 𝑃 ) ⊕ ( 𝐼 ‘ 𝑄 ) ) )

Proof

Step Hyp Ref Expression
1 dihjatcclem.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 dihjatcclem.l ⊢ ≤ = ( le ‘ 𝐾 )
3 dihjatcclem.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
4 dihjatcclem.j ⊢ ∨ = ( join ‘ 𝐾 )
5 dihjatcclem.m ⊢ ∧ = ( meet ‘ 𝐾 )
6 dihjatcclem.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
7 dihjatcclem.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
8 dihjatcclem.s ⊢ ⊕ = ( LSSum ‘ 𝑈 )
9 dihjatcclem.i ⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
10 dihjatcclem.v ⊢ 𝑉 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 )
11 dihjatcclem.k ⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
12 dihjatcclem.p ⊢ ( 𝜑 → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) )
13 dihjatcclem.q ⊢ ( 𝜑 → ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) )
14 dihjatcclem2.c ⊢ ( 𝜑 → ( 𝐼 ‘ 𝑉 ) ⊆ ( ( 𝐼 ‘ 𝑃 ) ⊕ ( 𝐼 ‘ 𝑄 ) ) )
15 1 2 3 4 5 6 7 8 9 10 11 12 13 dihjatcclem1 ⊢ ( 𝜑 → ( 𝐼 ‘ ( 𝑃 ∨ 𝑄 ) ) = ( ( ( 𝐼 ‘ 𝑃 ) ⊕ ( 𝐼 ‘ 𝑄 ) ) ⊕ ( 𝐼 ‘ 𝑉 ) ) )
16 3 7 11 dvhlmod ⊢ ( 𝜑 → 𝑈 ∈ LMod )
17 eqid ⊢ ( LSubSp ‘ 𝑈 ) = ( LSubSp ‘ 𝑈 )
18 17 lsssssubg ⊢ ( 𝑈 ∈ LMod → ( LSubSp ‘ 𝑈 ) ⊆ ( SubGrp ‘ 𝑈 ) )
19 16 18 syl ⊢ ( 𝜑 → ( LSubSp ‘ 𝑈 ) ⊆ ( SubGrp ‘ 𝑈 ) )
20 12 simpld ⊢ ( 𝜑 → 𝑃 ∈ 𝐴 )
21 1 6 atbase ⊢ ( 𝑃 ∈ 𝐴 → 𝑃 ∈ 𝐵 )
22 20 21 syl ⊢ ( 𝜑 → 𝑃 ∈ 𝐵 )
23 1 3 9 7 17 dihlss ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ∈ 𝐵 ) → ( 𝐼 ‘ 𝑃 ) ∈ ( LSubSp ‘ 𝑈 ) )
24 11 22 23 syl2anc ⊢ ( 𝜑 → ( 𝐼 ‘ 𝑃 ) ∈ ( LSubSp ‘ 𝑈 ) )
25 13 simpld ⊢ ( 𝜑 → 𝑄 ∈ 𝐴 )
26 1 6 atbase ⊢ ( 𝑄 ∈ 𝐴 → 𝑄 ∈ 𝐵 )
27 25 26 syl ⊢ ( 𝜑 → 𝑄 ∈ 𝐵 )
28 1 3 9 7 17 dihlss ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑄 ∈ 𝐵 ) → ( 𝐼 ‘ 𝑄 ) ∈ ( LSubSp ‘ 𝑈 ) )
29 11 27 28 syl2anc ⊢ ( 𝜑 → ( 𝐼 ‘ 𝑄 ) ∈ ( LSubSp ‘ 𝑈 ) )
30 17 8 lsmcl ⊢ ( ( 𝑈 ∈ LMod ∧ ( 𝐼 ‘ 𝑃 ) ∈ ( LSubSp ‘ 𝑈 ) ∧ ( 𝐼 ‘ 𝑄 ) ∈ ( LSubSp ‘ 𝑈 ) ) → ( ( 𝐼 ‘ 𝑃 ) ⊕ ( 𝐼 ‘ 𝑄 ) ) ∈ ( LSubSp ‘ 𝑈 ) )
31 16 24 29 30 syl3anc ⊢ ( 𝜑 → ( ( 𝐼 ‘ 𝑃 ) ⊕ ( 𝐼 ‘ 𝑄 ) ) ∈ ( LSubSp ‘ 𝑈 ) )
32 19 31 sseldd ⊢ ( 𝜑 → ( ( 𝐼 ‘ 𝑃 ) ⊕ ( 𝐼 ‘ 𝑄 ) ) ∈ ( SubGrp ‘ 𝑈 ) )
33 10 fveq2i ⊢ ( 𝐼 ‘ 𝑉 ) = ( 𝐼 ‘ ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) )
34 11 simpld ⊢ ( 𝜑 → 𝐾 ∈ HL )
35 34 hllatd ⊢ ( 𝜑 → 𝐾 ∈ Lat )
36 1 4 6 hlatjcl ⊢ ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ( 𝑃 ∨ 𝑄 ) ∈ 𝐵 )
37 34 20 25 36 syl3anc ⊢ ( 𝜑 → ( 𝑃 ∨ 𝑄 ) ∈ 𝐵 )
38 11 simprd ⊢ ( 𝜑 → 𝑊 ∈ 𝐻 )
39 1 3 lhpbase ⊢ ( 𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵 )
40 38 39 syl ⊢ ( 𝜑 → 𝑊 ∈ 𝐵 )
41 1 5 latmcl ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑃 ∨ 𝑄 ) ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) → ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ∈ 𝐵 )
42 35 37 40 41 syl3anc ⊢ ( 𝜑 → ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ∈ 𝐵 )
43 1 3 9 7 17 dihlss ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ∈ 𝐵 ) → ( 𝐼 ‘ ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ) ∈ ( LSubSp ‘ 𝑈 ) )
44 11 42 43 syl2anc ⊢ ( 𝜑 → ( 𝐼 ‘ ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ) ∈ ( LSubSp ‘ 𝑈 ) )
45 33 44 eqeltrid ⊢ ( 𝜑 → ( 𝐼 ‘ 𝑉 ) ∈ ( LSubSp ‘ 𝑈 ) )
46 19 45 sseldd ⊢ ( 𝜑 → ( 𝐼 ‘ 𝑉 ) ∈ ( SubGrp ‘ 𝑈 ) )
47 8 lsmss2 ⊢ ( ( ( ( 𝐼 ‘ 𝑃 ) ⊕ ( 𝐼 ‘ 𝑄 ) ) ∈ ( SubGrp ‘ 𝑈 ) ∧ ( 𝐼 ‘ 𝑉 ) ∈ ( SubGrp ‘ 𝑈 ) ∧ ( 𝐼 ‘ 𝑉 ) ⊆ ( ( 𝐼 ‘ 𝑃 ) ⊕ ( 𝐼 ‘ 𝑄 ) ) ) → ( ( ( 𝐼 ‘ 𝑃 ) ⊕ ( 𝐼 ‘ 𝑄 ) ) ⊕ ( 𝐼 ‘ 𝑉 ) ) = ( ( 𝐼 ‘ 𝑃 ) ⊕ ( 𝐼 ‘ 𝑄 ) ) )
48 32 46 14 47 syl3anc ⊢ ( 𝜑 → ( ( ( 𝐼 ‘ 𝑃 ) ⊕ ( 𝐼 ‘ 𝑄 ) ) ⊕ ( 𝐼 ‘ 𝑉 ) ) = ( ( 𝐼 ‘ 𝑃 ) ⊕ ( 𝐼 ‘ 𝑄 ) ) )
49 15 48 eqtrd ⊢ ( 𝜑 → ( 𝐼 ‘ ( 𝑃 ∨ 𝑄 ) ) = ( ( 𝐼 ‘ 𝑃 ) ⊕ ( 𝐼 ‘ 𝑄 ) ) )