Metamath Proof Explorer


Theorem dihjatcclem3

Description: Lemma for dihjatcc . (Contributed by NM, 28-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 ∧ 𝑊𝐻 ) )
dihjatcclem.p ( 𝜑 → ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) )
dihjatcclem.q ( 𝜑 → ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) )
dihjatcc.w 𝐶 = ( ( oc ‘ 𝐾 ) ‘ 𝑊 )
dihjatcc.t 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
dihjatcc.r 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
dihjatcc.e 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
dihjatcc.g 𝐺 = ( 𝑑𝑇 ( 𝑑𝐶 ) = 𝑃 )
dihjatcc.dd 𝐷 = ( 𝑑𝑇 ( 𝑑𝐶 ) = 𝑄 )
Assertion dihjatcclem3 ( 𝜑 → ( 𝑅 ‘ ( 𝐺 𝐷 ) ) = 𝑉 )

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 ∧ 𝑊𝐻 ) )
12 dihjatcclem.p ( 𝜑 → ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) )
13 dihjatcclem.q ( 𝜑 → ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) )
14 dihjatcc.w 𝐶 = ( ( oc ‘ 𝐾 ) ‘ 𝑊 )
15 dihjatcc.t 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
16 dihjatcc.r 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
17 dihjatcc.e 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
18 dihjatcc.g 𝐺 = ( 𝑑𝑇 ( 𝑑𝐶 ) = 𝑃 )
19 dihjatcc.dd 𝐷 = ( 𝑑𝑇 ( 𝑑𝐶 ) = 𝑄 )
20 2 6 3 14 lhpocnel2 ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) → ( 𝐶𝐴 ∧ ¬ 𝐶 𝑊 ) )
21 11 20 syl ( 𝜑 → ( 𝐶𝐴 ∧ ¬ 𝐶 𝑊 ) )
22 2 6 3 15 18 ltrniotacl ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐶𝐴 ∧ ¬ 𝐶 𝑊 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) → 𝐺𝑇 )
23 11 21 12 22 syl3anc ( 𝜑𝐺𝑇 )
24 2 6 3 15 19 ltrniotacl ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐶𝐴 ∧ ¬ 𝐶 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) → 𝐷𝑇 )
25 11 21 13 24 syl3anc ( 𝜑𝐷𝑇 )
26 3 15 ltrncnv ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝐷𝑇 ) → 𝐷𝑇 )
27 11 25 26 syl2anc ( 𝜑 𝐷𝑇 )
28 3 15 ltrnco ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝐺𝑇 𝐷𝑇 ) → ( 𝐺 𝐷 ) ∈ 𝑇 )
29 11 23 27 28 syl3anc ( 𝜑 → ( 𝐺 𝐷 ) ∈ 𝑇 )
30 2 4 5 6 3 15 16 trlval2 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐺 𝐷 ) ∈ 𝑇 ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) → ( 𝑅 ‘ ( 𝐺 𝐷 ) ) = ( ( 𝑄 ( ( 𝐺 𝐷 ) ‘ 𝑄 ) ) 𝑊 ) )
31 11 29 13 30 syl3anc ( 𝜑 → ( 𝑅 ‘ ( 𝐺 𝐷 ) ) = ( ( 𝑄 ( ( 𝐺 𝐷 ) ‘ 𝑄 ) ) 𝑊 ) )
32 13 simpld ( 𝜑𝑄𝐴 )
33 2 6 3 15 ltrncoval ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐺𝑇 𝐷𝑇 ) ∧ 𝑄𝐴 ) → ( ( 𝐺 𝐷 ) ‘ 𝑄 ) = ( 𝐺 ‘ ( 𝐷𝑄 ) ) )
34 11 23 27 32 33 syl121anc ( 𝜑 → ( ( 𝐺 𝐷 ) ‘ 𝑄 ) = ( 𝐺 ‘ ( 𝐷𝑄 ) ) )
35 2 6 3 15 19 ltrniotacnvval ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐶𝐴 ∧ ¬ 𝐶 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) → ( 𝐷𝑄 ) = 𝐶 )
36 11 21 13 35 syl3anc ( 𝜑 → ( 𝐷𝑄 ) = 𝐶 )
37 36 fveq2d ( 𝜑 → ( 𝐺 ‘ ( 𝐷𝑄 ) ) = ( 𝐺𝐶 ) )
38 2 6 3 15 18 ltrniotaval ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐶𝐴 ∧ ¬ 𝐶 𝑊 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) → ( 𝐺𝐶 ) = 𝑃 )
39 11 21 12 38 syl3anc ( 𝜑 → ( 𝐺𝐶 ) = 𝑃 )
40 37 39 eqtrd ( 𝜑 → ( 𝐺 ‘ ( 𝐷𝑄 ) ) = 𝑃 )
41 34 40 eqtrd ( 𝜑 → ( ( 𝐺 𝐷 ) ‘ 𝑄 ) = 𝑃 )
42 41 oveq2d ( 𝜑 → ( 𝑄 ( ( 𝐺 𝐷 ) ‘ 𝑄 ) ) = ( 𝑄 𝑃 ) )
43 11 simpld ( 𝜑𝐾 ∈ HL )
44 12 simpld ( 𝜑𝑃𝐴 )
45 4 6 hlatjcom ( ( 𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴 ) → ( 𝑃 𝑄 ) = ( 𝑄 𝑃 ) )
46 43 44 32 45 syl3anc ( 𝜑 → ( 𝑃 𝑄 ) = ( 𝑄 𝑃 ) )
47 42 46 eqtr4d ( 𝜑 → ( 𝑄 ( ( 𝐺 𝐷 ) ‘ 𝑄 ) ) = ( 𝑃 𝑄 ) )
48 47 oveq1d ( 𝜑 → ( ( 𝑄 ( ( 𝐺 𝐷 ) ‘ 𝑄 ) ) 𝑊 ) = ( ( 𝑃 𝑄 ) 𝑊 ) )
49 48 10 eqtr4di ( 𝜑 → ( ( 𝑄 ( ( 𝐺 𝐷 ) ‘ 𝑄 ) ) 𝑊 ) = 𝑉 )
50 31 49 eqtrd ( 𝜑 → ( 𝑅 ‘ ( 𝐺 𝐷 ) ) = 𝑉 )