Step |
Hyp |
Ref |
Expression |
1 |
|
dia2dimlem6.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
2 |
|
dia2dimlem6.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
3 |
|
dia2dimlem6.m |
⊢ ∧ = ( meet ‘ 𝐾 ) |
4 |
|
dia2dimlem6.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
5 |
|
dia2dimlem6.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
6 |
|
dia2dimlem6.t |
⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) |
7 |
|
dia2dimlem6.r |
⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) |
8 |
|
dia2dimlem6.y |
⊢ 𝑌 = ( ( DVecA ‘ 𝐾 ) ‘ 𝑊 ) |
9 |
|
dia2dimlem6.s |
⊢ 𝑆 = ( LSubSp ‘ 𝑌 ) |
10 |
|
dia2dimlem6.pl |
⊢ ⊕ = ( LSSum ‘ 𝑌 ) |
11 |
|
dia2dimlem6.n |
⊢ 𝑁 = ( LSpan ‘ 𝑌 ) |
12 |
|
dia2dimlem6.i |
⊢ 𝐼 = ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) |
13 |
|
dia2dimlem6.q |
⊢ 𝑄 = ( ( 𝑃 ∨ 𝑈 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∨ 𝑉 ) ) |
14 |
|
dia2dimlem6.k |
⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
15 |
|
dia2dimlem6.u |
⊢ ( 𝜑 → ( 𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊 ) ) |
16 |
|
dia2dimlem6.v |
⊢ ( 𝜑 → ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) |
17 |
|
dia2dimlem6.p |
⊢ ( 𝜑 → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) |
18 |
|
dia2dimlem6.f |
⊢ ( 𝜑 → ( 𝐹 ∈ 𝑇 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) ) |
19 |
|
dia2dimlem6.rf |
⊢ ( 𝜑 → ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑈 ∨ 𝑉 ) ) |
20 |
|
dia2dimlem6.uv |
⊢ ( 𝜑 → 𝑈 ≠ 𝑉 ) |
21 |
|
dia2dimlem6.ru |
⊢ ( 𝜑 → ( 𝑅 ‘ 𝐹 ) ≠ 𝑈 ) |
22 |
|
dia2dimlem6.rv |
⊢ ( 𝜑 → ( 𝑅 ‘ 𝐹 ) ≠ 𝑉 ) |
23 |
1 2 3 4 5 6 7 13 14 15 16 17 18 19 20 21
|
dia2dimlem1 |
⊢ ( 𝜑 → ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) |
24 |
18
|
simpld |
⊢ ( 𝜑 → 𝐹 ∈ 𝑇 ) |
25 |
1 4 5 6
|
ltrnel |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ( 𝐹 ‘ 𝑃 ) ∈ 𝐴 ∧ ¬ ( 𝐹 ‘ 𝑃 ) ≤ 𝑊 ) ) |
26 |
14 24 17 25
|
syl3anc |
⊢ ( 𝜑 → ( ( 𝐹 ‘ 𝑃 ) ∈ 𝐴 ∧ ¬ ( 𝐹 ‘ 𝑃 ) ≤ 𝑊 ) ) |
27 |
1 4 5 6
|
cdleme50ex |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ 𝐴 ∧ ¬ ( 𝐹 ‘ 𝑃 ) ≤ 𝑊 ) ) → ∃ 𝑑 ∈ 𝑇 ( 𝑑 ‘ 𝑄 ) = ( 𝐹 ‘ 𝑃 ) ) |
28 |
14 23 26 27
|
syl3anc |
⊢ ( 𝜑 → ∃ 𝑑 ∈ 𝑇 ( 𝑑 ‘ 𝑄 ) = ( 𝐹 ‘ 𝑃 ) ) |
29 |
1 4 5 6
|
cdleme50ex |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ∃ 𝑔 ∈ 𝑇 ( 𝑔 ‘ 𝑃 ) = 𝑄 ) |
30 |
14 17 23 29
|
syl3anc |
⊢ ( 𝜑 → ∃ 𝑔 ∈ 𝑇 ( 𝑔 ‘ 𝑃 ) = 𝑄 ) |
31 |
14
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ ( 𝑔 ∈ 𝑇 ∧ ( 𝑔 ‘ 𝑃 ) = 𝑄 ∧ 𝑑 ∈ 𝑇 ) ∧ ( 𝑑 ‘ 𝑄 ) = ( 𝐹 ‘ 𝑃 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
32 |
15
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ ( 𝑔 ∈ 𝑇 ∧ ( 𝑔 ‘ 𝑃 ) = 𝑄 ∧ 𝑑 ∈ 𝑇 ) ∧ ( 𝑑 ‘ 𝑄 ) = ( 𝐹 ‘ 𝑃 ) ) → ( 𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊 ) ) |
33 |
16
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ ( 𝑔 ∈ 𝑇 ∧ ( 𝑔 ‘ 𝑃 ) = 𝑄 ∧ 𝑑 ∈ 𝑇 ) ∧ ( 𝑑 ‘ 𝑄 ) = ( 𝐹 ‘ 𝑃 ) ) → ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) |
34 |
17
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ ( 𝑔 ∈ 𝑇 ∧ ( 𝑔 ‘ 𝑃 ) = 𝑄 ∧ 𝑑 ∈ 𝑇 ) ∧ ( 𝑑 ‘ 𝑄 ) = ( 𝐹 ‘ 𝑃 ) ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) |
35 |
18
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ ( 𝑔 ∈ 𝑇 ∧ ( 𝑔 ‘ 𝑃 ) = 𝑄 ∧ 𝑑 ∈ 𝑇 ) ∧ ( 𝑑 ‘ 𝑄 ) = ( 𝐹 ‘ 𝑃 ) ) → ( 𝐹 ∈ 𝑇 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) ) |
36 |
19
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ ( 𝑔 ∈ 𝑇 ∧ ( 𝑔 ‘ 𝑃 ) = 𝑄 ∧ 𝑑 ∈ 𝑇 ) ∧ ( 𝑑 ‘ 𝑄 ) = ( 𝐹 ‘ 𝑃 ) ) → ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑈 ∨ 𝑉 ) ) |
37 |
20
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ ( 𝑔 ∈ 𝑇 ∧ ( 𝑔 ‘ 𝑃 ) = 𝑄 ∧ 𝑑 ∈ 𝑇 ) ∧ ( 𝑑 ‘ 𝑄 ) = ( 𝐹 ‘ 𝑃 ) ) → 𝑈 ≠ 𝑉 ) |
38 |
21
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ ( 𝑔 ∈ 𝑇 ∧ ( 𝑔 ‘ 𝑃 ) = 𝑄 ∧ 𝑑 ∈ 𝑇 ) ∧ ( 𝑑 ‘ 𝑄 ) = ( 𝐹 ‘ 𝑃 ) ) → ( 𝑅 ‘ 𝐹 ) ≠ 𝑈 ) |
39 |
22
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ ( 𝑔 ∈ 𝑇 ∧ ( 𝑔 ‘ 𝑃 ) = 𝑄 ∧ 𝑑 ∈ 𝑇 ) ∧ ( 𝑑 ‘ 𝑄 ) = ( 𝐹 ‘ 𝑃 ) ) → ( 𝑅 ‘ 𝐹 ) ≠ 𝑉 ) |
40 |
|
simp21 |
⊢ ( ( 𝜑 ∧ ( 𝑔 ∈ 𝑇 ∧ ( 𝑔 ‘ 𝑃 ) = 𝑄 ∧ 𝑑 ∈ 𝑇 ) ∧ ( 𝑑 ‘ 𝑄 ) = ( 𝐹 ‘ 𝑃 ) ) → 𝑔 ∈ 𝑇 ) |
41 |
|
simp22 |
⊢ ( ( 𝜑 ∧ ( 𝑔 ∈ 𝑇 ∧ ( 𝑔 ‘ 𝑃 ) = 𝑄 ∧ 𝑑 ∈ 𝑇 ) ∧ ( 𝑑 ‘ 𝑄 ) = ( 𝐹 ‘ 𝑃 ) ) → ( 𝑔 ‘ 𝑃 ) = 𝑄 ) |
42 |
|
simp23 |
⊢ ( ( 𝜑 ∧ ( 𝑔 ∈ 𝑇 ∧ ( 𝑔 ‘ 𝑃 ) = 𝑄 ∧ 𝑑 ∈ 𝑇 ) ∧ ( 𝑑 ‘ 𝑄 ) = ( 𝐹 ‘ 𝑃 ) ) → 𝑑 ∈ 𝑇 ) |
43 |
|
simp3 |
⊢ ( ( 𝜑 ∧ ( 𝑔 ∈ 𝑇 ∧ ( 𝑔 ‘ 𝑃 ) = 𝑄 ∧ 𝑑 ∈ 𝑇 ) ∧ ( 𝑑 ‘ 𝑄 ) = ( 𝐹 ‘ 𝑃 ) ) → ( 𝑑 ‘ 𝑄 ) = ( 𝐹 ‘ 𝑃 ) ) |
44 |
1 2 3 4 5 6 7 8 9 10 11 12 13 31 32 33 34 35 36 37 38 39 40 41 42 43
|
dia2dimlem5 |
⊢ ( ( 𝜑 ∧ ( 𝑔 ∈ 𝑇 ∧ ( 𝑔 ‘ 𝑃 ) = 𝑄 ∧ 𝑑 ∈ 𝑇 ) ∧ ( 𝑑 ‘ 𝑄 ) = ( 𝐹 ‘ 𝑃 ) ) → 𝐹 ∈ ( ( 𝐼 ‘ 𝑈 ) ⊕ ( 𝐼 ‘ 𝑉 ) ) ) |
45 |
44
|
3exp |
⊢ ( 𝜑 → ( ( 𝑔 ∈ 𝑇 ∧ ( 𝑔 ‘ 𝑃 ) = 𝑄 ∧ 𝑑 ∈ 𝑇 ) → ( ( 𝑑 ‘ 𝑄 ) = ( 𝐹 ‘ 𝑃 ) → 𝐹 ∈ ( ( 𝐼 ‘ 𝑈 ) ⊕ ( 𝐼 ‘ 𝑉 ) ) ) ) ) |
46 |
45
|
3expd |
⊢ ( 𝜑 → ( 𝑔 ∈ 𝑇 → ( ( 𝑔 ‘ 𝑃 ) = 𝑄 → ( 𝑑 ∈ 𝑇 → ( ( 𝑑 ‘ 𝑄 ) = ( 𝐹 ‘ 𝑃 ) → 𝐹 ∈ ( ( 𝐼 ‘ 𝑈 ) ⊕ ( 𝐼 ‘ 𝑉 ) ) ) ) ) ) ) |
47 |
46
|
rexlimdv |
⊢ ( 𝜑 → ( ∃ 𝑔 ∈ 𝑇 ( 𝑔 ‘ 𝑃 ) = 𝑄 → ( 𝑑 ∈ 𝑇 → ( ( 𝑑 ‘ 𝑄 ) = ( 𝐹 ‘ 𝑃 ) → 𝐹 ∈ ( ( 𝐼 ‘ 𝑈 ) ⊕ ( 𝐼 ‘ 𝑉 ) ) ) ) ) ) |
48 |
30 47
|
mpd |
⊢ ( 𝜑 → ( 𝑑 ∈ 𝑇 → ( ( 𝑑 ‘ 𝑄 ) = ( 𝐹 ‘ 𝑃 ) → 𝐹 ∈ ( ( 𝐼 ‘ 𝑈 ) ⊕ ( 𝐼 ‘ 𝑉 ) ) ) ) ) |
49 |
48
|
rexlimdv |
⊢ ( 𝜑 → ( ∃ 𝑑 ∈ 𝑇 ( 𝑑 ‘ 𝑄 ) = ( 𝐹 ‘ 𝑃 ) → 𝐹 ∈ ( ( 𝐼 ‘ 𝑈 ) ⊕ ( 𝐼 ‘ 𝑉 ) ) ) ) |
50 |
28 49
|
mpd |
⊢ ( 𝜑 → 𝐹 ∈ ( ( 𝐼 ‘ 𝑈 ) ⊕ ( 𝐼 ‘ 𝑉 ) ) ) |