Step |
Hyp |
Ref |
Expression |
1 |
|
cdleme40.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
cdleme40.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
3 |
|
cdleme40.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
4 |
|
cdleme40.m |
⊢ ∧ = ( meet ‘ 𝐾 ) |
5 |
|
cdleme40.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
6 |
|
cdleme40.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
7 |
|
cdleme40.u |
⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) |
8 |
|
cdleme40.e |
⊢ 𝐸 = ( ( 𝑡 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑡 ) ∧ 𝑊 ) ) ) |
9 |
|
cdleme40.g |
⊢ 𝐺 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐸 ∨ ( ( 𝑠 ∨ 𝑡 ) ∧ 𝑊 ) ) ) |
10 |
|
cdleme40.i |
⊢ 𝐼 = ( ℩ 𝑦 ∈ 𝐵 ∀ 𝑡 ∈ 𝐴 ( ( ¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑦 = 𝐺 ) ) |
11 |
|
cdleme40.n |
⊢ 𝑁 = if ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) , 𝐼 , 𝐷 ) |
12 |
|
cdleme40.d |
⊢ 𝐷 = ( ( 𝑠 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑠 ) ∧ 𝑊 ) ) ) |
13 |
|
cdleme40r.y |
⊢ 𝑌 = ( ( 𝑢 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑢 ) ∧ 𝑊 ) ) ) |
14 |
|
cdleme40r.t |
⊢ 𝑇 = ( ( 𝑣 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑣 ) ∧ 𝑊 ) ) ) |
15 |
|
cdleme40r.x |
⊢ 𝑋 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑇 ∨ ( ( 𝑢 ∨ 𝑣 ) ∧ 𝑊 ) ) ) |
16 |
|
cdleme40r.o |
⊢ 𝑂 = ( ℩ 𝑧 ∈ 𝐵 ∀ 𝑣 ∈ 𝐴 ( ( ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑧 = 𝑋 ) ) |
17 |
|
cdleme40r.v |
⊢ 𝑉 = if ( 𝑢 ≤ ( 𝑃 ∨ 𝑄 ) , 𝑂 , 𝑌 ) |
18 |
|
breq1 |
⊢ ( 𝑠 = 𝑢 → ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ↔ 𝑢 ≤ ( 𝑃 ∨ 𝑄 ) ) ) |
19 |
|
oveq1 |
⊢ ( 𝑠 = 𝑢 → ( 𝑠 ∨ 𝑡 ) = ( 𝑢 ∨ 𝑡 ) ) |
20 |
19
|
oveq1d |
⊢ ( 𝑠 = 𝑢 → ( ( 𝑠 ∨ 𝑡 ) ∧ 𝑊 ) = ( ( 𝑢 ∨ 𝑡 ) ∧ 𝑊 ) ) |
21 |
20
|
oveq2d |
⊢ ( 𝑠 = 𝑢 → ( 𝐸 ∨ ( ( 𝑠 ∨ 𝑡 ) ∧ 𝑊 ) ) = ( 𝐸 ∨ ( ( 𝑢 ∨ 𝑡 ) ∧ 𝑊 ) ) ) |
22 |
21
|
oveq2d |
⊢ ( 𝑠 = 𝑢 → ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐸 ∨ ( ( 𝑠 ∨ 𝑡 ) ∧ 𝑊 ) ) ) = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐸 ∨ ( ( 𝑢 ∨ 𝑡 ) ∧ 𝑊 ) ) ) ) |
23 |
9 22
|
syl5eq |
⊢ ( 𝑠 = 𝑢 → 𝐺 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐸 ∨ ( ( 𝑢 ∨ 𝑡 ) ∧ 𝑊 ) ) ) ) |
24 |
23
|
eqeq2d |
⊢ ( 𝑠 = 𝑢 → ( 𝑦 = 𝐺 ↔ 𝑦 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐸 ∨ ( ( 𝑢 ∨ 𝑡 ) ∧ 𝑊 ) ) ) ) ) |
25 |
24
|
imbi2d |
⊢ ( 𝑠 = 𝑢 → ( ( ( ¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑦 = 𝐺 ) ↔ ( ( ¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑦 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐸 ∨ ( ( 𝑢 ∨ 𝑡 ) ∧ 𝑊 ) ) ) ) ) ) |
26 |
25
|
ralbidv |
⊢ ( 𝑠 = 𝑢 → ( ∀ 𝑡 ∈ 𝐴 ( ( ¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑦 = 𝐺 ) ↔ ∀ 𝑡 ∈ 𝐴 ( ( ¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑦 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐸 ∨ ( ( 𝑢 ∨ 𝑡 ) ∧ 𝑊 ) ) ) ) ) ) |
27 |
26
|
riotabidv |
⊢ ( 𝑠 = 𝑢 → ( ℩ 𝑦 ∈ 𝐵 ∀ 𝑡 ∈ 𝐴 ( ( ¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑦 = 𝐺 ) ) = ( ℩ 𝑦 ∈ 𝐵 ∀ 𝑡 ∈ 𝐴 ( ( ¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑦 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐸 ∨ ( ( 𝑢 ∨ 𝑡 ) ∧ 𝑊 ) ) ) ) ) ) |
28 |
|
eqeq1 |
⊢ ( 𝑦 = 𝑧 → ( 𝑦 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐸 ∨ ( ( 𝑢 ∨ 𝑡 ) ∧ 𝑊 ) ) ) ↔ 𝑧 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐸 ∨ ( ( 𝑢 ∨ 𝑡 ) ∧ 𝑊 ) ) ) ) ) |
29 |
28
|
imbi2d |
⊢ ( 𝑦 = 𝑧 → ( ( ( ¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑦 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐸 ∨ ( ( 𝑢 ∨ 𝑡 ) ∧ 𝑊 ) ) ) ) ↔ ( ( ¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑧 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐸 ∨ ( ( 𝑢 ∨ 𝑡 ) ∧ 𝑊 ) ) ) ) ) ) |
30 |
29
|
ralbidv |
⊢ ( 𝑦 = 𝑧 → ( ∀ 𝑡 ∈ 𝐴 ( ( ¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑦 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐸 ∨ ( ( 𝑢 ∨ 𝑡 ) ∧ 𝑊 ) ) ) ) ↔ ∀ 𝑡 ∈ 𝐴 ( ( ¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑧 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐸 ∨ ( ( 𝑢 ∨ 𝑡 ) ∧ 𝑊 ) ) ) ) ) ) |
31 |
|
breq1 |
⊢ ( 𝑡 = 𝑣 → ( 𝑡 ≤ 𝑊 ↔ 𝑣 ≤ 𝑊 ) ) |
32 |
31
|
notbid |
⊢ ( 𝑡 = 𝑣 → ( ¬ 𝑡 ≤ 𝑊 ↔ ¬ 𝑣 ≤ 𝑊 ) ) |
33 |
|
breq1 |
⊢ ( 𝑡 = 𝑣 → ( 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ↔ 𝑣 ≤ ( 𝑃 ∨ 𝑄 ) ) ) |
34 |
33
|
notbid |
⊢ ( 𝑡 = 𝑣 → ( ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ↔ ¬ 𝑣 ≤ ( 𝑃 ∨ 𝑄 ) ) ) |
35 |
32 34
|
anbi12d |
⊢ ( 𝑡 = 𝑣 → ( ( ¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ↔ ( ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) |
36 |
|
oveq1 |
⊢ ( 𝑡 = 𝑣 → ( 𝑡 ∨ 𝑈 ) = ( 𝑣 ∨ 𝑈 ) ) |
37 |
|
oveq2 |
⊢ ( 𝑡 = 𝑣 → ( 𝑃 ∨ 𝑡 ) = ( 𝑃 ∨ 𝑣 ) ) |
38 |
37
|
oveq1d |
⊢ ( 𝑡 = 𝑣 → ( ( 𝑃 ∨ 𝑡 ) ∧ 𝑊 ) = ( ( 𝑃 ∨ 𝑣 ) ∧ 𝑊 ) ) |
39 |
38
|
oveq2d |
⊢ ( 𝑡 = 𝑣 → ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑡 ) ∧ 𝑊 ) ) = ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑣 ) ∧ 𝑊 ) ) ) |
40 |
36 39
|
oveq12d |
⊢ ( 𝑡 = 𝑣 → ( ( 𝑡 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑡 ) ∧ 𝑊 ) ) ) = ( ( 𝑣 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑣 ) ∧ 𝑊 ) ) ) ) |
41 |
40 8 14
|
3eqtr4g |
⊢ ( 𝑡 = 𝑣 → 𝐸 = 𝑇 ) |
42 |
|
oveq2 |
⊢ ( 𝑡 = 𝑣 → ( 𝑢 ∨ 𝑡 ) = ( 𝑢 ∨ 𝑣 ) ) |
43 |
42
|
oveq1d |
⊢ ( 𝑡 = 𝑣 → ( ( 𝑢 ∨ 𝑡 ) ∧ 𝑊 ) = ( ( 𝑢 ∨ 𝑣 ) ∧ 𝑊 ) ) |
44 |
41 43
|
oveq12d |
⊢ ( 𝑡 = 𝑣 → ( 𝐸 ∨ ( ( 𝑢 ∨ 𝑡 ) ∧ 𝑊 ) ) = ( 𝑇 ∨ ( ( 𝑢 ∨ 𝑣 ) ∧ 𝑊 ) ) ) |
45 |
44
|
oveq2d |
⊢ ( 𝑡 = 𝑣 → ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐸 ∨ ( ( 𝑢 ∨ 𝑡 ) ∧ 𝑊 ) ) ) = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑇 ∨ ( ( 𝑢 ∨ 𝑣 ) ∧ 𝑊 ) ) ) ) |
46 |
45 15
|
eqtr4di |
⊢ ( 𝑡 = 𝑣 → ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐸 ∨ ( ( 𝑢 ∨ 𝑡 ) ∧ 𝑊 ) ) ) = 𝑋 ) |
47 |
46
|
eqeq2d |
⊢ ( 𝑡 = 𝑣 → ( 𝑧 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐸 ∨ ( ( 𝑢 ∨ 𝑡 ) ∧ 𝑊 ) ) ) ↔ 𝑧 = 𝑋 ) ) |
48 |
35 47
|
imbi12d |
⊢ ( 𝑡 = 𝑣 → ( ( ( ¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑧 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐸 ∨ ( ( 𝑢 ∨ 𝑡 ) ∧ 𝑊 ) ) ) ) ↔ ( ( ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑧 = 𝑋 ) ) ) |
49 |
48
|
cbvralvw |
⊢ ( ∀ 𝑡 ∈ 𝐴 ( ( ¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑧 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐸 ∨ ( ( 𝑢 ∨ 𝑡 ) ∧ 𝑊 ) ) ) ) ↔ ∀ 𝑣 ∈ 𝐴 ( ( ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑧 = 𝑋 ) ) |
50 |
30 49
|
bitrdi |
⊢ ( 𝑦 = 𝑧 → ( ∀ 𝑡 ∈ 𝐴 ( ( ¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑦 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐸 ∨ ( ( 𝑢 ∨ 𝑡 ) ∧ 𝑊 ) ) ) ) ↔ ∀ 𝑣 ∈ 𝐴 ( ( ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑧 = 𝑋 ) ) ) |
51 |
50
|
cbvriotavw |
⊢ ( ℩ 𝑦 ∈ 𝐵 ∀ 𝑡 ∈ 𝐴 ( ( ¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑦 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐸 ∨ ( ( 𝑢 ∨ 𝑡 ) ∧ 𝑊 ) ) ) ) ) = ( ℩ 𝑧 ∈ 𝐵 ∀ 𝑣 ∈ 𝐴 ( ( ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑧 = 𝑋 ) ) |
52 |
27 51
|
eqtrdi |
⊢ ( 𝑠 = 𝑢 → ( ℩ 𝑦 ∈ 𝐵 ∀ 𝑡 ∈ 𝐴 ( ( ¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑦 = 𝐺 ) ) = ( ℩ 𝑧 ∈ 𝐵 ∀ 𝑣 ∈ 𝐴 ( ( ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑧 = 𝑋 ) ) ) |
53 |
52 10 16
|
3eqtr4g |
⊢ ( 𝑠 = 𝑢 → 𝐼 = 𝑂 ) |
54 |
|
oveq1 |
⊢ ( 𝑠 = 𝑢 → ( 𝑠 ∨ 𝑈 ) = ( 𝑢 ∨ 𝑈 ) ) |
55 |
|
oveq2 |
⊢ ( 𝑠 = 𝑢 → ( 𝑃 ∨ 𝑠 ) = ( 𝑃 ∨ 𝑢 ) ) |
56 |
55
|
oveq1d |
⊢ ( 𝑠 = 𝑢 → ( ( 𝑃 ∨ 𝑠 ) ∧ 𝑊 ) = ( ( 𝑃 ∨ 𝑢 ) ∧ 𝑊 ) ) |
57 |
56
|
oveq2d |
⊢ ( 𝑠 = 𝑢 → ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑠 ) ∧ 𝑊 ) ) = ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑢 ) ∧ 𝑊 ) ) ) |
58 |
54 57
|
oveq12d |
⊢ ( 𝑠 = 𝑢 → ( ( 𝑠 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑠 ) ∧ 𝑊 ) ) ) = ( ( 𝑢 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑢 ) ∧ 𝑊 ) ) ) ) |
59 |
58 12 13
|
3eqtr4g |
⊢ ( 𝑠 = 𝑢 → 𝐷 = 𝑌 ) |
60 |
18 53 59
|
ifbieq12d |
⊢ ( 𝑠 = 𝑢 → if ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) , 𝐼 , 𝐷 ) = if ( 𝑢 ≤ ( 𝑃 ∨ 𝑄 ) , 𝑂 , 𝑌 ) ) |
61 |
60 11 17
|
3eqtr4g |
⊢ ( 𝑠 = 𝑢 → 𝑁 = 𝑉 ) |
62 |
61
|
cbvcsbv |
⊢ ⦋ 𝑅 / 𝑠 ⦌ 𝑁 = ⦋ 𝑅 / 𝑢 ⦌ 𝑉 |
63 |
62
|
a1i |
⊢ ( 𝑅 ∈ 𝐴 → ⦋ 𝑅 / 𝑠 ⦌ 𝑁 = ⦋ 𝑅 / 𝑢 ⦌ 𝑉 ) |