Proof of Theorem mulsasslem3
Step | Hyp | Ref
| Expression |
1 | | oveq1 7411 |
. . . . . . . . . . 11
⊢ (𝑥𝑂 = 𝑥 → (𝑥𝑂 ·s
𝐵) = (𝑥 ·s 𝐵)) |
2 | 1 | oveq1d 7419 |
. . . . . . . . . 10
⊢ (𝑥𝑂 = 𝑥 → ((𝑥𝑂 ·s
𝐵) ·s
𝐶) = ((𝑥 ·s 𝐵) ·s 𝐶)) |
3 | | oveq1 7411 |
. . . . . . . . . 10
⊢ (𝑥𝑂 = 𝑥 → (𝑥𝑂 ·s
(𝐵 ·s
𝐶)) = (𝑥 ·s (𝐵 ·s 𝐶))) |
4 | 2, 3 | eqeq12d 2749 |
. . . . . . . . 9
⊢ (𝑥𝑂 = 𝑥 → (((𝑥𝑂 ·s
𝐵) ·s
𝐶) = (𝑥𝑂 ·s
(𝐵 ·s
𝐶)) ↔ ((𝑥 ·s 𝐵) ·s 𝐶) = (𝑥 ·s (𝐵 ·s 𝐶)))) |
5 | | mulsasslem3.11 |
. . . . . . . . . 10
⊢ (𝜑 → ∀𝑥𝑂 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))((𝑥𝑂 ·s
𝐵) ·s
𝐶) = (𝑥𝑂 ·s
(𝐵 ·s
𝐶))) |
6 | 5 | adantr 482 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → ∀𝑥𝑂 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))((𝑥𝑂 ·s
𝐵) ·s
𝐶) = (𝑥𝑂 ·s
(𝐵 ·s
𝐶))) |
7 | | mulsasslem3.4 |
. . . . . . . . . 10
⊢ 𝑃 ⊆ (( L ‘𝐴) ∪ ( R ‘𝐴)) |
8 | | simprll 778 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → 𝑥 ∈ 𝑃) |
9 | 7, 8 | sselid 3979 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → 𝑥 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))) |
10 | 4, 6, 9 | rspcdva 3613 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → ((𝑥 ·s 𝐵) ·s 𝐶) = (𝑥 ·s (𝐵 ·s 𝐶))) |
11 | | oveq2 7412 |
. . . . . . . . . . . . 13
⊢ (𝑦𝑂 = 𝑦 → (𝐴 ·s 𝑦𝑂) = (𝐴 ·s 𝑦)) |
12 | 11 | oveq1d 7419 |
. . . . . . . . . . . 12
⊢ (𝑦𝑂 = 𝑦 → ((𝐴 ·s 𝑦𝑂) ·s
𝐶) = ((𝐴 ·s 𝑦) ·s 𝐶)) |
13 | | oveq1 7411 |
. . . . . . . . . . . . 13
⊢ (𝑦𝑂 = 𝑦 → (𝑦𝑂 ·s
𝐶) = (𝑦 ·s 𝐶)) |
14 | 13 | oveq2d 7420 |
. . . . . . . . . . . 12
⊢ (𝑦𝑂 = 𝑦 → (𝐴 ·s (𝑦𝑂 ·s
𝐶)) = (𝐴 ·s (𝑦 ·s 𝐶))) |
15 | 12, 14 | eqeq12d 2749 |
. . . . . . . . . . 11
⊢ (𝑦𝑂 = 𝑦 → (((𝐴 ·s 𝑦𝑂) ·s
𝐶) = (𝐴 ·s (𝑦𝑂 ·s
𝐶)) ↔ ((𝐴 ·s 𝑦) ·s 𝐶) = (𝐴 ·s (𝑦 ·s 𝐶)))) |
16 | | mulsasslem3.12 |
. . . . . . . . . . . 12
⊢ (𝜑 → ∀𝑦𝑂 ∈ (( L ‘𝐵) ∪ ( R ‘𝐵))((𝐴 ·s 𝑦𝑂) ·s
𝐶) = (𝐴 ·s (𝑦𝑂 ·s
𝐶))) |
17 | 16 | adantr 482 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → ∀𝑦𝑂 ∈ (( L ‘𝐵) ∪ ( R ‘𝐵))((𝐴 ·s 𝑦𝑂) ·s
𝐶) = (𝐴 ·s (𝑦𝑂 ·s
𝐶))) |
18 | | mulsasslem3.5 |
. . . . . . . . . . . 12
⊢ 𝑄 ⊆ (( L ‘𝐵) ∪ ( R ‘𝐵)) |
19 | | simprlr 779 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → 𝑦 ∈ 𝑄) |
20 | 18, 19 | sselid 3979 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → 𝑦 ∈ (( L ‘𝐵) ∪ ( R ‘𝐵))) |
21 | 15, 17, 20 | rspcdva 3613 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → ((𝐴 ·s 𝑦) ·s 𝐶) = (𝐴 ·s (𝑦 ·s 𝐶))) |
22 | | oveq2 7412 |
. . . . . . . . . . . . . 14
⊢ (𝑧𝑂 = 𝑧 → ((𝐴 ·s 𝐵) ·s 𝑧𝑂) = ((𝐴 ·s 𝐵) ·s 𝑧)) |
23 | | oveq2 7412 |
. . . . . . . . . . . . . . 15
⊢ (𝑧𝑂 = 𝑧 → (𝐵 ·s 𝑧𝑂) = (𝐵 ·s 𝑧)) |
24 | 23 | oveq2d 7420 |
. . . . . . . . . . . . . 14
⊢ (𝑧𝑂 = 𝑧 → (𝐴 ·s (𝐵 ·s 𝑧𝑂)) = (𝐴 ·s (𝐵 ·s 𝑧))) |
25 | 22, 24 | eqeq12d 2749 |
. . . . . . . . . . . . 13
⊢ (𝑧𝑂 = 𝑧 → (((𝐴 ·s 𝐵) ·s 𝑧𝑂) = (𝐴 ·s (𝐵 ·s 𝑧𝑂)) ↔ ((𝐴 ·s 𝐵) ·s 𝑧) = (𝐴 ·s (𝐵 ·s 𝑧)))) |
26 | | mulsasslem3.13 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ∀𝑧𝑂 ∈ (( L ‘𝐶) ∪ ( R ‘𝐶))((𝐴 ·s 𝐵) ·s 𝑧𝑂) = (𝐴 ·s (𝐵 ·s 𝑧𝑂))) |
27 | 26 | adantr 482 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → ∀𝑧𝑂 ∈ (( L ‘𝐶) ∪ ( R ‘𝐶))((𝐴 ·s 𝐵) ·s 𝑧𝑂) = (𝐴 ·s (𝐵 ·s 𝑧𝑂))) |
28 | | mulsasslem3.6 |
. . . . . . . . . . . . . 14
⊢ 𝑅 ⊆ (( L ‘𝐶) ∪ ( R ‘𝐶)) |
29 | | simprr 772 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → 𝑧 ∈ 𝑅) |
30 | 28, 29 | sselid 3979 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → 𝑧 ∈ (( L ‘𝐶) ∪ ( R ‘𝐶))) |
31 | 25, 27, 30 | rspcdva 3613 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → ((𝐴 ·s 𝐵) ·s 𝑧) = (𝐴 ·s (𝐵 ·s 𝑧))) |
32 | | leftssno 27355 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ( L
‘𝐴) ⊆ No |
33 | | rightssno 27356 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ( R
‘𝐴) ⊆ No |
34 | 32, 33 | unssi 4184 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (( L
‘𝐴) ∪ ( R
‘𝐴)) ⊆ No |
35 | 7, 34 | sstri 3990 |
. . . . . . . . . . . . . . . . . . . 20
⊢ 𝑃 ⊆
No |
36 | 35, 8 | sselid 3979 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → 𝑥 ∈ No
) |
37 | | mulsasslem3.2 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → 𝐵 ∈ No
) |
38 | 37 | adantr 482 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → 𝐵 ∈ No
) |
39 | 36, 38 | mulscld 27571 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → (𝑥 ·s 𝐵) ∈ No
) |
40 | | leftssno 27355 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ( L
‘𝐶) ⊆ No |
41 | | rightssno 27356 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ( R
‘𝐶) ⊆ No |
42 | 40, 41 | unssi 4184 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (( L
‘𝐶) ∪ ( R
‘𝐶)) ⊆ No |
43 | 28, 42 | sstri 3990 |
. . . . . . . . . . . . . . . . . . 19
⊢ 𝑅 ⊆
No |
44 | 43, 29 | sselid 3979 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → 𝑧 ∈ No
) |
45 | 39, 44 | mulscld 27571 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → ((𝑥 ·s 𝐵) ·s 𝑧) ∈ No
) |
46 | | mulsasslem3.1 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → 𝐴 ∈ No
) |
47 | 46 | adantr 482 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → 𝐴 ∈ No
) |
48 | | leftssno 27355 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ( L
‘𝐵) ⊆ No |
49 | | rightssno 27356 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ( R
‘𝐵) ⊆ No |
50 | 48, 49 | unssi 4184 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (( L
‘𝐵) ∪ ( R
‘𝐵)) ⊆ No |
51 | 18, 50 | sstri 3990 |
. . . . . . . . . . . . . . . . . . . 20
⊢ 𝑄 ⊆
No |
52 | 51, 19 | sselid 3979 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → 𝑦 ∈ No
) |
53 | 47, 52 | mulscld 27571 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → (𝐴 ·s 𝑦) ∈ No
) |
54 | 53, 44 | mulscld 27571 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → ((𝐴 ·s 𝑦) ·s 𝑧) ∈ No
) |
55 | 45, 54 | addscomd 27431 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → (((𝑥 ·s 𝐵) ·s 𝑧) +s ((𝐴 ·s 𝑦) ·s 𝑧)) = (((𝐴 ·s 𝑦) ·s 𝑧) +s ((𝑥 ·s 𝐵) ·s 𝑧))) |
56 | 55 | oveq1d 7419 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → ((((𝑥 ·s 𝐵) ·s 𝑧) +s ((𝐴 ·s 𝑦) ·s 𝑧)) -s ((𝑥 ·s 𝑦) ·s 𝑧)) = ((((𝐴 ·s 𝑦) ·s 𝑧) +s ((𝑥 ·s 𝐵) ·s 𝑧)) -s ((𝑥 ·s 𝑦) ·s 𝑧))) |
57 | 36, 52 | mulscld 27571 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → (𝑥 ·s 𝑦) ∈ No
) |
58 | 57, 44 | mulscld 27571 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → ((𝑥 ·s 𝑦) ·s 𝑧) ∈ No
) |
59 | 54, 45, 58 | addsubsassd 27528 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → ((((𝐴 ·s 𝑦) ·s 𝑧) +s ((𝑥 ·s 𝐵) ·s 𝑧)) -s ((𝑥 ·s 𝑦) ·s 𝑧)) = (((𝐴 ·s 𝑦) ·s 𝑧) +s (((𝑥 ·s 𝐵) ·s 𝑧) -s ((𝑥 ·s 𝑦) ·s 𝑧)))) |
60 | 56, 59 | eqtrd 2773 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → ((((𝑥 ·s 𝐵) ·s 𝑧) +s ((𝐴 ·s 𝑦) ·s 𝑧)) -s ((𝑥 ·s 𝑦) ·s 𝑧)) = (((𝐴 ·s 𝑦) ·s 𝑧) +s (((𝑥 ·s 𝐵) ·s 𝑧) -s ((𝑥 ·s 𝑦) ·s 𝑧)))) |
61 | 60 | oveq1d 7419 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → (((((𝑥 ·s 𝐵) ·s 𝑧) +s ((𝐴 ·s 𝑦) ·s 𝑧)) -s ((𝑥 ·s 𝑦) ·s 𝑧)) +s ((𝑥 ·s 𝑦) ·s 𝐶)) = ((((𝐴 ·s 𝑦) ·s 𝑧) +s (((𝑥 ·s 𝐵) ·s 𝑧) -s ((𝑥 ·s 𝑦) ·s 𝑧))) +s ((𝑥 ·s 𝑦) ·s 𝐶))) |
62 | 45, 58 | subscld 27515 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → (((𝑥 ·s 𝐵) ·s 𝑧) -s ((𝑥 ·s 𝑦) ·s 𝑧)) ∈ No
) |
63 | | mulsasslem3.3 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝐶 ∈ No
) |
64 | 63 | adantr 482 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → 𝐶 ∈ No
) |
65 | 57, 64 | mulscld 27571 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → ((𝑥 ·s 𝑦) ·s 𝐶) ∈ No
) |
66 | 54, 62, 65 | addsassd 27469 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → ((((𝐴 ·s 𝑦) ·s 𝑧) +s (((𝑥 ·s 𝐵) ·s 𝑧) -s ((𝑥 ·s 𝑦) ·s 𝑧))) +s ((𝑥 ·s 𝑦) ·s 𝐶)) = (((𝐴 ·s 𝑦) ·s 𝑧) +s ((((𝑥 ·s 𝐵) ·s 𝑧) -s ((𝑥 ·s 𝑦) ·s 𝑧)) +s ((𝑥 ·s 𝑦) ·s 𝐶)))) |
67 | 11 | oveq1d 7419 |
. . . . . . . . . . . . . . . 16
⊢ (𝑦𝑂 = 𝑦 → ((𝐴 ·s 𝑦𝑂) ·s
𝑧𝑂) =
((𝐴 ·s
𝑦) ·s
𝑧𝑂)) |
68 | | oveq1 7411 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑦𝑂 = 𝑦 → (𝑦𝑂 ·s
𝑧𝑂) =
(𝑦 ·s
𝑧𝑂)) |
69 | 68 | oveq2d 7420 |
. . . . . . . . . . . . . . . 16
⊢ (𝑦𝑂 = 𝑦 → (𝐴 ·s (𝑦𝑂 ·s
𝑧𝑂)) =
(𝐴 ·s
(𝑦 ·s
𝑧𝑂))) |
70 | 67, 69 | eqeq12d 2749 |
. . . . . . . . . . . . . . 15
⊢ (𝑦𝑂 = 𝑦 → (((𝐴 ·s 𝑦𝑂) ·s
𝑧𝑂) =
(𝐴 ·s
(𝑦𝑂
·s 𝑧𝑂)) ↔ ((𝐴 ·s 𝑦) ·s 𝑧𝑂) = (𝐴 ·s (𝑦 ·s 𝑧𝑂)))) |
71 | | oveq2 7412 |
. . . . . . . . . . . . . . . 16
⊢ (𝑧𝑂 = 𝑧 → ((𝐴 ·s 𝑦) ·s 𝑧𝑂) = ((𝐴 ·s 𝑦) ·s 𝑧)) |
72 | | oveq2 7412 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑧𝑂 = 𝑧 → (𝑦 ·s 𝑧𝑂) = (𝑦 ·s 𝑧)) |
73 | 72 | oveq2d 7420 |
. . . . . . . . . . . . . . . 16
⊢ (𝑧𝑂 = 𝑧 → (𝐴 ·s (𝑦 ·s 𝑧𝑂)) = (𝐴 ·s (𝑦 ·s 𝑧))) |
74 | 71, 73 | eqeq12d 2749 |
. . . . . . . . . . . . . . 15
⊢ (𝑧𝑂 = 𝑧 → (((𝐴 ·s 𝑦) ·s 𝑧𝑂) = (𝐴 ·s (𝑦 ·s 𝑧𝑂)) ↔ ((𝐴 ·s 𝑦) ·s 𝑧) = (𝐴 ·s (𝑦 ·s 𝑧)))) |
75 | | mulsasslem3.10 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ∀𝑦𝑂 ∈ (( L ‘𝐵) ∪ ( R ‘𝐵))∀𝑧𝑂 ∈ (( L ‘𝐶) ∪ ( R ‘𝐶))((𝐴 ·s 𝑦𝑂) ·s
𝑧𝑂) =
(𝐴 ·s
(𝑦𝑂
·s 𝑧𝑂))) |
76 | 75 | adantr 482 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → ∀𝑦𝑂 ∈ (( L ‘𝐵) ∪ ( R ‘𝐵))∀𝑧𝑂 ∈ (( L ‘𝐶) ∪ ( R ‘𝐶))((𝐴 ·s 𝑦𝑂) ·s
𝑧𝑂) =
(𝐴 ·s
(𝑦𝑂
·s 𝑧𝑂))) |
77 | 70, 74, 76, 20, 30 | rspc2dv 3625 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → ((𝐴 ·s 𝑦) ·s 𝑧) = (𝐴 ·s (𝑦 ·s 𝑧))) |
78 | 45, 65, 58 | addsubsd 27529 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → ((((𝑥 ·s 𝐵) ·s 𝑧) +s ((𝑥 ·s 𝑦) ·s 𝐶)) -s ((𝑥 ·s 𝑦) ·s 𝑧)) = ((((𝑥 ·s 𝐵) ·s 𝑧) -s ((𝑥 ·s 𝑦) ·s 𝑧)) +s ((𝑥 ·s 𝑦) ·s 𝐶))) |
79 | 1 | oveq1d 7419 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑥𝑂 = 𝑥 → ((𝑥𝑂 ·s
𝐵) ·s
𝑧𝑂) =
((𝑥 ·s
𝐵) ·s
𝑧𝑂)) |
80 | | oveq1 7411 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑥𝑂 = 𝑥 → (𝑥𝑂 ·s
(𝐵 ·s
𝑧𝑂)) =
(𝑥 ·s
(𝐵 ·s
𝑧𝑂))) |
81 | 79, 80 | eqeq12d 2749 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑥𝑂 = 𝑥 → (((𝑥𝑂 ·s
𝐵) ·s
𝑧𝑂) =
(𝑥𝑂
·s (𝐵
·s 𝑧𝑂)) ↔ ((𝑥 ·s 𝐵) ·s 𝑧𝑂) = (𝑥 ·s (𝐵 ·s 𝑧𝑂)))) |
82 | | oveq2 7412 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑧𝑂 = 𝑧 → ((𝑥 ·s 𝐵) ·s 𝑧𝑂) = ((𝑥 ·s 𝐵) ·s 𝑧)) |
83 | 23 | oveq2d 7420 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑧𝑂 = 𝑧 → (𝑥 ·s (𝐵 ·s 𝑧𝑂)) = (𝑥 ·s (𝐵 ·s 𝑧))) |
84 | 82, 83 | eqeq12d 2749 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑧𝑂 = 𝑧 → (((𝑥 ·s 𝐵) ·s 𝑧𝑂) = (𝑥 ·s (𝐵 ·s 𝑧𝑂)) ↔ ((𝑥 ·s 𝐵) ·s 𝑧) = (𝑥 ·s (𝐵 ·s 𝑧)))) |
85 | | mulsasslem3.9 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → ∀𝑥𝑂 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))∀𝑧𝑂 ∈ (( L ‘𝐶) ∪ ( R ‘𝐶))((𝑥𝑂 ·s
𝐵) ·s
𝑧𝑂) =
(𝑥𝑂
·s (𝐵
·s 𝑧𝑂))) |
86 | 85 | adantr 482 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → ∀𝑥𝑂 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))∀𝑧𝑂 ∈ (( L ‘𝐶) ∪ ( R ‘𝐶))((𝑥𝑂 ·s
𝐵) ·s
𝑧𝑂) =
(𝑥𝑂
·s (𝐵
·s 𝑧𝑂))) |
87 | 81, 84, 86, 9, 30 | rspc2dv 3625 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → ((𝑥 ·s 𝐵) ·s 𝑧) = (𝑥 ·s (𝐵 ·s 𝑧))) |
88 | | oveq1 7411 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑥𝑂 = 𝑥 → (𝑥𝑂 ·s
𝑦𝑂) =
(𝑥 ·s
𝑦𝑂)) |
89 | 88 | oveq1d 7419 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑥𝑂 = 𝑥 → ((𝑥𝑂 ·s
𝑦𝑂)
·s 𝐶) =
((𝑥 ·s
𝑦𝑂)
·s 𝐶)) |
90 | | oveq1 7411 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑥𝑂 = 𝑥 → (𝑥𝑂 ·s
(𝑦𝑂
·s 𝐶)) =
(𝑥 ·s
(𝑦𝑂
·s 𝐶))) |
91 | 89, 90 | eqeq12d 2749 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑥𝑂 = 𝑥 → (((𝑥𝑂 ·s
𝑦𝑂)
·s 𝐶) =
(𝑥𝑂
·s (𝑦𝑂 ·s
𝐶)) ↔ ((𝑥 ·s 𝑦𝑂)
·s 𝐶) =
(𝑥 ·s
(𝑦𝑂
·s 𝐶)))) |
92 | | oveq2 7412 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑦𝑂 = 𝑦 → (𝑥 ·s 𝑦𝑂) = (𝑥 ·s 𝑦)) |
93 | 92 | oveq1d 7419 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑦𝑂 = 𝑦 → ((𝑥 ·s 𝑦𝑂) ·s
𝐶) = ((𝑥 ·s 𝑦) ·s 𝐶)) |
94 | 13 | oveq2d 7420 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑦𝑂 = 𝑦 → (𝑥 ·s (𝑦𝑂 ·s
𝐶)) = (𝑥 ·s (𝑦 ·s 𝐶))) |
95 | 93, 94 | eqeq12d 2749 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑦𝑂 = 𝑦 → (((𝑥 ·s 𝑦𝑂) ·s
𝐶) = (𝑥 ·s (𝑦𝑂 ·s
𝐶)) ↔ ((𝑥 ·s 𝑦) ·s 𝐶) = (𝑥 ·s (𝑦 ·s 𝐶)))) |
96 | | mulsasslem3.8 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → ∀𝑥𝑂 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))∀𝑦𝑂 ∈ (( L ‘𝐵) ∪ ( R ‘𝐵))((𝑥𝑂 ·s
𝑦𝑂)
·s 𝐶) =
(𝑥𝑂
·s (𝑦𝑂 ·s
𝐶))) |
97 | 96 | adantr 482 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → ∀𝑥𝑂 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))∀𝑦𝑂 ∈ (( L ‘𝐵) ∪ ( R ‘𝐵))((𝑥𝑂 ·s
𝑦𝑂)
·s 𝐶) =
(𝑥𝑂
·s (𝑦𝑂 ·s
𝐶))) |
98 | 91, 95, 97, 9, 20 | rspc2dv 3625 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → ((𝑥 ·s 𝑦) ·s 𝐶) = (𝑥 ·s (𝑦 ·s 𝐶))) |
99 | 87, 98 | oveq12d 7422 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → (((𝑥 ·s 𝐵) ·s 𝑧) +s ((𝑥 ·s 𝑦) ·s 𝐶)) = ((𝑥 ·s (𝐵 ·s 𝑧)) +s (𝑥 ·s (𝑦 ·s 𝐶)))) |
100 | 38, 44 | mulscld 27571 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → (𝐵 ·s 𝑧) ∈ No
) |
101 | 36, 100 | mulscld 27571 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → (𝑥 ·s (𝐵 ·s 𝑧)) ∈ No
) |
102 | 52, 64 | mulscld 27571 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → (𝑦 ·s 𝐶) ∈ No
) |
103 | 36, 102 | mulscld 27571 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → (𝑥 ·s (𝑦 ·s 𝐶)) ∈ No
) |
104 | 101, 103 | addscomd 27431 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → ((𝑥 ·s (𝐵 ·s 𝑧)) +s (𝑥 ·s (𝑦 ·s 𝐶))) = ((𝑥 ·s (𝑦 ·s 𝐶)) +s (𝑥 ·s (𝐵 ·s 𝑧)))) |
105 | 99, 104 | eqtrd 2773 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → (((𝑥 ·s 𝐵) ·s 𝑧) +s ((𝑥 ·s 𝑦) ·s 𝐶)) = ((𝑥 ·s (𝑦 ·s 𝐶)) +s (𝑥 ·s (𝐵 ·s 𝑧)))) |
106 | 88 | oveq1d 7419 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑥𝑂 = 𝑥 → ((𝑥𝑂 ·s
𝑦𝑂)
·s 𝑧𝑂) = ((𝑥 ·s 𝑦𝑂) ·s
𝑧𝑂)) |
107 | | oveq1 7411 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑥𝑂 = 𝑥 → (𝑥𝑂 ·s
(𝑦𝑂
·s 𝑧𝑂)) = (𝑥 ·s (𝑦𝑂 ·s
𝑧𝑂))) |
108 | 106, 107 | eqeq12d 2749 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥𝑂 = 𝑥 → (((𝑥𝑂 ·s
𝑦𝑂)
·s 𝑧𝑂) = (𝑥𝑂 ·s
(𝑦𝑂
·s 𝑧𝑂)) ↔ ((𝑥 ·s 𝑦𝑂)
·s 𝑧𝑂) = (𝑥 ·s (𝑦𝑂 ·s
𝑧𝑂)))) |
109 | 92 | oveq1d 7419 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑦𝑂 = 𝑦 → ((𝑥 ·s 𝑦𝑂) ·s
𝑧𝑂) =
((𝑥 ·s
𝑦) ·s
𝑧𝑂)) |
110 | 68 | oveq2d 7420 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑦𝑂 = 𝑦 → (𝑥 ·s (𝑦𝑂 ·s
𝑧𝑂)) =
(𝑥 ·s
(𝑦 ·s
𝑧𝑂))) |
111 | 109, 110 | eqeq12d 2749 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑦𝑂 = 𝑦 → (((𝑥 ·s 𝑦𝑂) ·s
𝑧𝑂) =
(𝑥 ·s
(𝑦𝑂
·s 𝑧𝑂)) ↔ ((𝑥 ·s 𝑦) ·s 𝑧𝑂) = (𝑥 ·s (𝑦 ·s 𝑧𝑂)))) |
112 | | oveq2 7412 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑧𝑂 = 𝑧 → ((𝑥 ·s 𝑦) ·s 𝑧𝑂) = ((𝑥 ·s 𝑦) ·s 𝑧)) |
113 | 72 | oveq2d 7420 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑧𝑂 = 𝑧 → (𝑥 ·s (𝑦 ·s 𝑧𝑂)) = (𝑥 ·s (𝑦 ·s 𝑧))) |
114 | 112, 113 | eqeq12d 2749 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑧𝑂 = 𝑧 → (((𝑥 ·s 𝑦) ·s 𝑧𝑂) = (𝑥 ·s (𝑦 ·s 𝑧𝑂)) ↔ ((𝑥 ·s 𝑦) ·s 𝑧) = (𝑥 ·s (𝑦 ·s 𝑧)))) |
115 | | mulsasslem3.7 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → ∀𝑥𝑂 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))∀𝑦𝑂 ∈ (( L ‘𝐵) ∪ ( R ‘𝐵))∀𝑧𝑂 ∈ (( L ‘𝐶) ∪ ( R ‘𝐶))((𝑥𝑂 ·s
𝑦𝑂)
·s 𝑧𝑂) = (𝑥𝑂 ·s
(𝑦𝑂
·s 𝑧𝑂))) |
116 | 115 | adantr 482 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → ∀𝑥𝑂 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))∀𝑦𝑂 ∈ (( L ‘𝐵) ∪ ( R ‘𝐵))∀𝑧𝑂 ∈ (( L ‘𝐶) ∪ ( R ‘𝐶))((𝑥𝑂 ·s
𝑦𝑂)
·s 𝑧𝑂) = (𝑥𝑂 ·s
(𝑦𝑂
·s 𝑧𝑂))) |
117 | 108, 111,
114, 116, 9, 20, 30 | rspc3dv 3628 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → ((𝑥 ·s 𝑦) ·s 𝑧) = (𝑥 ·s (𝑦 ·s 𝑧))) |
118 | 105, 117 | oveq12d 7422 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → ((((𝑥 ·s 𝐵) ·s 𝑧) +s ((𝑥 ·s 𝑦) ·s 𝐶)) -s ((𝑥 ·s 𝑦) ·s 𝑧)) = (((𝑥 ·s (𝑦 ·s 𝐶)) +s (𝑥 ·s (𝐵 ·s 𝑧))) -s (𝑥 ·s (𝑦 ·s 𝑧)))) |
119 | 78, 118 | eqtr3d 2775 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → ((((𝑥 ·s 𝐵) ·s 𝑧) -s ((𝑥 ·s 𝑦) ·s 𝑧)) +s ((𝑥 ·s 𝑦) ·s 𝐶)) = (((𝑥 ·s (𝑦 ·s 𝐶)) +s (𝑥 ·s (𝐵 ·s 𝑧))) -s (𝑥 ·s (𝑦 ·s 𝑧)))) |
120 | 77, 119 | oveq12d 7422 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → (((𝐴 ·s 𝑦) ·s 𝑧) +s ((((𝑥 ·s 𝐵) ·s 𝑧) -s ((𝑥 ·s 𝑦) ·s 𝑧)) +s ((𝑥 ·s 𝑦) ·s 𝐶))) = ((𝐴 ·s (𝑦 ·s 𝑧)) +s (((𝑥 ·s (𝑦 ·s 𝐶)) +s (𝑥 ·s (𝐵 ·s 𝑧))) -s (𝑥 ·s (𝑦 ·s 𝑧))))) |
121 | 61, 66, 120 | 3eqtrd 2777 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → (((((𝑥 ·s 𝐵) ·s 𝑧) +s ((𝐴 ·s 𝑦) ·s 𝑧)) -s ((𝑥 ·s 𝑦) ·s 𝑧)) +s ((𝑥 ·s 𝑦) ·s 𝐶)) = ((𝐴 ·s (𝑦 ·s 𝑧)) +s (((𝑥 ·s (𝑦 ·s 𝐶)) +s (𝑥 ·s (𝐵 ·s 𝑧))) -s (𝑥 ·s (𝑦 ·s 𝑧))))) |
122 | 31, 121 | oveq12d 7422 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → (((𝐴 ·s 𝐵) ·s 𝑧) -s (((((𝑥 ·s 𝐵) ·s 𝑧) +s ((𝐴 ·s 𝑦) ·s 𝑧)) -s ((𝑥 ·s 𝑦) ·s 𝑧)) +s ((𝑥 ·s 𝑦) ·s 𝐶))) = ((𝐴 ·s (𝐵 ·s 𝑧)) -s ((𝐴 ·s (𝑦 ·s 𝑧)) +s (((𝑥 ·s (𝑦 ·s 𝐶)) +s (𝑥 ·s (𝐵 ·s 𝑧))) -s (𝑥 ·s (𝑦 ·s 𝑧)))))) |
123 | 47, 38 | mulscld 27571 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → (𝐴 ·s 𝐵) ∈ No
) |
124 | 123, 44 | mulscld 27571 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → ((𝐴 ·s 𝐵) ·s 𝑧) ∈ No
) |
125 | 45, 54 | addscld 27444 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → (((𝑥 ·s 𝐵) ·s 𝑧) +s ((𝐴 ·s 𝑦) ·s 𝑧)) ∈ No
) |
126 | 125, 58 | subscld 27515 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → ((((𝑥 ·s 𝐵) ·s 𝑧) +s ((𝐴 ·s 𝑦) ·s 𝑧)) -s ((𝑥 ·s 𝑦) ·s 𝑧)) ∈ No
) |
127 | 124, 126,
65 | subsubs4d 27540 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → ((((𝐴 ·s 𝐵) ·s 𝑧) -s ((((𝑥 ·s 𝐵) ·s 𝑧) +s ((𝐴 ·s 𝑦) ·s 𝑧)) -s ((𝑥 ·s 𝑦) ·s 𝑧))) -s ((𝑥 ·s 𝑦) ·s 𝐶)) = (((𝐴 ·s 𝐵) ·s 𝑧) -s (((((𝑥 ·s 𝐵) ·s 𝑧) +s ((𝐴 ·s 𝑦) ·s 𝑧)) -s ((𝑥 ·s 𝑦) ·s 𝑧)) +s ((𝑥 ·s 𝑦) ·s 𝐶)))) |
128 | 47, 100 | mulscld 27571 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → (𝐴 ·s (𝐵 ·s 𝑧)) ∈ No
) |
129 | 52, 44 | mulscld 27571 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → (𝑦 ·s 𝑧) ∈ No
) |
130 | 47, 129 | mulscld 27571 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → (𝐴 ·s (𝑦 ·s 𝑧)) ∈ No
) |
131 | 103, 101 | addscld 27444 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → ((𝑥 ·s (𝑦 ·s 𝐶)) +s (𝑥 ·s (𝐵 ·s 𝑧))) ∈ No
) |
132 | 36, 129 | mulscld 27571 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → (𝑥 ·s (𝑦 ·s 𝑧)) ∈ No
) |
133 | 131, 132 | subscld 27515 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → (((𝑥 ·s (𝑦 ·s 𝐶)) +s (𝑥 ·s (𝐵 ·s 𝑧))) -s (𝑥 ·s (𝑦 ·s 𝑧))) ∈ No
) |
134 | 128, 130,
133 | subsubs4d 27540 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → (((𝐴 ·s (𝐵 ·s 𝑧)) -s (𝐴 ·s (𝑦 ·s 𝑧))) -s (((𝑥 ·s (𝑦 ·s 𝐶)) +s (𝑥 ·s (𝐵 ·s 𝑧))) -s (𝑥 ·s (𝑦 ·s 𝑧)))) = ((𝐴 ·s (𝐵 ·s 𝑧)) -s ((𝐴 ·s (𝑦 ·s 𝑧)) +s (((𝑥 ·s (𝑦 ·s 𝐶)) +s (𝑥 ·s (𝐵 ·s 𝑧))) -s (𝑥 ·s (𝑦 ·s 𝑧)))))) |
135 | 122, 127,
134 | 3eqtr4d 2783 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → ((((𝐴 ·s 𝐵) ·s 𝑧) -s ((((𝑥 ·s 𝐵) ·s 𝑧) +s ((𝐴 ·s 𝑦) ·s 𝑧)) -s ((𝑥 ·s 𝑦) ·s 𝑧))) -s ((𝑥 ·s 𝑦) ·s 𝐶)) = (((𝐴 ·s (𝐵 ·s 𝑧)) -s (𝐴 ·s (𝑦 ·s 𝑧))) -s (((𝑥 ·s (𝑦 ·s 𝐶)) +s (𝑥 ·s (𝐵 ·s 𝑧))) -s (𝑥 ·s (𝑦 ·s 𝑧))))) |
136 | 21, 135 | oveq12d 7422 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → (((𝐴 ·s 𝑦) ·s 𝐶) +s ((((𝐴 ·s 𝐵) ·s 𝑧) -s ((((𝑥 ·s 𝐵) ·s 𝑧) +s ((𝐴 ·s 𝑦) ·s 𝑧)) -s ((𝑥 ·s 𝑦) ·s 𝑧))) -s ((𝑥 ·s 𝑦) ·s 𝐶))) = ((𝐴 ·s (𝑦 ·s 𝐶)) +s (((𝐴 ·s (𝐵 ·s 𝑧)) -s (𝐴 ·s (𝑦 ·s 𝑧))) -s (((𝑥 ·s (𝑦 ·s 𝐶)) +s (𝑥 ·s (𝐵 ·s 𝑧))) -s (𝑥 ·s (𝑦 ·s 𝑧)))))) |
137 | 53, 64 | mulscld 27571 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → ((𝐴 ·s 𝑦) ·s 𝐶) ∈ No
) |
138 | 124, 126 | subscld 27515 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → (((𝐴 ·s 𝐵) ·s 𝑧) -s ((((𝑥 ·s 𝐵) ·s 𝑧) +s ((𝐴 ·s 𝑦) ·s 𝑧)) -s ((𝑥 ·s 𝑦) ·s 𝑧))) ∈ No
) |
139 | 137, 138,
65 | addsubsd 27529 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → ((((𝐴 ·s 𝑦) ·s 𝐶) +s (((𝐴 ·s 𝐵) ·s 𝑧) -s ((((𝑥 ·s 𝐵) ·s 𝑧) +s ((𝐴 ·s 𝑦) ·s 𝑧)) -s ((𝑥 ·s 𝑦) ·s 𝑧)))) -s ((𝑥 ·s 𝑦) ·s 𝐶)) = ((((𝐴 ·s 𝑦) ·s 𝐶) -s ((𝑥 ·s 𝑦) ·s 𝐶)) +s (((𝐴 ·s 𝐵) ·s 𝑧) -s ((((𝑥 ·s 𝐵) ·s 𝑧) +s ((𝐴 ·s 𝑦) ·s 𝑧)) -s ((𝑥 ·s 𝑦) ·s 𝑧))))) |
140 | 137, 138,
65 | addsubsassd 27528 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → ((((𝐴 ·s 𝑦) ·s 𝐶) +s (((𝐴 ·s 𝐵) ·s 𝑧) -s ((((𝑥 ·s 𝐵) ·s 𝑧) +s ((𝐴 ·s 𝑦) ·s 𝑧)) -s ((𝑥 ·s 𝑦) ·s 𝑧)))) -s ((𝑥 ·s 𝑦) ·s 𝐶)) = (((𝐴 ·s 𝑦) ·s 𝐶) +s ((((𝐴 ·s 𝐵) ·s 𝑧) -s ((((𝑥 ·s 𝐵) ·s 𝑧) +s ((𝐴 ·s 𝑦) ·s 𝑧)) -s ((𝑥 ·s 𝑦) ·s 𝑧))) -s ((𝑥 ·s 𝑦) ·s 𝐶)))) |
141 | 139, 140 | eqtr3d 2775 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → ((((𝐴 ·s 𝑦) ·s 𝐶) -s ((𝑥 ·s 𝑦) ·s 𝐶)) +s (((𝐴 ·s 𝐵) ·s 𝑧) -s ((((𝑥 ·s 𝐵) ·s 𝑧) +s ((𝐴 ·s 𝑦) ·s 𝑧)) -s ((𝑥 ·s 𝑦) ·s 𝑧)))) = (((𝐴 ·s 𝑦) ·s 𝐶) +s ((((𝐴 ·s 𝐵) ·s 𝑧) -s ((((𝑥 ·s 𝐵) ·s 𝑧) +s ((𝐴 ·s 𝑦) ·s 𝑧)) -s ((𝑥 ·s 𝑦) ·s 𝑧))) -s ((𝑥 ·s 𝑦) ·s 𝐶)))) |
142 | 47, 102 | mulscld 27571 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → (𝐴 ·s (𝑦 ·s 𝐶)) ∈ No
) |
143 | 142, 128,
130 | addsubsassd 27528 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → (((𝐴 ·s (𝑦 ·s 𝐶)) +s (𝐴 ·s (𝐵 ·s 𝑧))) -s (𝐴 ·s (𝑦 ·s 𝑧))) = ((𝐴 ·s (𝑦 ·s 𝐶)) +s ((𝐴 ·s (𝐵 ·s 𝑧)) -s (𝐴 ·s (𝑦 ·s 𝑧))))) |
144 | 143 | oveq1d 7419 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → ((((𝐴 ·s (𝑦 ·s 𝐶)) +s (𝐴 ·s (𝐵 ·s 𝑧))) -s (𝐴 ·s (𝑦 ·s 𝑧))) -s (((𝑥 ·s (𝑦 ·s 𝐶)) +s (𝑥 ·s (𝐵 ·s 𝑧))) -s (𝑥 ·s (𝑦 ·s 𝑧)))) = (((𝐴 ·s (𝑦 ·s 𝐶)) +s ((𝐴 ·s (𝐵 ·s 𝑧)) -s (𝐴 ·s (𝑦 ·s 𝑧)))) -s (((𝑥 ·s (𝑦 ·s 𝐶)) +s (𝑥 ·s (𝐵 ·s 𝑧))) -s (𝑥 ·s (𝑦 ·s 𝑧))))) |
145 | 128, 130 | subscld 27515 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → ((𝐴 ·s (𝐵 ·s 𝑧)) -s (𝐴 ·s (𝑦 ·s 𝑧))) ∈ No
) |
146 | 142, 145,
133 | addsubsassd 27528 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → (((𝐴 ·s (𝑦 ·s 𝐶)) +s ((𝐴 ·s (𝐵 ·s 𝑧)) -s (𝐴 ·s (𝑦 ·s 𝑧)))) -s (((𝑥 ·s (𝑦 ·s 𝐶)) +s (𝑥 ·s (𝐵 ·s 𝑧))) -s (𝑥 ·s (𝑦 ·s 𝑧)))) = ((𝐴 ·s (𝑦 ·s 𝐶)) +s (((𝐴 ·s (𝐵 ·s 𝑧)) -s (𝐴 ·s (𝑦 ·s 𝑧))) -s (((𝑥 ·s (𝑦 ·s 𝐶)) +s (𝑥 ·s (𝐵 ·s 𝑧))) -s (𝑥 ·s (𝑦 ·s 𝑧)))))) |
147 | 144, 146 | eqtrd 2773 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → ((((𝐴 ·s (𝑦 ·s 𝐶)) +s (𝐴 ·s (𝐵 ·s 𝑧))) -s (𝐴 ·s (𝑦 ·s 𝑧))) -s (((𝑥 ·s (𝑦 ·s 𝐶)) +s (𝑥 ·s (𝐵 ·s 𝑧))) -s (𝑥 ·s (𝑦 ·s 𝑧)))) = ((𝐴 ·s (𝑦 ·s 𝐶)) +s (((𝐴 ·s (𝐵 ·s 𝑧)) -s (𝐴 ·s (𝑦 ·s 𝑧))) -s (((𝑥 ·s (𝑦 ·s 𝐶)) +s (𝑥 ·s (𝐵 ·s 𝑧))) -s (𝑥 ·s (𝑦 ·s 𝑧)))))) |
148 | 136, 141,
147 | 3eqtr4d 2783 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → ((((𝐴 ·s 𝑦) ·s 𝐶) -s ((𝑥 ·s 𝑦) ·s 𝐶)) +s (((𝐴 ·s 𝐵) ·s 𝑧) -s ((((𝑥 ·s 𝐵) ·s 𝑧) +s ((𝐴 ·s 𝑦) ·s 𝑧)) -s ((𝑥 ·s 𝑦) ·s 𝑧)))) = ((((𝐴 ·s (𝑦 ·s 𝐶)) +s (𝐴 ·s (𝐵 ·s 𝑧))) -s (𝐴 ·s (𝑦 ·s 𝑧))) -s (((𝑥 ·s (𝑦 ·s 𝐶)) +s (𝑥 ·s (𝐵 ·s 𝑧))) -s (𝑥 ·s (𝑦 ·s 𝑧))))) |
149 | 10, 148 | oveq12d 7422 |
. . . . . . 7
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → (((𝑥 ·s 𝐵) ·s 𝐶) +s ((((𝐴 ·s 𝑦) ·s 𝐶) -s ((𝑥 ·s 𝑦) ·s 𝐶)) +s (((𝐴 ·s 𝐵) ·s 𝑧) -s ((((𝑥 ·s 𝐵) ·s 𝑧) +s ((𝐴 ·s 𝑦) ·s 𝑧)) -s ((𝑥 ·s 𝑦) ·s 𝑧))))) = ((𝑥 ·s (𝐵 ·s 𝐶)) +s ((((𝐴 ·s (𝑦 ·s 𝐶)) +s (𝐴 ·s (𝐵 ·s 𝑧))) -s (𝐴 ·s (𝑦 ·s 𝑧))) -s (((𝑥 ·s (𝑦 ·s 𝐶)) +s (𝑥 ·s (𝐵 ·s 𝑧))) -s (𝑥 ·s (𝑦 ·s 𝑧)))))) |
150 | 39, 64 | mulscld 27571 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → ((𝑥 ·s 𝐵) ·s 𝐶) ∈ No
) |
151 | 150, 137 | addscld 27444 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → (((𝑥 ·s 𝐵) ·s 𝐶) +s ((𝐴 ·s 𝑦) ·s 𝐶)) ∈ No
) |
152 | 151, 65 | subscld 27515 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → ((((𝑥 ·s 𝐵) ·s 𝐶) +s ((𝐴 ·s 𝑦) ·s 𝐶)) -s ((𝑥 ·s 𝑦) ·s 𝐶)) ∈ No
) |
153 | 152, 124,
126 | addsubsassd 27528 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → ((((((𝑥 ·s 𝐵) ·s 𝐶) +s ((𝐴 ·s 𝑦) ·s 𝐶)) -s ((𝑥 ·s 𝑦) ·s 𝐶)) +s ((𝐴 ·s 𝐵) ·s 𝑧)) -s ((((𝑥 ·s 𝐵) ·s 𝑧) +s ((𝐴 ·s 𝑦) ·s 𝑧)) -s ((𝑥 ·s 𝑦) ·s 𝑧))) = (((((𝑥 ·s 𝐵) ·s 𝐶) +s ((𝐴 ·s 𝑦) ·s 𝐶)) -s ((𝑥 ·s 𝑦) ·s 𝐶)) +s (((𝐴 ·s 𝐵) ·s 𝑧) -s ((((𝑥 ·s 𝐵) ·s 𝑧) +s ((𝐴 ·s 𝑦) ·s 𝑧)) -s ((𝑥 ·s 𝑦) ·s 𝑧))))) |
154 | 150, 137,
65 | addsubsassd 27528 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → ((((𝑥 ·s 𝐵) ·s 𝐶) +s ((𝐴 ·s 𝑦) ·s 𝐶)) -s ((𝑥 ·s 𝑦) ·s 𝐶)) = (((𝑥 ·s 𝐵) ·s 𝐶) +s (((𝐴 ·s 𝑦) ·s 𝐶) -s ((𝑥 ·s 𝑦) ·s 𝐶)))) |
155 | 154 | oveq1d 7419 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → (((((𝑥 ·s 𝐵) ·s 𝐶) +s ((𝐴 ·s 𝑦) ·s 𝐶)) -s ((𝑥 ·s 𝑦) ·s 𝐶)) +s (((𝐴 ·s 𝐵) ·s 𝑧) -s ((((𝑥 ·s 𝐵) ·s 𝑧) +s ((𝐴 ·s 𝑦) ·s 𝑧)) -s ((𝑥 ·s 𝑦) ·s 𝑧)))) = ((((𝑥 ·s 𝐵) ·s 𝐶) +s (((𝐴 ·s 𝑦) ·s 𝐶) -s ((𝑥 ·s 𝑦) ·s 𝐶))) +s (((𝐴 ·s 𝐵) ·s 𝑧) -s ((((𝑥 ·s 𝐵) ·s 𝑧) +s ((𝐴 ·s 𝑦) ·s 𝑧)) -s ((𝑥 ·s 𝑦) ·s 𝑧))))) |
156 | 137, 65 | subscld 27515 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → (((𝐴 ·s 𝑦) ·s 𝐶) -s ((𝑥 ·s 𝑦) ·s 𝐶)) ∈ No
) |
157 | 150, 156,
138 | addsassd 27469 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → ((((𝑥 ·s 𝐵) ·s 𝐶) +s (((𝐴 ·s 𝑦) ·s 𝐶) -s ((𝑥 ·s 𝑦) ·s 𝐶))) +s (((𝐴 ·s 𝐵) ·s 𝑧) -s ((((𝑥 ·s 𝐵) ·s 𝑧) +s ((𝐴 ·s 𝑦) ·s 𝑧)) -s ((𝑥 ·s 𝑦) ·s 𝑧)))) = (((𝑥 ·s 𝐵) ·s 𝐶) +s ((((𝐴 ·s 𝑦) ·s 𝐶) -s ((𝑥 ·s 𝑦) ·s 𝐶)) +s (((𝐴 ·s 𝐵) ·s 𝑧) -s ((((𝑥 ·s 𝐵) ·s 𝑧) +s ((𝐴 ·s 𝑦) ·s 𝑧)) -s ((𝑥 ·s 𝑦) ·s 𝑧)))))) |
158 | 153, 155,
157 | 3eqtrd 2777 |
. . . . . . 7
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → ((((((𝑥 ·s 𝐵) ·s 𝐶) +s ((𝐴 ·s 𝑦) ·s 𝐶)) -s ((𝑥 ·s 𝑦) ·s 𝐶)) +s ((𝐴 ·s 𝐵) ·s 𝑧)) -s ((((𝑥 ·s 𝐵) ·s 𝑧) +s ((𝐴 ·s 𝑦) ·s 𝑧)) -s ((𝑥 ·s 𝑦) ·s 𝑧))) = (((𝑥 ·s 𝐵) ·s 𝐶) +s ((((𝐴 ·s 𝑦) ·s 𝐶) -s ((𝑥 ·s 𝑦) ·s 𝐶)) +s (((𝐴 ·s 𝐵) ·s 𝑧) -s ((((𝑥 ·s 𝐵) ·s 𝑧) +s ((𝐴 ·s 𝑦) ·s 𝑧)) -s ((𝑥 ·s 𝑦) ·s 𝑧)))))) |
159 | 38, 64 | mulscld 27571 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → (𝐵 ·s 𝐶) ∈ No
) |
160 | 36, 159 | mulscld 27571 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → (𝑥 ·s (𝐵 ·s 𝐶)) ∈ No
) |
161 | 142, 128 | addscld 27444 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → ((𝐴 ·s (𝑦 ·s 𝐶)) +s (𝐴 ·s (𝐵 ·s 𝑧))) ∈ No
) |
162 | 161, 130 | subscld 27515 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → (((𝐴 ·s (𝑦 ·s 𝐶)) +s (𝐴 ·s (𝐵 ·s 𝑧))) -s (𝐴 ·s (𝑦 ·s 𝑧))) ∈ No
) |
163 | 160, 162,
133 | addsubsassd 27528 |
. . . . . . 7
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → (((𝑥 ·s (𝐵 ·s 𝐶)) +s (((𝐴 ·s (𝑦 ·s 𝐶)) +s (𝐴 ·s (𝐵 ·s 𝑧))) -s (𝐴 ·s (𝑦 ·s 𝑧)))) -s (((𝑥 ·s (𝑦 ·s 𝐶)) +s (𝑥 ·s (𝐵 ·s 𝑧))) -s (𝑥 ·s (𝑦 ·s 𝑧)))) = ((𝑥 ·s (𝐵 ·s 𝐶)) +s ((((𝐴 ·s (𝑦 ·s 𝐶)) +s (𝐴 ·s (𝐵 ·s 𝑧))) -s (𝐴 ·s (𝑦 ·s 𝑧))) -s (((𝑥 ·s (𝑦 ·s 𝐶)) +s (𝑥 ·s (𝐵 ·s 𝑧))) -s (𝑥 ·s (𝑦 ·s 𝑧)))))) |
164 | 149, 158,
163 | 3eqtr4d 2783 |
. . . . . 6
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → ((((((𝑥 ·s 𝐵) ·s 𝐶) +s ((𝐴 ·s 𝑦) ·s 𝐶)) -s ((𝑥 ·s 𝑦) ·s 𝐶)) +s ((𝐴 ·s 𝐵) ·s 𝑧)) -s ((((𝑥 ·s 𝐵) ·s 𝑧) +s ((𝐴 ·s 𝑦) ·s 𝑧)) -s ((𝑥 ·s 𝑦) ·s 𝑧))) = (((𝑥 ·s (𝐵 ·s 𝐶)) +s (((𝐴 ·s (𝑦 ·s 𝐶)) +s (𝐴 ·s (𝐵 ·s 𝑧))) -s (𝐴 ·s (𝑦 ·s 𝑧)))) -s (((𝑥 ·s (𝑦 ·s 𝐶)) +s (𝑥 ·s (𝐵 ·s 𝑧))) -s (𝑥 ·s (𝑦 ·s 𝑧))))) |
165 | 39, 53 | addscld 27444 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → ((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) ∈ No
) |
166 | 165, 57, 64 | subsdird 27594 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → ((((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)) ·s 𝐶) = ((((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) ·s 𝐶) -s ((𝑥 ·s 𝑦) ·s 𝐶))) |
167 | 39, 53, 64 | addsdird 27592 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) ·s 𝐶) = (((𝑥 ·s 𝐵) ·s 𝐶) +s ((𝐴 ·s 𝑦) ·s 𝐶))) |
168 | 167 | oveq1d 7419 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → ((((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) ·s 𝐶) -s ((𝑥 ·s 𝑦) ·s 𝐶)) = ((((𝑥 ·s 𝐵) ·s 𝐶) +s ((𝐴 ·s 𝑦) ·s 𝐶)) -s ((𝑥 ·s 𝑦) ·s 𝐶))) |
169 | 166, 168 | eqtrd 2773 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → ((((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)) ·s 𝐶) = ((((𝑥 ·s 𝐵) ·s 𝐶) +s ((𝐴 ·s 𝑦) ·s 𝐶)) -s ((𝑥 ·s 𝑦) ·s 𝐶))) |
170 | 169 | oveq1d 7419 |
. . . . . . 7
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → (((((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧)) = (((((𝑥 ·s 𝐵) ·s 𝐶) +s ((𝐴 ·s 𝑦) ·s 𝐶)) -s ((𝑥 ·s 𝑦) ·s 𝐶)) +s ((𝐴 ·s 𝐵) ·s 𝑧))) |
171 | 165, 57, 44 | subsdird 27594 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → ((((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)) ·s 𝑧) = ((((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) ·s 𝑧) -s ((𝑥 ·s 𝑦) ·s 𝑧))) |
172 | 39, 53, 44 | addsdird 27592 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) ·s 𝑧) = (((𝑥 ·s 𝐵) ·s 𝑧) +s ((𝐴 ·s 𝑦) ·s 𝑧))) |
173 | 172 | oveq1d 7419 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → ((((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) ·s 𝑧) -s ((𝑥 ·s 𝑦) ·s 𝑧)) = ((((𝑥 ·s 𝐵) ·s 𝑧) +s ((𝐴 ·s 𝑦) ·s 𝑧)) -s ((𝑥 ·s 𝑦) ·s 𝑧))) |
174 | 171, 173 | eqtrd 2773 |
. . . . . . 7
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → ((((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)) ·s 𝑧) = ((((𝑥 ·s 𝐵) ·s 𝑧) +s ((𝐴 ·s 𝑦) ·s 𝑧)) -s ((𝑥 ·s 𝑦) ·s 𝑧))) |
175 | 170, 174 | oveq12d 7422 |
. . . . . 6
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → ((((((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧)) -s ((((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)) ·s 𝑧)) = ((((((𝑥 ·s 𝐵) ·s 𝐶) +s ((𝐴 ·s 𝑦) ·s 𝐶)) -s ((𝑥 ·s 𝑦) ·s 𝐶)) +s ((𝐴 ·s 𝐵) ·s 𝑧)) -s ((((𝑥 ·s 𝐵) ·s 𝑧) +s ((𝐴 ·s 𝑦) ·s 𝑧)) -s ((𝑥 ·s 𝑦) ·s 𝑧)))) |
176 | 102, 100 | addscld 27444 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → ((𝑦 ·s 𝐶) +s (𝐵 ·s 𝑧)) ∈ No
) |
177 | 47, 176, 129 | subsdid 27593 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → (𝐴 ·s (((𝑦 ·s 𝐶) +s (𝐵 ·s 𝑧)) -s (𝑦 ·s 𝑧))) = ((𝐴 ·s ((𝑦 ·s 𝐶) +s (𝐵 ·s 𝑧))) -s (𝐴 ·s (𝑦 ·s 𝑧)))) |
178 | 47, 102, 100 | addsdid 27591 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → (𝐴 ·s ((𝑦 ·s 𝐶) +s (𝐵 ·s 𝑧))) = ((𝐴 ·s (𝑦 ·s 𝐶)) +s (𝐴 ·s (𝐵 ·s 𝑧)))) |
179 | 178 | oveq1d 7419 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → ((𝐴 ·s ((𝑦 ·s 𝐶) +s (𝐵 ·s 𝑧))) -s (𝐴 ·s (𝑦 ·s 𝑧))) = (((𝐴 ·s (𝑦 ·s 𝐶)) +s (𝐴 ·s (𝐵 ·s 𝑧))) -s (𝐴 ·s (𝑦 ·s 𝑧)))) |
180 | 177, 179 | eqtrd 2773 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → (𝐴 ·s (((𝑦 ·s 𝐶) +s (𝐵 ·s 𝑧)) -s (𝑦 ·s 𝑧))) = (((𝐴 ·s (𝑦 ·s 𝐶)) +s (𝐴 ·s (𝐵 ·s 𝑧))) -s (𝐴 ·s (𝑦 ·s 𝑧)))) |
181 | 180 | oveq2d 7420 |
. . . . . . 7
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → ((𝑥 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦 ·s 𝐶) +s (𝐵 ·s 𝑧)) -s (𝑦 ·s 𝑧)))) = ((𝑥 ·s (𝐵 ·s 𝐶)) +s (((𝐴 ·s (𝑦 ·s 𝐶)) +s (𝐴 ·s (𝐵 ·s 𝑧))) -s (𝐴 ·s (𝑦 ·s 𝑧))))) |
182 | 36, 176, 129 | subsdid 27593 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → (𝑥 ·s (((𝑦 ·s 𝐶) +s (𝐵 ·s 𝑧)) -s (𝑦 ·s 𝑧))) = ((𝑥 ·s ((𝑦 ·s 𝐶) +s (𝐵 ·s 𝑧))) -s (𝑥 ·s (𝑦 ·s 𝑧)))) |
183 | 36, 102, 100 | addsdid 27591 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → (𝑥 ·s ((𝑦 ·s 𝐶) +s (𝐵 ·s 𝑧))) = ((𝑥 ·s (𝑦 ·s 𝐶)) +s (𝑥 ·s (𝐵 ·s 𝑧)))) |
184 | 183 | oveq1d 7419 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → ((𝑥 ·s ((𝑦 ·s 𝐶) +s (𝐵 ·s 𝑧))) -s (𝑥 ·s (𝑦 ·s 𝑧))) = (((𝑥 ·s (𝑦 ·s 𝐶)) +s (𝑥 ·s (𝐵 ·s 𝑧))) -s (𝑥 ·s (𝑦 ·s 𝑧)))) |
185 | 182, 184 | eqtrd 2773 |
. . . . . . 7
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → (𝑥 ·s (((𝑦 ·s 𝐶) +s (𝐵 ·s 𝑧)) -s (𝑦 ·s 𝑧))) = (((𝑥 ·s (𝑦 ·s 𝐶)) +s (𝑥 ·s (𝐵 ·s 𝑧))) -s (𝑥 ·s (𝑦 ·s 𝑧)))) |
186 | 181, 185 | oveq12d 7422 |
. . . . . 6
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → (((𝑥 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦 ·s 𝐶) +s (𝐵 ·s 𝑧)) -s (𝑦 ·s 𝑧)))) -s (𝑥 ·s (((𝑦 ·s 𝐶) +s (𝐵 ·s 𝑧)) -s (𝑦 ·s 𝑧)))) = (((𝑥 ·s (𝐵 ·s 𝐶)) +s (((𝐴 ·s (𝑦 ·s 𝐶)) +s (𝐴 ·s (𝐵 ·s 𝑧))) -s (𝐴 ·s (𝑦 ·s 𝑧)))) -s (((𝑥 ·s (𝑦 ·s 𝐶)) +s (𝑥 ·s (𝐵 ·s 𝑧))) -s (𝑥 ·s (𝑦 ·s 𝑧))))) |
187 | 164, 175,
186 | 3eqtr4d 2783 |
. . . . 5
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → ((((((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧)) -s ((((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)) ·s 𝑧)) = (((𝑥 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦 ·s 𝐶) +s (𝐵 ·s 𝑧)) -s (𝑦 ·s 𝑧)))) -s (𝑥 ·s (((𝑦 ·s 𝐶) +s (𝐵 ·s 𝑧)) -s (𝑦 ·s 𝑧))))) |
188 | 187 | eqeq2d 2744 |
. . . 4
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄) ∧ 𝑧 ∈ 𝑅)) → (𝑎 = ((((((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧)) -s ((((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)) ·s 𝑧)) ↔ 𝑎 = (((𝑥 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦 ·s 𝐶) +s (𝐵 ·s 𝑧)) -s (𝑦 ·s 𝑧)))) -s (𝑥 ·s (((𝑦 ·s 𝐶) +s (𝐵 ·s 𝑧)) -s (𝑦 ·s 𝑧)))))) |
189 | 188 | anassrs 469 |
. . 3
⊢ (((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄)) ∧ 𝑧 ∈ 𝑅) → (𝑎 = ((((((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧)) -s ((((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)) ·s 𝑧)) ↔ 𝑎 = (((𝑥 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦 ·s 𝐶) +s (𝐵 ·s 𝑧)) -s (𝑦 ·s 𝑧)))) -s (𝑥 ·s (((𝑦 ·s 𝐶) +s (𝐵 ·s 𝑧)) -s (𝑦 ·s 𝑧)))))) |
190 | 189 | rexbidva 3177 |
. 2
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑄)) → (∃𝑧 ∈ 𝑅 𝑎 = ((((((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧)) -s ((((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)) ·s 𝑧)) ↔ ∃𝑧 ∈ 𝑅 𝑎 = (((𝑥 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦 ·s 𝐶) +s (𝐵 ·s 𝑧)) -s (𝑦 ·s 𝑧)))) -s (𝑥 ·s (((𝑦 ·s 𝐶) +s (𝐵 ·s 𝑧)) -s (𝑦 ·s 𝑧)))))) |
191 | 190 | 2rexbidva 3218 |
1
⊢ (𝜑 → (∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑄 ∃𝑧 ∈ 𝑅 𝑎 = ((((((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧)) -s ((((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)) ·s 𝑧)) ↔ ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑄 ∃𝑧 ∈ 𝑅 𝑎 = (((𝑥 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦 ·s 𝐶) +s (𝐵 ·s 𝑧)) -s (𝑦 ·s 𝑧)))) -s (𝑥 ·s (((𝑦 ·s 𝐶) +s (𝐵 ·s 𝑧)) -s (𝑦 ·s 𝑧)))))) |