| Step |
Hyp |
Ref |
Expression |
| 1 |
|
oveq1 |
⊢ ( 𝑎 = 𝑑 → ( 𝑎 ·no ( 𝑏 +no 𝑐 ) ) = ( 𝑑 ·no ( 𝑏 +no 𝑐 ) ) ) |
| 2 |
|
oveq1 |
⊢ ( 𝑎 = 𝑑 → ( 𝑎 ·no 𝑏 ) = ( 𝑑 ·no 𝑏 ) ) |
| 3 |
|
oveq1 |
⊢ ( 𝑎 = 𝑑 → ( 𝑎 ·no 𝑐 ) = ( 𝑑 ·no 𝑐 ) ) |
| 4 |
2 3
|
oveq12d |
⊢ ( 𝑎 = 𝑑 → ( ( 𝑎 ·no 𝑏 ) +no ( 𝑎 ·no 𝑐 ) ) = ( ( 𝑑 ·no 𝑏 ) +no ( 𝑑 ·no 𝑐 ) ) ) |
| 5 |
1 4
|
eqeq12d |
⊢ ( 𝑎 = 𝑑 → ( ( 𝑎 ·no ( 𝑏 +no 𝑐 ) ) = ( ( 𝑎 ·no 𝑏 ) +no ( 𝑎 ·no 𝑐 ) ) ↔ ( 𝑑 ·no ( 𝑏 +no 𝑐 ) ) = ( ( 𝑑 ·no 𝑏 ) +no ( 𝑑 ·no 𝑐 ) ) ) ) |
| 6 |
|
oveq1 |
⊢ ( 𝑏 = 𝑒 → ( 𝑏 +no 𝑐 ) = ( 𝑒 +no 𝑐 ) ) |
| 7 |
6
|
oveq2d |
⊢ ( 𝑏 = 𝑒 → ( 𝑑 ·no ( 𝑏 +no 𝑐 ) ) = ( 𝑑 ·no ( 𝑒 +no 𝑐 ) ) ) |
| 8 |
|
oveq2 |
⊢ ( 𝑏 = 𝑒 → ( 𝑑 ·no 𝑏 ) = ( 𝑑 ·no 𝑒 ) ) |
| 9 |
8
|
oveq1d |
⊢ ( 𝑏 = 𝑒 → ( ( 𝑑 ·no 𝑏 ) +no ( 𝑑 ·no 𝑐 ) ) = ( ( 𝑑 ·no 𝑒 ) +no ( 𝑑 ·no 𝑐 ) ) ) |
| 10 |
7 9
|
eqeq12d |
⊢ ( 𝑏 = 𝑒 → ( ( 𝑑 ·no ( 𝑏 +no 𝑐 ) ) = ( ( 𝑑 ·no 𝑏 ) +no ( 𝑑 ·no 𝑐 ) ) ↔ ( 𝑑 ·no ( 𝑒 +no 𝑐 ) ) = ( ( 𝑑 ·no 𝑒 ) +no ( 𝑑 ·no 𝑐 ) ) ) ) |
| 11 |
|
oveq2 |
⊢ ( 𝑐 = 𝑓 → ( 𝑒 +no 𝑐 ) = ( 𝑒 +no 𝑓 ) ) |
| 12 |
11
|
oveq2d |
⊢ ( 𝑐 = 𝑓 → ( 𝑑 ·no ( 𝑒 +no 𝑐 ) ) = ( 𝑑 ·no ( 𝑒 +no 𝑓 ) ) ) |
| 13 |
|
oveq2 |
⊢ ( 𝑐 = 𝑓 → ( 𝑑 ·no 𝑐 ) = ( 𝑑 ·no 𝑓 ) ) |
| 14 |
13
|
oveq2d |
⊢ ( 𝑐 = 𝑓 → ( ( 𝑑 ·no 𝑒 ) +no ( 𝑑 ·no 𝑐 ) ) = ( ( 𝑑 ·no 𝑒 ) +no ( 𝑑 ·no 𝑓 ) ) ) |
| 15 |
12 14
|
eqeq12d |
⊢ ( 𝑐 = 𝑓 → ( ( 𝑑 ·no ( 𝑒 +no 𝑐 ) ) = ( ( 𝑑 ·no 𝑒 ) +no ( 𝑑 ·no 𝑐 ) ) ↔ ( 𝑑 ·no ( 𝑒 +no 𝑓 ) ) = ( ( 𝑑 ·no 𝑒 ) +no ( 𝑑 ·no 𝑓 ) ) ) ) |
| 16 |
|
oveq1 |
⊢ ( 𝑎 = 𝑑 → ( 𝑎 ·no ( 𝑒 +no 𝑓 ) ) = ( 𝑑 ·no ( 𝑒 +no 𝑓 ) ) ) |
| 17 |
|
oveq1 |
⊢ ( 𝑎 = 𝑑 → ( 𝑎 ·no 𝑒 ) = ( 𝑑 ·no 𝑒 ) ) |
| 18 |
|
oveq1 |
⊢ ( 𝑎 = 𝑑 → ( 𝑎 ·no 𝑓 ) = ( 𝑑 ·no 𝑓 ) ) |
| 19 |
17 18
|
oveq12d |
⊢ ( 𝑎 = 𝑑 → ( ( 𝑎 ·no 𝑒 ) +no ( 𝑎 ·no 𝑓 ) ) = ( ( 𝑑 ·no 𝑒 ) +no ( 𝑑 ·no 𝑓 ) ) ) |
| 20 |
16 19
|
eqeq12d |
⊢ ( 𝑎 = 𝑑 → ( ( 𝑎 ·no ( 𝑒 +no 𝑓 ) ) = ( ( 𝑎 ·no 𝑒 ) +no ( 𝑎 ·no 𝑓 ) ) ↔ ( 𝑑 ·no ( 𝑒 +no 𝑓 ) ) = ( ( 𝑑 ·no 𝑒 ) +no ( 𝑑 ·no 𝑓 ) ) ) ) |
| 21 |
|
oveq1 |
⊢ ( 𝑏 = 𝑒 → ( 𝑏 +no 𝑓 ) = ( 𝑒 +no 𝑓 ) ) |
| 22 |
21
|
oveq2d |
⊢ ( 𝑏 = 𝑒 → ( 𝑎 ·no ( 𝑏 +no 𝑓 ) ) = ( 𝑎 ·no ( 𝑒 +no 𝑓 ) ) ) |
| 23 |
|
oveq2 |
⊢ ( 𝑏 = 𝑒 → ( 𝑎 ·no 𝑏 ) = ( 𝑎 ·no 𝑒 ) ) |
| 24 |
23
|
oveq1d |
⊢ ( 𝑏 = 𝑒 → ( ( 𝑎 ·no 𝑏 ) +no ( 𝑎 ·no 𝑓 ) ) = ( ( 𝑎 ·no 𝑒 ) +no ( 𝑎 ·no 𝑓 ) ) ) |
| 25 |
22 24
|
eqeq12d |
⊢ ( 𝑏 = 𝑒 → ( ( 𝑎 ·no ( 𝑏 +no 𝑓 ) ) = ( ( 𝑎 ·no 𝑏 ) +no ( 𝑎 ·no 𝑓 ) ) ↔ ( 𝑎 ·no ( 𝑒 +no 𝑓 ) ) = ( ( 𝑎 ·no 𝑒 ) +no ( 𝑎 ·no 𝑓 ) ) ) ) |
| 26 |
21
|
oveq2d |
⊢ ( 𝑏 = 𝑒 → ( 𝑑 ·no ( 𝑏 +no 𝑓 ) ) = ( 𝑑 ·no ( 𝑒 +no 𝑓 ) ) ) |
| 27 |
8
|
oveq1d |
⊢ ( 𝑏 = 𝑒 → ( ( 𝑑 ·no 𝑏 ) +no ( 𝑑 ·no 𝑓 ) ) = ( ( 𝑑 ·no 𝑒 ) +no ( 𝑑 ·no 𝑓 ) ) ) |
| 28 |
26 27
|
eqeq12d |
⊢ ( 𝑏 = 𝑒 → ( ( 𝑑 ·no ( 𝑏 +no 𝑓 ) ) = ( ( 𝑑 ·no 𝑏 ) +no ( 𝑑 ·no 𝑓 ) ) ↔ ( 𝑑 ·no ( 𝑒 +no 𝑓 ) ) = ( ( 𝑑 ·no 𝑒 ) +no ( 𝑑 ·no 𝑓 ) ) ) ) |
| 29 |
11
|
oveq2d |
⊢ ( 𝑐 = 𝑓 → ( 𝑎 ·no ( 𝑒 +no 𝑐 ) ) = ( 𝑎 ·no ( 𝑒 +no 𝑓 ) ) ) |
| 30 |
|
oveq2 |
⊢ ( 𝑐 = 𝑓 → ( 𝑎 ·no 𝑐 ) = ( 𝑎 ·no 𝑓 ) ) |
| 31 |
30
|
oveq2d |
⊢ ( 𝑐 = 𝑓 → ( ( 𝑎 ·no 𝑒 ) +no ( 𝑎 ·no 𝑐 ) ) = ( ( 𝑎 ·no 𝑒 ) +no ( 𝑎 ·no 𝑓 ) ) ) |
| 32 |
29 31
|
eqeq12d |
⊢ ( 𝑐 = 𝑓 → ( ( 𝑎 ·no ( 𝑒 +no 𝑐 ) ) = ( ( 𝑎 ·no 𝑒 ) +no ( 𝑎 ·no 𝑐 ) ) ↔ ( 𝑎 ·no ( 𝑒 +no 𝑓 ) ) = ( ( 𝑎 ·no 𝑒 ) +no ( 𝑎 ·no 𝑓 ) ) ) ) |
| 33 |
|
oveq1 |
⊢ ( 𝑎 = 𝐴 → ( 𝑎 ·no ( 𝑏 +no 𝑐 ) ) = ( 𝐴 ·no ( 𝑏 +no 𝑐 ) ) ) |
| 34 |
|
oveq1 |
⊢ ( 𝑎 = 𝐴 → ( 𝑎 ·no 𝑏 ) = ( 𝐴 ·no 𝑏 ) ) |
| 35 |
|
oveq1 |
⊢ ( 𝑎 = 𝐴 → ( 𝑎 ·no 𝑐 ) = ( 𝐴 ·no 𝑐 ) ) |
| 36 |
34 35
|
oveq12d |
⊢ ( 𝑎 = 𝐴 → ( ( 𝑎 ·no 𝑏 ) +no ( 𝑎 ·no 𝑐 ) ) = ( ( 𝐴 ·no 𝑏 ) +no ( 𝐴 ·no 𝑐 ) ) ) |
| 37 |
33 36
|
eqeq12d |
⊢ ( 𝑎 = 𝐴 → ( ( 𝑎 ·no ( 𝑏 +no 𝑐 ) ) = ( ( 𝑎 ·no 𝑏 ) +no ( 𝑎 ·no 𝑐 ) ) ↔ ( 𝐴 ·no ( 𝑏 +no 𝑐 ) ) = ( ( 𝐴 ·no 𝑏 ) +no ( 𝐴 ·no 𝑐 ) ) ) ) |
| 38 |
|
oveq1 |
⊢ ( 𝑏 = 𝐵 → ( 𝑏 +no 𝑐 ) = ( 𝐵 +no 𝑐 ) ) |
| 39 |
38
|
oveq2d |
⊢ ( 𝑏 = 𝐵 → ( 𝐴 ·no ( 𝑏 +no 𝑐 ) ) = ( 𝐴 ·no ( 𝐵 +no 𝑐 ) ) ) |
| 40 |
|
oveq2 |
⊢ ( 𝑏 = 𝐵 → ( 𝐴 ·no 𝑏 ) = ( 𝐴 ·no 𝐵 ) ) |
| 41 |
40
|
oveq1d |
⊢ ( 𝑏 = 𝐵 → ( ( 𝐴 ·no 𝑏 ) +no ( 𝐴 ·no 𝑐 ) ) = ( ( 𝐴 ·no 𝐵 ) +no ( 𝐴 ·no 𝑐 ) ) ) |
| 42 |
39 41
|
eqeq12d |
⊢ ( 𝑏 = 𝐵 → ( ( 𝐴 ·no ( 𝑏 +no 𝑐 ) ) = ( ( 𝐴 ·no 𝑏 ) +no ( 𝐴 ·no 𝑐 ) ) ↔ ( 𝐴 ·no ( 𝐵 +no 𝑐 ) ) = ( ( 𝐴 ·no 𝐵 ) +no ( 𝐴 ·no 𝑐 ) ) ) ) |
| 43 |
|
oveq2 |
⊢ ( 𝑐 = 𝐶 → ( 𝐵 +no 𝑐 ) = ( 𝐵 +no 𝐶 ) ) |
| 44 |
43
|
oveq2d |
⊢ ( 𝑐 = 𝐶 → ( 𝐴 ·no ( 𝐵 +no 𝑐 ) ) = ( 𝐴 ·no ( 𝐵 +no 𝐶 ) ) ) |
| 45 |
|
oveq2 |
⊢ ( 𝑐 = 𝐶 → ( 𝐴 ·no 𝑐 ) = ( 𝐴 ·no 𝐶 ) ) |
| 46 |
45
|
oveq2d |
⊢ ( 𝑐 = 𝐶 → ( ( 𝐴 ·no 𝐵 ) +no ( 𝐴 ·no 𝑐 ) ) = ( ( 𝐴 ·no 𝐵 ) +no ( 𝐴 ·no 𝐶 ) ) ) |
| 47 |
44 46
|
eqeq12d |
⊢ ( 𝑐 = 𝐶 → ( ( 𝐴 ·no ( 𝐵 +no 𝑐 ) ) = ( ( 𝐴 ·no 𝐵 ) +no ( 𝐴 ·no 𝑐 ) ) ↔ ( 𝐴 ·no ( 𝐵 +no 𝐶 ) ) = ( ( 𝐴 ·no 𝐵 ) +no ( 𝐴 ·no 𝐶 ) ) ) ) |
| 48 |
|
simpl1 |
⊢ ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On ) ∧ ( ( ∀ 𝑑 ∈ 𝑎 ∀ 𝑒 ∈ 𝑏 ∀ 𝑓 ∈ 𝑐 ( 𝑑 ·no ( 𝑒 +no 𝑓 ) ) = ( ( 𝑑 ·no 𝑒 ) +no ( 𝑑 ·no 𝑓 ) ) ∧ ∀ 𝑑 ∈ 𝑎 ∀ 𝑒 ∈ 𝑏 ( 𝑑 ·no ( 𝑒 +no 𝑐 ) ) = ( ( 𝑑 ·no 𝑒 ) +no ( 𝑑 ·no 𝑐 ) ) ∧ ∀ 𝑑 ∈ 𝑎 ∀ 𝑓 ∈ 𝑐 ( 𝑑 ·no ( 𝑏 +no 𝑓 ) ) = ( ( 𝑑 ·no 𝑏 ) +no ( 𝑑 ·no 𝑓 ) ) ) ∧ ( ∀ 𝑑 ∈ 𝑎 ( 𝑑 ·no ( 𝑏 +no 𝑐 ) ) = ( ( 𝑑 ·no 𝑏 ) +no ( 𝑑 ·no 𝑐 ) ) ∧ ∀ 𝑒 ∈ 𝑏 ∀ 𝑓 ∈ 𝑐 ( 𝑎 ·no ( 𝑒 +no 𝑓 ) ) = ( ( 𝑎 ·no 𝑒 ) +no ( 𝑎 ·no 𝑓 ) ) ∧ ∀ 𝑒 ∈ 𝑏 ( 𝑎 ·no ( 𝑒 +no 𝑐 ) ) = ( ( 𝑎 ·no 𝑒 ) +no ( 𝑎 ·no 𝑐 ) ) ) ∧ ∀ 𝑓 ∈ 𝑐 ( 𝑎 ·no ( 𝑏 +no 𝑓 ) ) = ( ( 𝑎 ·no 𝑏 ) +no ( 𝑎 ·no 𝑓 ) ) ) ) → 𝑎 ∈ On ) |
| 49 |
|
simpl2 |
⊢ ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On ) ∧ ( ( ∀ 𝑑 ∈ 𝑎 ∀ 𝑒 ∈ 𝑏 ∀ 𝑓 ∈ 𝑐 ( 𝑑 ·no ( 𝑒 +no 𝑓 ) ) = ( ( 𝑑 ·no 𝑒 ) +no ( 𝑑 ·no 𝑓 ) ) ∧ ∀ 𝑑 ∈ 𝑎 ∀ 𝑒 ∈ 𝑏 ( 𝑑 ·no ( 𝑒 +no 𝑐 ) ) = ( ( 𝑑 ·no 𝑒 ) +no ( 𝑑 ·no 𝑐 ) ) ∧ ∀ 𝑑 ∈ 𝑎 ∀ 𝑓 ∈ 𝑐 ( 𝑑 ·no ( 𝑏 +no 𝑓 ) ) = ( ( 𝑑 ·no 𝑏 ) +no ( 𝑑 ·no 𝑓 ) ) ) ∧ ( ∀ 𝑑 ∈ 𝑎 ( 𝑑 ·no ( 𝑏 +no 𝑐 ) ) = ( ( 𝑑 ·no 𝑏 ) +no ( 𝑑 ·no 𝑐 ) ) ∧ ∀ 𝑒 ∈ 𝑏 ∀ 𝑓 ∈ 𝑐 ( 𝑎 ·no ( 𝑒 +no 𝑓 ) ) = ( ( 𝑎 ·no 𝑒 ) +no ( 𝑎 ·no 𝑓 ) ) ∧ ∀ 𝑒 ∈ 𝑏 ( 𝑎 ·no ( 𝑒 +no 𝑐 ) ) = ( ( 𝑎 ·no 𝑒 ) +no ( 𝑎 ·no 𝑐 ) ) ) ∧ ∀ 𝑓 ∈ 𝑐 ( 𝑎 ·no ( 𝑏 +no 𝑓 ) ) = ( ( 𝑎 ·no 𝑏 ) +no ( 𝑎 ·no 𝑓 ) ) ) ) → 𝑏 ∈ On ) |
| 50 |
|
simpl3 |
⊢ ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On ) ∧ ( ( ∀ 𝑑 ∈ 𝑎 ∀ 𝑒 ∈ 𝑏 ∀ 𝑓 ∈ 𝑐 ( 𝑑 ·no ( 𝑒 +no 𝑓 ) ) = ( ( 𝑑 ·no 𝑒 ) +no ( 𝑑 ·no 𝑓 ) ) ∧ ∀ 𝑑 ∈ 𝑎 ∀ 𝑒 ∈ 𝑏 ( 𝑑 ·no ( 𝑒 +no 𝑐 ) ) = ( ( 𝑑 ·no 𝑒 ) +no ( 𝑑 ·no 𝑐 ) ) ∧ ∀ 𝑑 ∈ 𝑎 ∀ 𝑓 ∈ 𝑐 ( 𝑑 ·no ( 𝑏 +no 𝑓 ) ) = ( ( 𝑑 ·no 𝑏 ) +no ( 𝑑 ·no 𝑓 ) ) ) ∧ ( ∀ 𝑑 ∈ 𝑎 ( 𝑑 ·no ( 𝑏 +no 𝑐 ) ) = ( ( 𝑑 ·no 𝑏 ) +no ( 𝑑 ·no 𝑐 ) ) ∧ ∀ 𝑒 ∈ 𝑏 ∀ 𝑓 ∈ 𝑐 ( 𝑎 ·no ( 𝑒 +no 𝑓 ) ) = ( ( 𝑎 ·no 𝑒 ) +no ( 𝑎 ·no 𝑓 ) ) ∧ ∀ 𝑒 ∈ 𝑏 ( 𝑎 ·no ( 𝑒 +no 𝑐 ) ) = ( ( 𝑎 ·no 𝑒 ) +no ( 𝑎 ·no 𝑐 ) ) ) ∧ ∀ 𝑓 ∈ 𝑐 ( 𝑎 ·no ( 𝑏 +no 𝑓 ) ) = ( ( 𝑎 ·no 𝑏 ) +no ( 𝑎 ·no 𝑓 ) ) ) ) → 𝑐 ∈ On ) |
| 51 |
|
simpr21 |
⊢ ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On ) ∧ ( ( ∀ 𝑑 ∈ 𝑎 ∀ 𝑒 ∈ 𝑏 ∀ 𝑓 ∈ 𝑐 ( 𝑑 ·no ( 𝑒 +no 𝑓 ) ) = ( ( 𝑑 ·no 𝑒 ) +no ( 𝑑 ·no 𝑓 ) ) ∧ ∀ 𝑑 ∈ 𝑎 ∀ 𝑒 ∈ 𝑏 ( 𝑑 ·no ( 𝑒 +no 𝑐 ) ) = ( ( 𝑑 ·no 𝑒 ) +no ( 𝑑 ·no 𝑐 ) ) ∧ ∀ 𝑑 ∈ 𝑎 ∀ 𝑓 ∈ 𝑐 ( 𝑑 ·no ( 𝑏 +no 𝑓 ) ) = ( ( 𝑑 ·no 𝑏 ) +no ( 𝑑 ·no 𝑓 ) ) ) ∧ ( ∀ 𝑑 ∈ 𝑎 ( 𝑑 ·no ( 𝑏 +no 𝑐 ) ) = ( ( 𝑑 ·no 𝑏 ) +no ( 𝑑 ·no 𝑐 ) ) ∧ ∀ 𝑒 ∈ 𝑏 ∀ 𝑓 ∈ 𝑐 ( 𝑎 ·no ( 𝑒 +no 𝑓 ) ) = ( ( 𝑎 ·no 𝑒 ) +no ( 𝑎 ·no 𝑓 ) ) ∧ ∀ 𝑒 ∈ 𝑏 ( 𝑎 ·no ( 𝑒 +no 𝑐 ) ) = ( ( 𝑎 ·no 𝑒 ) +no ( 𝑎 ·no 𝑐 ) ) ) ∧ ∀ 𝑓 ∈ 𝑐 ( 𝑎 ·no ( 𝑏 +no 𝑓 ) ) = ( ( 𝑎 ·no 𝑏 ) +no ( 𝑎 ·no 𝑓 ) ) ) ) → ∀ 𝑑 ∈ 𝑎 ( 𝑑 ·no ( 𝑏 +no 𝑐 ) ) = ( ( 𝑑 ·no 𝑏 ) +no ( 𝑑 ·no 𝑐 ) ) ) |
| 52 |
|
simpr23 |
⊢ ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On ) ∧ ( ( ∀ 𝑑 ∈ 𝑎 ∀ 𝑒 ∈ 𝑏 ∀ 𝑓 ∈ 𝑐 ( 𝑑 ·no ( 𝑒 +no 𝑓 ) ) = ( ( 𝑑 ·no 𝑒 ) +no ( 𝑑 ·no 𝑓 ) ) ∧ ∀ 𝑑 ∈ 𝑎 ∀ 𝑒 ∈ 𝑏 ( 𝑑 ·no ( 𝑒 +no 𝑐 ) ) = ( ( 𝑑 ·no 𝑒 ) +no ( 𝑑 ·no 𝑐 ) ) ∧ ∀ 𝑑 ∈ 𝑎 ∀ 𝑓 ∈ 𝑐 ( 𝑑 ·no ( 𝑏 +no 𝑓 ) ) = ( ( 𝑑 ·no 𝑏 ) +no ( 𝑑 ·no 𝑓 ) ) ) ∧ ( ∀ 𝑑 ∈ 𝑎 ( 𝑑 ·no ( 𝑏 +no 𝑐 ) ) = ( ( 𝑑 ·no 𝑏 ) +no ( 𝑑 ·no 𝑐 ) ) ∧ ∀ 𝑒 ∈ 𝑏 ∀ 𝑓 ∈ 𝑐 ( 𝑎 ·no ( 𝑒 +no 𝑓 ) ) = ( ( 𝑎 ·no 𝑒 ) +no ( 𝑎 ·no 𝑓 ) ) ∧ ∀ 𝑒 ∈ 𝑏 ( 𝑎 ·no ( 𝑒 +no 𝑐 ) ) = ( ( 𝑎 ·no 𝑒 ) +no ( 𝑎 ·no 𝑐 ) ) ) ∧ ∀ 𝑓 ∈ 𝑐 ( 𝑎 ·no ( 𝑏 +no 𝑓 ) ) = ( ( 𝑎 ·no 𝑏 ) +no ( 𝑎 ·no 𝑓 ) ) ) ) → ∀ 𝑒 ∈ 𝑏 ( 𝑎 ·no ( 𝑒 +no 𝑐 ) ) = ( ( 𝑎 ·no 𝑒 ) +no ( 𝑎 ·no 𝑐 ) ) ) |
| 53 |
|
simpr3 |
⊢ ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On ) ∧ ( ( ∀ 𝑑 ∈ 𝑎 ∀ 𝑒 ∈ 𝑏 ∀ 𝑓 ∈ 𝑐 ( 𝑑 ·no ( 𝑒 +no 𝑓 ) ) = ( ( 𝑑 ·no 𝑒 ) +no ( 𝑑 ·no 𝑓 ) ) ∧ ∀ 𝑑 ∈ 𝑎 ∀ 𝑒 ∈ 𝑏 ( 𝑑 ·no ( 𝑒 +no 𝑐 ) ) = ( ( 𝑑 ·no 𝑒 ) +no ( 𝑑 ·no 𝑐 ) ) ∧ ∀ 𝑑 ∈ 𝑎 ∀ 𝑓 ∈ 𝑐 ( 𝑑 ·no ( 𝑏 +no 𝑓 ) ) = ( ( 𝑑 ·no 𝑏 ) +no ( 𝑑 ·no 𝑓 ) ) ) ∧ ( ∀ 𝑑 ∈ 𝑎 ( 𝑑 ·no ( 𝑏 +no 𝑐 ) ) = ( ( 𝑑 ·no 𝑏 ) +no ( 𝑑 ·no 𝑐 ) ) ∧ ∀ 𝑒 ∈ 𝑏 ∀ 𝑓 ∈ 𝑐 ( 𝑎 ·no ( 𝑒 +no 𝑓 ) ) = ( ( 𝑎 ·no 𝑒 ) +no ( 𝑎 ·no 𝑓 ) ) ∧ ∀ 𝑒 ∈ 𝑏 ( 𝑎 ·no ( 𝑒 +no 𝑐 ) ) = ( ( 𝑎 ·no 𝑒 ) +no ( 𝑎 ·no 𝑐 ) ) ) ∧ ∀ 𝑓 ∈ 𝑐 ( 𝑎 ·no ( 𝑏 +no 𝑓 ) ) = ( ( 𝑎 ·no 𝑏 ) +no ( 𝑎 ·no 𝑓 ) ) ) ) → ∀ 𝑓 ∈ 𝑐 ( 𝑎 ·no ( 𝑏 +no 𝑓 ) ) = ( ( 𝑎 ·no 𝑏 ) +no ( 𝑎 ·no 𝑓 ) ) ) |
| 54 |
|
simpr12 |
⊢ ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On ) ∧ ( ( ∀ 𝑑 ∈ 𝑎 ∀ 𝑒 ∈ 𝑏 ∀ 𝑓 ∈ 𝑐 ( 𝑑 ·no ( 𝑒 +no 𝑓 ) ) = ( ( 𝑑 ·no 𝑒 ) +no ( 𝑑 ·no 𝑓 ) ) ∧ ∀ 𝑑 ∈ 𝑎 ∀ 𝑒 ∈ 𝑏 ( 𝑑 ·no ( 𝑒 +no 𝑐 ) ) = ( ( 𝑑 ·no 𝑒 ) +no ( 𝑑 ·no 𝑐 ) ) ∧ ∀ 𝑑 ∈ 𝑎 ∀ 𝑓 ∈ 𝑐 ( 𝑑 ·no ( 𝑏 +no 𝑓 ) ) = ( ( 𝑑 ·no 𝑏 ) +no ( 𝑑 ·no 𝑓 ) ) ) ∧ ( ∀ 𝑑 ∈ 𝑎 ( 𝑑 ·no ( 𝑏 +no 𝑐 ) ) = ( ( 𝑑 ·no 𝑏 ) +no ( 𝑑 ·no 𝑐 ) ) ∧ ∀ 𝑒 ∈ 𝑏 ∀ 𝑓 ∈ 𝑐 ( 𝑎 ·no ( 𝑒 +no 𝑓 ) ) = ( ( 𝑎 ·no 𝑒 ) +no ( 𝑎 ·no 𝑓 ) ) ∧ ∀ 𝑒 ∈ 𝑏 ( 𝑎 ·no ( 𝑒 +no 𝑐 ) ) = ( ( 𝑎 ·no 𝑒 ) +no ( 𝑎 ·no 𝑐 ) ) ) ∧ ∀ 𝑓 ∈ 𝑐 ( 𝑎 ·no ( 𝑏 +no 𝑓 ) ) = ( ( 𝑎 ·no 𝑏 ) +no ( 𝑎 ·no 𝑓 ) ) ) ) → ∀ 𝑑 ∈ 𝑎 ∀ 𝑒 ∈ 𝑏 ( 𝑑 ·no ( 𝑒 +no 𝑐 ) ) = ( ( 𝑑 ·no 𝑒 ) +no ( 𝑑 ·no 𝑐 ) ) ) |
| 55 |
|
simpr13 |
⊢ ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On ) ∧ ( ( ∀ 𝑑 ∈ 𝑎 ∀ 𝑒 ∈ 𝑏 ∀ 𝑓 ∈ 𝑐 ( 𝑑 ·no ( 𝑒 +no 𝑓 ) ) = ( ( 𝑑 ·no 𝑒 ) +no ( 𝑑 ·no 𝑓 ) ) ∧ ∀ 𝑑 ∈ 𝑎 ∀ 𝑒 ∈ 𝑏 ( 𝑑 ·no ( 𝑒 +no 𝑐 ) ) = ( ( 𝑑 ·no 𝑒 ) +no ( 𝑑 ·no 𝑐 ) ) ∧ ∀ 𝑑 ∈ 𝑎 ∀ 𝑓 ∈ 𝑐 ( 𝑑 ·no ( 𝑏 +no 𝑓 ) ) = ( ( 𝑑 ·no 𝑏 ) +no ( 𝑑 ·no 𝑓 ) ) ) ∧ ( ∀ 𝑑 ∈ 𝑎 ( 𝑑 ·no ( 𝑏 +no 𝑐 ) ) = ( ( 𝑑 ·no 𝑏 ) +no ( 𝑑 ·no 𝑐 ) ) ∧ ∀ 𝑒 ∈ 𝑏 ∀ 𝑓 ∈ 𝑐 ( 𝑎 ·no ( 𝑒 +no 𝑓 ) ) = ( ( 𝑎 ·no 𝑒 ) +no ( 𝑎 ·no 𝑓 ) ) ∧ ∀ 𝑒 ∈ 𝑏 ( 𝑎 ·no ( 𝑒 +no 𝑐 ) ) = ( ( 𝑎 ·no 𝑒 ) +no ( 𝑎 ·no 𝑐 ) ) ) ∧ ∀ 𝑓 ∈ 𝑐 ( 𝑎 ·no ( 𝑏 +no 𝑓 ) ) = ( ( 𝑎 ·no 𝑏 ) +no ( 𝑎 ·no 𝑓 ) ) ) ) → ∀ 𝑑 ∈ 𝑎 ∀ 𝑓 ∈ 𝑐 ( 𝑑 ·no ( 𝑏 +no 𝑓 ) ) = ( ( 𝑑 ·no 𝑏 ) +no ( 𝑑 ·no 𝑓 ) ) ) |
| 56 |
48 49 50 51 52 53 54 55
|
nadddilem4 |
⊢ ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On ) ∧ ( ( ∀ 𝑑 ∈ 𝑎 ∀ 𝑒 ∈ 𝑏 ∀ 𝑓 ∈ 𝑐 ( 𝑑 ·no ( 𝑒 +no 𝑓 ) ) = ( ( 𝑑 ·no 𝑒 ) +no ( 𝑑 ·no 𝑓 ) ) ∧ ∀ 𝑑 ∈ 𝑎 ∀ 𝑒 ∈ 𝑏 ( 𝑑 ·no ( 𝑒 +no 𝑐 ) ) = ( ( 𝑑 ·no 𝑒 ) +no ( 𝑑 ·no 𝑐 ) ) ∧ ∀ 𝑑 ∈ 𝑎 ∀ 𝑓 ∈ 𝑐 ( 𝑑 ·no ( 𝑏 +no 𝑓 ) ) = ( ( 𝑑 ·no 𝑏 ) +no ( 𝑑 ·no 𝑓 ) ) ) ∧ ( ∀ 𝑑 ∈ 𝑎 ( 𝑑 ·no ( 𝑏 +no 𝑐 ) ) = ( ( 𝑑 ·no 𝑏 ) +no ( 𝑑 ·no 𝑐 ) ) ∧ ∀ 𝑒 ∈ 𝑏 ∀ 𝑓 ∈ 𝑐 ( 𝑎 ·no ( 𝑒 +no 𝑓 ) ) = ( ( 𝑎 ·no 𝑒 ) +no ( 𝑎 ·no 𝑓 ) ) ∧ ∀ 𝑒 ∈ 𝑏 ( 𝑎 ·no ( 𝑒 +no 𝑐 ) ) = ( ( 𝑎 ·no 𝑒 ) +no ( 𝑎 ·no 𝑐 ) ) ) ∧ ∀ 𝑓 ∈ 𝑐 ( 𝑎 ·no ( 𝑏 +no 𝑓 ) ) = ( ( 𝑎 ·no 𝑏 ) +no ( 𝑎 ·no 𝑓 ) ) ) ) → ( 𝑎 ·no ( 𝑏 +no 𝑐 ) ) ⊆ ( ( 𝑎 ·no 𝑏 ) +no ( 𝑎 ·no 𝑐 ) ) ) |
| 57 |
48 49 50 51 52 53 54 55
|
nadddilem2 |
⊢ ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On ) ∧ ( ( ∀ 𝑑 ∈ 𝑎 ∀ 𝑒 ∈ 𝑏 ∀ 𝑓 ∈ 𝑐 ( 𝑑 ·no ( 𝑒 +no 𝑓 ) ) = ( ( 𝑑 ·no 𝑒 ) +no ( 𝑑 ·no 𝑓 ) ) ∧ ∀ 𝑑 ∈ 𝑎 ∀ 𝑒 ∈ 𝑏 ( 𝑑 ·no ( 𝑒 +no 𝑐 ) ) = ( ( 𝑑 ·no 𝑒 ) +no ( 𝑑 ·no 𝑐 ) ) ∧ ∀ 𝑑 ∈ 𝑎 ∀ 𝑓 ∈ 𝑐 ( 𝑑 ·no ( 𝑏 +no 𝑓 ) ) = ( ( 𝑑 ·no 𝑏 ) +no ( 𝑑 ·no 𝑓 ) ) ) ∧ ( ∀ 𝑑 ∈ 𝑎 ( 𝑑 ·no ( 𝑏 +no 𝑐 ) ) = ( ( 𝑑 ·no 𝑏 ) +no ( 𝑑 ·no 𝑐 ) ) ∧ ∀ 𝑒 ∈ 𝑏 ∀ 𝑓 ∈ 𝑐 ( 𝑎 ·no ( 𝑒 +no 𝑓 ) ) = ( ( 𝑎 ·no 𝑒 ) +no ( 𝑎 ·no 𝑓 ) ) ∧ ∀ 𝑒 ∈ 𝑏 ( 𝑎 ·no ( 𝑒 +no 𝑐 ) ) = ( ( 𝑎 ·no 𝑒 ) +no ( 𝑎 ·no 𝑐 ) ) ) ∧ ∀ 𝑓 ∈ 𝑐 ( 𝑎 ·no ( 𝑏 +no 𝑓 ) ) = ( ( 𝑎 ·no 𝑏 ) +no ( 𝑎 ·no 𝑓 ) ) ) ) → ( ( 𝑎 ·no 𝑏 ) +no ( 𝑎 ·no 𝑐 ) ) ⊆ ( 𝑎 ·no ( 𝑏 +no 𝑐 ) ) ) |
| 58 |
56 57
|
eqssd |
⊢ ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On ) ∧ ( ( ∀ 𝑑 ∈ 𝑎 ∀ 𝑒 ∈ 𝑏 ∀ 𝑓 ∈ 𝑐 ( 𝑑 ·no ( 𝑒 +no 𝑓 ) ) = ( ( 𝑑 ·no 𝑒 ) +no ( 𝑑 ·no 𝑓 ) ) ∧ ∀ 𝑑 ∈ 𝑎 ∀ 𝑒 ∈ 𝑏 ( 𝑑 ·no ( 𝑒 +no 𝑐 ) ) = ( ( 𝑑 ·no 𝑒 ) +no ( 𝑑 ·no 𝑐 ) ) ∧ ∀ 𝑑 ∈ 𝑎 ∀ 𝑓 ∈ 𝑐 ( 𝑑 ·no ( 𝑏 +no 𝑓 ) ) = ( ( 𝑑 ·no 𝑏 ) +no ( 𝑑 ·no 𝑓 ) ) ) ∧ ( ∀ 𝑑 ∈ 𝑎 ( 𝑑 ·no ( 𝑏 +no 𝑐 ) ) = ( ( 𝑑 ·no 𝑏 ) +no ( 𝑑 ·no 𝑐 ) ) ∧ ∀ 𝑒 ∈ 𝑏 ∀ 𝑓 ∈ 𝑐 ( 𝑎 ·no ( 𝑒 +no 𝑓 ) ) = ( ( 𝑎 ·no 𝑒 ) +no ( 𝑎 ·no 𝑓 ) ) ∧ ∀ 𝑒 ∈ 𝑏 ( 𝑎 ·no ( 𝑒 +no 𝑐 ) ) = ( ( 𝑎 ·no 𝑒 ) +no ( 𝑎 ·no 𝑐 ) ) ) ∧ ∀ 𝑓 ∈ 𝑐 ( 𝑎 ·no ( 𝑏 +no 𝑓 ) ) = ( ( 𝑎 ·no 𝑏 ) +no ( 𝑎 ·no 𝑓 ) ) ) ) → ( 𝑎 ·no ( 𝑏 +no 𝑐 ) ) = ( ( 𝑎 ·no 𝑏 ) +no ( 𝑎 ·no 𝑐 ) ) ) |
| 59 |
58
|
ex |
⊢ ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On ) → ( ( ( ∀ 𝑑 ∈ 𝑎 ∀ 𝑒 ∈ 𝑏 ∀ 𝑓 ∈ 𝑐 ( 𝑑 ·no ( 𝑒 +no 𝑓 ) ) = ( ( 𝑑 ·no 𝑒 ) +no ( 𝑑 ·no 𝑓 ) ) ∧ ∀ 𝑑 ∈ 𝑎 ∀ 𝑒 ∈ 𝑏 ( 𝑑 ·no ( 𝑒 +no 𝑐 ) ) = ( ( 𝑑 ·no 𝑒 ) +no ( 𝑑 ·no 𝑐 ) ) ∧ ∀ 𝑑 ∈ 𝑎 ∀ 𝑓 ∈ 𝑐 ( 𝑑 ·no ( 𝑏 +no 𝑓 ) ) = ( ( 𝑑 ·no 𝑏 ) +no ( 𝑑 ·no 𝑓 ) ) ) ∧ ( ∀ 𝑑 ∈ 𝑎 ( 𝑑 ·no ( 𝑏 +no 𝑐 ) ) = ( ( 𝑑 ·no 𝑏 ) +no ( 𝑑 ·no 𝑐 ) ) ∧ ∀ 𝑒 ∈ 𝑏 ∀ 𝑓 ∈ 𝑐 ( 𝑎 ·no ( 𝑒 +no 𝑓 ) ) = ( ( 𝑎 ·no 𝑒 ) +no ( 𝑎 ·no 𝑓 ) ) ∧ ∀ 𝑒 ∈ 𝑏 ( 𝑎 ·no ( 𝑒 +no 𝑐 ) ) = ( ( 𝑎 ·no 𝑒 ) +no ( 𝑎 ·no 𝑐 ) ) ) ∧ ∀ 𝑓 ∈ 𝑐 ( 𝑎 ·no ( 𝑏 +no 𝑓 ) ) = ( ( 𝑎 ·no 𝑏 ) +no ( 𝑎 ·no 𝑓 ) ) ) → ( 𝑎 ·no ( 𝑏 +no 𝑐 ) ) = ( ( 𝑎 ·no 𝑏 ) +no ( 𝑎 ·no 𝑐 ) ) ) ) |
| 60 |
5 10 15 20 25 28 32 37 42 47 59
|
on3ind |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) → ( 𝐴 ·no ( 𝐵 +no 𝐶 ) ) = ( ( 𝐴 ·no 𝐵 ) +no ( 𝐴 ·no 𝐶 ) ) ) |