| Step |
Hyp |
Ref |
Expression |
| 1 |
|
oveq1 |
⊢ ( 𝑥 = 𝑥𝑂 → ( 𝑥 ·s ( 𝑦 +s 𝑧 ) ) = ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧 ) ) ) |
| 2 |
|
oveq1 |
⊢ ( 𝑥 = 𝑥𝑂 → ( 𝑥 ·s 𝑦 ) = ( 𝑥𝑂 ·s 𝑦 ) ) |
| 3 |
|
oveq1 |
⊢ ( 𝑥 = 𝑥𝑂 → ( 𝑥 ·s 𝑧 ) = ( 𝑥𝑂 ·s 𝑧 ) ) |
| 4 |
2 3
|
oveq12d |
⊢ ( 𝑥 = 𝑥𝑂 → ( ( 𝑥 ·s 𝑦 ) +s ( 𝑥 ·s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ) |
| 5 |
1 4
|
eqeq12d |
⊢ ( 𝑥 = 𝑥𝑂 → ( ( 𝑥 ·s ( 𝑦 +s 𝑧 ) ) = ( ( 𝑥 ·s 𝑦 ) +s ( 𝑥 ·s 𝑧 ) ) ↔ ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ) ) |
| 6 |
|
oveq1 |
⊢ ( 𝑦 = 𝑦𝑂 → ( 𝑦 +s 𝑧 ) = ( 𝑦𝑂 +s 𝑧 ) ) |
| 7 |
6
|
oveq2d |
⊢ ( 𝑦 = 𝑦𝑂 → ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧 ) ) = ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧 ) ) ) |
| 8 |
|
oveq2 |
⊢ ( 𝑦 = 𝑦𝑂 → ( 𝑥𝑂 ·s 𝑦 ) = ( 𝑥𝑂 ·s 𝑦𝑂 ) ) |
| 9 |
8
|
oveq1d |
⊢ ( 𝑦 = 𝑦𝑂 → ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ) |
| 10 |
7 9
|
eqeq12d |
⊢ ( 𝑦 = 𝑦𝑂 → ( ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ↔ ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ) ) |
| 11 |
|
oveq2 |
⊢ ( 𝑧 = 𝑧𝑂 → ( 𝑦𝑂 +s 𝑧 ) = ( 𝑦𝑂 +s 𝑧𝑂 ) ) |
| 12 |
11
|
oveq2d |
⊢ ( 𝑧 = 𝑧𝑂 → ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) ) |
| 13 |
|
oveq2 |
⊢ ( 𝑧 = 𝑧𝑂 → ( 𝑥𝑂 ·s 𝑧 ) = ( 𝑥𝑂 ·s 𝑧𝑂 ) ) |
| 14 |
13
|
oveq2d |
⊢ ( 𝑧 = 𝑧𝑂 → ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ) |
| 15 |
12 14
|
eqeq12d |
⊢ ( 𝑧 = 𝑧𝑂 → ( ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ↔ ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ) ) |
| 16 |
|
oveq1 |
⊢ ( 𝑥 = 𝑥𝑂 → ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) ) |
| 17 |
|
oveq1 |
⊢ ( 𝑥 = 𝑥𝑂 → ( 𝑥 ·s 𝑦𝑂 ) = ( 𝑥𝑂 ·s 𝑦𝑂 ) ) |
| 18 |
|
oveq1 |
⊢ ( 𝑥 = 𝑥𝑂 → ( 𝑥 ·s 𝑧𝑂 ) = ( 𝑥𝑂 ·s 𝑧𝑂 ) ) |
| 19 |
17 18
|
oveq12d |
⊢ ( 𝑥 = 𝑥𝑂 → ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ) |
| 20 |
16 19
|
eqeq12d |
⊢ ( 𝑥 = 𝑥𝑂 → ( ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ↔ ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ) ) |
| 21 |
|
oveq1 |
⊢ ( 𝑦 = 𝑦𝑂 → ( 𝑦 +s 𝑧𝑂 ) = ( 𝑦𝑂 +s 𝑧𝑂 ) ) |
| 22 |
21
|
oveq2d |
⊢ ( 𝑦 = 𝑦𝑂 → ( 𝑥 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) ) |
| 23 |
|
oveq2 |
⊢ ( 𝑦 = 𝑦𝑂 → ( 𝑥 ·s 𝑦 ) = ( 𝑥 ·s 𝑦𝑂 ) ) |
| 24 |
23
|
oveq1d |
⊢ ( 𝑦 = 𝑦𝑂 → ( ( 𝑥 ·s 𝑦 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ) |
| 25 |
22 24
|
eqeq12d |
⊢ ( 𝑦 = 𝑦𝑂 → ( ( 𝑥 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ↔ ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ) ) |
| 26 |
21
|
oveq2d |
⊢ ( 𝑦 = 𝑦𝑂 → ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) ) |
| 27 |
8
|
oveq1d |
⊢ ( 𝑦 = 𝑦𝑂 → ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ) |
| 28 |
26 27
|
eqeq12d |
⊢ ( 𝑦 = 𝑦𝑂 → ( ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ↔ ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ) ) |
| 29 |
11
|
oveq2d |
⊢ ( 𝑧 = 𝑧𝑂 → ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) ) |
| 30 |
|
oveq2 |
⊢ ( 𝑧 = 𝑧𝑂 → ( 𝑥 ·s 𝑧 ) = ( 𝑥 ·s 𝑧𝑂 ) ) |
| 31 |
30
|
oveq2d |
⊢ ( 𝑧 = 𝑧𝑂 → ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ) |
| 32 |
29 31
|
eqeq12d |
⊢ ( 𝑧 = 𝑧𝑂 → ( ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧 ) ) ↔ ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ) ) |
| 33 |
|
oveq1 |
⊢ ( 𝑥 = 𝐴 → ( 𝑥 ·s ( 𝑦 +s 𝑧 ) ) = ( 𝐴 ·s ( 𝑦 +s 𝑧 ) ) ) |
| 34 |
|
oveq1 |
⊢ ( 𝑥 = 𝐴 → ( 𝑥 ·s 𝑦 ) = ( 𝐴 ·s 𝑦 ) ) |
| 35 |
|
oveq1 |
⊢ ( 𝑥 = 𝐴 → ( 𝑥 ·s 𝑧 ) = ( 𝐴 ·s 𝑧 ) ) |
| 36 |
34 35
|
oveq12d |
⊢ ( 𝑥 = 𝐴 → ( ( 𝑥 ·s 𝑦 ) +s ( 𝑥 ·s 𝑧 ) ) = ( ( 𝐴 ·s 𝑦 ) +s ( 𝐴 ·s 𝑧 ) ) ) |
| 37 |
33 36
|
eqeq12d |
⊢ ( 𝑥 = 𝐴 → ( ( 𝑥 ·s ( 𝑦 +s 𝑧 ) ) = ( ( 𝑥 ·s 𝑦 ) +s ( 𝑥 ·s 𝑧 ) ) ↔ ( 𝐴 ·s ( 𝑦 +s 𝑧 ) ) = ( ( 𝐴 ·s 𝑦 ) +s ( 𝐴 ·s 𝑧 ) ) ) ) |
| 38 |
|
oveq1 |
⊢ ( 𝑦 = 𝐵 → ( 𝑦 +s 𝑧 ) = ( 𝐵 +s 𝑧 ) ) |
| 39 |
38
|
oveq2d |
⊢ ( 𝑦 = 𝐵 → ( 𝐴 ·s ( 𝑦 +s 𝑧 ) ) = ( 𝐴 ·s ( 𝐵 +s 𝑧 ) ) ) |
| 40 |
|
oveq2 |
⊢ ( 𝑦 = 𝐵 → ( 𝐴 ·s 𝑦 ) = ( 𝐴 ·s 𝐵 ) ) |
| 41 |
40
|
oveq1d |
⊢ ( 𝑦 = 𝐵 → ( ( 𝐴 ·s 𝑦 ) +s ( 𝐴 ·s 𝑧 ) ) = ( ( 𝐴 ·s 𝐵 ) +s ( 𝐴 ·s 𝑧 ) ) ) |
| 42 |
39 41
|
eqeq12d |
⊢ ( 𝑦 = 𝐵 → ( ( 𝐴 ·s ( 𝑦 +s 𝑧 ) ) = ( ( 𝐴 ·s 𝑦 ) +s ( 𝐴 ·s 𝑧 ) ) ↔ ( 𝐴 ·s ( 𝐵 +s 𝑧 ) ) = ( ( 𝐴 ·s 𝐵 ) +s ( 𝐴 ·s 𝑧 ) ) ) ) |
| 43 |
|
oveq2 |
⊢ ( 𝑧 = 𝐶 → ( 𝐵 +s 𝑧 ) = ( 𝐵 +s 𝐶 ) ) |
| 44 |
43
|
oveq2d |
⊢ ( 𝑧 = 𝐶 → ( 𝐴 ·s ( 𝐵 +s 𝑧 ) ) = ( 𝐴 ·s ( 𝐵 +s 𝐶 ) ) ) |
| 45 |
|
oveq2 |
⊢ ( 𝑧 = 𝐶 → ( 𝐴 ·s 𝑧 ) = ( 𝐴 ·s 𝐶 ) ) |
| 46 |
45
|
oveq2d |
⊢ ( 𝑧 = 𝐶 → ( ( 𝐴 ·s 𝐵 ) +s ( 𝐴 ·s 𝑧 ) ) = ( ( 𝐴 ·s 𝐵 ) +s ( 𝐴 ·s 𝐶 ) ) ) |
| 47 |
44 46
|
eqeq12d |
⊢ ( 𝑧 = 𝐶 → ( ( 𝐴 ·s ( 𝐵 +s 𝑧 ) ) = ( ( 𝐴 ·s 𝐵 ) +s ( 𝐴 ·s 𝑧 ) ) ↔ ( 𝐴 ·s ( 𝐵 +s 𝐶 ) ) = ( ( 𝐴 ·s 𝐵 ) +s ( 𝐴 ·s 𝐶 ) ) ) ) |
| 48 |
|
simpl1 |
⊢ ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧 ) ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ) ) → 𝑥 ∈ No ) |
| 49 |
|
simpl2 |
⊢ ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧 ) ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ) ) → 𝑦 ∈ No ) |
| 50 |
|
simpl3 |
⊢ ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧 ) ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ) ) → 𝑧 ∈ No ) |
| 51 |
|
simpr21 |
⊢ ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧 ) ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ) ) → ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ) |
| 52 |
|
simpr23 |
⊢ ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧 ) ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ) ) → ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧 ) ) ) |
| 53 |
|
simpr12 |
⊢ ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧 ) ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ) ) → ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ) |
| 54 |
|
elun1 |
⊢ ( 𝑥𝐿 ∈ ( L ‘ 𝑥 ) → 𝑥𝐿 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ) |
| 55 |
54
|
adantr |
⊢ ( ( 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∧ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) ) → 𝑥𝐿 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ) |
| 56 |
|
elun1 |
⊢ ( 𝑦𝐿 ∈ ( L ‘ 𝑦 ) → 𝑦𝐿 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ) |
| 57 |
56
|
adantl |
⊢ ( ( 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∧ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) ) → 𝑦𝐿 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ) |
| 58 |
48 49 50 51 52 53 55 57
|
addsdilem3 |
⊢ ( ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧 ) ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ) ) ∧ ( 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∧ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) ) ) → ( ( ( 𝑥𝐿 ·s ( 𝑦 +s 𝑧 ) ) +s ( 𝑥 ·s ( 𝑦𝐿 +s 𝑧 ) ) ) -s ( 𝑥𝐿 ·s ( 𝑦𝐿 +s 𝑧 ) ) ) = ( ( ( ( 𝑥𝐿 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑦𝐿 ) ) +s ( 𝑥 ·s 𝑧 ) ) ) |
| 59 |
58
|
eqeq2d |
⊢ ( ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧 ) ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ) ) ∧ ( 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∧ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) ) ) → ( 𝑎 = ( ( ( 𝑥𝐿 ·s ( 𝑦 +s 𝑧 ) ) +s ( 𝑥 ·s ( 𝑦𝐿 +s 𝑧 ) ) ) -s ( 𝑥𝐿 ·s ( 𝑦𝐿 +s 𝑧 ) ) ) ↔ 𝑎 = ( ( ( ( 𝑥𝐿 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑦𝐿 ) ) +s ( 𝑥 ·s 𝑧 ) ) ) ) |
| 60 |
59
|
2rexbidva |
⊢ ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧 ) ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ) ) → ( ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) 𝑎 = ( ( ( 𝑥𝐿 ·s ( 𝑦 +s 𝑧 ) ) +s ( 𝑥 ·s ( 𝑦𝐿 +s 𝑧 ) ) ) -s ( 𝑥𝐿 ·s ( 𝑦𝐿 +s 𝑧 ) ) ) ↔ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) 𝑎 = ( ( ( ( 𝑥𝐿 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑦𝐿 ) ) +s ( 𝑥 ·s 𝑧 ) ) ) ) |
| 61 |
60
|
abbidv |
⊢ ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧 ) ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ) ) → { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) 𝑎 = ( ( ( 𝑥𝐿 ·s ( 𝑦 +s 𝑧 ) ) +s ( 𝑥 ·s ( 𝑦𝐿 +s 𝑧 ) ) ) -s ( 𝑥𝐿 ·s ( 𝑦𝐿 +s 𝑧 ) ) ) } = { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) 𝑎 = ( ( ( ( 𝑥𝐿 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑦𝐿 ) ) +s ( 𝑥 ·s 𝑧 ) ) } ) |
| 62 |
|
simpr3 |
⊢ ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧 ) ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ) ) → ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ) |
| 63 |
|
simpr13 |
⊢ ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧 ) ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ) ) → ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ) |
| 64 |
54
|
adantr |
⊢ ( ( 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∧ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) ) → 𝑥𝐿 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ) |
| 65 |
|
elun1 |
⊢ ( 𝑧𝐿 ∈ ( L ‘ 𝑧 ) → 𝑧𝐿 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ) |
| 66 |
65
|
adantl |
⊢ ( ( 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∧ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) ) → 𝑧𝐿 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ) |
| 67 |
48 49 50 51 62 63 64 66
|
addsdilem4 |
⊢ ( ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧 ) ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ) ) ∧ ( 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∧ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) ) ) → ( ( ( 𝑥𝐿 ·s ( 𝑦 +s 𝑧 ) ) +s ( 𝑥 ·s ( 𝑦 +s 𝑧𝐿 ) ) ) -s ( 𝑥𝐿 ·s ( 𝑦 +s 𝑧𝐿 ) ) ) = ( ( 𝑥 ·s 𝑦 ) +s ( ( ( 𝑥𝐿 ·s 𝑧 ) +s ( 𝑥 ·s 𝑧𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑧𝐿 ) ) ) ) |
| 68 |
67
|
eqeq2d |
⊢ ( ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧 ) ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ) ) ∧ ( 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∧ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) ) ) → ( 𝑎 = ( ( ( 𝑥𝐿 ·s ( 𝑦 +s 𝑧 ) ) +s ( 𝑥 ·s ( 𝑦 +s 𝑧𝐿 ) ) ) -s ( 𝑥𝐿 ·s ( 𝑦 +s 𝑧𝐿 ) ) ) ↔ 𝑎 = ( ( 𝑥 ·s 𝑦 ) +s ( ( ( 𝑥𝐿 ·s 𝑧 ) +s ( 𝑥 ·s 𝑧𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑧𝐿 ) ) ) ) ) |
| 69 |
68
|
2rexbidva |
⊢ ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧 ) ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ) ) → ( ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝐿 ·s ( 𝑦 +s 𝑧 ) ) +s ( 𝑥 ·s ( 𝑦 +s 𝑧𝐿 ) ) ) -s ( 𝑥𝐿 ·s ( 𝑦 +s 𝑧𝐿 ) ) ) ↔ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( 𝑥 ·s 𝑦 ) +s ( ( ( 𝑥𝐿 ·s 𝑧 ) +s ( 𝑥 ·s 𝑧𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑧𝐿 ) ) ) ) ) |
| 70 |
69
|
abbidv |
⊢ ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧 ) ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ) ) → { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝐿 ·s ( 𝑦 +s 𝑧 ) ) +s ( 𝑥 ·s ( 𝑦 +s 𝑧𝐿 ) ) ) -s ( 𝑥𝐿 ·s ( 𝑦 +s 𝑧𝐿 ) ) ) } = { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( 𝑥 ·s 𝑦 ) +s ( ( ( 𝑥𝐿 ·s 𝑧 ) +s ( 𝑥 ·s 𝑧𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑧𝐿 ) ) ) } ) |
| 71 |
61 70
|
uneq12d |
⊢ ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧 ) ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ) ) → ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) 𝑎 = ( ( ( 𝑥𝐿 ·s ( 𝑦 +s 𝑧 ) ) +s ( 𝑥 ·s ( 𝑦𝐿 +s 𝑧 ) ) ) -s ( 𝑥𝐿 ·s ( 𝑦𝐿 +s 𝑧 ) ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝐿 ·s ( 𝑦 +s 𝑧 ) ) +s ( 𝑥 ·s ( 𝑦 +s 𝑧𝐿 ) ) ) -s ( 𝑥𝐿 ·s ( 𝑦 +s 𝑧𝐿 ) ) ) } ) = ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) 𝑎 = ( ( ( ( 𝑥𝐿 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑦𝐿 ) ) +s ( 𝑥 ·s 𝑧 ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( 𝑥 ·s 𝑦 ) +s ( ( ( 𝑥𝐿 ·s 𝑧 ) +s ( 𝑥 ·s 𝑧𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑧𝐿 ) ) ) } ) ) |
| 72 |
|
elun2 |
⊢ ( 𝑥𝑅 ∈ ( R ‘ 𝑥 ) → 𝑥𝑅 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ) |
| 73 |
72
|
adantr |
⊢ ( ( 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∧ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ) → 𝑥𝑅 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ) |
| 74 |
|
elun2 |
⊢ ( 𝑦𝑅 ∈ ( R ‘ 𝑦 ) → 𝑦𝑅 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ) |
| 75 |
74
|
adantl |
⊢ ( ( 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∧ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ) → 𝑦𝑅 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ) |
| 76 |
48 49 50 51 52 53 73 75
|
addsdilem3 |
⊢ ( ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧 ) ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ) ) ∧ ( 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∧ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ) ) → ( ( ( 𝑥𝑅 ·s ( 𝑦 +s 𝑧 ) ) +s ( 𝑥 ·s ( 𝑦𝑅 +s 𝑧 ) ) ) -s ( 𝑥𝑅 ·s ( 𝑦𝑅 +s 𝑧 ) ) ) = ( ( ( ( 𝑥𝑅 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑦𝑅 ) ) +s ( 𝑥 ·s 𝑧 ) ) ) |
| 77 |
76
|
eqeq2d |
⊢ ( ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧 ) ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ) ) ∧ ( 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∧ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ) ) → ( 𝑎 = ( ( ( 𝑥𝑅 ·s ( 𝑦 +s 𝑧 ) ) +s ( 𝑥 ·s ( 𝑦𝑅 +s 𝑧 ) ) ) -s ( 𝑥𝑅 ·s ( 𝑦𝑅 +s 𝑧 ) ) ) ↔ 𝑎 = ( ( ( ( 𝑥𝑅 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑦𝑅 ) ) +s ( 𝑥 ·s 𝑧 ) ) ) ) |
| 78 |
77
|
2rexbidva |
⊢ ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧 ) ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ) ) → ( ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) 𝑎 = ( ( ( 𝑥𝑅 ·s ( 𝑦 +s 𝑧 ) ) +s ( 𝑥 ·s ( 𝑦𝑅 +s 𝑧 ) ) ) -s ( 𝑥𝑅 ·s ( 𝑦𝑅 +s 𝑧 ) ) ) ↔ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) 𝑎 = ( ( ( ( 𝑥𝑅 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑦𝑅 ) ) +s ( 𝑥 ·s 𝑧 ) ) ) ) |
| 79 |
78
|
abbidv |
⊢ ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧 ) ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ) ) → { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) 𝑎 = ( ( ( 𝑥𝑅 ·s ( 𝑦 +s 𝑧 ) ) +s ( 𝑥 ·s ( 𝑦𝑅 +s 𝑧 ) ) ) -s ( 𝑥𝑅 ·s ( 𝑦𝑅 +s 𝑧 ) ) ) } = { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) 𝑎 = ( ( ( ( 𝑥𝑅 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑦𝑅 ) ) +s ( 𝑥 ·s 𝑧 ) ) } ) |
| 80 |
72
|
adantr |
⊢ ( ( 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∧ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) ) → 𝑥𝑅 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ) |
| 81 |
|
elun2 |
⊢ ( 𝑧𝑅 ∈ ( R ‘ 𝑧 ) → 𝑧𝑅 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ) |
| 82 |
81
|
adantl |
⊢ ( ( 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∧ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) ) → 𝑧𝑅 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ) |
| 83 |
48 49 50 51 62 63 80 82
|
addsdilem4 |
⊢ ( ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧 ) ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ) ) ∧ ( 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∧ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) ) ) → ( ( ( 𝑥𝑅 ·s ( 𝑦 +s 𝑧 ) ) +s ( 𝑥 ·s ( 𝑦 +s 𝑧𝑅 ) ) ) -s ( 𝑥𝑅 ·s ( 𝑦 +s 𝑧𝑅 ) ) ) = ( ( 𝑥 ·s 𝑦 ) +s ( ( ( 𝑥𝑅 ·s 𝑧 ) +s ( 𝑥 ·s 𝑧𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑧𝑅 ) ) ) ) |
| 84 |
83
|
eqeq2d |
⊢ ( ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧 ) ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ) ) ∧ ( 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∧ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) ) ) → ( 𝑎 = ( ( ( 𝑥𝑅 ·s ( 𝑦 +s 𝑧 ) ) +s ( 𝑥 ·s ( 𝑦 +s 𝑧𝑅 ) ) ) -s ( 𝑥𝑅 ·s ( 𝑦 +s 𝑧𝑅 ) ) ) ↔ 𝑎 = ( ( 𝑥 ·s 𝑦 ) +s ( ( ( 𝑥𝑅 ·s 𝑧 ) +s ( 𝑥 ·s 𝑧𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑧𝑅 ) ) ) ) ) |
| 85 |
84
|
2rexbidva |
⊢ ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧 ) ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ) ) → ( ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝑅 ·s ( 𝑦 +s 𝑧 ) ) +s ( 𝑥 ·s ( 𝑦 +s 𝑧𝑅 ) ) ) -s ( 𝑥𝑅 ·s ( 𝑦 +s 𝑧𝑅 ) ) ) ↔ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( 𝑥 ·s 𝑦 ) +s ( ( ( 𝑥𝑅 ·s 𝑧 ) +s ( 𝑥 ·s 𝑧𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑧𝑅 ) ) ) ) ) |
| 86 |
85
|
abbidv |
⊢ ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧 ) ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ) ) → { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝑅 ·s ( 𝑦 +s 𝑧 ) ) +s ( 𝑥 ·s ( 𝑦 +s 𝑧𝑅 ) ) ) -s ( 𝑥𝑅 ·s ( 𝑦 +s 𝑧𝑅 ) ) ) } = { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( 𝑥 ·s 𝑦 ) +s ( ( ( 𝑥𝑅 ·s 𝑧 ) +s ( 𝑥 ·s 𝑧𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑧𝑅 ) ) ) } ) |
| 87 |
79 86
|
uneq12d |
⊢ ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧 ) ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ) ) → ( { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) 𝑎 = ( ( ( 𝑥𝑅 ·s ( 𝑦 +s 𝑧 ) ) +s ( 𝑥 ·s ( 𝑦𝑅 +s 𝑧 ) ) ) -s ( 𝑥𝑅 ·s ( 𝑦𝑅 +s 𝑧 ) ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝑅 ·s ( 𝑦 +s 𝑧 ) ) +s ( 𝑥 ·s ( 𝑦 +s 𝑧𝑅 ) ) ) -s ( 𝑥𝑅 ·s ( 𝑦 +s 𝑧𝑅 ) ) ) } ) = ( { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) 𝑎 = ( ( ( ( 𝑥𝑅 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑦𝑅 ) ) +s ( 𝑥 ·s 𝑧 ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( 𝑥 ·s 𝑦 ) +s ( ( ( 𝑥𝑅 ·s 𝑧 ) +s ( 𝑥 ·s 𝑧𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑧𝑅 ) ) ) } ) ) |
| 88 |
71 87
|
uneq12d |
⊢ ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧 ) ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ) ) → ( ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) 𝑎 = ( ( ( 𝑥𝐿 ·s ( 𝑦 +s 𝑧 ) ) +s ( 𝑥 ·s ( 𝑦𝐿 +s 𝑧 ) ) ) -s ( 𝑥𝐿 ·s ( 𝑦𝐿 +s 𝑧 ) ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝐿 ·s ( 𝑦 +s 𝑧 ) ) +s ( 𝑥 ·s ( 𝑦 +s 𝑧𝐿 ) ) ) -s ( 𝑥𝐿 ·s ( 𝑦 +s 𝑧𝐿 ) ) ) } ) ∪ ( { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) 𝑎 = ( ( ( 𝑥𝑅 ·s ( 𝑦 +s 𝑧 ) ) +s ( 𝑥 ·s ( 𝑦𝑅 +s 𝑧 ) ) ) -s ( 𝑥𝑅 ·s ( 𝑦𝑅 +s 𝑧 ) ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝑅 ·s ( 𝑦 +s 𝑧 ) ) +s ( 𝑥 ·s ( 𝑦 +s 𝑧𝑅 ) ) ) -s ( 𝑥𝑅 ·s ( 𝑦 +s 𝑧𝑅 ) ) ) } ) ) = ( ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) 𝑎 = ( ( ( ( 𝑥𝐿 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑦𝐿 ) ) +s ( 𝑥 ·s 𝑧 ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( 𝑥 ·s 𝑦 ) +s ( ( ( 𝑥𝐿 ·s 𝑧 ) +s ( 𝑥 ·s 𝑧𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑧𝐿 ) ) ) } ) ∪ ( { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) 𝑎 = ( ( ( ( 𝑥𝑅 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑦𝑅 ) ) +s ( 𝑥 ·s 𝑧 ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( 𝑥 ·s 𝑦 ) +s ( ( ( 𝑥𝑅 ·s 𝑧 ) +s ( 𝑥 ·s 𝑧𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑧𝑅 ) ) ) } ) ) ) |
| 89 |
|
un4 |
⊢ ( ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) 𝑎 = ( ( ( ( 𝑥𝐿 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑦𝐿 ) ) +s ( 𝑥 ·s 𝑧 ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( 𝑥 ·s 𝑦 ) +s ( ( ( 𝑥𝐿 ·s 𝑧 ) +s ( 𝑥 ·s 𝑧𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑧𝐿 ) ) ) } ) ∪ ( { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) 𝑎 = ( ( ( ( 𝑥𝑅 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑦𝑅 ) ) +s ( 𝑥 ·s 𝑧 ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( 𝑥 ·s 𝑦 ) +s ( ( ( 𝑥𝑅 ·s 𝑧 ) +s ( 𝑥 ·s 𝑧𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑧𝑅 ) ) ) } ) ) = ( ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) 𝑎 = ( ( ( ( 𝑥𝐿 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑦𝐿 ) ) +s ( 𝑥 ·s 𝑧 ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) 𝑎 = ( ( ( ( 𝑥𝑅 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑦𝑅 ) ) +s ( 𝑥 ·s 𝑧 ) ) } ) ∪ ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( 𝑥 ·s 𝑦 ) +s ( ( ( 𝑥𝐿 ·s 𝑧 ) +s ( 𝑥 ·s 𝑧𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑧𝐿 ) ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( 𝑥 ·s 𝑦 ) +s ( ( ( 𝑥𝑅 ·s 𝑧 ) +s ( 𝑥 ·s 𝑧𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑧𝑅 ) ) ) } ) ) |
| 90 |
88 89
|
eqtrdi |
⊢ ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧 ) ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ) ) → ( ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) 𝑎 = ( ( ( 𝑥𝐿 ·s ( 𝑦 +s 𝑧 ) ) +s ( 𝑥 ·s ( 𝑦𝐿 +s 𝑧 ) ) ) -s ( 𝑥𝐿 ·s ( 𝑦𝐿 +s 𝑧 ) ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝐿 ·s ( 𝑦 +s 𝑧 ) ) +s ( 𝑥 ·s ( 𝑦 +s 𝑧𝐿 ) ) ) -s ( 𝑥𝐿 ·s ( 𝑦 +s 𝑧𝐿 ) ) ) } ) ∪ ( { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) 𝑎 = ( ( ( 𝑥𝑅 ·s ( 𝑦 +s 𝑧 ) ) +s ( 𝑥 ·s ( 𝑦𝑅 +s 𝑧 ) ) ) -s ( 𝑥𝑅 ·s ( 𝑦𝑅 +s 𝑧 ) ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝑅 ·s ( 𝑦 +s 𝑧 ) ) +s ( 𝑥 ·s ( 𝑦 +s 𝑧𝑅 ) ) ) -s ( 𝑥𝑅 ·s ( 𝑦 +s 𝑧𝑅 ) ) ) } ) ) = ( ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) 𝑎 = ( ( ( ( 𝑥𝐿 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑦𝐿 ) ) +s ( 𝑥 ·s 𝑧 ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) 𝑎 = ( ( ( ( 𝑥𝑅 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑦𝑅 ) ) +s ( 𝑥 ·s 𝑧 ) ) } ) ∪ ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( 𝑥 ·s 𝑦 ) +s ( ( ( 𝑥𝐿 ·s 𝑧 ) +s ( 𝑥 ·s 𝑧𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑧𝐿 ) ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( 𝑥 ·s 𝑦 ) +s ( ( ( 𝑥𝑅 ·s 𝑧 ) +s ( 𝑥 ·s 𝑧𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑧𝑅 ) ) ) } ) ) ) |
| 91 |
54
|
adantr |
⊢ ( ( 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∧ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ) → 𝑥𝐿 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ) |
| 92 |
74
|
adantl |
⊢ ( ( 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∧ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ) → 𝑦𝑅 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ) |
| 93 |
48 49 50 51 52 53 91 92
|
addsdilem3 |
⊢ ( ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧 ) ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ) ) ∧ ( 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∧ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ) ) → ( ( ( 𝑥𝐿 ·s ( 𝑦 +s 𝑧 ) ) +s ( 𝑥 ·s ( 𝑦𝑅 +s 𝑧 ) ) ) -s ( 𝑥𝐿 ·s ( 𝑦𝑅 +s 𝑧 ) ) ) = ( ( ( ( 𝑥𝐿 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑦𝑅 ) ) +s ( 𝑥 ·s 𝑧 ) ) ) |
| 94 |
93
|
eqeq2d |
⊢ ( ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧 ) ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ) ) ∧ ( 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∧ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ) ) → ( 𝑎 = ( ( ( 𝑥𝐿 ·s ( 𝑦 +s 𝑧 ) ) +s ( 𝑥 ·s ( 𝑦𝑅 +s 𝑧 ) ) ) -s ( 𝑥𝐿 ·s ( 𝑦𝑅 +s 𝑧 ) ) ) ↔ 𝑎 = ( ( ( ( 𝑥𝐿 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑦𝑅 ) ) +s ( 𝑥 ·s 𝑧 ) ) ) ) |
| 95 |
94
|
2rexbidva |
⊢ ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧 ) ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ) ) → ( ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) 𝑎 = ( ( ( 𝑥𝐿 ·s ( 𝑦 +s 𝑧 ) ) +s ( 𝑥 ·s ( 𝑦𝑅 +s 𝑧 ) ) ) -s ( 𝑥𝐿 ·s ( 𝑦𝑅 +s 𝑧 ) ) ) ↔ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) 𝑎 = ( ( ( ( 𝑥𝐿 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑦𝑅 ) ) +s ( 𝑥 ·s 𝑧 ) ) ) ) |
| 96 |
95
|
abbidv |
⊢ ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧 ) ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ) ) → { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) 𝑎 = ( ( ( 𝑥𝐿 ·s ( 𝑦 +s 𝑧 ) ) +s ( 𝑥 ·s ( 𝑦𝑅 +s 𝑧 ) ) ) -s ( 𝑥𝐿 ·s ( 𝑦𝑅 +s 𝑧 ) ) ) } = { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) 𝑎 = ( ( ( ( 𝑥𝐿 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑦𝑅 ) ) +s ( 𝑥 ·s 𝑧 ) ) } ) |
| 97 |
54
|
adantr |
⊢ ( ( 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∧ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) ) → 𝑥𝐿 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ) |
| 98 |
81
|
adantl |
⊢ ( ( 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∧ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) ) → 𝑧𝑅 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ) |
| 99 |
48 49 50 51 62 63 97 98
|
addsdilem4 |
⊢ ( ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧 ) ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ) ) ∧ ( 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∧ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) ) ) → ( ( ( 𝑥𝐿 ·s ( 𝑦 +s 𝑧 ) ) +s ( 𝑥 ·s ( 𝑦 +s 𝑧𝑅 ) ) ) -s ( 𝑥𝐿 ·s ( 𝑦 +s 𝑧𝑅 ) ) ) = ( ( 𝑥 ·s 𝑦 ) +s ( ( ( 𝑥𝐿 ·s 𝑧 ) +s ( 𝑥 ·s 𝑧𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑧𝑅 ) ) ) ) |
| 100 |
99
|
eqeq2d |
⊢ ( ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧 ) ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ) ) ∧ ( 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∧ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) ) ) → ( 𝑎 = ( ( ( 𝑥𝐿 ·s ( 𝑦 +s 𝑧 ) ) +s ( 𝑥 ·s ( 𝑦 +s 𝑧𝑅 ) ) ) -s ( 𝑥𝐿 ·s ( 𝑦 +s 𝑧𝑅 ) ) ) ↔ 𝑎 = ( ( 𝑥 ·s 𝑦 ) +s ( ( ( 𝑥𝐿 ·s 𝑧 ) +s ( 𝑥 ·s 𝑧𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑧𝑅 ) ) ) ) ) |
| 101 |
100
|
2rexbidva |
⊢ ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧 ) ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ) ) → ( ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝐿 ·s ( 𝑦 +s 𝑧 ) ) +s ( 𝑥 ·s ( 𝑦 +s 𝑧𝑅 ) ) ) -s ( 𝑥𝐿 ·s ( 𝑦 +s 𝑧𝑅 ) ) ) ↔ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( 𝑥 ·s 𝑦 ) +s ( ( ( 𝑥𝐿 ·s 𝑧 ) +s ( 𝑥 ·s 𝑧𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑧𝑅 ) ) ) ) ) |
| 102 |
101
|
abbidv |
⊢ ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧 ) ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ) ) → { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝐿 ·s ( 𝑦 +s 𝑧 ) ) +s ( 𝑥 ·s ( 𝑦 +s 𝑧𝑅 ) ) ) -s ( 𝑥𝐿 ·s ( 𝑦 +s 𝑧𝑅 ) ) ) } = { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( 𝑥 ·s 𝑦 ) +s ( ( ( 𝑥𝐿 ·s 𝑧 ) +s ( 𝑥 ·s 𝑧𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑧𝑅 ) ) ) } ) |
| 103 |
96 102
|
uneq12d |
⊢ ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧 ) ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ) ) → ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) 𝑎 = ( ( ( 𝑥𝐿 ·s ( 𝑦 +s 𝑧 ) ) +s ( 𝑥 ·s ( 𝑦𝑅 +s 𝑧 ) ) ) -s ( 𝑥𝐿 ·s ( 𝑦𝑅 +s 𝑧 ) ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝐿 ·s ( 𝑦 +s 𝑧 ) ) +s ( 𝑥 ·s ( 𝑦 +s 𝑧𝑅 ) ) ) -s ( 𝑥𝐿 ·s ( 𝑦 +s 𝑧𝑅 ) ) ) } ) = ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) 𝑎 = ( ( ( ( 𝑥𝐿 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑦𝑅 ) ) +s ( 𝑥 ·s 𝑧 ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( 𝑥 ·s 𝑦 ) +s ( ( ( 𝑥𝐿 ·s 𝑧 ) +s ( 𝑥 ·s 𝑧𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑧𝑅 ) ) ) } ) ) |
| 104 |
72
|
adantr |
⊢ ( ( 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∧ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) ) → 𝑥𝑅 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ) |
| 105 |
56
|
adantl |
⊢ ( ( 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∧ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) ) → 𝑦𝐿 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ) |
| 106 |
48 49 50 51 52 53 104 105
|
addsdilem3 |
⊢ ( ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧 ) ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ) ) ∧ ( 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∧ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) ) ) → ( ( ( 𝑥𝑅 ·s ( 𝑦 +s 𝑧 ) ) +s ( 𝑥 ·s ( 𝑦𝐿 +s 𝑧 ) ) ) -s ( 𝑥𝑅 ·s ( 𝑦𝐿 +s 𝑧 ) ) ) = ( ( ( ( 𝑥𝑅 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑦𝐿 ) ) +s ( 𝑥 ·s 𝑧 ) ) ) |
| 107 |
106
|
eqeq2d |
⊢ ( ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧 ) ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ) ) ∧ ( 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∧ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) ) ) → ( 𝑎 = ( ( ( 𝑥𝑅 ·s ( 𝑦 +s 𝑧 ) ) +s ( 𝑥 ·s ( 𝑦𝐿 +s 𝑧 ) ) ) -s ( 𝑥𝑅 ·s ( 𝑦𝐿 +s 𝑧 ) ) ) ↔ 𝑎 = ( ( ( ( 𝑥𝑅 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑦𝐿 ) ) +s ( 𝑥 ·s 𝑧 ) ) ) ) |
| 108 |
107
|
2rexbidva |
⊢ ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧 ) ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ) ) → ( ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) 𝑎 = ( ( ( 𝑥𝑅 ·s ( 𝑦 +s 𝑧 ) ) +s ( 𝑥 ·s ( 𝑦𝐿 +s 𝑧 ) ) ) -s ( 𝑥𝑅 ·s ( 𝑦𝐿 +s 𝑧 ) ) ) ↔ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) 𝑎 = ( ( ( ( 𝑥𝑅 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑦𝐿 ) ) +s ( 𝑥 ·s 𝑧 ) ) ) ) |
| 109 |
108
|
abbidv |
⊢ ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧 ) ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ) ) → { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) 𝑎 = ( ( ( 𝑥𝑅 ·s ( 𝑦 +s 𝑧 ) ) +s ( 𝑥 ·s ( 𝑦𝐿 +s 𝑧 ) ) ) -s ( 𝑥𝑅 ·s ( 𝑦𝐿 +s 𝑧 ) ) ) } = { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) 𝑎 = ( ( ( ( 𝑥𝑅 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑦𝐿 ) ) +s ( 𝑥 ·s 𝑧 ) ) } ) |
| 110 |
72
|
adantr |
⊢ ( ( 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∧ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) ) → 𝑥𝑅 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ) |
| 111 |
65
|
adantl |
⊢ ( ( 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∧ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) ) → 𝑧𝐿 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ) |
| 112 |
48 49 50 51 62 63 110 111
|
addsdilem4 |
⊢ ( ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧 ) ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ) ) ∧ ( 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∧ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) ) ) → ( ( ( 𝑥𝑅 ·s ( 𝑦 +s 𝑧 ) ) +s ( 𝑥 ·s ( 𝑦 +s 𝑧𝐿 ) ) ) -s ( 𝑥𝑅 ·s ( 𝑦 +s 𝑧𝐿 ) ) ) = ( ( 𝑥 ·s 𝑦 ) +s ( ( ( 𝑥𝑅 ·s 𝑧 ) +s ( 𝑥 ·s 𝑧𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑧𝐿 ) ) ) ) |
| 113 |
112
|
eqeq2d |
⊢ ( ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧 ) ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ) ) ∧ ( 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∧ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) ) ) → ( 𝑎 = ( ( ( 𝑥𝑅 ·s ( 𝑦 +s 𝑧 ) ) +s ( 𝑥 ·s ( 𝑦 +s 𝑧𝐿 ) ) ) -s ( 𝑥𝑅 ·s ( 𝑦 +s 𝑧𝐿 ) ) ) ↔ 𝑎 = ( ( 𝑥 ·s 𝑦 ) +s ( ( ( 𝑥𝑅 ·s 𝑧 ) +s ( 𝑥 ·s 𝑧𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑧𝐿 ) ) ) ) ) |
| 114 |
113
|
2rexbidva |
⊢ ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧 ) ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ) ) → ( ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝑅 ·s ( 𝑦 +s 𝑧 ) ) +s ( 𝑥 ·s ( 𝑦 +s 𝑧𝐿 ) ) ) -s ( 𝑥𝑅 ·s ( 𝑦 +s 𝑧𝐿 ) ) ) ↔ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( 𝑥 ·s 𝑦 ) +s ( ( ( 𝑥𝑅 ·s 𝑧 ) +s ( 𝑥 ·s 𝑧𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑧𝐿 ) ) ) ) ) |
| 115 |
114
|
abbidv |
⊢ ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧 ) ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ) ) → { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝑅 ·s ( 𝑦 +s 𝑧 ) ) +s ( 𝑥 ·s ( 𝑦 +s 𝑧𝐿 ) ) ) -s ( 𝑥𝑅 ·s ( 𝑦 +s 𝑧𝐿 ) ) ) } = { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( 𝑥 ·s 𝑦 ) +s ( ( ( 𝑥𝑅 ·s 𝑧 ) +s ( 𝑥 ·s 𝑧𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑧𝐿 ) ) ) } ) |
| 116 |
109 115
|
uneq12d |
⊢ ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧 ) ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ) ) → ( { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) 𝑎 = ( ( ( 𝑥𝑅 ·s ( 𝑦 +s 𝑧 ) ) +s ( 𝑥 ·s ( 𝑦𝐿 +s 𝑧 ) ) ) -s ( 𝑥𝑅 ·s ( 𝑦𝐿 +s 𝑧 ) ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝑅 ·s ( 𝑦 +s 𝑧 ) ) +s ( 𝑥 ·s ( 𝑦 +s 𝑧𝐿 ) ) ) -s ( 𝑥𝑅 ·s ( 𝑦 +s 𝑧𝐿 ) ) ) } ) = ( { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) 𝑎 = ( ( ( ( 𝑥𝑅 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑦𝐿 ) ) +s ( 𝑥 ·s 𝑧 ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( 𝑥 ·s 𝑦 ) +s ( ( ( 𝑥𝑅 ·s 𝑧 ) +s ( 𝑥 ·s 𝑧𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑧𝐿 ) ) ) } ) ) |
| 117 |
103 116
|
uneq12d |
⊢ ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧 ) ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ) ) → ( ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) 𝑎 = ( ( ( 𝑥𝐿 ·s ( 𝑦 +s 𝑧 ) ) +s ( 𝑥 ·s ( 𝑦𝑅 +s 𝑧 ) ) ) -s ( 𝑥𝐿 ·s ( 𝑦𝑅 +s 𝑧 ) ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝐿 ·s ( 𝑦 +s 𝑧 ) ) +s ( 𝑥 ·s ( 𝑦 +s 𝑧𝑅 ) ) ) -s ( 𝑥𝐿 ·s ( 𝑦 +s 𝑧𝑅 ) ) ) } ) ∪ ( { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) 𝑎 = ( ( ( 𝑥𝑅 ·s ( 𝑦 +s 𝑧 ) ) +s ( 𝑥 ·s ( 𝑦𝐿 +s 𝑧 ) ) ) -s ( 𝑥𝑅 ·s ( 𝑦𝐿 +s 𝑧 ) ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝑅 ·s ( 𝑦 +s 𝑧 ) ) +s ( 𝑥 ·s ( 𝑦 +s 𝑧𝐿 ) ) ) -s ( 𝑥𝑅 ·s ( 𝑦 +s 𝑧𝐿 ) ) ) } ) ) = ( ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) 𝑎 = ( ( ( ( 𝑥𝐿 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑦𝑅 ) ) +s ( 𝑥 ·s 𝑧 ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( 𝑥 ·s 𝑦 ) +s ( ( ( 𝑥𝐿 ·s 𝑧 ) +s ( 𝑥 ·s 𝑧𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑧𝑅 ) ) ) } ) ∪ ( { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) 𝑎 = ( ( ( ( 𝑥𝑅 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑦𝐿 ) ) +s ( 𝑥 ·s 𝑧 ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( 𝑥 ·s 𝑦 ) +s ( ( ( 𝑥𝑅 ·s 𝑧 ) +s ( 𝑥 ·s 𝑧𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑧𝐿 ) ) ) } ) ) ) |
| 118 |
|
un4 |
⊢ ( ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) 𝑎 = ( ( ( ( 𝑥𝐿 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑦𝑅 ) ) +s ( 𝑥 ·s 𝑧 ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( 𝑥 ·s 𝑦 ) +s ( ( ( 𝑥𝐿 ·s 𝑧 ) +s ( 𝑥 ·s 𝑧𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑧𝑅 ) ) ) } ) ∪ ( { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) 𝑎 = ( ( ( ( 𝑥𝑅 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑦𝐿 ) ) +s ( 𝑥 ·s 𝑧 ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( 𝑥 ·s 𝑦 ) +s ( ( ( 𝑥𝑅 ·s 𝑧 ) +s ( 𝑥 ·s 𝑧𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑧𝐿 ) ) ) } ) ) = ( ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) 𝑎 = ( ( ( ( 𝑥𝐿 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑦𝑅 ) ) +s ( 𝑥 ·s 𝑧 ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) 𝑎 = ( ( ( ( 𝑥𝑅 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑦𝐿 ) ) +s ( 𝑥 ·s 𝑧 ) ) } ) ∪ ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( 𝑥 ·s 𝑦 ) +s ( ( ( 𝑥𝐿 ·s 𝑧 ) +s ( 𝑥 ·s 𝑧𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑧𝑅 ) ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( 𝑥 ·s 𝑦 ) +s ( ( ( 𝑥𝑅 ·s 𝑧 ) +s ( 𝑥 ·s 𝑧𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑧𝐿 ) ) ) } ) ) |
| 119 |
117 118
|
eqtrdi |
⊢ ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧 ) ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ) ) → ( ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) 𝑎 = ( ( ( 𝑥𝐿 ·s ( 𝑦 +s 𝑧 ) ) +s ( 𝑥 ·s ( 𝑦𝑅 +s 𝑧 ) ) ) -s ( 𝑥𝐿 ·s ( 𝑦𝑅 +s 𝑧 ) ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝐿 ·s ( 𝑦 +s 𝑧 ) ) +s ( 𝑥 ·s ( 𝑦 +s 𝑧𝑅 ) ) ) -s ( 𝑥𝐿 ·s ( 𝑦 +s 𝑧𝑅 ) ) ) } ) ∪ ( { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) 𝑎 = ( ( ( 𝑥𝑅 ·s ( 𝑦 +s 𝑧 ) ) +s ( 𝑥 ·s ( 𝑦𝐿 +s 𝑧 ) ) ) -s ( 𝑥𝑅 ·s ( 𝑦𝐿 +s 𝑧 ) ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝑅 ·s ( 𝑦 +s 𝑧 ) ) +s ( 𝑥 ·s ( 𝑦 +s 𝑧𝐿 ) ) ) -s ( 𝑥𝑅 ·s ( 𝑦 +s 𝑧𝐿 ) ) ) } ) ) = ( ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) 𝑎 = ( ( ( ( 𝑥𝐿 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑦𝑅 ) ) +s ( 𝑥 ·s 𝑧 ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) 𝑎 = ( ( ( ( 𝑥𝑅 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑦𝐿 ) ) +s ( 𝑥 ·s 𝑧 ) ) } ) ∪ ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( 𝑥 ·s 𝑦 ) +s ( ( ( 𝑥𝐿 ·s 𝑧 ) +s ( 𝑥 ·s 𝑧𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑧𝑅 ) ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( 𝑥 ·s 𝑦 ) +s ( ( ( 𝑥𝑅 ·s 𝑧 ) +s ( 𝑥 ·s 𝑧𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑧𝐿 ) ) ) } ) ) ) |
| 120 |
90 119
|
oveq12d |
⊢ ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧 ) ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ) ) → ( ( ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) 𝑎 = ( ( ( 𝑥𝐿 ·s ( 𝑦 +s 𝑧 ) ) +s ( 𝑥 ·s ( 𝑦𝐿 +s 𝑧 ) ) ) -s ( 𝑥𝐿 ·s ( 𝑦𝐿 +s 𝑧 ) ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝐿 ·s ( 𝑦 +s 𝑧 ) ) +s ( 𝑥 ·s ( 𝑦 +s 𝑧𝐿 ) ) ) -s ( 𝑥𝐿 ·s ( 𝑦 +s 𝑧𝐿 ) ) ) } ) ∪ ( { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) 𝑎 = ( ( ( 𝑥𝑅 ·s ( 𝑦 +s 𝑧 ) ) +s ( 𝑥 ·s ( 𝑦𝑅 +s 𝑧 ) ) ) -s ( 𝑥𝑅 ·s ( 𝑦𝑅 +s 𝑧 ) ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝑅 ·s ( 𝑦 +s 𝑧 ) ) +s ( 𝑥 ·s ( 𝑦 +s 𝑧𝑅 ) ) ) -s ( 𝑥𝑅 ·s ( 𝑦 +s 𝑧𝑅 ) ) ) } ) ) |s ( ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) 𝑎 = ( ( ( 𝑥𝐿 ·s ( 𝑦 +s 𝑧 ) ) +s ( 𝑥 ·s ( 𝑦𝑅 +s 𝑧 ) ) ) -s ( 𝑥𝐿 ·s ( 𝑦𝑅 +s 𝑧 ) ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝐿 ·s ( 𝑦 +s 𝑧 ) ) +s ( 𝑥 ·s ( 𝑦 +s 𝑧𝑅 ) ) ) -s ( 𝑥𝐿 ·s ( 𝑦 +s 𝑧𝑅 ) ) ) } ) ∪ ( { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) 𝑎 = ( ( ( 𝑥𝑅 ·s ( 𝑦 +s 𝑧 ) ) +s ( 𝑥 ·s ( 𝑦𝐿 +s 𝑧 ) ) ) -s ( 𝑥𝑅 ·s ( 𝑦𝐿 +s 𝑧 ) ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝑅 ·s ( 𝑦 +s 𝑧 ) ) +s ( 𝑥 ·s ( 𝑦 +s 𝑧𝐿 ) ) ) -s ( 𝑥𝑅 ·s ( 𝑦 +s 𝑧𝐿 ) ) ) } ) ) ) = ( ( ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) 𝑎 = ( ( ( ( 𝑥𝐿 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑦𝐿 ) ) +s ( 𝑥 ·s 𝑧 ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) 𝑎 = ( ( ( ( 𝑥𝑅 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑦𝑅 ) ) +s ( 𝑥 ·s 𝑧 ) ) } ) ∪ ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( 𝑥 ·s 𝑦 ) +s ( ( ( 𝑥𝐿 ·s 𝑧 ) +s ( 𝑥 ·s 𝑧𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑧𝐿 ) ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( 𝑥 ·s 𝑦 ) +s ( ( ( 𝑥𝑅 ·s 𝑧 ) +s ( 𝑥 ·s 𝑧𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑧𝑅 ) ) ) } ) ) |s ( ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) 𝑎 = ( ( ( ( 𝑥𝐿 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑦𝑅 ) ) +s ( 𝑥 ·s 𝑧 ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) 𝑎 = ( ( ( ( 𝑥𝑅 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑦𝐿 ) ) +s ( 𝑥 ·s 𝑧 ) ) } ) ∪ ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( 𝑥 ·s 𝑦 ) +s ( ( ( 𝑥𝐿 ·s 𝑧 ) +s ( 𝑥 ·s 𝑧𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑧𝑅 ) ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( 𝑥 ·s 𝑦 ) +s ( ( ( 𝑥𝑅 ·s 𝑧 ) +s ( 𝑥 ·s 𝑧𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑧𝐿 ) ) ) } ) ) ) ) |
| 121 |
48 49 50
|
addsdilem1 |
⊢ ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧 ) ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ) ) → ( 𝑥 ·s ( 𝑦 +s 𝑧 ) ) = ( ( ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) 𝑎 = ( ( ( 𝑥𝐿 ·s ( 𝑦 +s 𝑧 ) ) +s ( 𝑥 ·s ( 𝑦𝐿 +s 𝑧 ) ) ) -s ( 𝑥𝐿 ·s ( 𝑦𝐿 +s 𝑧 ) ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝐿 ·s ( 𝑦 +s 𝑧 ) ) +s ( 𝑥 ·s ( 𝑦 +s 𝑧𝐿 ) ) ) -s ( 𝑥𝐿 ·s ( 𝑦 +s 𝑧𝐿 ) ) ) } ) ∪ ( { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) 𝑎 = ( ( ( 𝑥𝑅 ·s ( 𝑦 +s 𝑧 ) ) +s ( 𝑥 ·s ( 𝑦𝑅 +s 𝑧 ) ) ) -s ( 𝑥𝑅 ·s ( 𝑦𝑅 +s 𝑧 ) ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝑅 ·s ( 𝑦 +s 𝑧 ) ) +s ( 𝑥 ·s ( 𝑦 +s 𝑧𝑅 ) ) ) -s ( 𝑥𝑅 ·s ( 𝑦 +s 𝑧𝑅 ) ) ) } ) ) |s ( ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) 𝑎 = ( ( ( 𝑥𝐿 ·s ( 𝑦 +s 𝑧 ) ) +s ( 𝑥 ·s ( 𝑦𝑅 +s 𝑧 ) ) ) -s ( 𝑥𝐿 ·s ( 𝑦𝑅 +s 𝑧 ) ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝐿 ·s ( 𝑦 +s 𝑧 ) ) +s ( 𝑥 ·s ( 𝑦 +s 𝑧𝑅 ) ) ) -s ( 𝑥𝐿 ·s ( 𝑦 +s 𝑧𝑅 ) ) ) } ) ∪ ( { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) 𝑎 = ( ( ( 𝑥𝑅 ·s ( 𝑦 +s 𝑧 ) ) +s ( 𝑥 ·s ( 𝑦𝐿 +s 𝑧 ) ) ) -s ( 𝑥𝑅 ·s ( 𝑦𝐿 +s 𝑧 ) ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝑅 ·s ( 𝑦 +s 𝑧 ) ) +s ( 𝑥 ·s ( 𝑦 +s 𝑧𝐿 ) ) ) -s ( 𝑥𝑅 ·s ( 𝑦 +s 𝑧𝐿 ) ) ) } ) ) ) ) |
| 122 |
48 49 50
|
addsdilem2 |
⊢ ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧 ) ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ) ) → ( ( 𝑥 ·s 𝑦 ) +s ( 𝑥 ·s 𝑧 ) ) = ( ( ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) 𝑎 = ( ( ( ( 𝑥𝐿 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑦𝐿 ) ) +s ( 𝑥 ·s 𝑧 ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) 𝑎 = ( ( ( ( 𝑥𝑅 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑦𝑅 ) ) +s ( 𝑥 ·s 𝑧 ) ) } ) ∪ ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( 𝑥 ·s 𝑦 ) +s ( ( ( 𝑥𝐿 ·s 𝑧 ) +s ( 𝑥 ·s 𝑧𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑧𝐿 ) ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( 𝑥 ·s 𝑦 ) +s ( ( ( 𝑥𝑅 ·s 𝑧 ) +s ( 𝑥 ·s 𝑧𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑧𝑅 ) ) ) } ) ) |s ( ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) 𝑎 = ( ( ( ( 𝑥𝐿 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑦𝑅 ) ) +s ( 𝑥 ·s 𝑧 ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) 𝑎 = ( ( ( ( 𝑥𝑅 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑦𝐿 ) ) +s ( 𝑥 ·s 𝑧 ) ) } ) ∪ ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( 𝑥 ·s 𝑦 ) +s ( ( ( 𝑥𝐿 ·s 𝑧 ) +s ( 𝑥 ·s 𝑧𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑧𝑅 ) ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( 𝑥 ·s 𝑦 ) +s ( ( ( 𝑥𝑅 ·s 𝑧 ) +s ( 𝑥 ·s 𝑧𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑧𝐿 ) ) ) } ) ) ) ) |
| 123 |
120 121 122
|
3eqtr4d |
⊢ ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧 ) ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ) ) → ( 𝑥 ·s ( 𝑦 +s 𝑧 ) ) = ( ( 𝑥 ·s 𝑦 ) +s ( 𝑥 ·s 𝑧 ) ) ) |
| 124 |
123
|
ex |
⊢ ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) → ( ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥𝑂 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧𝑂 ) ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( 𝑥𝑂 ·s ( 𝑦 +s 𝑧 ) ) = ( ( 𝑥𝑂 ·s 𝑦 ) +s ( 𝑥𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( 𝑥 ·s ( 𝑦𝑂 +s 𝑧 ) ) = ( ( 𝑥 ·s 𝑦𝑂 ) +s ( 𝑥 ·s 𝑧 ) ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( 𝑥 ·s ( 𝑦 +s 𝑧𝑂 ) ) = ( ( 𝑥 ·s 𝑦 ) +s ( 𝑥 ·s 𝑧𝑂 ) ) ) → ( 𝑥 ·s ( 𝑦 +s 𝑧 ) ) = ( ( 𝑥 ·s 𝑦 ) +s ( 𝑥 ·s 𝑧 ) ) ) ) |
| 125 |
5 10 15 20 25 28 32 37 42 47 124
|
no3inds |
⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( 𝐴 ·s ( 𝐵 +s 𝐶 ) ) = ( ( 𝐴 ·s 𝐵 ) +s ( 𝐴 ·s 𝐶 ) ) ) |