Step | Hyp | Ref
| Expression |
1 | | addsdilem.1 |
. . . 4
⊢ (𝜑 → 𝐴 ∈ No
) |
2 | | addsdilem.2 |
. . . 4
⊢ (𝜑 → 𝐵 ∈ No
) |
3 | 1, 2 | mulscut2 27518 |
. . 3
⊢ (𝜑 → ({𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (((𝑥𝐿 ·s
𝐵) +s (𝐴 ·s 𝑦𝐿))
-s (𝑥𝐿 ·s
𝑦𝐿))}
∪ {𝑏 ∣
∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (((𝑥𝑅 ·s
𝐵) +s (𝐴 ·s 𝑦𝑅))
-s (𝑥𝑅 ·s
𝑦𝑅))})
<<s ({𝑏 ∣
∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (((𝑥𝐿 ·s
𝐵) +s (𝐴 ·s 𝑦𝑅))
-s (𝑥𝐿 ·s
𝑦𝑅))}
∪ {𝑏 ∣
∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (((𝑥𝑅 ·s
𝐵) +s (𝐴 ·s 𝑦𝐿))
-s (𝑥𝑅 ·s
𝑦𝐿))})) |
4 | | addsdilem.3 |
. . . 4
⊢ (𝜑 → 𝐶 ∈ No
) |
5 | 1, 4 | mulscut2 27518 |
. . 3
⊢ (𝜑 → ({𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑥𝐿 ·s
𝐶) +s (𝐴 ·s 𝑧𝐿))
-s (𝑥𝐿 ·s
𝑧𝐿))}
∪ {𝑏 ∣
∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑥𝑅 ·s
𝐶) +s (𝐴 ·s 𝑧𝑅))
-s (𝑥𝑅 ·s
𝑧𝑅))})
<<s ({𝑏 ∣
∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑥𝐿 ·s
𝐶) +s (𝐴 ·s 𝑧𝑅))
-s (𝑥𝐿 ·s
𝑧𝑅))}
∪ {𝑏 ∣
∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑥𝑅 ·s
𝐶) +s (𝐴 ·s 𝑧𝐿))
-s (𝑥𝑅 ·s
𝑧𝐿))})) |
6 | | mulsval2 27496 |
. . . 4
⊢ ((𝐴 ∈
No ∧ 𝐵 ∈
No ) → (𝐴 ·s 𝐵) = (({𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (((𝑥𝐿 ·s
𝐵) +s (𝐴 ·s 𝑦𝐿))
-s (𝑥𝐿 ·s
𝑦𝐿))}
∪ {𝑏 ∣
∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (((𝑥𝑅 ·s
𝐵) +s (𝐴 ·s 𝑦𝑅))
-s (𝑥𝑅 ·s
𝑦𝑅))})
|s ({𝑏 ∣ ∃𝑥𝐿 ∈ ( L
‘𝐴)∃𝑦𝑅 ∈ ( R
‘𝐵)𝑏 = (((𝑥𝐿 ·s
𝐵) +s (𝐴 ·s 𝑦𝑅))
-s (𝑥𝐿 ·s
𝑦𝑅))}
∪ {𝑏 ∣
∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (((𝑥𝑅 ·s
𝐵) +s (𝐴 ·s 𝑦𝐿))
-s (𝑥𝑅 ·s
𝑦𝐿))}))) |
7 | 1, 2, 6 | syl2anc 584 |
. . 3
⊢ (𝜑 → (𝐴 ·s 𝐵) = (({𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (((𝑥𝐿 ·s
𝐵) +s (𝐴 ·s 𝑦𝐿))
-s (𝑥𝐿 ·s
𝑦𝐿))}
∪ {𝑏 ∣
∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (((𝑥𝑅 ·s
𝐵) +s (𝐴 ·s 𝑦𝑅))
-s (𝑥𝑅 ·s
𝑦𝑅))})
|s ({𝑏 ∣ ∃𝑥𝐿 ∈ ( L
‘𝐴)∃𝑦𝑅 ∈ ( R
‘𝐵)𝑏 = (((𝑥𝐿 ·s
𝐵) +s (𝐴 ·s 𝑦𝑅))
-s (𝑥𝐿 ·s
𝑦𝑅))}
∪ {𝑏 ∣
∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (((𝑥𝑅 ·s
𝐵) +s (𝐴 ·s 𝑦𝐿))
-s (𝑥𝑅 ·s
𝑦𝐿))}))) |
8 | | mulsval2 27496 |
. . . 4
⊢ ((𝐴 ∈
No ∧ 𝐶 ∈
No ) → (𝐴 ·s 𝐶) = (({𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑥𝐿 ·s
𝐶) +s (𝐴 ·s 𝑧𝐿))
-s (𝑥𝐿 ·s
𝑧𝐿))}
∪ {𝑏 ∣
∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑥𝑅 ·s
𝐶) +s (𝐴 ·s 𝑧𝑅))
-s (𝑥𝑅 ·s
𝑧𝑅))})
|s ({𝑏 ∣ ∃𝑥𝐿 ∈ ( L
‘𝐴)∃𝑧𝑅 ∈ ( R
‘𝐶)𝑏 = (((𝑥𝐿 ·s
𝐶) +s (𝐴 ·s 𝑧𝑅))
-s (𝑥𝐿 ·s
𝑧𝑅))}
∪ {𝑏 ∣
∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑥𝑅 ·s
𝐶) +s (𝐴 ·s 𝑧𝐿))
-s (𝑥𝑅 ·s
𝑧𝐿))}))) |
9 | 1, 4, 8 | syl2anc 584 |
. . 3
⊢ (𝜑 → (𝐴 ·s 𝐶) = (({𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑥𝐿 ·s
𝐶) +s (𝐴 ·s 𝑧𝐿))
-s (𝑥𝐿 ·s
𝑧𝐿))}
∪ {𝑏 ∣
∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑥𝑅 ·s
𝐶) +s (𝐴 ·s 𝑧𝑅))
-s (𝑥𝑅 ·s
𝑧𝑅))})
|s ({𝑏 ∣ ∃𝑥𝐿 ∈ ( L
‘𝐴)∃𝑧𝑅 ∈ ( R
‘𝐶)𝑏 = (((𝑥𝐿 ·s
𝐶) +s (𝐴 ·s 𝑧𝑅))
-s (𝑥𝐿 ·s
𝑧𝑅))}
∪ {𝑏 ∣
∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑥𝑅 ·s
𝐶) +s (𝐴 ·s 𝑧𝐿))
-s (𝑥𝑅 ·s
𝑧𝐿))}))) |
10 | 3, 5, 7, 9 | addsunif 27414 |
. 2
⊢ (𝜑 → ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝐶)) = (({𝑎 ∣ ∃𝑡 ∈ ({𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (((𝑥𝐿 ·s
𝐵) +s (𝐴 ·s 𝑦𝐿))
-s (𝑥𝐿 ·s
𝑦𝐿))}
∪ {𝑏 ∣
∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (((𝑥𝑅 ·s
𝐵) +s (𝐴 ·s 𝑦𝑅))
-s (𝑥𝑅 ·s
𝑦𝑅))})𝑎 = (𝑡 +s (𝐴 ·s 𝐶))} ∪ {𝑎 ∣ ∃𝑡 ∈ ({𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑥𝐿 ·s
𝐶) +s (𝐴 ·s 𝑧𝐿))
-s (𝑥𝐿 ·s
𝑧𝐿))}
∪ {𝑏 ∣
∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑥𝑅 ·s
𝐶) +s (𝐴 ·s 𝑧𝑅))
-s (𝑥𝑅 ·s
𝑧𝑅))})𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)}) |s ({𝑎 ∣ ∃𝑡 ∈ ({𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (((𝑥𝐿 ·s
𝐵) +s (𝐴 ·s 𝑦𝑅))
-s (𝑥𝐿 ·s
𝑦𝑅))}
∪ {𝑏 ∣
∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (((𝑥𝑅 ·s
𝐵) +s (𝐴 ·s 𝑦𝐿))
-s (𝑥𝑅 ·s
𝑦𝐿))})𝑎 = (𝑡 +s (𝐴 ·s 𝐶))} ∪ {𝑎 ∣ ∃𝑡 ∈ ({𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑥𝐿 ·s
𝐶) +s (𝐴 ·s 𝑧𝑅))
-s (𝑥𝐿 ·s
𝑧𝑅))}
∪ {𝑏 ∣
∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑥𝑅 ·s
𝐶) +s (𝐴 ·s 𝑧𝐿))
-s (𝑥𝑅 ·s
𝑧𝐿))})𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)}))) |
11 | | unab 4295 |
. . . . 5
⊢ ({𝑎 ∣ ∃𝑥𝐿 ∈ ( L
‘𝐴)∃𝑦𝐿 ∈ ( L
‘𝐵)𝑎 = ((((𝑥𝐿 ·s
𝐵) +s (𝐴 ·s 𝑦𝐿))
-s (𝑥𝐿 ·s
𝑦𝐿))
+s (𝐴
·s 𝐶))}
∪ {𝑎 ∣
∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = ((((𝑥𝑅 ·s
𝐵) +s (𝐴 ·s 𝑦𝑅))
-s (𝑥𝑅 ·s
𝑦𝑅))
+s (𝐴
·s 𝐶))})
= {𝑎 ∣ (∃𝑥𝐿 ∈ ( L
‘𝐴)∃𝑦𝐿 ∈ ( L
‘𝐵)𝑎 = ((((𝑥𝐿 ·s
𝐵) +s (𝐴 ·s 𝑦𝐿))
-s (𝑥𝐿 ·s
𝑦𝐿))
+s (𝐴
·s 𝐶))
∨ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = ((((𝑥𝑅 ·s
𝐵) +s (𝐴 ·s 𝑦𝑅))
-s (𝑥𝑅 ·s
𝑦𝑅))
+s (𝐴
·s 𝐶)))} |
12 | | rexun 4187 |
. . . . . . 7
⊢
(∃𝑡 ∈
({𝑏 ∣ ∃𝑥𝐿 ∈ ( L
‘𝐴)∃𝑦𝐿 ∈ ( L
‘𝐵)𝑏 = (((𝑥𝐿 ·s
𝐵) +s (𝐴 ·s 𝑦𝐿))
-s (𝑥𝐿 ·s
𝑦𝐿))}
∪ {𝑏 ∣
∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (((𝑥𝑅 ·s
𝐵) +s (𝐴 ·s 𝑦𝑅))
-s (𝑥𝑅 ·s
𝑦𝑅))})𝑎 = (𝑡 +s (𝐴 ·s 𝐶)) ↔ (∃𝑡 ∈ {𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (((𝑥𝐿 ·s
𝐵) +s (𝐴 ·s 𝑦𝐿))
-s (𝑥𝐿 ·s
𝑦𝐿))}𝑎 = (𝑡 +s (𝐴 ·s 𝐶)) ∨ ∃𝑡 ∈ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (((𝑥𝑅 ·s
𝐵) +s (𝐴 ·s 𝑦𝑅))
-s (𝑥𝑅 ·s
𝑦𝑅))}𝑎 = (𝑡 +s (𝐴 ·s 𝐶)))) |
13 | | eqeq1 2736 |
. . . . . . . . . . 11
⊢ (𝑏 = 𝑡 → (𝑏 = (((𝑥𝐿 ·s
𝐵) +s (𝐴 ·s 𝑦𝐿))
-s (𝑥𝐿 ·s
𝑦𝐿))
↔ 𝑡 = (((𝑥𝐿
·s 𝐵)
+s (𝐴
·s 𝑦𝐿)) -s (𝑥𝐿
·s 𝑦𝐿)))) |
14 | 13 | 2rexbidv 3219 |
. . . . . . . . . 10
⊢ (𝑏 = 𝑡 → (∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (((𝑥𝐿 ·s
𝐵) +s (𝐴 ·s 𝑦𝐿))
-s (𝑥𝐿 ·s
𝑦𝐿))
↔ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑡 = (((𝑥𝐿 ·s
𝐵) +s (𝐴 ·s 𝑦𝐿))
-s (𝑥𝐿 ·s
𝑦𝐿)))) |
15 | 14 | rexab 3687 |
. . . . . . . . 9
⊢
(∃𝑡 ∈
{𝑏 ∣ ∃𝑥𝐿 ∈ ( L
‘𝐴)∃𝑦𝐿 ∈ ( L
‘𝐵)𝑏 = (((𝑥𝐿 ·s
𝐵) +s (𝐴 ·s 𝑦𝐿))
-s (𝑥𝐿 ·s
𝑦𝐿))}𝑎 = (𝑡 +s (𝐴 ·s 𝐶)) ↔ ∃𝑡(∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑡 = (((𝑥𝐿 ·s
𝐵) +s (𝐴 ·s 𝑦𝐿))
-s (𝑥𝐿 ·s
𝑦𝐿))
∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶)))) |
16 | | rexcom4 3285 |
. . . . . . . . . 10
⊢
(∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑡∃𝑦𝐿 ∈ ( L ‘𝐵)(𝑡 = (((𝑥𝐿 ·s
𝐵) +s (𝐴 ·s 𝑦𝐿))
-s (𝑥𝐿 ·s
𝑦𝐿))
∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶))) ↔ ∃𝑡∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)(𝑡 = (((𝑥𝐿 ·s
𝐵) +s (𝐴 ·s 𝑦𝐿))
-s (𝑥𝐿 ·s
𝑦𝐿))
∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶)))) |
17 | | rexcom4 3285 |
. . . . . . . . . . . 12
⊢
(∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑡(𝑡 = (((𝑥𝐿 ·s
𝐵) +s (𝐴 ·s 𝑦𝐿))
-s (𝑥𝐿 ·s
𝑦𝐿))
∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶))) ↔ ∃𝑡∃𝑦𝐿 ∈ ( L ‘𝐵)(𝑡 = (((𝑥𝐿 ·s
𝐵) +s (𝐴 ·s 𝑦𝐿))
-s (𝑥𝐿 ·s
𝑦𝐿))
∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶)))) |
18 | | ovex 7427 |
. . . . . . . . . . . . . 14
⊢ (((𝑥𝐿
·s 𝐵)
+s (𝐴
·s 𝑦𝐿)) -s (𝑥𝐿
·s 𝑦𝐿)) ∈
V |
19 | | oveq1 7401 |
. . . . . . . . . . . . . . 15
⊢ (𝑡 = (((𝑥𝐿 ·s
𝐵) +s (𝐴 ·s 𝑦𝐿))
-s (𝑥𝐿 ·s
𝑦𝐿))
→ (𝑡 +s
(𝐴 ·s
𝐶)) = ((((𝑥𝐿
·s 𝐵)
+s (𝐴
·s 𝑦𝐿)) -s (𝑥𝐿
·s 𝑦𝐿)) +s (𝐴 ·s 𝐶))) |
20 | 19 | eqeq2d 2743 |
. . . . . . . . . . . . . 14
⊢ (𝑡 = (((𝑥𝐿 ·s
𝐵) +s (𝐴 ·s 𝑦𝐿))
-s (𝑥𝐿 ·s
𝑦𝐿))
→ (𝑎 = (𝑡 +s (𝐴 ·s 𝐶)) ↔ 𝑎 = ((((𝑥𝐿 ·s
𝐵) +s (𝐴 ·s 𝑦𝐿))
-s (𝑥𝐿 ·s
𝑦𝐿))
+s (𝐴
·s 𝐶)))) |
21 | 18, 20 | ceqsexv 3523 |
. . . . . . . . . . . . 13
⊢
(∃𝑡(𝑡 = (((𝑥𝐿 ·s
𝐵) +s (𝐴 ·s 𝑦𝐿))
-s (𝑥𝐿 ·s
𝑦𝐿))
∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶))) ↔ 𝑎 = ((((𝑥𝐿 ·s
𝐵) +s (𝐴 ·s 𝑦𝐿))
-s (𝑥𝐿 ·s
𝑦𝐿))
+s (𝐴
·s 𝐶))) |
22 | 21 | rexbii 3094 |
. . . . . . . . . . . 12
⊢
(∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑡(𝑡 = (((𝑥𝐿 ·s
𝐵) +s (𝐴 ·s 𝑦𝐿))
-s (𝑥𝐿 ·s
𝑦𝐿))
∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶))) ↔ ∃𝑦𝐿 ∈ ( L
‘𝐵)𝑎 = ((((𝑥𝐿 ·s
𝐵) +s (𝐴 ·s 𝑦𝐿))
-s (𝑥𝐿 ·s
𝑦𝐿))
+s (𝐴
·s 𝐶))) |
23 | 17, 22 | bitr3i 276 |
. . . . . . . . . . 11
⊢
(∃𝑡∃𝑦𝐿 ∈ ( L ‘𝐵)(𝑡 = (((𝑥𝐿 ·s
𝐵) +s (𝐴 ·s 𝑦𝐿))
-s (𝑥𝐿 ·s
𝑦𝐿))
∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶))) ↔ ∃𝑦𝐿 ∈ ( L
‘𝐵)𝑎 = ((((𝑥𝐿 ·s
𝐵) +s (𝐴 ·s 𝑦𝐿))
-s (𝑥𝐿 ·s
𝑦𝐿))
+s (𝐴
·s 𝐶))) |
24 | 23 | rexbii 3094 |
. . . . . . . . . 10
⊢
(∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑡∃𝑦𝐿 ∈ ( L ‘𝐵)(𝑡 = (((𝑥𝐿 ·s
𝐵) +s (𝐴 ·s 𝑦𝐿))
-s (𝑥𝐿 ·s
𝑦𝐿))
∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶))) ↔ ∃𝑥𝐿 ∈ ( L
‘𝐴)∃𝑦𝐿 ∈ ( L
‘𝐵)𝑎 = ((((𝑥𝐿 ·s
𝐵) +s (𝐴 ·s 𝑦𝐿))
-s (𝑥𝐿 ·s
𝑦𝐿))
+s (𝐴
·s 𝐶))) |
25 | | r19.41vv 3224 |
. . . . . . . . . . 11
⊢
(∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)(𝑡 = (((𝑥𝐿 ·s
𝐵) +s (𝐴 ·s 𝑦𝐿))
-s (𝑥𝐿 ·s
𝑦𝐿))
∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶))) ↔ (∃𝑥𝐿 ∈ ( L
‘𝐴)∃𝑦𝐿 ∈ ( L
‘𝐵)𝑡 = (((𝑥𝐿 ·s
𝐵) +s (𝐴 ·s 𝑦𝐿))
-s (𝑥𝐿 ·s
𝑦𝐿))
∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶)))) |
26 | 25 | exbii 1850 |
. . . . . . . . . 10
⊢
(∃𝑡∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)(𝑡 = (((𝑥𝐿 ·s
𝐵) +s (𝐴 ·s 𝑦𝐿))
-s (𝑥𝐿 ·s
𝑦𝐿))
∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶))) ↔ ∃𝑡(∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑡 = (((𝑥𝐿 ·s
𝐵) +s (𝐴 ·s 𝑦𝐿))
-s (𝑥𝐿 ·s
𝑦𝐿))
∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶)))) |
27 | 16, 24, 26 | 3bitr3ri 301 |
. . . . . . . . 9
⊢
(∃𝑡(∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑡 = (((𝑥𝐿 ·s
𝐵) +s (𝐴 ·s 𝑦𝐿))
-s (𝑥𝐿 ·s
𝑦𝐿))
∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶))) ↔ ∃𝑥𝐿 ∈ ( L
‘𝐴)∃𝑦𝐿 ∈ ( L
‘𝐵)𝑎 = ((((𝑥𝐿 ·s
𝐵) +s (𝐴 ·s 𝑦𝐿))
-s (𝑥𝐿 ·s
𝑦𝐿))
+s (𝐴
·s 𝐶))) |
28 | 15, 27 | bitri 274 |
. . . . . . . 8
⊢
(∃𝑡 ∈
{𝑏 ∣ ∃𝑥𝐿 ∈ ( L
‘𝐴)∃𝑦𝐿 ∈ ( L
‘𝐵)𝑏 = (((𝑥𝐿 ·s
𝐵) +s (𝐴 ·s 𝑦𝐿))
-s (𝑥𝐿 ·s
𝑦𝐿))}𝑎 = (𝑡 +s (𝐴 ·s 𝐶)) ↔ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = ((((𝑥𝐿 ·s
𝐵) +s (𝐴 ·s 𝑦𝐿))
-s (𝑥𝐿 ·s
𝑦𝐿))
+s (𝐴
·s 𝐶))) |
29 | | eqeq1 2736 |
. . . . . . . . . . 11
⊢ (𝑏 = 𝑡 → (𝑏 = (((𝑥𝑅 ·s
𝐵) +s (𝐴 ·s 𝑦𝑅))
-s (𝑥𝑅 ·s
𝑦𝑅))
↔ 𝑡 = (((𝑥𝑅
·s 𝐵)
+s (𝐴
·s 𝑦𝑅)) -s (𝑥𝑅
·s 𝑦𝑅)))) |
30 | 29 | 2rexbidv 3219 |
. . . . . . . . . 10
⊢ (𝑏 = 𝑡 → (∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (((𝑥𝑅 ·s
𝐵) +s (𝐴 ·s 𝑦𝑅))
-s (𝑥𝑅 ·s
𝑦𝑅))
↔ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑡 = (((𝑥𝑅 ·s
𝐵) +s (𝐴 ·s 𝑦𝑅))
-s (𝑥𝑅 ·s
𝑦𝑅)))) |
31 | 30 | rexab 3687 |
. . . . . . . . 9
⊢
(∃𝑡 ∈
{𝑏 ∣ ∃𝑥𝑅 ∈ ( R
‘𝐴)∃𝑦𝑅 ∈ ( R
‘𝐵)𝑏 = (((𝑥𝑅 ·s
𝐵) +s (𝐴 ·s 𝑦𝑅))
-s (𝑥𝑅 ·s
𝑦𝑅))}𝑎 = (𝑡 +s (𝐴 ·s 𝐶)) ↔ ∃𝑡(∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑡 = (((𝑥𝑅 ·s
𝐵) +s (𝐴 ·s 𝑦𝑅))
-s (𝑥𝑅 ·s
𝑦𝑅))
∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶)))) |
32 | | rexcom4 3285 |
. . . . . . . . . 10
⊢
(∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑡∃𝑦𝑅 ∈ ( R ‘𝐵)(𝑡 = (((𝑥𝑅 ·s
𝐵) +s (𝐴 ·s 𝑦𝑅))
-s (𝑥𝑅 ·s
𝑦𝑅))
∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶))) ↔ ∃𝑡∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)(𝑡 = (((𝑥𝑅 ·s
𝐵) +s (𝐴 ·s 𝑦𝑅))
-s (𝑥𝑅 ·s
𝑦𝑅))
∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶)))) |
33 | | rexcom4 3285 |
. . . . . . . . . . . 12
⊢
(∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑡(𝑡 = (((𝑥𝑅 ·s
𝐵) +s (𝐴 ·s 𝑦𝑅))
-s (𝑥𝑅 ·s
𝑦𝑅))
∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶))) ↔ ∃𝑡∃𝑦𝑅 ∈ ( R ‘𝐵)(𝑡 = (((𝑥𝑅 ·s
𝐵) +s (𝐴 ·s 𝑦𝑅))
-s (𝑥𝑅 ·s
𝑦𝑅))
∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶)))) |
34 | | ovex 7427 |
. . . . . . . . . . . . . 14
⊢ (((𝑥𝑅
·s 𝐵)
+s (𝐴
·s 𝑦𝑅)) -s (𝑥𝑅
·s 𝑦𝑅)) ∈
V |
35 | | oveq1 7401 |
. . . . . . . . . . . . . . 15
⊢ (𝑡 = (((𝑥𝑅 ·s
𝐵) +s (𝐴 ·s 𝑦𝑅))
-s (𝑥𝑅 ·s
𝑦𝑅))
→ (𝑡 +s
(𝐴 ·s
𝐶)) = ((((𝑥𝑅
·s 𝐵)
+s (𝐴
·s 𝑦𝑅)) -s (𝑥𝑅
·s 𝑦𝑅)) +s (𝐴 ·s 𝐶))) |
36 | 35 | eqeq2d 2743 |
. . . . . . . . . . . . . 14
⊢ (𝑡 = (((𝑥𝑅 ·s
𝐵) +s (𝐴 ·s 𝑦𝑅))
-s (𝑥𝑅 ·s
𝑦𝑅))
→ (𝑎 = (𝑡 +s (𝐴 ·s 𝐶)) ↔ 𝑎 = ((((𝑥𝑅 ·s
𝐵) +s (𝐴 ·s 𝑦𝑅))
-s (𝑥𝑅 ·s
𝑦𝑅))
+s (𝐴
·s 𝐶)))) |
37 | 34, 36 | ceqsexv 3523 |
. . . . . . . . . . . . 13
⊢
(∃𝑡(𝑡 = (((𝑥𝑅 ·s
𝐵) +s (𝐴 ·s 𝑦𝑅))
-s (𝑥𝑅 ·s
𝑦𝑅))
∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶))) ↔ 𝑎 = ((((𝑥𝑅 ·s
𝐵) +s (𝐴 ·s 𝑦𝑅))
-s (𝑥𝑅 ·s
𝑦𝑅))
+s (𝐴
·s 𝐶))) |
38 | 37 | rexbii 3094 |
. . . . . . . . . . . 12
⊢
(∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑡(𝑡 = (((𝑥𝑅 ·s
𝐵) +s (𝐴 ·s 𝑦𝑅))
-s (𝑥𝑅 ·s
𝑦𝑅))
∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶))) ↔ ∃𝑦𝑅 ∈ ( R
‘𝐵)𝑎 = ((((𝑥𝑅 ·s
𝐵) +s (𝐴 ·s 𝑦𝑅))
-s (𝑥𝑅 ·s
𝑦𝑅))
+s (𝐴
·s 𝐶))) |
39 | 33, 38 | bitr3i 276 |
. . . . . . . . . . 11
⊢
(∃𝑡∃𝑦𝑅 ∈ ( R ‘𝐵)(𝑡 = (((𝑥𝑅 ·s
𝐵) +s (𝐴 ·s 𝑦𝑅))
-s (𝑥𝑅 ·s
𝑦𝑅))
∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶))) ↔ ∃𝑦𝑅 ∈ ( R
‘𝐵)𝑎 = ((((𝑥𝑅 ·s
𝐵) +s (𝐴 ·s 𝑦𝑅))
-s (𝑥𝑅 ·s
𝑦𝑅))
+s (𝐴
·s 𝐶))) |
40 | 39 | rexbii 3094 |
. . . . . . . . . 10
⊢
(∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑡∃𝑦𝑅 ∈ ( R ‘𝐵)(𝑡 = (((𝑥𝑅 ·s
𝐵) +s (𝐴 ·s 𝑦𝑅))
-s (𝑥𝑅 ·s
𝑦𝑅))
∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶))) ↔ ∃𝑥𝑅 ∈ ( R
‘𝐴)∃𝑦𝑅 ∈ ( R
‘𝐵)𝑎 = ((((𝑥𝑅 ·s
𝐵) +s (𝐴 ·s 𝑦𝑅))
-s (𝑥𝑅 ·s
𝑦𝑅))
+s (𝐴
·s 𝐶))) |
41 | | r19.41vv 3224 |
. . . . . . . . . . 11
⊢
(∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)(𝑡 = (((𝑥𝑅 ·s
𝐵) +s (𝐴 ·s 𝑦𝑅))
-s (𝑥𝑅 ·s
𝑦𝑅))
∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶))) ↔ (∃𝑥𝑅 ∈ ( R
‘𝐴)∃𝑦𝑅 ∈ ( R
‘𝐵)𝑡 = (((𝑥𝑅 ·s
𝐵) +s (𝐴 ·s 𝑦𝑅))
-s (𝑥𝑅 ·s
𝑦𝑅))
∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶)))) |
42 | 41 | exbii 1850 |
. . . . . . . . . 10
⊢
(∃𝑡∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)(𝑡 = (((𝑥𝑅 ·s
𝐵) +s (𝐴 ·s 𝑦𝑅))
-s (𝑥𝑅 ·s
𝑦𝑅))
∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶))) ↔ ∃𝑡(∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑡 = (((𝑥𝑅 ·s
𝐵) +s (𝐴 ·s 𝑦𝑅))
-s (𝑥𝑅 ·s
𝑦𝑅))
∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶)))) |
43 | 32, 40, 42 | 3bitr3ri 301 |
. . . . . . . . 9
⊢
(∃𝑡(∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑡 = (((𝑥𝑅 ·s
𝐵) +s (𝐴 ·s 𝑦𝑅))
-s (𝑥𝑅 ·s
𝑦𝑅))
∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶))) ↔ ∃𝑥𝑅 ∈ ( R
‘𝐴)∃𝑦𝑅 ∈ ( R
‘𝐵)𝑎 = ((((𝑥𝑅 ·s
𝐵) +s (𝐴 ·s 𝑦𝑅))
-s (𝑥𝑅 ·s
𝑦𝑅))
+s (𝐴
·s 𝐶))) |
44 | 31, 43 | bitri 274 |
. . . . . . . 8
⊢
(∃𝑡 ∈
{𝑏 ∣ ∃𝑥𝑅 ∈ ( R
‘𝐴)∃𝑦𝑅 ∈ ( R
‘𝐵)𝑏 = (((𝑥𝑅 ·s
𝐵) +s (𝐴 ·s 𝑦𝑅))
-s (𝑥𝑅 ·s
𝑦𝑅))}𝑎 = (𝑡 +s (𝐴 ·s 𝐶)) ↔ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = ((((𝑥𝑅 ·s
𝐵) +s (𝐴 ·s 𝑦𝑅))
-s (𝑥𝑅 ·s
𝑦𝑅))
+s (𝐴
·s 𝐶))) |
45 | 28, 44 | orbi12i 913 |
. . . . . . 7
⊢
((∃𝑡 ∈
{𝑏 ∣ ∃𝑥𝐿 ∈ ( L
‘𝐴)∃𝑦𝐿 ∈ ( L
‘𝐵)𝑏 = (((𝑥𝐿 ·s
𝐵) +s (𝐴 ·s 𝑦𝐿))
-s (𝑥𝐿 ·s
𝑦𝐿))}𝑎 = (𝑡 +s (𝐴 ·s 𝐶)) ∨ ∃𝑡 ∈ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (((𝑥𝑅 ·s
𝐵) +s (𝐴 ·s 𝑦𝑅))
-s (𝑥𝑅 ·s
𝑦𝑅))}𝑎 = (𝑡 +s (𝐴 ·s 𝐶))) ↔ (∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = ((((𝑥𝐿 ·s
𝐵) +s (𝐴 ·s 𝑦𝐿))
-s (𝑥𝐿 ·s
𝑦𝐿))
+s (𝐴
·s 𝐶))
∨ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = ((((𝑥𝑅 ·s
𝐵) +s (𝐴 ·s 𝑦𝑅))
-s (𝑥𝑅 ·s
𝑦𝑅))
+s (𝐴
·s 𝐶)))) |
46 | 12, 45 | bitr2i 275 |
. . . . . 6
⊢
((∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = ((((𝑥𝐿 ·s
𝐵) +s (𝐴 ·s 𝑦𝐿))
-s (𝑥𝐿 ·s
𝑦𝐿))
+s (𝐴
·s 𝐶))
∨ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = ((((𝑥𝑅 ·s
𝐵) +s (𝐴 ·s 𝑦𝑅))
-s (𝑥𝑅 ·s
𝑦𝑅))
+s (𝐴
·s 𝐶)))
↔ ∃𝑡 ∈
({𝑏 ∣ ∃𝑥𝐿 ∈ ( L
‘𝐴)∃𝑦𝐿 ∈ ( L
‘𝐵)𝑏 = (((𝑥𝐿 ·s
𝐵) +s (𝐴 ·s 𝑦𝐿))
-s (𝑥𝐿 ·s
𝑦𝐿))}
∪ {𝑏 ∣
∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (((𝑥𝑅 ·s
𝐵) +s (𝐴 ·s 𝑦𝑅))
-s (𝑥𝑅 ·s
𝑦𝑅))})𝑎 = (𝑡 +s (𝐴 ·s 𝐶))) |
47 | 46 | abbii 2802 |
. . . . 5
⊢ {𝑎 ∣ (∃𝑥𝐿 ∈ ( L
‘𝐴)∃𝑦𝐿 ∈ ( L
‘𝐵)𝑎 = ((((𝑥𝐿 ·s
𝐵) +s (𝐴 ·s 𝑦𝐿))
-s (𝑥𝐿 ·s
𝑦𝐿))
+s (𝐴
·s 𝐶))
∨ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = ((((𝑥𝑅 ·s
𝐵) +s (𝐴 ·s 𝑦𝑅))
-s (𝑥𝑅 ·s
𝑦𝑅))
+s (𝐴
·s 𝐶)))}
= {𝑎 ∣ ∃𝑡 ∈ ({𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (((𝑥𝐿 ·s
𝐵) +s (𝐴 ·s 𝑦𝐿))
-s (𝑥𝐿 ·s
𝑦𝐿))}
∪ {𝑏 ∣
∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (((𝑥𝑅 ·s
𝐵) +s (𝐴 ·s 𝑦𝑅))
-s (𝑥𝑅 ·s
𝑦𝑅))})𝑎 = (𝑡 +s (𝐴 ·s 𝐶))} |
48 | 11, 47 | eqtri 2760 |
. . . 4
⊢ ({𝑎 ∣ ∃𝑥𝐿 ∈ ( L
‘𝐴)∃𝑦𝐿 ∈ ( L
‘𝐵)𝑎 = ((((𝑥𝐿 ·s
𝐵) +s (𝐴 ·s 𝑦𝐿))
-s (𝑥𝐿 ·s
𝑦𝐿))
+s (𝐴
·s 𝐶))}
∪ {𝑎 ∣
∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = ((((𝑥𝑅 ·s
𝐵) +s (𝐴 ·s 𝑦𝑅))
-s (𝑥𝑅 ·s
𝑦𝑅))
+s (𝐴
·s 𝐶))})
= {𝑎 ∣ ∃𝑡 ∈ ({𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (((𝑥𝐿 ·s
𝐵) +s (𝐴 ·s 𝑦𝐿))
-s (𝑥𝐿 ·s
𝑦𝐿))}
∪ {𝑏 ∣
∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (((𝑥𝑅 ·s
𝐵) +s (𝐴 ·s 𝑦𝑅))
-s (𝑥𝑅 ·s
𝑦𝑅))})𝑎 = (𝑡 +s (𝐴 ·s 𝐶))} |
49 | | unab 4295 |
. . . . 5
⊢ ({𝑎 ∣ ∃𝑥𝐿 ∈ ( L
‘𝐴)∃𝑧𝐿 ∈ ( L
‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝐿 ·s
𝐶) +s (𝐴 ·s 𝑧𝐿))
-s (𝑥𝐿 ·s
𝑧𝐿)))}
∪ {𝑎 ∣
∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝑅 ·s
𝐶) +s (𝐴 ·s 𝑧𝑅))
-s (𝑥𝑅 ·s
𝑧𝑅)))})
= {𝑎 ∣ (∃𝑥𝐿 ∈ ( L
‘𝐴)∃𝑧𝐿 ∈ ( L
‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝐿 ·s
𝐶) +s (𝐴 ·s 𝑧𝐿))
-s (𝑥𝐿 ·s
𝑧𝐿)))
∨ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝑅 ·s
𝐶) +s (𝐴 ·s 𝑧𝑅))
-s (𝑥𝑅 ·s
𝑧𝑅))))} |
50 | | rexun 4187 |
. . . . . . 7
⊢
(∃𝑡 ∈
({𝑏 ∣ ∃𝑥𝐿 ∈ ( L
‘𝐴)∃𝑧𝐿 ∈ ( L
‘𝐶)𝑏 = (((𝑥𝐿 ·s
𝐶) +s (𝐴 ·s 𝑧𝐿))
-s (𝑥𝐿 ·s
𝑧𝐿))}
∪ {𝑏 ∣
∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑥𝑅 ·s
𝐶) +s (𝐴 ·s 𝑧𝑅))
-s (𝑥𝑅 ·s
𝑧𝑅))})𝑎 = ((𝐴 ·s 𝐵) +s 𝑡) ↔ (∃𝑡 ∈ {𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑥𝐿 ·s
𝐶) +s (𝐴 ·s 𝑧𝐿))
-s (𝑥𝐿 ·s
𝑧𝐿))}𝑎 = ((𝐴 ·s 𝐵) +s 𝑡) ∨ ∃𝑡 ∈ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑥𝑅 ·s
𝐶) +s (𝐴 ·s 𝑧𝑅))
-s (𝑥𝑅 ·s
𝑧𝑅))}𝑎 = ((𝐴 ·s 𝐵) +s 𝑡))) |
51 | | eqeq1 2736 |
. . . . . . . . . . 11
⊢ (𝑏 = 𝑡 → (𝑏 = (((𝑥𝐿 ·s
𝐶) +s (𝐴 ·s 𝑧𝐿))
-s (𝑥𝐿 ·s
𝑧𝐿))
↔ 𝑡 = (((𝑥𝐿
·s 𝐶)
+s (𝐴
·s 𝑧𝐿)) -s (𝑥𝐿
·s 𝑧𝐿)))) |
52 | 51 | 2rexbidv 3219 |
. . . . . . . . . 10
⊢ (𝑏 = 𝑡 → (∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑥𝐿 ·s
𝐶) +s (𝐴 ·s 𝑧𝐿))
-s (𝑥𝐿 ·s
𝑧𝐿))
↔ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑡 = (((𝑥𝐿 ·s
𝐶) +s (𝐴 ·s 𝑧𝐿))
-s (𝑥𝐿 ·s
𝑧𝐿)))) |
53 | 52 | rexab 3687 |
. . . . . . . . 9
⊢
(∃𝑡 ∈
{𝑏 ∣ ∃𝑥𝐿 ∈ ( L
‘𝐴)∃𝑧𝐿 ∈ ( L
‘𝐶)𝑏 = (((𝑥𝐿 ·s
𝐶) +s (𝐴 ·s 𝑧𝐿))
-s (𝑥𝐿 ·s
𝑧𝐿))}𝑎 = ((𝐴 ·s 𝐵) +s 𝑡) ↔ ∃𝑡(∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑡 = (((𝑥𝐿 ·s
𝐶) +s (𝐴 ·s 𝑧𝐿))
-s (𝑥𝐿 ·s
𝑧𝐿))
∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡))) |
54 | | rexcom4 3285 |
. . . . . . . . . 10
⊢
(∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑡∃𝑧𝐿 ∈ ( L ‘𝐶)(𝑡 = (((𝑥𝐿 ·s
𝐶) +s (𝐴 ·s 𝑧𝐿))
-s (𝑥𝐿 ·s
𝑧𝐿))
∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)) ↔ ∃𝑡∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)(𝑡 = (((𝑥𝐿 ·s
𝐶) +s (𝐴 ·s 𝑧𝐿))
-s (𝑥𝐿 ·s
𝑧𝐿))
∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡))) |
55 | | rexcom4 3285 |
. . . . . . . . . . . 12
⊢
(∃𝑧𝐿 ∈ ( L ‘𝐶)∃𝑡(𝑡 = (((𝑥𝐿 ·s
𝐶) +s (𝐴 ·s 𝑧𝐿))
-s (𝑥𝐿 ·s
𝑧𝐿))
∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)) ↔ ∃𝑡∃𝑧𝐿 ∈ ( L ‘𝐶)(𝑡 = (((𝑥𝐿 ·s
𝐶) +s (𝐴 ·s 𝑧𝐿))
-s (𝑥𝐿 ·s
𝑧𝐿))
∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡))) |
56 | | ovex 7427 |
. . . . . . . . . . . . . 14
⊢ (((𝑥𝐿
·s 𝐶)
+s (𝐴
·s 𝑧𝐿)) -s (𝑥𝐿
·s 𝑧𝐿)) ∈
V |
57 | | oveq2 7402 |
. . . . . . . . . . . . . . 15
⊢ (𝑡 = (((𝑥𝐿 ·s
𝐶) +s (𝐴 ·s 𝑧𝐿))
-s (𝑥𝐿 ·s
𝑧𝐿))
→ ((𝐴
·s 𝐵)
+s 𝑡) = ((𝐴 ·s 𝐵) +s (((𝑥𝐿
·s 𝐶)
+s (𝐴
·s 𝑧𝐿)) -s (𝑥𝐿
·s 𝑧𝐿)))) |
58 | 57 | eqeq2d 2743 |
. . . . . . . . . . . . . 14
⊢ (𝑡 = (((𝑥𝐿 ·s
𝐶) +s (𝐴 ·s 𝑧𝐿))
-s (𝑥𝐿 ·s
𝑧𝐿))
→ (𝑎 = ((𝐴 ·s 𝐵) +s 𝑡) ↔ 𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝐿 ·s
𝐶) +s (𝐴 ·s 𝑧𝐿))
-s (𝑥𝐿 ·s
𝑧𝐿))))) |
59 | 56, 58 | ceqsexv 3523 |
. . . . . . . . . . . . 13
⊢
(∃𝑡(𝑡 = (((𝑥𝐿 ·s
𝐶) +s (𝐴 ·s 𝑧𝐿))
-s (𝑥𝐿 ·s
𝑧𝐿))
∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)) ↔ 𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝐿 ·s
𝐶) +s (𝐴 ·s 𝑧𝐿))
-s (𝑥𝐿 ·s
𝑧𝐿)))) |
60 | 59 | rexbii 3094 |
. . . . . . . . . . . 12
⊢
(∃𝑧𝐿 ∈ ( L ‘𝐶)∃𝑡(𝑡 = (((𝑥𝐿 ·s
𝐶) +s (𝐴 ·s 𝑧𝐿))
-s (𝑥𝐿 ·s
𝑧𝐿))
∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)) ↔ ∃𝑧𝐿 ∈ ( L
‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝐿 ·s
𝐶) +s (𝐴 ·s 𝑧𝐿))
-s (𝑥𝐿 ·s
𝑧𝐿)))) |
61 | 55, 60 | bitr3i 276 |
. . . . . . . . . . 11
⊢
(∃𝑡∃𝑧𝐿 ∈ ( L ‘𝐶)(𝑡 = (((𝑥𝐿 ·s
𝐶) +s (𝐴 ·s 𝑧𝐿))
-s (𝑥𝐿 ·s
𝑧𝐿))
∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)) ↔ ∃𝑧𝐿 ∈ ( L
‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝐿 ·s
𝐶) +s (𝐴 ·s 𝑧𝐿))
-s (𝑥𝐿 ·s
𝑧𝐿)))) |
62 | 61 | rexbii 3094 |
. . . . . . . . . 10
⊢
(∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑡∃𝑧𝐿 ∈ ( L ‘𝐶)(𝑡 = (((𝑥𝐿 ·s
𝐶) +s (𝐴 ·s 𝑧𝐿))
-s (𝑥𝐿 ·s
𝑧𝐿))
∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)) ↔ ∃𝑥𝐿 ∈ ( L
‘𝐴)∃𝑧𝐿 ∈ ( L
‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝐿 ·s
𝐶) +s (𝐴 ·s 𝑧𝐿))
-s (𝑥𝐿 ·s
𝑧𝐿)))) |
63 | | r19.41vv 3224 |
. . . . . . . . . . 11
⊢
(∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)(𝑡 = (((𝑥𝐿 ·s
𝐶) +s (𝐴 ·s 𝑧𝐿))
-s (𝑥𝐿 ·s
𝑧𝐿))
∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)) ↔ (∃𝑥𝐿 ∈ ( L
‘𝐴)∃𝑧𝐿 ∈ ( L
‘𝐶)𝑡 = (((𝑥𝐿 ·s
𝐶) +s (𝐴 ·s 𝑧𝐿))
-s (𝑥𝐿 ·s
𝑧𝐿))
∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡))) |
64 | 63 | exbii 1850 |
. . . . . . . . . 10
⊢
(∃𝑡∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)(𝑡 = (((𝑥𝐿 ·s
𝐶) +s (𝐴 ·s 𝑧𝐿))
-s (𝑥𝐿 ·s
𝑧𝐿))
∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)) ↔ ∃𝑡(∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑡 = (((𝑥𝐿 ·s
𝐶) +s (𝐴 ·s 𝑧𝐿))
-s (𝑥𝐿 ·s
𝑧𝐿))
∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡))) |
65 | 54, 62, 64 | 3bitr3ri 301 |
. . . . . . . . 9
⊢
(∃𝑡(∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑡 = (((𝑥𝐿 ·s
𝐶) +s (𝐴 ·s 𝑧𝐿))
-s (𝑥𝐿 ·s
𝑧𝐿))
∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)) ↔ ∃𝑥𝐿 ∈ ( L
‘𝐴)∃𝑧𝐿 ∈ ( L
‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝐿 ·s
𝐶) +s (𝐴 ·s 𝑧𝐿))
-s (𝑥𝐿 ·s
𝑧𝐿)))) |
66 | 53, 65 | bitri 274 |
. . . . . . . 8
⊢
(∃𝑡 ∈
{𝑏 ∣ ∃𝑥𝐿 ∈ ( L
‘𝐴)∃𝑧𝐿 ∈ ( L
‘𝐶)𝑏 = (((𝑥𝐿 ·s
𝐶) +s (𝐴 ·s 𝑧𝐿))
-s (𝑥𝐿 ·s
𝑧𝐿))}𝑎 = ((𝐴 ·s 𝐵) +s 𝑡) ↔ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝐿 ·s
𝐶) +s (𝐴 ·s 𝑧𝐿))
-s (𝑥𝐿 ·s
𝑧𝐿)))) |
67 | | eqeq1 2736 |
. . . . . . . . . . 11
⊢ (𝑏 = 𝑡 → (𝑏 = (((𝑥𝑅 ·s
𝐶) +s (𝐴 ·s 𝑧𝑅))
-s (𝑥𝑅 ·s
𝑧𝑅))
↔ 𝑡 = (((𝑥𝑅
·s 𝐶)
+s (𝐴
·s 𝑧𝑅)) -s (𝑥𝑅
·s 𝑧𝑅)))) |
68 | 67 | 2rexbidv 3219 |
. . . . . . . . . 10
⊢ (𝑏 = 𝑡 → (∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑥𝑅 ·s
𝐶) +s (𝐴 ·s 𝑧𝑅))
-s (𝑥𝑅 ·s
𝑧𝑅))
↔ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑡 = (((𝑥𝑅 ·s
𝐶) +s (𝐴 ·s 𝑧𝑅))
-s (𝑥𝑅 ·s
𝑧𝑅)))) |
69 | 68 | rexab 3687 |
. . . . . . . . 9
⊢
(∃𝑡 ∈
{𝑏 ∣ ∃𝑥𝑅 ∈ ( R
‘𝐴)∃𝑧𝑅 ∈ ( R
‘𝐶)𝑏 = (((𝑥𝑅 ·s
𝐶) +s (𝐴 ·s 𝑧𝑅))
-s (𝑥𝑅 ·s
𝑧𝑅))}𝑎 = ((𝐴 ·s 𝐵) +s 𝑡) ↔ ∃𝑡(∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑡 = (((𝑥𝑅 ·s
𝐶) +s (𝐴 ·s 𝑧𝑅))
-s (𝑥𝑅 ·s
𝑧𝑅))
∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡))) |
70 | | rexcom4 3285 |
. . . . . . . . . 10
⊢
(∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑡∃𝑧𝑅 ∈ ( R ‘𝐶)(𝑡 = (((𝑥𝑅 ·s
𝐶) +s (𝐴 ·s 𝑧𝑅))
-s (𝑥𝑅 ·s
𝑧𝑅))
∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)) ↔ ∃𝑡∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)(𝑡 = (((𝑥𝑅 ·s
𝐶) +s (𝐴 ·s 𝑧𝑅))
-s (𝑥𝑅 ·s
𝑧𝑅))
∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡))) |
71 | | rexcom4 3285 |
. . . . . . . . . . . 12
⊢
(∃𝑧𝑅 ∈ ( R ‘𝐶)∃𝑡(𝑡 = (((𝑥𝑅 ·s
𝐶) +s (𝐴 ·s 𝑧𝑅))
-s (𝑥𝑅 ·s
𝑧𝑅))
∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)) ↔ ∃𝑡∃𝑧𝑅 ∈ ( R ‘𝐶)(𝑡 = (((𝑥𝑅 ·s
𝐶) +s (𝐴 ·s 𝑧𝑅))
-s (𝑥𝑅 ·s
𝑧𝑅))
∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡))) |
72 | | ovex 7427 |
. . . . . . . . . . . . . 14
⊢ (((𝑥𝑅
·s 𝐶)
+s (𝐴
·s 𝑧𝑅)) -s (𝑥𝑅
·s 𝑧𝑅)) ∈
V |
73 | | oveq2 7402 |
. . . . . . . . . . . . . . 15
⊢ (𝑡 = (((𝑥𝑅 ·s
𝐶) +s (𝐴 ·s 𝑧𝑅))
-s (𝑥𝑅 ·s
𝑧𝑅))
→ ((𝐴
·s 𝐵)
+s 𝑡) = ((𝐴 ·s 𝐵) +s (((𝑥𝑅
·s 𝐶)
+s (𝐴
·s 𝑧𝑅)) -s (𝑥𝑅
·s 𝑧𝑅)))) |
74 | 73 | eqeq2d 2743 |
. . . . . . . . . . . . . 14
⊢ (𝑡 = (((𝑥𝑅 ·s
𝐶) +s (𝐴 ·s 𝑧𝑅))
-s (𝑥𝑅 ·s
𝑧𝑅))
→ (𝑎 = ((𝐴 ·s 𝐵) +s 𝑡) ↔ 𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝑅 ·s
𝐶) +s (𝐴 ·s 𝑧𝑅))
-s (𝑥𝑅 ·s
𝑧𝑅))))) |
75 | 72, 74 | ceqsexv 3523 |
. . . . . . . . . . . . 13
⊢
(∃𝑡(𝑡 = (((𝑥𝑅 ·s
𝐶) +s (𝐴 ·s 𝑧𝑅))
-s (𝑥𝑅 ·s
𝑧𝑅))
∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)) ↔ 𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝑅 ·s
𝐶) +s (𝐴 ·s 𝑧𝑅))
-s (𝑥𝑅 ·s
𝑧𝑅)))) |
76 | 75 | rexbii 3094 |
. . . . . . . . . . . 12
⊢
(∃𝑧𝑅 ∈ ( R ‘𝐶)∃𝑡(𝑡 = (((𝑥𝑅 ·s
𝐶) +s (𝐴 ·s 𝑧𝑅))
-s (𝑥𝑅 ·s
𝑧𝑅))
∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)) ↔ ∃𝑧𝑅 ∈ ( R
‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝑅 ·s
𝐶) +s (𝐴 ·s 𝑧𝑅))
-s (𝑥𝑅 ·s
𝑧𝑅)))) |
77 | 71, 76 | bitr3i 276 |
. . . . . . . . . . 11
⊢
(∃𝑡∃𝑧𝑅 ∈ ( R ‘𝐶)(𝑡 = (((𝑥𝑅 ·s
𝐶) +s (𝐴 ·s 𝑧𝑅))
-s (𝑥𝑅 ·s
𝑧𝑅))
∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)) ↔ ∃𝑧𝑅 ∈ ( R
‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝑅 ·s
𝐶) +s (𝐴 ·s 𝑧𝑅))
-s (𝑥𝑅 ·s
𝑧𝑅)))) |
78 | 77 | rexbii 3094 |
. . . . . . . . . 10
⊢
(∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑡∃𝑧𝑅 ∈ ( R ‘𝐶)(𝑡 = (((𝑥𝑅 ·s
𝐶) +s (𝐴 ·s 𝑧𝑅))
-s (𝑥𝑅 ·s
𝑧𝑅))
∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)) ↔ ∃𝑥𝑅 ∈ ( R
‘𝐴)∃𝑧𝑅 ∈ ( R
‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝑅 ·s
𝐶) +s (𝐴 ·s 𝑧𝑅))
-s (𝑥𝑅 ·s
𝑧𝑅)))) |
79 | | r19.41vv 3224 |
. . . . . . . . . . 11
⊢
(∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)(𝑡 = (((𝑥𝑅 ·s
𝐶) +s (𝐴 ·s 𝑧𝑅))
-s (𝑥𝑅 ·s
𝑧𝑅))
∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)) ↔ (∃𝑥𝑅 ∈ ( R
‘𝐴)∃𝑧𝑅 ∈ ( R
‘𝐶)𝑡 = (((𝑥𝑅 ·s
𝐶) +s (𝐴 ·s 𝑧𝑅))
-s (𝑥𝑅 ·s
𝑧𝑅))
∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡))) |
80 | 79 | exbii 1850 |
. . . . . . . . . 10
⊢
(∃𝑡∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)(𝑡 = (((𝑥𝑅 ·s
𝐶) +s (𝐴 ·s 𝑧𝑅))
-s (𝑥𝑅 ·s
𝑧𝑅))
∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)) ↔ ∃𝑡(∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑡 = (((𝑥𝑅 ·s
𝐶) +s (𝐴 ·s 𝑧𝑅))
-s (𝑥𝑅 ·s
𝑧𝑅))
∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡))) |
81 | 70, 78, 80 | 3bitr3ri 301 |
. . . . . . . . 9
⊢
(∃𝑡(∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑡 = (((𝑥𝑅 ·s
𝐶) +s (𝐴 ·s 𝑧𝑅))
-s (𝑥𝑅 ·s
𝑧𝑅))
∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)) ↔ ∃𝑥𝑅 ∈ ( R
‘𝐴)∃𝑧𝑅 ∈ ( R
‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝑅 ·s
𝐶) +s (𝐴 ·s 𝑧𝑅))
-s (𝑥𝑅 ·s
𝑧𝑅)))) |
82 | 69, 81 | bitri 274 |
. . . . . . . 8
⊢
(∃𝑡 ∈
{𝑏 ∣ ∃𝑥𝑅 ∈ ( R
‘𝐴)∃𝑧𝑅 ∈ ( R
‘𝐶)𝑏 = (((𝑥𝑅 ·s
𝐶) +s (𝐴 ·s 𝑧𝑅))
-s (𝑥𝑅 ·s
𝑧𝑅))}𝑎 = ((𝐴 ·s 𝐵) +s 𝑡) ↔ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝑅 ·s
𝐶) +s (𝐴 ·s 𝑧𝑅))
-s (𝑥𝑅 ·s
𝑧𝑅)))) |
83 | 66, 82 | orbi12i 913 |
. . . . . . 7
⊢
((∃𝑡 ∈
{𝑏 ∣ ∃𝑥𝐿 ∈ ( L
‘𝐴)∃𝑧𝐿 ∈ ( L
‘𝐶)𝑏 = (((𝑥𝐿 ·s
𝐶) +s (𝐴 ·s 𝑧𝐿))
-s (𝑥𝐿 ·s
𝑧𝐿))}𝑎 = ((𝐴 ·s 𝐵) +s 𝑡) ∨ ∃𝑡 ∈ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑥𝑅 ·s
𝐶) +s (𝐴 ·s 𝑧𝑅))
-s (𝑥𝑅 ·s
𝑧𝑅))}𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)) ↔ (∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝐿 ·s
𝐶) +s (𝐴 ·s 𝑧𝐿))
-s (𝑥𝐿 ·s
𝑧𝐿)))
∨ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝑅 ·s
𝐶) +s (𝐴 ·s 𝑧𝑅))
-s (𝑥𝑅 ·s
𝑧𝑅))))) |
84 | 50, 83 | bitr2i 275 |
. . . . . 6
⊢
((∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝐿 ·s
𝐶) +s (𝐴 ·s 𝑧𝐿))
-s (𝑥𝐿 ·s
𝑧𝐿)))
∨ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝑅 ·s
𝐶) +s (𝐴 ·s 𝑧𝑅))
-s (𝑥𝑅 ·s
𝑧𝑅))))
↔ ∃𝑡 ∈
({𝑏 ∣ ∃𝑥𝐿 ∈ ( L
‘𝐴)∃𝑧𝐿 ∈ ( L
‘𝐶)𝑏 = (((𝑥𝐿 ·s
𝐶) +s (𝐴 ·s 𝑧𝐿))
-s (𝑥𝐿 ·s
𝑧𝐿))}
∪ {𝑏 ∣
∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑥𝑅 ·s
𝐶) +s (𝐴 ·s 𝑧𝑅))
-s (𝑥𝑅 ·s
𝑧𝑅))})𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)) |
85 | 84 | abbii 2802 |
. . . . 5
⊢ {𝑎 ∣ (∃𝑥𝐿 ∈ ( L
‘𝐴)∃𝑧𝐿 ∈ ( L
‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝐿 ·s
𝐶) +s (𝐴 ·s 𝑧𝐿))
-s (𝑥𝐿 ·s
𝑧𝐿)))
∨ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝑅 ·s
𝐶) +s (𝐴 ·s 𝑧𝑅))
-s (𝑥𝑅 ·s
𝑧𝑅))))}
= {𝑎 ∣ ∃𝑡 ∈ ({𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑥𝐿 ·s
𝐶) +s (𝐴 ·s 𝑧𝐿))
-s (𝑥𝐿 ·s
𝑧𝐿))}
∪ {𝑏 ∣
∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑥𝑅 ·s
𝐶) +s (𝐴 ·s 𝑧𝑅))
-s (𝑥𝑅 ·s
𝑧𝑅))})𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)} |
86 | 49, 85 | eqtri 2760 |
. . . 4
⊢ ({𝑎 ∣ ∃𝑥𝐿 ∈ ( L
‘𝐴)∃𝑧𝐿 ∈ ( L
‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝐿 ·s
𝐶) +s (𝐴 ·s 𝑧𝐿))
-s (𝑥𝐿 ·s
𝑧𝐿)))}
∪ {𝑎 ∣
∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝑅 ·s
𝐶) +s (𝐴 ·s 𝑧𝑅))
-s (𝑥𝑅 ·s
𝑧𝑅)))})
= {𝑎 ∣ ∃𝑡 ∈ ({𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑥𝐿 ·s
𝐶) +s (𝐴 ·s 𝑧𝐿))
-s (𝑥𝐿 ·s
𝑧𝐿))}
∪ {𝑏 ∣
∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑥𝑅 ·s
𝐶) +s (𝐴 ·s 𝑧𝑅))
-s (𝑥𝑅 ·s
𝑧𝑅))})𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)} |
87 | 48, 86 | uneq12i 4158 |
. . 3
⊢ (({𝑎 ∣ ∃𝑥𝐿 ∈ ( L
‘𝐴)∃𝑦𝐿 ∈ ( L
‘𝐵)𝑎 = ((((𝑥𝐿 ·s
𝐵) +s (𝐴 ·s 𝑦𝐿))
-s (𝑥𝐿 ·s
𝑦𝐿))
+s (𝐴
·s 𝐶))}
∪ {𝑎 ∣
∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = ((((𝑥𝑅 ·s
𝐵) +s (𝐴 ·s 𝑦𝑅))
-s (𝑥𝑅 ·s
𝑦𝑅))
+s (𝐴
·s 𝐶))})
∪ ({𝑎 ∣
∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝐿 ·s
𝐶) +s (𝐴 ·s 𝑧𝐿))
-s (𝑥𝐿 ·s
𝑧𝐿)))}
∪ {𝑎 ∣
∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝑅 ·s
𝐶) +s (𝐴 ·s 𝑧𝑅))
-s (𝑥𝑅 ·s
𝑧𝑅)))}))
= ({𝑎 ∣ ∃𝑡 ∈ ({𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (((𝑥𝐿 ·s
𝐵) +s (𝐴 ·s 𝑦𝐿))
-s (𝑥𝐿 ·s
𝑦𝐿))}
∪ {𝑏 ∣
∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (((𝑥𝑅 ·s
𝐵) +s (𝐴 ·s 𝑦𝑅))
-s (𝑥𝑅 ·s
𝑦𝑅))})𝑎 = (𝑡 +s (𝐴 ·s 𝐶))} ∪ {𝑎 ∣ ∃𝑡 ∈ ({𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑥𝐿 ·s
𝐶) +s (𝐴 ·s 𝑧𝐿))
-s (𝑥𝐿 ·s
𝑧𝐿))}
∪ {𝑏 ∣
∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑥𝑅 ·s
𝐶) +s (𝐴 ·s 𝑧𝑅))
-s (𝑥𝑅 ·s
𝑧𝑅))})𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)}) |
88 | | unab 4295 |
. . . . 5
⊢ ({𝑎 ∣ ∃𝑥𝐿 ∈ ( L
‘𝐴)∃𝑦𝑅 ∈ ( R
‘𝐵)𝑎 = ((((𝑥𝐿 ·s
𝐵) +s (𝐴 ·s 𝑦𝑅))
-s (𝑥𝐿 ·s
𝑦𝑅))
+s (𝐴
·s 𝐶))}
∪ {𝑎 ∣
∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = ((((𝑥𝑅 ·s
𝐵) +s (𝐴 ·s 𝑦𝐿))
-s (𝑥𝑅 ·s
𝑦𝐿))
+s (𝐴
·s 𝐶))})
= {𝑎 ∣ (∃𝑥𝐿 ∈ ( L
‘𝐴)∃𝑦𝑅 ∈ ( R
‘𝐵)𝑎 = ((((𝑥𝐿 ·s
𝐵) +s (𝐴 ·s 𝑦𝑅))
-s (𝑥𝐿 ·s
𝑦𝑅))
+s (𝐴
·s 𝐶))
∨ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = ((((𝑥𝑅 ·s
𝐵) +s (𝐴 ·s 𝑦𝐿))
-s (𝑥𝑅 ·s
𝑦𝐿))
+s (𝐴
·s 𝐶)))} |
89 | | rexun 4187 |
. . . . . . 7
⊢
(∃𝑡 ∈
({𝑏 ∣ ∃𝑥𝐿 ∈ ( L
‘𝐴)∃𝑦𝑅 ∈ ( R
‘𝐵)𝑏 = (((𝑥𝐿 ·s
𝐵) +s (𝐴 ·s 𝑦𝑅))
-s (𝑥𝐿 ·s
𝑦𝑅))}
∪ {𝑏 ∣
∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (((𝑥𝑅 ·s
𝐵) +s (𝐴 ·s 𝑦𝐿))
-s (𝑥𝑅 ·s
𝑦𝐿))})𝑎 = (𝑡 +s (𝐴 ·s 𝐶)) ↔ (∃𝑡 ∈ {𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (((𝑥𝐿 ·s
𝐵) +s (𝐴 ·s 𝑦𝑅))
-s (𝑥𝐿 ·s
𝑦𝑅))}𝑎 = (𝑡 +s (𝐴 ·s 𝐶)) ∨ ∃𝑡 ∈ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (((𝑥𝑅 ·s
𝐵) +s (𝐴 ·s 𝑦𝐿))
-s (𝑥𝑅 ·s
𝑦𝐿))}𝑎 = (𝑡 +s (𝐴 ·s 𝐶)))) |
90 | | eqeq1 2736 |
. . . . . . . . . . 11
⊢ (𝑏 = 𝑡 → (𝑏 = (((𝑥𝐿 ·s
𝐵) +s (𝐴 ·s 𝑦𝑅))
-s (𝑥𝐿 ·s
𝑦𝑅))
↔ 𝑡 = (((𝑥𝐿
·s 𝐵)
+s (𝐴
·s 𝑦𝑅)) -s (𝑥𝐿
·s 𝑦𝑅)))) |
91 | 90 | 2rexbidv 3219 |
. . . . . . . . . 10
⊢ (𝑏 = 𝑡 → (∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (((𝑥𝐿 ·s
𝐵) +s (𝐴 ·s 𝑦𝑅))
-s (𝑥𝐿 ·s
𝑦𝑅))
↔ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑡 = (((𝑥𝐿 ·s
𝐵) +s (𝐴 ·s 𝑦𝑅))
-s (𝑥𝐿 ·s
𝑦𝑅)))) |
92 | 91 | rexab 3687 |
. . . . . . . . 9
⊢
(∃𝑡 ∈
{𝑏 ∣ ∃𝑥𝐿 ∈ ( L
‘𝐴)∃𝑦𝑅 ∈ ( R
‘𝐵)𝑏 = (((𝑥𝐿 ·s
𝐵) +s (𝐴 ·s 𝑦𝑅))
-s (𝑥𝐿 ·s
𝑦𝑅))}𝑎 = (𝑡 +s (𝐴 ·s 𝐶)) ↔ ∃𝑡(∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑡 = (((𝑥𝐿 ·s
𝐵) +s (𝐴 ·s 𝑦𝑅))
-s (𝑥𝐿 ·s
𝑦𝑅))
∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶)))) |
93 | | rexcom4 3285 |
. . . . . . . . . 10
⊢
(∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑡∃𝑦𝑅 ∈ ( R ‘𝐵)(𝑡 = (((𝑥𝐿 ·s
𝐵) +s (𝐴 ·s 𝑦𝑅))
-s (𝑥𝐿 ·s
𝑦𝑅))
∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶))) ↔ ∃𝑡∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)(𝑡 = (((𝑥𝐿 ·s
𝐵) +s (𝐴 ·s 𝑦𝑅))
-s (𝑥𝐿 ·s
𝑦𝑅))
∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶)))) |
94 | | rexcom4 3285 |
. . . . . . . . . . . 12
⊢
(∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑡(𝑡 = (((𝑥𝐿 ·s
𝐵) +s (𝐴 ·s 𝑦𝑅))
-s (𝑥𝐿 ·s
𝑦𝑅))
∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶))) ↔ ∃𝑡∃𝑦𝑅 ∈ ( R ‘𝐵)(𝑡 = (((𝑥𝐿 ·s
𝐵) +s (𝐴 ·s 𝑦𝑅))
-s (𝑥𝐿 ·s
𝑦𝑅))
∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶)))) |
95 | | ovex 7427 |
. . . . . . . . . . . . . 14
⊢ (((𝑥𝐿
·s 𝐵)
+s (𝐴
·s 𝑦𝑅)) -s (𝑥𝐿
·s 𝑦𝑅)) ∈
V |
96 | | oveq1 7401 |
. . . . . . . . . . . . . . 15
⊢ (𝑡 = (((𝑥𝐿 ·s
𝐵) +s (𝐴 ·s 𝑦𝑅))
-s (𝑥𝐿 ·s
𝑦𝑅))
→ (𝑡 +s
(𝐴 ·s
𝐶)) = ((((𝑥𝐿
·s 𝐵)
+s (𝐴
·s 𝑦𝑅)) -s (𝑥𝐿
·s 𝑦𝑅)) +s (𝐴 ·s 𝐶))) |
97 | 96 | eqeq2d 2743 |
. . . . . . . . . . . . . 14
⊢ (𝑡 = (((𝑥𝐿 ·s
𝐵) +s (𝐴 ·s 𝑦𝑅))
-s (𝑥𝐿 ·s
𝑦𝑅))
→ (𝑎 = (𝑡 +s (𝐴 ·s 𝐶)) ↔ 𝑎 = ((((𝑥𝐿 ·s
𝐵) +s (𝐴 ·s 𝑦𝑅))
-s (𝑥𝐿 ·s
𝑦𝑅))
+s (𝐴
·s 𝐶)))) |
98 | 95, 97 | ceqsexv 3523 |
. . . . . . . . . . . . 13
⊢
(∃𝑡(𝑡 = (((𝑥𝐿 ·s
𝐵) +s (𝐴 ·s 𝑦𝑅))
-s (𝑥𝐿 ·s
𝑦𝑅))
∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶))) ↔ 𝑎 = ((((𝑥𝐿 ·s
𝐵) +s (𝐴 ·s 𝑦𝑅))
-s (𝑥𝐿 ·s
𝑦𝑅))
+s (𝐴
·s 𝐶))) |
99 | 98 | rexbii 3094 |
. . . . . . . . . . . 12
⊢
(∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑡(𝑡 = (((𝑥𝐿 ·s
𝐵) +s (𝐴 ·s 𝑦𝑅))
-s (𝑥𝐿 ·s
𝑦𝑅))
∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶))) ↔ ∃𝑦𝑅 ∈ ( R
‘𝐵)𝑎 = ((((𝑥𝐿 ·s
𝐵) +s (𝐴 ·s 𝑦𝑅))
-s (𝑥𝐿 ·s
𝑦𝑅))
+s (𝐴
·s 𝐶))) |
100 | 94, 99 | bitr3i 276 |
. . . . . . . . . . 11
⊢
(∃𝑡∃𝑦𝑅 ∈ ( R ‘𝐵)(𝑡 = (((𝑥𝐿 ·s
𝐵) +s (𝐴 ·s 𝑦𝑅))
-s (𝑥𝐿 ·s
𝑦𝑅))
∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶))) ↔ ∃𝑦𝑅 ∈ ( R
‘𝐵)𝑎 = ((((𝑥𝐿 ·s
𝐵) +s (𝐴 ·s 𝑦𝑅))
-s (𝑥𝐿 ·s
𝑦𝑅))
+s (𝐴
·s 𝐶))) |
101 | 100 | rexbii 3094 |
. . . . . . . . . 10
⊢
(∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑡∃𝑦𝑅 ∈ ( R ‘𝐵)(𝑡 = (((𝑥𝐿 ·s
𝐵) +s (𝐴 ·s 𝑦𝑅))
-s (𝑥𝐿 ·s
𝑦𝑅))
∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶))) ↔ ∃𝑥𝐿 ∈ ( L
‘𝐴)∃𝑦𝑅 ∈ ( R
‘𝐵)𝑎 = ((((𝑥𝐿 ·s
𝐵) +s (𝐴 ·s 𝑦𝑅))
-s (𝑥𝐿 ·s
𝑦𝑅))
+s (𝐴
·s 𝐶))) |
102 | | r19.41vv 3224 |
. . . . . . . . . . 11
⊢
(∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)(𝑡 = (((𝑥𝐿 ·s
𝐵) +s (𝐴 ·s 𝑦𝑅))
-s (𝑥𝐿 ·s
𝑦𝑅))
∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶))) ↔ (∃𝑥𝐿 ∈ ( L
‘𝐴)∃𝑦𝑅 ∈ ( R
‘𝐵)𝑡 = (((𝑥𝐿 ·s
𝐵) +s (𝐴 ·s 𝑦𝑅))
-s (𝑥𝐿 ·s
𝑦𝑅))
∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶)))) |
103 | 102 | exbii 1850 |
. . . . . . . . . 10
⊢
(∃𝑡∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)(𝑡 = (((𝑥𝐿 ·s
𝐵) +s (𝐴 ·s 𝑦𝑅))
-s (𝑥𝐿 ·s
𝑦𝑅))
∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶))) ↔ ∃𝑡(∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑡 = (((𝑥𝐿 ·s
𝐵) +s (𝐴 ·s 𝑦𝑅))
-s (𝑥𝐿 ·s
𝑦𝑅))
∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶)))) |
104 | 93, 101, 103 | 3bitr3ri 301 |
. . . . . . . . 9
⊢
(∃𝑡(∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑡 = (((𝑥𝐿 ·s
𝐵) +s (𝐴 ·s 𝑦𝑅))
-s (𝑥𝐿 ·s
𝑦𝑅))
∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶))) ↔ ∃𝑥𝐿 ∈ ( L
‘𝐴)∃𝑦𝑅 ∈ ( R
‘𝐵)𝑎 = ((((𝑥𝐿 ·s
𝐵) +s (𝐴 ·s 𝑦𝑅))
-s (𝑥𝐿 ·s
𝑦𝑅))
+s (𝐴
·s 𝐶))) |
105 | 92, 104 | bitri 274 |
. . . . . . . 8
⊢
(∃𝑡 ∈
{𝑏 ∣ ∃𝑥𝐿 ∈ ( L
‘𝐴)∃𝑦𝑅 ∈ ( R
‘𝐵)𝑏 = (((𝑥𝐿 ·s
𝐵) +s (𝐴 ·s 𝑦𝑅))
-s (𝑥𝐿 ·s
𝑦𝑅))}𝑎 = (𝑡 +s (𝐴 ·s 𝐶)) ↔ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = ((((𝑥𝐿 ·s
𝐵) +s (𝐴 ·s 𝑦𝑅))
-s (𝑥𝐿 ·s
𝑦𝑅))
+s (𝐴
·s 𝐶))) |
106 | | eqeq1 2736 |
. . . . . . . . . . 11
⊢ (𝑏 = 𝑡 → (𝑏 = (((𝑥𝑅 ·s
𝐵) +s (𝐴 ·s 𝑦𝐿))
-s (𝑥𝑅 ·s
𝑦𝐿))
↔ 𝑡 = (((𝑥𝑅
·s 𝐵)
+s (𝐴
·s 𝑦𝐿)) -s (𝑥𝑅
·s 𝑦𝐿)))) |
107 | 106 | 2rexbidv 3219 |
. . . . . . . . . 10
⊢ (𝑏 = 𝑡 → (∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (((𝑥𝑅 ·s
𝐵) +s (𝐴 ·s 𝑦𝐿))
-s (𝑥𝑅 ·s
𝑦𝐿))
↔ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑡 = (((𝑥𝑅 ·s
𝐵) +s (𝐴 ·s 𝑦𝐿))
-s (𝑥𝑅 ·s
𝑦𝐿)))) |
108 | 107 | rexab 3687 |
. . . . . . . . 9
⊢
(∃𝑡 ∈
{𝑏 ∣ ∃𝑥𝑅 ∈ ( R
‘𝐴)∃𝑦𝐿 ∈ ( L
‘𝐵)𝑏 = (((𝑥𝑅 ·s
𝐵) +s (𝐴 ·s 𝑦𝐿))
-s (𝑥𝑅 ·s
𝑦𝐿))}𝑎 = (𝑡 +s (𝐴 ·s 𝐶)) ↔ ∃𝑡(∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑡 = (((𝑥𝑅 ·s
𝐵) +s (𝐴 ·s 𝑦𝐿))
-s (𝑥𝑅 ·s
𝑦𝐿))
∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶)))) |
109 | | rexcom4 3285 |
. . . . . . . . . 10
⊢
(∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑡∃𝑦𝐿 ∈ ( L ‘𝐵)(𝑡 = (((𝑥𝑅 ·s
𝐵) +s (𝐴 ·s 𝑦𝐿))
-s (𝑥𝑅 ·s
𝑦𝐿))
∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶))) ↔ ∃𝑡∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)(𝑡 = (((𝑥𝑅 ·s
𝐵) +s (𝐴 ·s 𝑦𝐿))
-s (𝑥𝑅 ·s
𝑦𝐿))
∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶)))) |
110 | | rexcom4 3285 |
. . . . . . . . . . . 12
⊢
(∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑡(𝑡 = (((𝑥𝑅 ·s
𝐵) +s (𝐴 ·s 𝑦𝐿))
-s (𝑥𝑅 ·s
𝑦𝐿))
∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶))) ↔ ∃𝑡∃𝑦𝐿 ∈ ( L ‘𝐵)(𝑡 = (((𝑥𝑅 ·s
𝐵) +s (𝐴 ·s 𝑦𝐿))
-s (𝑥𝑅 ·s
𝑦𝐿))
∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶)))) |
111 | | ovex 7427 |
. . . . . . . . . . . . . 14
⊢ (((𝑥𝑅
·s 𝐵)
+s (𝐴
·s 𝑦𝐿)) -s (𝑥𝑅
·s 𝑦𝐿)) ∈
V |
112 | | oveq1 7401 |
. . . . . . . . . . . . . . 15
⊢ (𝑡 = (((𝑥𝑅 ·s
𝐵) +s (𝐴 ·s 𝑦𝐿))
-s (𝑥𝑅 ·s
𝑦𝐿))
→ (𝑡 +s
(𝐴 ·s
𝐶)) = ((((𝑥𝑅
·s 𝐵)
+s (𝐴
·s 𝑦𝐿)) -s (𝑥𝑅
·s 𝑦𝐿)) +s (𝐴 ·s 𝐶))) |
113 | 112 | eqeq2d 2743 |
. . . . . . . . . . . . . 14
⊢ (𝑡 = (((𝑥𝑅 ·s
𝐵) +s (𝐴 ·s 𝑦𝐿))
-s (𝑥𝑅 ·s
𝑦𝐿))
→ (𝑎 = (𝑡 +s (𝐴 ·s 𝐶)) ↔ 𝑎 = ((((𝑥𝑅 ·s
𝐵) +s (𝐴 ·s 𝑦𝐿))
-s (𝑥𝑅 ·s
𝑦𝐿))
+s (𝐴
·s 𝐶)))) |
114 | 111, 113 | ceqsexv 3523 |
. . . . . . . . . . . . 13
⊢
(∃𝑡(𝑡 = (((𝑥𝑅 ·s
𝐵) +s (𝐴 ·s 𝑦𝐿))
-s (𝑥𝑅 ·s
𝑦𝐿))
∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶))) ↔ 𝑎 = ((((𝑥𝑅 ·s
𝐵) +s (𝐴 ·s 𝑦𝐿))
-s (𝑥𝑅 ·s
𝑦𝐿))
+s (𝐴
·s 𝐶))) |
115 | 114 | rexbii 3094 |
. . . . . . . . . . . 12
⊢
(∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑡(𝑡 = (((𝑥𝑅 ·s
𝐵) +s (𝐴 ·s 𝑦𝐿))
-s (𝑥𝑅 ·s
𝑦𝐿))
∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶))) ↔ ∃𝑦𝐿 ∈ ( L
‘𝐵)𝑎 = ((((𝑥𝑅 ·s
𝐵) +s (𝐴 ·s 𝑦𝐿))
-s (𝑥𝑅 ·s
𝑦𝐿))
+s (𝐴
·s 𝐶))) |
116 | 110, 115 | bitr3i 276 |
. . . . . . . . . . 11
⊢
(∃𝑡∃𝑦𝐿 ∈ ( L ‘𝐵)(𝑡 = (((𝑥𝑅 ·s
𝐵) +s (𝐴 ·s 𝑦𝐿))
-s (𝑥𝑅 ·s
𝑦𝐿))
∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶))) ↔ ∃𝑦𝐿 ∈ ( L
‘𝐵)𝑎 = ((((𝑥𝑅 ·s
𝐵) +s (𝐴 ·s 𝑦𝐿))
-s (𝑥𝑅 ·s
𝑦𝐿))
+s (𝐴
·s 𝐶))) |
117 | 116 | rexbii 3094 |
. . . . . . . . . 10
⊢
(∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑡∃𝑦𝐿 ∈ ( L ‘𝐵)(𝑡 = (((𝑥𝑅 ·s
𝐵) +s (𝐴 ·s 𝑦𝐿))
-s (𝑥𝑅 ·s
𝑦𝐿))
∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶))) ↔ ∃𝑥𝑅 ∈ ( R
‘𝐴)∃𝑦𝐿 ∈ ( L
‘𝐵)𝑎 = ((((𝑥𝑅 ·s
𝐵) +s (𝐴 ·s 𝑦𝐿))
-s (𝑥𝑅 ·s
𝑦𝐿))
+s (𝐴
·s 𝐶))) |
118 | | r19.41vv 3224 |
. . . . . . . . . . 11
⊢
(∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)(𝑡 = (((𝑥𝑅 ·s
𝐵) +s (𝐴 ·s 𝑦𝐿))
-s (𝑥𝑅 ·s
𝑦𝐿))
∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶))) ↔ (∃𝑥𝑅 ∈ ( R
‘𝐴)∃𝑦𝐿 ∈ ( L
‘𝐵)𝑡 = (((𝑥𝑅 ·s
𝐵) +s (𝐴 ·s 𝑦𝐿))
-s (𝑥𝑅 ·s
𝑦𝐿))
∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶)))) |
119 | 118 | exbii 1850 |
. . . . . . . . . 10
⊢
(∃𝑡∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)(𝑡 = (((𝑥𝑅 ·s
𝐵) +s (𝐴 ·s 𝑦𝐿))
-s (𝑥𝑅 ·s
𝑦𝐿))
∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶))) ↔ ∃𝑡(∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑡 = (((𝑥𝑅 ·s
𝐵) +s (𝐴 ·s 𝑦𝐿))
-s (𝑥𝑅 ·s
𝑦𝐿))
∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶)))) |
120 | 109, 117,
119 | 3bitr3ri 301 |
. . . . . . . . 9
⊢
(∃𝑡(∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑡 = (((𝑥𝑅 ·s
𝐵) +s (𝐴 ·s 𝑦𝐿))
-s (𝑥𝑅 ·s
𝑦𝐿))
∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶))) ↔ ∃𝑥𝑅 ∈ ( R
‘𝐴)∃𝑦𝐿 ∈ ( L
‘𝐵)𝑎 = ((((𝑥𝑅 ·s
𝐵) +s (𝐴 ·s 𝑦𝐿))
-s (𝑥𝑅 ·s
𝑦𝐿))
+s (𝐴
·s 𝐶))) |
121 | 108, 120 | bitri 274 |
. . . . . . . 8
⊢
(∃𝑡 ∈
{𝑏 ∣ ∃𝑥𝑅 ∈ ( R
‘𝐴)∃𝑦𝐿 ∈ ( L
‘𝐵)𝑏 = (((𝑥𝑅 ·s
𝐵) +s (𝐴 ·s 𝑦𝐿))
-s (𝑥𝑅 ·s
𝑦𝐿))}𝑎 = (𝑡 +s (𝐴 ·s 𝐶)) ↔ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = ((((𝑥𝑅 ·s
𝐵) +s (𝐴 ·s 𝑦𝐿))
-s (𝑥𝑅 ·s
𝑦𝐿))
+s (𝐴
·s 𝐶))) |
122 | 105, 121 | orbi12i 913 |
. . . . . . 7
⊢
((∃𝑡 ∈
{𝑏 ∣ ∃𝑥𝐿 ∈ ( L
‘𝐴)∃𝑦𝑅 ∈ ( R
‘𝐵)𝑏 = (((𝑥𝐿 ·s
𝐵) +s (𝐴 ·s 𝑦𝑅))
-s (𝑥𝐿 ·s
𝑦𝑅))}𝑎 = (𝑡 +s (𝐴 ·s 𝐶)) ∨ ∃𝑡 ∈ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (((𝑥𝑅 ·s
𝐵) +s (𝐴 ·s 𝑦𝐿))
-s (𝑥𝑅 ·s
𝑦𝐿))}𝑎 = (𝑡 +s (𝐴 ·s 𝐶))) ↔ (∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = ((((𝑥𝐿 ·s
𝐵) +s (𝐴 ·s 𝑦𝑅))
-s (𝑥𝐿 ·s
𝑦𝑅))
+s (𝐴
·s 𝐶))
∨ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = ((((𝑥𝑅 ·s
𝐵) +s (𝐴 ·s 𝑦𝐿))
-s (𝑥𝑅 ·s
𝑦𝐿))
+s (𝐴
·s 𝐶)))) |
123 | 89, 122 | bitr2i 275 |
. . . . . 6
⊢
((∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = ((((𝑥𝐿 ·s
𝐵) +s (𝐴 ·s 𝑦𝑅))
-s (𝑥𝐿 ·s
𝑦𝑅))
+s (𝐴
·s 𝐶))
∨ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = ((((𝑥𝑅 ·s
𝐵) +s (𝐴 ·s 𝑦𝐿))
-s (𝑥𝑅 ·s
𝑦𝐿))
+s (𝐴
·s 𝐶)))
↔ ∃𝑡 ∈
({𝑏 ∣ ∃𝑥𝐿 ∈ ( L
‘𝐴)∃𝑦𝑅 ∈ ( R
‘𝐵)𝑏 = (((𝑥𝐿 ·s
𝐵) +s (𝐴 ·s 𝑦𝑅))
-s (𝑥𝐿 ·s
𝑦𝑅))}
∪ {𝑏 ∣
∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (((𝑥𝑅 ·s
𝐵) +s (𝐴 ·s 𝑦𝐿))
-s (𝑥𝑅 ·s
𝑦𝐿))})𝑎 = (𝑡 +s (𝐴 ·s 𝐶))) |
124 | 123 | abbii 2802 |
. . . . 5
⊢ {𝑎 ∣ (∃𝑥𝐿 ∈ ( L
‘𝐴)∃𝑦𝑅 ∈ ( R
‘𝐵)𝑎 = ((((𝑥𝐿 ·s
𝐵) +s (𝐴 ·s 𝑦𝑅))
-s (𝑥𝐿 ·s
𝑦𝑅))
+s (𝐴
·s 𝐶))
∨ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = ((((𝑥𝑅 ·s
𝐵) +s (𝐴 ·s 𝑦𝐿))
-s (𝑥𝑅 ·s
𝑦𝐿))
+s (𝐴
·s 𝐶)))}
= {𝑎 ∣ ∃𝑡 ∈ ({𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (((𝑥𝐿 ·s
𝐵) +s (𝐴 ·s 𝑦𝑅))
-s (𝑥𝐿 ·s
𝑦𝑅))}
∪ {𝑏 ∣
∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (((𝑥𝑅 ·s
𝐵) +s (𝐴 ·s 𝑦𝐿))
-s (𝑥𝑅 ·s
𝑦𝐿))})𝑎 = (𝑡 +s (𝐴 ·s 𝐶))} |
125 | 88, 124 | eqtri 2760 |
. . . 4
⊢ ({𝑎 ∣ ∃𝑥𝐿 ∈ ( L
‘𝐴)∃𝑦𝑅 ∈ ( R
‘𝐵)𝑎 = ((((𝑥𝐿 ·s
𝐵) +s (𝐴 ·s 𝑦𝑅))
-s (𝑥𝐿 ·s
𝑦𝑅))
+s (𝐴
·s 𝐶))}
∪ {𝑎 ∣
∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = ((((𝑥𝑅 ·s
𝐵) +s (𝐴 ·s 𝑦𝐿))
-s (𝑥𝑅 ·s
𝑦𝐿))
+s (𝐴
·s 𝐶))})
= {𝑎 ∣ ∃𝑡 ∈ ({𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (((𝑥𝐿 ·s
𝐵) +s (𝐴 ·s 𝑦𝑅))
-s (𝑥𝐿 ·s
𝑦𝑅))}
∪ {𝑏 ∣
∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (((𝑥𝑅 ·s
𝐵) +s (𝐴 ·s 𝑦𝐿))
-s (𝑥𝑅 ·s
𝑦𝐿))})𝑎 = (𝑡 +s (𝐴 ·s 𝐶))} |
126 | | unab 4295 |
. . . . 5
⊢ ({𝑎 ∣ ∃𝑥𝐿 ∈ ( L
‘𝐴)∃𝑧𝑅 ∈ ( R
‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝐿 ·s
𝐶) +s (𝐴 ·s 𝑧𝑅))
-s (𝑥𝐿 ·s
𝑧𝑅)))}
∪ {𝑎 ∣
∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝑅 ·s
𝐶) +s (𝐴 ·s 𝑧𝐿))
-s (𝑥𝑅 ·s
𝑧𝐿)))})
= {𝑎 ∣ (∃𝑥𝐿 ∈ ( L
‘𝐴)∃𝑧𝑅 ∈ ( R
‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝐿 ·s
𝐶) +s (𝐴 ·s 𝑧𝑅))
-s (𝑥𝐿 ·s
𝑧𝑅)))
∨ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝑅 ·s
𝐶) +s (𝐴 ·s 𝑧𝐿))
-s (𝑥𝑅 ·s
𝑧𝐿))))} |
127 | | rexun 4187 |
. . . . . . 7
⊢
(∃𝑡 ∈
({𝑏 ∣ ∃𝑥𝐿 ∈ ( L
‘𝐴)∃𝑧𝑅 ∈ ( R
‘𝐶)𝑏 = (((𝑥𝐿 ·s
𝐶) +s (𝐴 ·s 𝑧𝑅))
-s (𝑥𝐿 ·s
𝑧𝑅))}
∪ {𝑏 ∣
∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑥𝑅 ·s
𝐶) +s (𝐴 ·s 𝑧𝐿))
-s (𝑥𝑅 ·s
𝑧𝐿))})𝑎 = ((𝐴 ·s 𝐵) +s 𝑡) ↔ (∃𝑡 ∈ {𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑥𝐿 ·s
𝐶) +s (𝐴 ·s 𝑧𝑅))
-s (𝑥𝐿 ·s
𝑧𝑅))}𝑎 = ((𝐴 ·s 𝐵) +s 𝑡) ∨ ∃𝑡 ∈ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑥𝑅 ·s
𝐶) +s (𝐴 ·s 𝑧𝐿))
-s (𝑥𝑅 ·s
𝑧𝐿))}𝑎 = ((𝐴 ·s 𝐵) +s 𝑡))) |
128 | | eqeq1 2736 |
. . . . . . . . . . 11
⊢ (𝑏 = 𝑡 → (𝑏 = (((𝑥𝐿 ·s
𝐶) +s (𝐴 ·s 𝑧𝑅))
-s (𝑥𝐿 ·s
𝑧𝑅))
↔ 𝑡 = (((𝑥𝐿
·s 𝐶)
+s (𝐴
·s 𝑧𝑅)) -s (𝑥𝐿
·s 𝑧𝑅)))) |
129 | 128 | 2rexbidv 3219 |
. . . . . . . . . 10
⊢ (𝑏 = 𝑡 → (∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑥𝐿 ·s
𝐶) +s (𝐴 ·s 𝑧𝑅))
-s (𝑥𝐿 ·s
𝑧𝑅))
↔ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑡 = (((𝑥𝐿 ·s
𝐶) +s (𝐴 ·s 𝑧𝑅))
-s (𝑥𝐿 ·s
𝑧𝑅)))) |
130 | 129 | rexab 3687 |
. . . . . . . . 9
⊢
(∃𝑡 ∈
{𝑏 ∣ ∃𝑥𝐿 ∈ ( L
‘𝐴)∃𝑧𝑅 ∈ ( R
‘𝐶)𝑏 = (((𝑥𝐿 ·s
𝐶) +s (𝐴 ·s 𝑧𝑅))
-s (𝑥𝐿 ·s
𝑧𝑅))}𝑎 = ((𝐴 ·s 𝐵) +s 𝑡) ↔ ∃𝑡(∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑡 = (((𝑥𝐿 ·s
𝐶) +s (𝐴 ·s 𝑧𝑅))
-s (𝑥𝐿 ·s
𝑧𝑅))
∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡))) |
131 | | rexcom4 3285 |
. . . . . . . . . 10
⊢
(∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑡∃𝑧𝑅 ∈ ( R ‘𝐶)(𝑡 = (((𝑥𝐿 ·s
𝐶) +s (𝐴 ·s 𝑧𝑅))
-s (𝑥𝐿 ·s
𝑧𝑅))
∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)) ↔ ∃𝑡∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)(𝑡 = (((𝑥𝐿 ·s
𝐶) +s (𝐴 ·s 𝑧𝑅))
-s (𝑥𝐿 ·s
𝑧𝑅))
∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡))) |
132 | | rexcom4 3285 |
. . . . . . . . . . . 12
⊢
(∃𝑧𝑅 ∈ ( R ‘𝐶)∃𝑡(𝑡 = (((𝑥𝐿 ·s
𝐶) +s (𝐴 ·s 𝑧𝑅))
-s (𝑥𝐿 ·s
𝑧𝑅))
∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)) ↔ ∃𝑡∃𝑧𝑅 ∈ ( R ‘𝐶)(𝑡 = (((𝑥𝐿 ·s
𝐶) +s (𝐴 ·s 𝑧𝑅))
-s (𝑥𝐿 ·s
𝑧𝑅))
∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡))) |
133 | | ovex 7427 |
. . . . . . . . . . . . . 14
⊢ (((𝑥𝐿
·s 𝐶)
+s (𝐴
·s 𝑧𝑅)) -s (𝑥𝐿
·s 𝑧𝑅)) ∈
V |
134 | | oveq2 7402 |
. . . . . . . . . . . . . . 15
⊢ (𝑡 = (((𝑥𝐿 ·s
𝐶) +s (𝐴 ·s 𝑧𝑅))
-s (𝑥𝐿 ·s
𝑧𝑅))
→ ((𝐴
·s 𝐵)
+s 𝑡) = ((𝐴 ·s 𝐵) +s (((𝑥𝐿
·s 𝐶)
+s (𝐴
·s 𝑧𝑅)) -s (𝑥𝐿
·s 𝑧𝑅)))) |
135 | 134 | eqeq2d 2743 |
. . . . . . . . . . . . . 14
⊢ (𝑡 = (((𝑥𝐿 ·s
𝐶) +s (𝐴 ·s 𝑧𝑅))
-s (𝑥𝐿 ·s
𝑧𝑅))
→ (𝑎 = ((𝐴 ·s 𝐵) +s 𝑡) ↔ 𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝐿 ·s
𝐶) +s (𝐴 ·s 𝑧𝑅))
-s (𝑥𝐿 ·s
𝑧𝑅))))) |
136 | 133, 135 | ceqsexv 3523 |
. . . . . . . . . . . . 13
⊢
(∃𝑡(𝑡 = (((𝑥𝐿 ·s
𝐶) +s (𝐴 ·s 𝑧𝑅))
-s (𝑥𝐿 ·s
𝑧𝑅))
∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)) ↔ 𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝐿 ·s
𝐶) +s (𝐴 ·s 𝑧𝑅))
-s (𝑥𝐿 ·s
𝑧𝑅)))) |
137 | 136 | rexbii 3094 |
. . . . . . . . . . . 12
⊢
(∃𝑧𝑅 ∈ ( R ‘𝐶)∃𝑡(𝑡 = (((𝑥𝐿 ·s
𝐶) +s (𝐴 ·s 𝑧𝑅))
-s (𝑥𝐿 ·s
𝑧𝑅))
∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)) ↔ ∃𝑧𝑅 ∈ ( R
‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝐿 ·s
𝐶) +s (𝐴 ·s 𝑧𝑅))
-s (𝑥𝐿 ·s
𝑧𝑅)))) |
138 | 132, 137 | bitr3i 276 |
. . . . . . . . . . 11
⊢
(∃𝑡∃𝑧𝑅 ∈ ( R ‘𝐶)(𝑡 = (((𝑥𝐿 ·s
𝐶) +s (𝐴 ·s 𝑧𝑅))
-s (𝑥𝐿 ·s
𝑧𝑅))
∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)) ↔ ∃𝑧𝑅 ∈ ( R
‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝐿 ·s
𝐶) +s (𝐴 ·s 𝑧𝑅))
-s (𝑥𝐿 ·s
𝑧𝑅)))) |
139 | 138 | rexbii 3094 |
. . . . . . . . . 10
⊢
(∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑡∃𝑧𝑅 ∈ ( R ‘𝐶)(𝑡 = (((𝑥𝐿 ·s
𝐶) +s (𝐴 ·s 𝑧𝑅))
-s (𝑥𝐿 ·s
𝑧𝑅))
∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)) ↔ ∃𝑥𝐿 ∈ ( L
‘𝐴)∃𝑧𝑅 ∈ ( R
‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝐿 ·s
𝐶) +s (𝐴 ·s 𝑧𝑅))
-s (𝑥𝐿 ·s
𝑧𝑅)))) |
140 | | r19.41vv 3224 |
. . . . . . . . . . 11
⊢
(∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)(𝑡 = (((𝑥𝐿 ·s
𝐶) +s (𝐴 ·s 𝑧𝑅))
-s (𝑥𝐿 ·s
𝑧𝑅))
∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)) ↔ (∃𝑥𝐿 ∈ ( L
‘𝐴)∃𝑧𝑅 ∈ ( R
‘𝐶)𝑡 = (((𝑥𝐿 ·s
𝐶) +s (𝐴 ·s 𝑧𝑅))
-s (𝑥𝐿 ·s
𝑧𝑅))
∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡))) |
141 | 140 | exbii 1850 |
. . . . . . . . . 10
⊢
(∃𝑡∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)(𝑡 = (((𝑥𝐿 ·s
𝐶) +s (𝐴 ·s 𝑧𝑅))
-s (𝑥𝐿 ·s
𝑧𝑅))
∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)) ↔ ∃𝑡(∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑡 = (((𝑥𝐿 ·s
𝐶) +s (𝐴 ·s 𝑧𝑅))
-s (𝑥𝐿 ·s
𝑧𝑅))
∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡))) |
142 | 131, 139,
141 | 3bitr3ri 301 |
. . . . . . . . 9
⊢
(∃𝑡(∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑡 = (((𝑥𝐿 ·s
𝐶) +s (𝐴 ·s 𝑧𝑅))
-s (𝑥𝐿 ·s
𝑧𝑅))
∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)) ↔ ∃𝑥𝐿 ∈ ( L
‘𝐴)∃𝑧𝑅 ∈ ( R
‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝐿 ·s
𝐶) +s (𝐴 ·s 𝑧𝑅))
-s (𝑥𝐿 ·s
𝑧𝑅)))) |
143 | 130, 142 | bitri 274 |
. . . . . . . 8
⊢
(∃𝑡 ∈
{𝑏 ∣ ∃𝑥𝐿 ∈ ( L
‘𝐴)∃𝑧𝑅 ∈ ( R
‘𝐶)𝑏 = (((𝑥𝐿 ·s
𝐶) +s (𝐴 ·s 𝑧𝑅))
-s (𝑥𝐿 ·s
𝑧𝑅))}𝑎 = ((𝐴 ·s 𝐵) +s 𝑡) ↔ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝐿 ·s
𝐶) +s (𝐴 ·s 𝑧𝑅))
-s (𝑥𝐿 ·s
𝑧𝑅)))) |
144 | | eqeq1 2736 |
. . . . . . . . . . 11
⊢ (𝑏 = 𝑡 → (𝑏 = (((𝑥𝑅 ·s
𝐶) +s (𝐴 ·s 𝑧𝐿))
-s (𝑥𝑅 ·s
𝑧𝐿))
↔ 𝑡 = (((𝑥𝑅
·s 𝐶)
+s (𝐴
·s 𝑧𝐿)) -s (𝑥𝑅
·s 𝑧𝐿)))) |
145 | 144 | 2rexbidv 3219 |
. . . . . . . . . 10
⊢ (𝑏 = 𝑡 → (∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑥𝑅 ·s
𝐶) +s (𝐴 ·s 𝑧𝐿))
-s (𝑥𝑅 ·s
𝑧𝐿))
↔ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑡 = (((𝑥𝑅 ·s
𝐶) +s (𝐴 ·s 𝑧𝐿))
-s (𝑥𝑅 ·s
𝑧𝐿)))) |
146 | 145 | rexab 3687 |
. . . . . . . . 9
⊢
(∃𝑡 ∈
{𝑏 ∣ ∃𝑥𝑅 ∈ ( R
‘𝐴)∃𝑧𝐿 ∈ ( L
‘𝐶)𝑏 = (((𝑥𝑅 ·s
𝐶) +s (𝐴 ·s 𝑧𝐿))
-s (𝑥𝑅 ·s
𝑧𝐿))}𝑎 = ((𝐴 ·s 𝐵) +s 𝑡) ↔ ∃𝑡(∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑡 = (((𝑥𝑅 ·s
𝐶) +s (𝐴 ·s 𝑧𝐿))
-s (𝑥𝑅 ·s
𝑧𝐿))
∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡))) |
147 | | rexcom4 3285 |
. . . . . . . . . 10
⊢
(∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑡∃𝑧𝐿 ∈ ( L ‘𝐶)(𝑡 = (((𝑥𝑅 ·s
𝐶) +s (𝐴 ·s 𝑧𝐿))
-s (𝑥𝑅 ·s
𝑧𝐿))
∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)) ↔ ∃𝑡∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)(𝑡 = (((𝑥𝑅 ·s
𝐶) +s (𝐴 ·s 𝑧𝐿))
-s (𝑥𝑅 ·s
𝑧𝐿))
∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡))) |
148 | | rexcom4 3285 |
. . . . . . . . . . . 12
⊢
(∃𝑧𝐿 ∈ ( L ‘𝐶)∃𝑡(𝑡 = (((𝑥𝑅 ·s
𝐶) +s (𝐴 ·s 𝑧𝐿))
-s (𝑥𝑅 ·s
𝑧𝐿))
∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)) ↔ ∃𝑡∃𝑧𝐿 ∈ ( L ‘𝐶)(𝑡 = (((𝑥𝑅 ·s
𝐶) +s (𝐴 ·s 𝑧𝐿))
-s (𝑥𝑅 ·s
𝑧𝐿))
∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡))) |
149 | | ovex 7427 |
. . . . . . . . . . . . . 14
⊢ (((𝑥𝑅
·s 𝐶)
+s (𝐴
·s 𝑧𝐿)) -s (𝑥𝑅
·s 𝑧𝐿)) ∈
V |
150 | | oveq2 7402 |
. . . . . . . . . . . . . . 15
⊢ (𝑡 = (((𝑥𝑅 ·s
𝐶) +s (𝐴 ·s 𝑧𝐿))
-s (𝑥𝑅 ·s
𝑧𝐿))
→ ((𝐴
·s 𝐵)
+s 𝑡) = ((𝐴 ·s 𝐵) +s (((𝑥𝑅
·s 𝐶)
+s (𝐴
·s 𝑧𝐿)) -s (𝑥𝑅
·s 𝑧𝐿)))) |
151 | 150 | eqeq2d 2743 |
. . . . . . . . . . . . . 14
⊢ (𝑡 = (((𝑥𝑅 ·s
𝐶) +s (𝐴 ·s 𝑧𝐿))
-s (𝑥𝑅 ·s
𝑧𝐿))
→ (𝑎 = ((𝐴 ·s 𝐵) +s 𝑡) ↔ 𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝑅 ·s
𝐶) +s (𝐴 ·s 𝑧𝐿))
-s (𝑥𝑅 ·s
𝑧𝐿))))) |
152 | 149, 151 | ceqsexv 3523 |
. . . . . . . . . . . . 13
⊢
(∃𝑡(𝑡 = (((𝑥𝑅 ·s
𝐶) +s (𝐴 ·s 𝑧𝐿))
-s (𝑥𝑅 ·s
𝑧𝐿))
∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)) ↔ 𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝑅 ·s
𝐶) +s (𝐴 ·s 𝑧𝐿))
-s (𝑥𝑅 ·s
𝑧𝐿)))) |
153 | 152 | rexbii 3094 |
. . . . . . . . . . . 12
⊢
(∃𝑧𝐿 ∈ ( L ‘𝐶)∃𝑡(𝑡 = (((𝑥𝑅 ·s
𝐶) +s (𝐴 ·s 𝑧𝐿))
-s (𝑥𝑅 ·s
𝑧𝐿))
∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)) ↔ ∃𝑧𝐿 ∈ ( L
‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝑅 ·s
𝐶) +s (𝐴 ·s 𝑧𝐿))
-s (𝑥𝑅 ·s
𝑧𝐿)))) |
154 | 148, 153 | bitr3i 276 |
. . . . . . . . . . 11
⊢
(∃𝑡∃𝑧𝐿 ∈ ( L ‘𝐶)(𝑡 = (((𝑥𝑅 ·s
𝐶) +s (𝐴 ·s 𝑧𝐿))
-s (𝑥𝑅 ·s
𝑧𝐿))
∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)) ↔ ∃𝑧𝐿 ∈ ( L
‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝑅 ·s
𝐶) +s (𝐴 ·s 𝑧𝐿))
-s (𝑥𝑅 ·s
𝑧𝐿)))) |
155 | 154 | rexbii 3094 |
. . . . . . . . . 10
⊢
(∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑡∃𝑧𝐿 ∈ ( L ‘𝐶)(𝑡 = (((𝑥𝑅 ·s
𝐶) +s (𝐴 ·s 𝑧𝐿))
-s (𝑥𝑅 ·s
𝑧𝐿))
∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)) ↔ ∃𝑥𝑅 ∈ ( R
‘𝐴)∃𝑧𝐿 ∈ ( L
‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝑅 ·s
𝐶) +s (𝐴 ·s 𝑧𝐿))
-s (𝑥𝑅 ·s
𝑧𝐿)))) |
156 | | r19.41vv 3224 |
. . . . . . . . . . 11
⊢
(∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)(𝑡 = (((𝑥𝑅 ·s
𝐶) +s (𝐴 ·s 𝑧𝐿))
-s (𝑥𝑅 ·s
𝑧𝐿))
∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)) ↔ (∃𝑥𝑅 ∈ ( R
‘𝐴)∃𝑧𝐿 ∈ ( L
‘𝐶)𝑡 = (((𝑥𝑅 ·s
𝐶) +s (𝐴 ·s 𝑧𝐿))
-s (𝑥𝑅 ·s
𝑧𝐿))
∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡))) |
157 | 156 | exbii 1850 |
. . . . . . . . . 10
⊢
(∃𝑡∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)(𝑡 = (((𝑥𝑅 ·s
𝐶) +s (𝐴 ·s 𝑧𝐿))
-s (𝑥𝑅 ·s
𝑧𝐿))
∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)) ↔ ∃𝑡(∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑡 = (((𝑥𝑅 ·s
𝐶) +s (𝐴 ·s 𝑧𝐿))
-s (𝑥𝑅 ·s
𝑧𝐿))
∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡))) |
158 | 147, 155,
157 | 3bitr3ri 301 |
. . . . . . . . 9
⊢
(∃𝑡(∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑡 = (((𝑥𝑅 ·s
𝐶) +s (𝐴 ·s 𝑧𝐿))
-s (𝑥𝑅 ·s
𝑧𝐿))
∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)) ↔ ∃𝑥𝑅 ∈ ( R
‘𝐴)∃𝑧𝐿 ∈ ( L
‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝑅 ·s
𝐶) +s (𝐴 ·s 𝑧𝐿))
-s (𝑥𝑅 ·s
𝑧𝐿)))) |
159 | 146, 158 | bitri 274 |
. . . . . . . 8
⊢
(∃𝑡 ∈
{𝑏 ∣ ∃𝑥𝑅 ∈ ( R
‘𝐴)∃𝑧𝐿 ∈ ( L
‘𝐶)𝑏 = (((𝑥𝑅 ·s
𝐶) +s (𝐴 ·s 𝑧𝐿))
-s (𝑥𝑅 ·s
𝑧𝐿))}𝑎 = ((𝐴 ·s 𝐵) +s 𝑡) ↔ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝑅 ·s
𝐶) +s (𝐴 ·s 𝑧𝐿))
-s (𝑥𝑅 ·s
𝑧𝐿)))) |
160 | 143, 159 | orbi12i 913 |
. . . . . . 7
⊢
((∃𝑡 ∈
{𝑏 ∣ ∃𝑥𝐿 ∈ ( L
‘𝐴)∃𝑧𝑅 ∈ ( R
‘𝐶)𝑏 = (((𝑥𝐿 ·s
𝐶) +s (𝐴 ·s 𝑧𝑅))
-s (𝑥𝐿 ·s
𝑧𝑅))}𝑎 = ((𝐴 ·s 𝐵) +s 𝑡) ∨ ∃𝑡 ∈ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑥𝑅 ·s
𝐶) +s (𝐴 ·s 𝑧𝐿))
-s (𝑥𝑅 ·s
𝑧𝐿))}𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)) ↔ (∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝐿 ·s
𝐶) +s (𝐴 ·s 𝑧𝑅))
-s (𝑥𝐿 ·s
𝑧𝑅)))
∨ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝑅 ·s
𝐶) +s (𝐴 ·s 𝑧𝐿))
-s (𝑥𝑅 ·s
𝑧𝐿))))) |
161 | 127, 160 | bitr2i 275 |
. . . . . 6
⊢
((∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝐿 ·s
𝐶) +s (𝐴 ·s 𝑧𝑅))
-s (𝑥𝐿 ·s
𝑧𝑅)))
∨ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝑅 ·s
𝐶) +s (𝐴 ·s 𝑧𝐿))
-s (𝑥𝑅 ·s
𝑧𝐿))))
↔ ∃𝑡 ∈
({𝑏 ∣ ∃𝑥𝐿 ∈ ( L
‘𝐴)∃𝑧𝑅 ∈ ( R
‘𝐶)𝑏 = (((𝑥𝐿 ·s
𝐶) +s (𝐴 ·s 𝑧𝑅))
-s (𝑥𝐿 ·s
𝑧𝑅))}
∪ {𝑏 ∣
∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑥𝑅 ·s
𝐶) +s (𝐴 ·s 𝑧𝐿))
-s (𝑥𝑅 ·s
𝑧𝐿))})𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)) |
162 | 161 | abbii 2802 |
. . . . 5
⊢ {𝑎 ∣ (∃𝑥𝐿 ∈ ( L
‘𝐴)∃𝑧𝑅 ∈ ( R
‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝐿 ·s
𝐶) +s (𝐴 ·s 𝑧𝑅))
-s (𝑥𝐿 ·s
𝑧𝑅)))
∨ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝑅 ·s
𝐶) +s (𝐴 ·s 𝑧𝐿))
-s (𝑥𝑅 ·s
𝑧𝐿))))}
= {𝑎 ∣ ∃𝑡 ∈ ({𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑥𝐿 ·s
𝐶) +s (𝐴 ·s 𝑧𝑅))
-s (𝑥𝐿 ·s
𝑧𝑅))}
∪ {𝑏 ∣
∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑥𝑅 ·s
𝐶) +s (𝐴 ·s 𝑧𝐿))
-s (𝑥𝑅 ·s
𝑧𝐿))})𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)} |
163 | 126, 162 | eqtri 2760 |
. . . 4
⊢ ({𝑎 ∣ ∃𝑥𝐿 ∈ ( L
‘𝐴)∃𝑧𝑅 ∈ ( R
‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝐿 ·s
𝐶) +s (𝐴 ·s 𝑧𝑅))
-s (𝑥𝐿 ·s
𝑧𝑅)))}
∪ {𝑎 ∣
∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝑅 ·s
𝐶) +s (𝐴 ·s 𝑧𝐿))
-s (𝑥𝑅 ·s
𝑧𝐿)))})
= {𝑎 ∣ ∃𝑡 ∈ ({𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑥𝐿 ·s
𝐶) +s (𝐴 ·s 𝑧𝑅))
-s (𝑥𝐿 ·s
𝑧𝑅))}
∪ {𝑏 ∣
∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑥𝑅 ·s
𝐶) +s (𝐴 ·s 𝑧𝐿))
-s (𝑥𝑅 ·s
𝑧𝐿))})𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)} |
164 | 125, 163 | uneq12i 4158 |
. . 3
⊢ (({𝑎 ∣ ∃𝑥𝐿 ∈ ( L
‘𝐴)∃𝑦𝑅 ∈ ( R
‘𝐵)𝑎 = ((((𝑥𝐿 ·s
𝐵) +s (𝐴 ·s 𝑦𝑅))
-s (𝑥𝐿 ·s
𝑦𝑅))
+s (𝐴
·s 𝐶))}
∪ {𝑎 ∣
∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = ((((𝑥𝑅 ·s
𝐵) +s (𝐴 ·s 𝑦𝐿))
-s (𝑥𝑅 ·s
𝑦𝐿))
+s (𝐴
·s 𝐶))})
∪ ({𝑎 ∣
∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝐿 ·s
𝐶) +s (𝐴 ·s 𝑧𝑅))
-s (𝑥𝐿 ·s
𝑧𝑅)))}
∪ {𝑎 ∣
∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝑅 ·s
𝐶) +s (𝐴 ·s 𝑧𝐿))
-s (𝑥𝑅 ·s
𝑧𝐿)))}))
= ({𝑎 ∣ ∃𝑡 ∈ ({𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (((𝑥𝐿 ·s
𝐵) +s (𝐴 ·s 𝑦𝑅))
-s (𝑥𝐿 ·s
𝑦𝑅))}
∪ {𝑏 ∣
∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (((𝑥𝑅 ·s
𝐵) +s (𝐴 ·s 𝑦𝐿))
-s (𝑥𝑅 ·s
𝑦𝐿))})𝑎 = (𝑡 +s (𝐴 ·s 𝐶))} ∪ {𝑎 ∣ ∃𝑡 ∈ ({𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑥𝐿 ·s
𝐶) +s (𝐴 ·s 𝑧𝑅))
-s (𝑥𝐿 ·s
𝑧𝑅))}
∪ {𝑏 ∣
∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑥𝑅 ·s
𝐶) +s (𝐴 ·s 𝑧𝐿))
-s (𝑥𝑅 ·s
𝑧𝐿))})𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)}) |
165 | 87, 164 | oveq12i 7406 |
. 2
⊢ ((({𝑎 ∣ ∃𝑥𝐿 ∈ ( L
‘𝐴)∃𝑦𝐿 ∈ ( L
‘𝐵)𝑎 = ((((𝑥𝐿 ·s
𝐵) +s (𝐴 ·s 𝑦𝐿))
-s (𝑥𝐿 ·s
𝑦𝐿))
+s (𝐴
·s 𝐶))}
∪ {𝑎 ∣
∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = ((((𝑥𝑅 ·s
𝐵) +s (𝐴 ·s 𝑦𝑅))
-s (𝑥𝑅 ·s
𝑦𝑅))
+s (𝐴
·s 𝐶))})
∪ ({𝑎 ∣
∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝐿 ·s
𝐶) +s (𝐴 ·s 𝑧𝐿))
-s (𝑥𝐿 ·s
𝑧𝐿)))}
∪ {𝑎 ∣
∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝑅 ·s
𝐶) +s (𝐴 ·s 𝑧𝑅))
-s (𝑥𝑅 ·s
𝑧𝑅)))}))
|s (({𝑎 ∣
∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = ((((𝑥𝐿 ·s
𝐵) +s (𝐴 ·s 𝑦𝑅))
-s (𝑥𝐿 ·s
𝑦𝑅))
+s (𝐴
·s 𝐶))}
∪ {𝑎 ∣
∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = ((((𝑥𝑅 ·s
𝐵) +s (𝐴 ·s 𝑦𝐿))
-s (𝑥𝑅 ·s
𝑦𝐿))
+s (𝐴
·s 𝐶))})
∪ ({𝑎 ∣
∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝐿 ·s
𝐶) +s (𝐴 ·s 𝑧𝑅))
-s (𝑥𝐿 ·s
𝑧𝑅)))}
∪ {𝑎 ∣
∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝑅 ·s
𝐶) +s (𝐴 ·s 𝑧𝐿))
-s (𝑥𝑅 ·s
𝑧𝐿)))}))) = (({𝑎 ∣ ∃𝑡 ∈ ({𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (((𝑥𝐿 ·s
𝐵) +s (𝐴 ·s 𝑦𝐿))
-s (𝑥𝐿 ·s
𝑦𝐿))}
∪ {𝑏 ∣
∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (((𝑥𝑅 ·s
𝐵) +s (𝐴 ·s 𝑦𝑅))
-s (𝑥𝑅 ·s
𝑦𝑅))})𝑎 = (𝑡 +s (𝐴 ·s 𝐶))} ∪ {𝑎 ∣ ∃𝑡 ∈ ({𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑥𝐿 ·s
𝐶) +s (𝐴 ·s 𝑧𝐿))
-s (𝑥𝐿 ·s
𝑧𝐿))}
∪ {𝑏 ∣
∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑥𝑅 ·s
𝐶) +s (𝐴 ·s 𝑧𝑅))
-s (𝑥𝑅 ·s
𝑧𝑅))})𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)}) |s ({𝑎 ∣ ∃𝑡 ∈ ({𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (((𝑥𝐿 ·s
𝐵) +s (𝐴 ·s 𝑦𝑅))
-s (𝑥𝐿 ·s
𝑦𝑅))}
∪ {𝑏 ∣
∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (((𝑥𝑅 ·s
𝐵) +s (𝐴 ·s 𝑦𝐿))
-s (𝑥𝑅 ·s
𝑦𝐿))})𝑎 = (𝑡 +s (𝐴 ·s 𝐶))} ∪ {𝑎 ∣ ∃𝑡 ∈ ({𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑥𝐿 ·s
𝐶) +s (𝐴 ·s 𝑧𝑅))
-s (𝑥𝐿 ·s
𝑧𝑅))}
∪ {𝑏 ∣
∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑥𝑅 ·s
𝐶) +s (𝐴 ·s 𝑧𝐿))
-s (𝑥𝑅 ·s
𝑧𝐿))})𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)})) |
166 | 10, 165 | eqtr4di 2790 |
1
⊢ (𝜑 → ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝐶)) = ((({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = ((((𝑥𝐿 ·s
𝐵) +s (𝐴 ·s 𝑦𝐿))
-s (𝑥𝐿 ·s
𝑦𝐿))
+s (𝐴
·s 𝐶))}
∪ {𝑎 ∣
∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = ((((𝑥𝑅 ·s
𝐵) +s (𝐴 ·s 𝑦𝑅))
-s (𝑥𝑅 ·s
𝑦𝑅))
+s (𝐴
·s 𝐶))})
∪ ({𝑎 ∣
∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝐿 ·s
𝐶) +s (𝐴 ·s 𝑧𝐿))
-s (𝑥𝐿 ·s
𝑧𝐿)))}
∪ {𝑎 ∣
∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝑅 ·s
𝐶) +s (𝐴 ·s 𝑧𝑅))
-s (𝑥𝑅 ·s
𝑧𝑅)))}))
|s (({𝑎 ∣
∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = ((((𝑥𝐿 ·s
𝐵) +s (𝐴 ·s 𝑦𝑅))
-s (𝑥𝐿 ·s
𝑦𝑅))
+s (𝐴
·s 𝐶))}
∪ {𝑎 ∣
∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = ((((𝑥𝑅 ·s
𝐵) +s (𝐴 ·s 𝑦𝐿))
-s (𝑥𝑅 ·s
𝑦𝐿))
+s (𝐴
·s 𝐶))})
∪ ({𝑎 ∣
∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝐿 ·s
𝐶) +s (𝐴 ·s 𝑧𝑅))
-s (𝑥𝐿 ·s
𝑧𝑅)))}
∪ {𝑎 ∣
∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝑅 ·s
𝐶) +s (𝐴 ·s 𝑧𝐿))
-s (𝑥𝑅 ·s
𝑧𝐿)))})))) |