| Step | Hyp | Ref
| Expression |
| 1 | | addsdilem.1 |
. . . 4
⊢ (𝜑 → 𝐴 ∈ No
) |
| 2 | | addsdilem.2 |
. . . 4
⊢ (𝜑 → 𝐵 ∈ No
) |
| 3 | 1, 2 | mulscut2 28159 |
. . 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 28159 |
. . 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 28137 |
. . . 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 28137 |
. . . 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 28035 |
. 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 4308 |
. . . . 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 4196 |
. . . . . . 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 2741 |
. . . . . . . . . . 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 3222 |
. . . . . . . . . 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 3700 |
. . . . . . . . 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 3288 |
. . . . . . . . . 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 3288 |
. . . . . . . . . . . 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 7464 |
. . . . . . . . . . . . . 14
⊢ (((𝑥𝐿
·s 𝐵)
+s (𝐴
·s 𝑦𝐿)) -s (𝑥𝐿
·s 𝑦𝐿)) ∈
V |
| 19 | | oveq1 7438 |
. . . . . . . . . . . . . . 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 2748 |
. . . . . . . . . . . . . 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 3532 |
. . . . . . . . . . . . 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 277 |
. . . . . . . . . . 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 3227 |
. . . . . . . . . . 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 1848 |
. . . . . . . . . 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 302 |
. . . . . . . . 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 275 |
. . . . . . . 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 2741 |
. . . . . . . . . . 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 3222 |
. . . . . . . . . 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 3700 |
. . . . . . . . 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 3288 |
. . . . . . . . . 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 3288 |
. . . . . . . . . . . 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 7464 |
. . . . . . . . . . . . . 14
⊢ (((𝑥𝑅
·s 𝐵)
+s (𝐴
·s 𝑦𝑅)) -s (𝑥𝑅
·s 𝑦𝑅)) ∈
V |
| 35 | | oveq1 7438 |
. . . . . . . . . . . . . . 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 2748 |
. . . . . . . . . . . . . 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 3532 |
. . . . . . . . . . . . 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 277 |
. . . . . . . . . . 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 3227 |
. . . . . . . . . . 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 1848 |
. . . . . . . . . 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 302 |
. . . . . . . . 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 275 |
. . . . . . . 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 915 |
. . . . . . 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 276 |
. . . . . 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 2809 |
. . . . 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 2765 |
. . . 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 4308 |
. . . . 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 4196 |
. . . . . . 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 2741 |
. . . . . . . . . . 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 3222 |
. . . . . . . . . 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 3700 |
. . . . . . . . 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 3288 |
. . . . . . . . . 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 3288 |
. . . . . . . . . . . 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 7464 |
. . . . . . . . . . . . . 14
⊢ (((𝑥𝐿
·s 𝐶)
+s (𝐴
·s 𝑧𝐿)) -s (𝑥𝐿
·s 𝑧𝐿)) ∈
V |
| 57 | | oveq2 7439 |
. . . . . . . . . . . . . . 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 2748 |
. . . . . . . . . . . . . 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 3532 |
. . . . . . . . . . . . 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 277 |
. . . . . . . . . . 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 3227 |
. . . . . . . . . . 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 1848 |
. . . . . . . . . 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 302 |
. . . . . . . . 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 275 |
. . . . . . . 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 2741 |
. . . . . . . . . . 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 3222 |
. . . . . . . . . 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 3700 |
. . . . . . . . 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 3288 |
. . . . . . . . . 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 3288 |
. . . . . . . . . . . 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 7464 |
. . . . . . . . . . . . . 14
⊢ (((𝑥𝑅
·s 𝐶)
+s (𝐴
·s 𝑧𝑅)) -s (𝑥𝑅
·s 𝑧𝑅)) ∈
V |
| 73 | | oveq2 7439 |
. . . . . . . . . . . . . . 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 2748 |
. . . . . . . . . . . . . 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 3532 |
. . . . . . . . . . . . 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 277 |
. . . . . . . . . . 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 3227 |
. . . . . . . . . . 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 1848 |
. . . . . . . . . 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 302 |
. . . . . . . . 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 275 |
. . . . . . . 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 915 |
. . . . . . 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 276 |
. . . . . 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 2809 |
. . . . 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 2765 |
. . . 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 4166 |
. . 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 4308 |
. . . . 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 4196 |
. . . . . . 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 2741 |
. . . . . . . . . . 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 3222 |
. . . . . . . . . 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 3700 |
. . . . . . . . 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 3288 |
. . . . . . . . . 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 3288 |
. . . . . . . . . . . 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 7464 |
. . . . . . . . . . . . . 14
⊢ (((𝑥𝐿
·s 𝐵)
+s (𝐴
·s 𝑦𝑅)) -s (𝑥𝐿
·s 𝑦𝑅)) ∈
V |
| 96 | | oveq1 7438 |
. . . . . . . . . . . . . . 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 2748 |
. . . . . . . . . . . . . 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 3532 |
. . . . . . . . . . . . 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 277 |
. . . . . . . . . . 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 3227 |
. . . . . . . . . . 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 1848 |
. . . . . . . . . 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 302 |
. . . . . . . . 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 275 |
. . . . . . . 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 2741 |
. . . . . . . . . . 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 3222 |
. . . . . . . . . 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 3700 |
. . . . . . . . 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 3288 |
. . . . . . . . . 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 3288 |
. . . . . . . . . . . 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 7464 |
. . . . . . . . . . . . . 14
⊢ (((𝑥𝑅
·s 𝐵)
+s (𝐴
·s 𝑦𝐿)) -s (𝑥𝑅
·s 𝑦𝐿)) ∈
V |
| 112 | | oveq1 7438 |
. . . . . . . . . . . . . . 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 2748 |
. . . . . . . . . . . . . 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 3532 |
. . . . . . . . . . . . 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 277 |
. . . . . . . . . . 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 3227 |
. . . . . . . . . . 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 1848 |
. . . . . . . . . 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 302 |
. . . . . . . . 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 275 |
. . . . . . . 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 915 |
. . . . . . 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 276 |
. . . . . 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 2809 |
. . . . 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 2765 |
. . . 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 4308 |
. . . . 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 4196 |
. . . . . . 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 2741 |
. . . . . . . . . . 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 3222 |
. . . . . . . . . 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 3700 |
. . . . . . . . 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 3288 |
. . . . . . . . . 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 3288 |
. . . . . . . . . . . 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 7464 |
. . . . . . . . . . . . . 14
⊢ (((𝑥𝐿
·s 𝐶)
+s (𝐴
·s 𝑧𝑅)) -s (𝑥𝐿
·s 𝑧𝑅)) ∈
V |
| 134 | | oveq2 7439 |
. . . . . . . . . . . . . . 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 2748 |
. . . . . . . . . . . . . 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 3532 |
. . . . . . . . . . . . 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 277 |
. . . . . . . . . . 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 3227 |
. . . . . . . . . . 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 1848 |
. . . . . . . . . 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 302 |
. . . . . . . . 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 275 |
. . . . . . . 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 2741 |
. . . . . . . . . . 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 3222 |
. . . . . . . . . 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 3700 |
. . . . . . . . 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 3288 |
. . . . . . . . . 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 3288 |
. . . . . . . . . . . 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 7464 |
. . . . . . . . . . . . . 14
⊢ (((𝑥𝑅
·s 𝐶)
+s (𝐴
·s 𝑧𝐿)) -s (𝑥𝑅
·s 𝑧𝐿)) ∈
V |
| 150 | | oveq2 7439 |
. . . . . . . . . . . . . . 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 2748 |
. . . . . . . . . . . . . 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 3532 |
. . . . . . . . . . . . 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 277 |
. . . . . . . . . . 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 3227 |
. . . . . . . . . . 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 1848 |
. . . . . . . . . 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 302 |
. . . . . . . . 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 275 |
. . . . . . . 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 915 |
. . . . . . 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 276 |
. . . . . 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 2809 |
. . . . 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 2765 |
. . . 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 4166 |
. . 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 7443 |
. 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 2795 |
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)))})))) |