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Theorem oeoelem 7630
 Description: Lemma for oeoe 7631. (Contributed by Eric Schmidt, 26-May-2009.)
Hypotheses
Ref Expression
oeoelem.1 𝐴 ∈ On
oeoelem.2 ∅ ∈ 𝐴
Assertion
Ref Expression
oeoelem ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴𝑜 𝐵) ↑𝑜 𝐶) = (𝐴𝑜 (𝐵 ·𝑜 𝐶)))

Proof of Theorem oeoelem
Dummy variables 𝑥 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 6618 . . . 4 (𝑥 = ∅ → ((𝐴𝑜 𝐵) ↑𝑜 𝑥) = ((𝐴𝑜 𝐵) ↑𝑜 ∅))
2 oveq2 6618 . . . . 5 (𝑥 = ∅ → (𝐵 ·𝑜 𝑥) = (𝐵 ·𝑜 ∅))
32oveq2d 6626 . . . 4 (𝑥 = ∅ → (𝐴𝑜 (𝐵 ·𝑜 𝑥)) = (𝐴𝑜 (𝐵 ·𝑜 ∅)))
41, 3eqeq12d 2636 . . 3 (𝑥 = ∅ → (((𝐴𝑜 𝐵) ↑𝑜 𝑥) = (𝐴𝑜 (𝐵 ·𝑜 𝑥)) ↔ ((𝐴𝑜 𝐵) ↑𝑜 ∅) = (𝐴𝑜 (𝐵 ·𝑜 ∅))))
5 oveq2 6618 . . . 4 (𝑥 = 𝑦 → ((𝐴𝑜 𝐵) ↑𝑜 𝑥) = ((𝐴𝑜 𝐵) ↑𝑜 𝑦))
6 oveq2 6618 . . . . 5 (𝑥 = 𝑦 → (𝐵 ·𝑜 𝑥) = (𝐵 ·𝑜 𝑦))
76oveq2d 6626 . . . 4 (𝑥 = 𝑦 → (𝐴𝑜 (𝐵 ·𝑜 𝑥)) = (𝐴𝑜 (𝐵 ·𝑜 𝑦)))
85, 7eqeq12d 2636 . . 3 (𝑥 = 𝑦 → (((𝐴𝑜 𝐵) ↑𝑜 𝑥) = (𝐴𝑜 (𝐵 ·𝑜 𝑥)) ↔ ((𝐴𝑜 𝐵) ↑𝑜 𝑦) = (𝐴𝑜 (𝐵 ·𝑜 𝑦))))
9 oveq2 6618 . . . 4 (𝑥 = suc 𝑦 → ((𝐴𝑜 𝐵) ↑𝑜 𝑥) = ((𝐴𝑜 𝐵) ↑𝑜 suc 𝑦))
10 oveq2 6618 . . . . 5 (𝑥 = suc 𝑦 → (𝐵 ·𝑜 𝑥) = (𝐵 ·𝑜 suc 𝑦))
1110oveq2d 6626 . . . 4 (𝑥 = suc 𝑦 → (𝐴𝑜 (𝐵 ·𝑜 𝑥)) = (𝐴𝑜 (𝐵 ·𝑜 suc 𝑦)))
129, 11eqeq12d 2636 . . 3 (𝑥 = suc 𝑦 → (((𝐴𝑜 𝐵) ↑𝑜 𝑥) = (𝐴𝑜 (𝐵 ·𝑜 𝑥)) ↔ ((𝐴𝑜 𝐵) ↑𝑜 suc 𝑦) = (𝐴𝑜 (𝐵 ·𝑜 suc 𝑦))))
13 oveq2 6618 . . . 4 (𝑥 = 𝐶 → ((𝐴𝑜 𝐵) ↑𝑜 𝑥) = ((𝐴𝑜 𝐵) ↑𝑜 𝐶))
14 oveq2 6618 . . . . 5 (𝑥 = 𝐶 → (𝐵 ·𝑜 𝑥) = (𝐵 ·𝑜 𝐶))
1514oveq2d 6626 . . . 4 (𝑥 = 𝐶 → (𝐴𝑜 (𝐵 ·𝑜 𝑥)) = (𝐴𝑜 (𝐵 ·𝑜 𝐶)))
1613, 15eqeq12d 2636 . . 3 (𝑥 = 𝐶 → (((𝐴𝑜 𝐵) ↑𝑜 𝑥) = (𝐴𝑜 (𝐵 ·𝑜 𝑥)) ↔ ((𝐴𝑜 𝐵) ↑𝑜 𝐶) = (𝐴𝑜 (𝐵 ·𝑜 𝐶))))
17 oeoelem.1 . . . . . 6 𝐴 ∈ On
18 oecl 7569 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝑜 𝐵) ∈ On)
1917, 18mpan 705 . . . . 5 (𝐵 ∈ On → (𝐴𝑜 𝐵) ∈ On)
20 oe0 7554 . . . . 5 ((𝐴𝑜 𝐵) ∈ On → ((𝐴𝑜 𝐵) ↑𝑜 ∅) = 1𝑜)
2119, 20syl 17 . . . 4 (𝐵 ∈ On → ((𝐴𝑜 𝐵) ↑𝑜 ∅) = 1𝑜)
22 om0 7549 . . . . . 6 (𝐵 ∈ On → (𝐵 ·𝑜 ∅) = ∅)
2322oveq2d 6626 . . . . 5 (𝐵 ∈ On → (𝐴𝑜 (𝐵 ·𝑜 ∅)) = (𝐴𝑜 ∅))
24 oe0 7554 . . . . . 6 (𝐴 ∈ On → (𝐴𝑜 ∅) = 1𝑜)
2517, 24ax-mp 5 . . . . 5 (𝐴𝑜 ∅) = 1𝑜
2623, 25syl6eq 2671 . . . 4 (𝐵 ∈ On → (𝐴𝑜 (𝐵 ·𝑜 ∅)) = 1𝑜)
2721, 26eqtr4d 2658 . . 3 (𝐵 ∈ On → ((𝐴𝑜 𝐵) ↑𝑜 ∅) = (𝐴𝑜 (𝐵 ·𝑜 ∅)))
28 oveq1 6617 . . . . 5 (((𝐴𝑜 𝐵) ↑𝑜 𝑦) = (𝐴𝑜 (𝐵 ·𝑜 𝑦)) → (((𝐴𝑜 𝐵) ↑𝑜 𝑦) ·𝑜 (𝐴𝑜 𝐵)) = ((𝐴𝑜 (𝐵 ·𝑜 𝑦)) ·𝑜 (𝐴𝑜 𝐵)))
29 oesuc 7559 . . . . . . 7 (((𝐴𝑜 𝐵) ∈ On ∧ 𝑦 ∈ On) → ((𝐴𝑜 𝐵) ↑𝑜 suc 𝑦) = (((𝐴𝑜 𝐵) ↑𝑜 𝑦) ·𝑜 (𝐴𝑜 𝐵)))
3019, 29sylan 488 . . . . . 6 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → ((𝐴𝑜 𝐵) ↑𝑜 suc 𝑦) = (((𝐴𝑜 𝐵) ↑𝑜 𝑦) ·𝑜 (𝐴𝑜 𝐵)))
31 omsuc 7558 . . . . . . . 8 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ·𝑜 suc 𝑦) = ((𝐵 ·𝑜 𝑦) +𝑜 𝐵))
3231oveq2d 6626 . . . . . . 7 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴𝑜 (𝐵 ·𝑜 suc 𝑦)) = (𝐴𝑜 ((𝐵 ·𝑜 𝑦) +𝑜 𝐵)))
33 omcl 7568 . . . . . . . . 9 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ·𝑜 𝑦) ∈ On)
34 oeoa 7629 . . . . . . . . . 10 ((𝐴 ∈ On ∧ (𝐵 ·𝑜 𝑦) ∈ On ∧ 𝐵 ∈ On) → (𝐴𝑜 ((𝐵 ·𝑜 𝑦) +𝑜 𝐵)) = ((𝐴𝑜 (𝐵 ·𝑜 𝑦)) ·𝑜 (𝐴𝑜 𝐵)))
3517, 34mp3an1 1408 . . . . . . . . 9 (((𝐵 ·𝑜 𝑦) ∈ On ∧ 𝐵 ∈ On) → (𝐴𝑜 ((𝐵 ·𝑜 𝑦) +𝑜 𝐵)) = ((𝐴𝑜 (𝐵 ·𝑜 𝑦)) ·𝑜 (𝐴𝑜 𝐵)))
3633, 35sylan 488 . . . . . . . 8 (((𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ 𝐵 ∈ On) → (𝐴𝑜 ((𝐵 ·𝑜 𝑦) +𝑜 𝐵)) = ((𝐴𝑜 (𝐵 ·𝑜 𝑦)) ·𝑜 (𝐴𝑜 𝐵)))
3736anabss1 854 . . . . . . 7 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴𝑜 ((𝐵 ·𝑜 𝑦) +𝑜 𝐵)) = ((𝐴𝑜 (𝐵 ·𝑜 𝑦)) ·𝑜 (𝐴𝑜 𝐵)))
3832, 37eqtrd 2655 . . . . . 6 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴𝑜 (𝐵 ·𝑜 suc 𝑦)) = ((𝐴𝑜 (𝐵 ·𝑜 𝑦)) ·𝑜 (𝐴𝑜 𝐵)))
3930, 38eqeq12d 2636 . . . . 5 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (((𝐴𝑜 𝐵) ↑𝑜 suc 𝑦) = (𝐴𝑜 (𝐵 ·𝑜 suc 𝑦)) ↔ (((𝐴𝑜 𝐵) ↑𝑜 𝑦) ·𝑜 (𝐴𝑜 𝐵)) = ((𝐴𝑜 (𝐵 ·𝑜 𝑦)) ·𝑜 (𝐴𝑜 𝐵))))
4028, 39syl5ibr 236 . . . 4 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (((𝐴𝑜 𝐵) ↑𝑜 𝑦) = (𝐴𝑜 (𝐵 ·𝑜 𝑦)) → ((𝐴𝑜 𝐵) ↑𝑜 suc 𝑦) = (𝐴𝑜 (𝐵 ·𝑜 suc 𝑦))))
4140expcom 451 . . 3 (𝑦 ∈ On → (𝐵 ∈ On → (((𝐴𝑜 𝐵) ↑𝑜 𝑦) = (𝐴𝑜 (𝐵 ·𝑜 𝑦)) → ((𝐴𝑜 𝐵) ↑𝑜 suc 𝑦) = (𝐴𝑜 (𝐵 ·𝑜 suc 𝑦)))))
42 iuneq2 4508 . . . . 5 (∀𝑦𝑥 ((𝐴𝑜 𝐵) ↑𝑜 𝑦) = (𝐴𝑜 (𝐵 ·𝑜 𝑦)) → 𝑦𝑥 ((𝐴𝑜 𝐵) ↑𝑜 𝑦) = 𝑦𝑥 (𝐴𝑜 (𝐵 ·𝑜 𝑦)))
43 vex 3192 . . . . . . 7 𝑥 ∈ V
44 oeoelem.2 . . . . . . . . . . 11 ∅ ∈ 𝐴
45 oen0 7618 . . . . . . . . . . 11 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → ∅ ∈ (𝐴𝑜 𝐵))
4644, 45mpan2 706 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ∅ ∈ (𝐴𝑜 𝐵))
47 oelim 7566 . . . . . . . . . . 11 ((((𝐴𝑜 𝐵) ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ (𝐴𝑜 𝐵)) → ((𝐴𝑜 𝐵) ↑𝑜 𝑥) = 𝑦𝑥 ((𝐴𝑜 𝐵) ↑𝑜 𝑦))
4818, 47sylanl1 681 . . . . . . . . . 10 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ (𝐴𝑜 𝐵)) → ((𝐴𝑜 𝐵) ↑𝑜 𝑥) = 𝑦𝑥 ((𝐴𝑜 𝐵) ↑𝑜 𝑦))
4946, 48sylan2 491 . . . . . . . . 9 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) → ((𝐴𝑜 𝐵) ↑𝑜 𝑥) = 𝑦𝑥 ((𝐴𝑜 𝐵) ↑𝑜 𝑦))
5049anabss1 854 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → ((𝐴𝑜 𝐵) ↑𝑜 𝑥) = 𝑦𝑥 ((𝐴𝑜 𝐵) ↑𝑜 𝑦))
5117, 50mpanl1 715 . . . . . . 7 ((𝐵 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → ((𝐴𝑜 𝐵) ↑𝑜 𝑥) = 𝑦𝑥 ((𝐴𝑜 𝐵) ↑𝑜 𝑦))
5243, 51mpanr1 718 . . . . . 6 ((𝐵 ∈ On ∧ Lim 𝑥) → ((𝐴𝑜 𝐵) ↑𝑜 𝑥) = 𝑦𝑥 ((𝐴𝑜 𝐵) ↑𝑜 𝑦))
53 omlim 7565 . . . . . . . . 9 ((𝐵 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝐵 ·𝑜 𝑥) = 𝑦𝑥 (𝐵 ·𝑜 𝑦))
5443, 53mpanr1 718 . . . . . . . 8 ((𝐵 ∈ On ∧ Lim 𝑥) → (𝐵 ·𝑜 𝑥) = 𝑦𝑥 (𝐵 ·𝑜 𝑦))
5554oveq2d 6626 . . . . . . 7 ((𝐵 ∈ On ∧ Lim 𝑥) → (𝐴𝑜 (𝐵 ·𝑜 𝑥)) = (𝐴𝑜 𝑦𝑥 (𝐵 ·𝑜 𝑦)))
5643a1i 11 . . . . . . . 8 ((𝐵 ∈ On ∧ Lim 𝑥) → 𝑥 ∈ V)
57 limord 5748 . . . . . . . . . . . 12 (Lim 𝑥 → Ord 𝑥)
58 ordelon 5711 . . . . . . . . . . . 12 ((Ord 𝑥𝑦𝑥) → 𝑦 ∈ On)
5957, 58sylan 488 . . . . . . . . . . 11 ((Lim 𝑥𝑦𝑥) → 𝑦 ∈ On)
6059, 33sylan2 491 . . . . . . . . . 10 ((𝐵 ∈ On ∧ (Lim 𝑥𝑦𝑥)) → (𝐵 ·𝑜 𝑦) ∈ On)
6160anassrs 679 . . . . . . . . 9 (((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝑦𝑥) → (𝐵 ·𝑜 𝑦) ∈ On)
6261ralrimiva 2961 . . . . . . . 8 ((𝐵 ∈ On ∧ Lim 𝑥) → ∀𝑦𝑥 (𝐵 ·𝑜 𝑦) ∈ On)
63 0ellim 5751 . . . . . . . . . 10 (Lim 𝑥 → ∅ ∈ 𝑥)
64 ne0i 3902 . . . . . . . . . 10 (∅ ∈ 𝑥𝑥 ≠ ∅)
6563, 64syl 17 . . . . . . . . 9 (Lim 𝑥𝑥 ≠ ∅)
6665adantl 482 . . . . . . . 8 ((𝐵 ∈ On ∧ Lim 𝑥) → 𝑥 ≠ ∅)
67 vex 3192 . . . . . . . . . 10 𝑤 ∈ V
68 oelim 7566 . . . . . . . . . . . 12 (((𝐴 ∈ On ∧ (𝑤 ∈ V ∧ Lim 𝑤)) ∧ ∅ ∈ 𝐴) → (𝐴𝑜 𝑤) = 𝑧𝑤 (𝐴𝑜 𝑧))
6944, 68mpan2 706 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ (𝑤 ∈ V ∧ Lim 𝑤)) → (𝐴𝑜 𝑤) = 𝑧𝑤 (𝐴𝑜 𝑧))
7017, 69mpan 705 . . . . . . . . . 10 ((𝑤 ∈ V ∧ Lim 𝑤) → (𝐴𝑜 𝑤) = 𝑧𝑤 (𝐴𝑜 𝑧))
7167, 70mpan 705 . . . . . . . . 9 (Lim 𝑤 → (𝐴𝑜 𝑤) = 𝑧𝑤 (𝐴𝑜 𝑧))
72 oewordi 7623 . . . . . . . . . . . 12 (((𝑧 ∈ On ∧ 𝑤 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → (𝑧𝑤 → (𝐴𝑜 𝑧) ⊆ (𝐴𝑜 𝑤)))
7344, 72mpan2 706 . . . . . . . . . . 11 ((𝑧 ∈ On ∧ 𝑤 ∈ On ∧ 𝐴 ∈ On) → (𝑧𝑤 → (𝐴𝑜 𝑧) ⊆ (𝐴𝑜 𝑤)))
7417, 73mp3an3 1410 . . . . . . . . . 10 ((𝑧 ∈ On ∧ 𝑤 ∈ On) → (𝑧𝑤 → (𝐴𝑜 𝑧) ⊆ (𝐴𝑜 𝑤)))
75743impia 1258 . . . . . . . . 9 ((𝑧 ∈ On ∧ 𝑤 ∈ On ∧ 𝑧𝑤) → (𝐴𝑜 𝑧) ⊆ (𝐴𝑜 𝑤))
7671, 75onoviun 7392 . . . . . . . 8 ((𝑥 ∈ V ∧ ∀𝑦𝑥 (𝐵 ·𝑜 𝑦) ∈ On ∧ 𝑥 ≠ ∅) → (𝐴𝑜 𝑦𝑥 (𝐵 ·𝑜 𝑦)) = 𝑦𝑥 (𝐴𝑜 (𝐵 ·𝑜 𝑦)))
7756, 62, 66, 76syl3anc 1323 . . . . . . 7 ((𝐵 ∈ On ∧ Lim 𝑥) → (𝐴𝑜 𝑦𝑥 (𝐵 ·𝑜 𝑦)) = 𝑦𝑥 (𝐴𝑜 (𝐵 ·𝑜 𝑦)))
7855, 77eqtrd 2655 . . . . . 6 ((𝐵 ∈ On ∧ Lim 𝑥) → (𝐴𝑜 (𝐵 ·𝑜 𝑥)) = 𝑦𝑥 (𝐴𝑜 (𝐵 ·𝑜 𝑦)))
7952, 78eqeq12d 2636 . . . . 5 ((𝐵 ∈ On ∧ Lim 𝑥) → (((𝐴𝑜 𝐵) ↑𝑜 𝑥) = (𝐴𝑜 (𝐵 ·𝑜 𝑥)) ↔ 𝑦𝑥 ((𝐴𝑜 𝐵) ↑𝑜 𝑦) = 𝑦𝑥 (𝐴𝑜 (𝐵 ·𝑜 𝑦))))
8042, 79syl5ibr 236 . . . 4 ((𝐵 ∈ On ∧ Lim 𝑥) → (∀𝑦𝑥 ((𝐴𝑜 𝐵) ↑𝑜 𝑦) = (𝐴𝑜 (𝐵 ·𝑜 𝑦)) → ((𝐴𝑜 𝐵) ↑𝑜 𝑥) = (𝐴𝑜 (𝐵 ·𝑜 𝑥))))
8180expcom 451 . . 3 (Lim 𝑥 → (𝐵 ∈ On → (∀𝑦𝑥 ((𝐴𝑜 𝐵) ↑𝑜 𝑦) = (𝐴𝑜 (𝐵 ·𝑜 𝑦)) → ((𝐴𝑜 𝐵) ↑𝑜 𝑥) = (𝐴𝑜 (𝐵 ·𝑜 𝑥)))))
824, 8, 12, 16, 27, 41, 81tfinds3 7018 . 2 (𝐶 ∈ On → (𝐵 ∈ On → ((𝐴𝑜 𝐵) ↑𝑜 𝐶) = (𝐴𝑜 (𝐵 ·𝑜 𝐶))))
8382impcom 446 1 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴𝑜 𝐵) ↑𝑜 𝐶) = (𝐴𝑜 (𝐵 ·𝑜 𝐶)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1036   = wceq 1480   ∈ wcel 1987   ≠ wne 2790  ∀wral 2907  Vcvv 3189   ⊆ wss 3559  ∅c0 3896  ∪ ciun 4490  Ord word 5686  Oncon0 5687  Lim wlim 5688  suc csuc 5689  (class class class)co 6610  1𝑜c1o 7505   +𝑜 coa 7509   ·𝑜 comu 7510   ↑𝑜 coe 7511 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-2o 7513  df-oadd 7516  df-omul 7517  df-oexp 7518 This theorem is referenced by:  oeoe  7631
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