Theorem oalim

Description: Ordinal addition with a limit ordinal. Definition 8.1 of TakeutiZaring p. 56. (Contributed by NM, 3-Aug-2004) (Revised by Mario Carneiro, 8-Sep-2013)

	Ref	Expression
Assertion	oalim	⊢ ( ( 𝐴 ∈ On ∧ ( 𝐵 ∈ 𝐶 ∧ Lim 𝐵 ) ) → ( 𝐴 +o 𝐵 ) = ∪ 𝑥 ∈ 𝐵 ( 𝐴 +o 𝑥 ) )

Proof

Step	Hyp	Ref	Expression
1		limelon	⊢ ( ( 𝐵 ∈ 𝐶 ∧ Lim 𝐵 ) → 𝐵 ∈ On )
2		simpr	⊢ ( ( 𝐵 ∈ 𝐶 ∧ Lim 𝐵 ) → Lim 𝐵 )
3	1 2	jca	⊢ ( ( 𝐵 ∈ 𝐶 ∧ Lim 𝐵 ) → ( 𝐵 ∈ On ∧ Lim 𝐵 ) )
4		rdglim2a	⊢ ( ( 𝐵 ∈ On ∧ Lim 𝐵 ) → ( rec ( ( 𝑦 ∈ V ↦ suc 𝑦 ) , 𝐴 ) ‘ 𝐵 ) = ∪ 𝑥 ∈ 𝐵 ( rec ( ( 𝑦 ∈ V ↦ suc 𝑦 ) , 𝐴 ) ‘ 𝑥 ) )
5	4	adantl	⊢ ( ( 𝐴 ∈ On ∧ ( 𝐵 ∈ On ∧ Lim 𝐵 ) ) → ( rec ( ( 𝑦 ∈ V ↦ suc 𝑦 ) , 𝐴 ) ‘ 𝐵 ) = ∪ 𝑥 ∈ 𝐵 ( rec ( ( 𝑦 ∈ V ↦ suc 𝑦 ) , 𝐴 ) ‘ 𝑥 ) )
6		oav	⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( 𝐴 +o 𝐵 ) = ( rec ( ( 𝑦 ∈ V ↦ suc 𝑦 ) , 𝐴 ) ‘ 𝐵 ) )
7		onelon	⊢ ( ( 𝐵 ∈ On ∧ 𝑥 ∈ 𝐵 ) → 𝑥 ∈ On )
8		oav	⊢ ( ( 𝐴 ∈ On ∧ 𝑥 ∈ On ) → ( 𝐴 +o 𝑥 ) = ( rec ( ( 𝑦 ∈ V ↦ suc 𝑦 ) , 𝐴 ) ‘ 𝑥 ) )
9	7 8	sylan2	⊢ ( ( 𝐴 ∈ On ∧ ( 𝐵 ∈ On ∧ 𝑥 ∈ 𝐵 ) ) → ( 𝐴 +o 𝑥 ) = ( rec ( ( 𝑦 ∈ V ↦ suc 𝑦 ) , 𝐴 ) ‘ 𝑥 ) )
10	9	anassrs	⊢ ( ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ∧ 𝑥 ∈ 𝐵 ) → ( 𝐴 +o 𝑥 ) = ( rec ( ( 𝑦 ∈ V ↦ suc 𝑦 ) , 𝐴 ) ‘ 𝑥 ) )
11	10	iuneq2dv	⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ∪ 𝑥 ∈ 𝐵 ( 𝐴 +o 𝑥 ) = ∪ 𝑥 ∈ 𝐵 ( rec ( ( 𝑦 ∈ V ↦ suc 𝑦 ) , 𝐴 ) ‘ 𝑥 ) )
12	6 11	eqeq12d	⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( ( 𝐴 +o 𝐵 ) = ∪ 𝑥 ∈ 𝐵 ( 𝐴 +o 𝑥 ) ↔ ( rec ( ( 𝑦 ∈ V ↦ suc 𝑦 ) , 𝐴 ) ‘ 𝐵 ) = ∪ 𝑥 ∈ 𝐵 ( rec ( ( 𝑦 ∈ V ↦ suc 𝑦 ) , 𝐴 ) ‘ 𝑥 ) ) )
13	12	adantrr	⊢ ( ( 𝐴 ∈ On ∧ ( 𝐵 ∈ On ∧ Lim 𝐵 ) ) → ( ( 𝐴 +o 𝐵 ) = ∪ 𝑥 ∈ 𝐵 ( 𝐴 +o 𝑥 ) ↔ ( rec ( ( 𝑦 ∈ V ↦ suc 𝑦 ) , 𝐴 ) ‘ 𝐵 ) = ∪ 𝑥 ∈ 𝐵 ( rec ( ( 𝑦 ∈ V ↦ suc 𝑦 ) , 𝐴 ) ‘ 𝑥 ) ) )
14	5 13	mpbird	⊢ ( ( 𝐴 ∈ On ∧ ( 𝐵 ∈ On ∧ Lim 𝐵 ) ) → ( 𝐴 +o 𝐵 ) = ∪ 𝑥 ∈ 𝐵 ( 𝐴 +o 𝑥 ) )
15	3 14	sylan2	⊢ ( ( 𝐴 ∈ On ∧ ( 𝐵 ∈ 𝐶 ∧ Lim 𝐵 ) ) → ( 𝐴 +o 𝐵 ) = ∪ 𝑥 ∈ 𝐵 ( 𝐴 +o 𝑥 ) )