Theorem oalim 8544

Description: Ordinal addition with a limit ordinal. Definition 8.1 of [TakeutiZaring] p. 56. Definition 2.3 of [Schloeder] p. 4. (Contributed by NM, 3-Aug-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)

Assertion

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

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem oalim

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		limelon 6417	. . 3 ⊢ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) → 𝐵 ∈ On)
2		simpr 484	. . 3 ⊢ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) → Lim 𝐵)
3	1, 2	jca 511	. 2 ⊢ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) → (𝐵 ∈ On ∧ Lim 𝐵))
4		rdglim2a 8447	. . . 4 ⊢ ((𝐵 ∈ On ∧ Lim 𝐵) → (rec((𝑦 ∈ V ↦ suc 𝑦), 𝐴)‘𝐵) = ∪ 𝑥 ∈ 𝐵 (rec((𝑦 ∈ V ↦ suc 𝑦), 𝐴)‘𝑥))
5	4	adantl 481	. . 3 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝐵)) → (rec((𝑦 ∈ V ↦ suc 𝑦), 𝐴)‘𝐵) = ∪ 𝑥 ∈ 𝐵 (rec((𝑦 ∈ V ↦ suc 𝑦), 𝐴)‘𝑥))
6		oav 8523	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) = (rec((𝑦 ∈ V ↦ suc 𝑦), 𝐴)‘𝐵))
7		onelon 6377	. . . . . . . 8 ⊢ ((𝐵 ∈ On ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ On)
8		oav 8523	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 +o 𝑥) = (rec((𝑦 ∈ V ↦ suc 𝑦), 𝐴)‘𝑥))
9	7, 8	sylan2 593	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝑥 ∈ 𝐵)) → (𝐴 +o 𝑥) = (rec((𝑦 ∈ V ↦ suc 𝑦), 𝐴)‘𝑥))
10	9	anassrs 467	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑥 ∈ 𝐵) → (𝐴 +o 𝑥) = (rec((𝑦 ∈ V ↦ suc 𝑦), 𝐴)‘𝑥))
11	10	iuneq2dv 4992	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ∪ 𝑥 ∈ 𝐵 (𝐴 +o 𝑥) = ∪ 𝑥 ∈ 𝐵 (rec((𝑦 ∈ V ↦ suc 𝑦), 𝐴)‘𝑥))
12	6, 11	eqeq12d 2751	. . . 4 ⊢ ((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 717	. . 3 ⊢ ((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 257	. 2 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝐵)) → (𝐴 +o 𝐵) = ∪ 𝑥 ∈ 𝐵 (𝐴 +o 𝑥))
15	3, 14	sylan2 593	1 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → (𝐴 +o 𝐵) = ∪ 𝑥 ∈ 𝐵 (𝐴 +o 𝑥))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108  Vcvv 3459  ∪ ciun 4967   ↦ cmpt 5201  Oncon0 6352  Lim wlim 6353  suc csuc 6354  ‘cfv 6531  (class class class)co 7405  reccrdg 8423   +_o coa 8477

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7408  df-oprab 7409  df-mpo 7410  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-oadd 8484

This theorem is referenced by:  oacl  8547  oa0r  8550  oaordi  8558  oawordri  8562  oawordeulem  8566  oalimcl  8572  oaass  8573  oarec  8574  odi  8591  oeoalem  8608  oaabslem  8659  oaabs2  8661  oa0suclim  43299  naddgeoa  43418  naddwordnexlem4  43425