Theorem oalim 8526

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 6425	. . 3 ⊢ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) → 𝐵 ∈ On)
2		simpr 490	. . 3 ⊢ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) → Lim 𝐵)
3	1, 2	jca 521	. 2 ⊢ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) → (𝐵 ∈ On ∧ Lim 𝐵))
4		rdglim2a 8427	. . . 4 ⊢ ((𝐵 ∈ On ∧ Lim 𝐵) → (rec((𝑦 ∈ V ↦ suc 𝑦), 𝐴)‘𝐵) = ∪ 𝑥 ∈ 𝐵 (rec((𝑦 ∈ V ↦ suc 𝑦), 𝐴)‘𝑥))
5	4	adantl 487	. . 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 8505	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) = (rec((𝑦 ∈ V ↦ suc 𝑦), 𝐴)‘𝐵))
7		onelon 6384	. . . . . . . 8 ⊢ ((𝐵 ∈ On ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ On)
8		oav 8505	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 +o 𝑥) = (rec((𝑦 ∈ V ↦ suc 𝑦), 𝐴)‘𝑥))
9	7, 8	sylan2 605	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝑥 ∈ 𝐵)) → (𝐴 +o 𝑥) = (rec((𝑦 ∈ V ↦ suc 𝑦), 𝐴)‘𝑥))
10	9	anassrs 473	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑥 ∈ 𝐵) → (𝐴 +o 𝑥) = (rec((𝑦 ∈ V ↦ suc 𝑦), 𝐴)‘𝑥))
11	10	iuneq2dv 4976	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ∪ 𝑥 ∈ 𝐵 (𝐴 +o 𝑥) = ∪ 𝑥 ∈ 𝐵 (rec((𝑦 ∈ V ↦ suc 𝑦), 𝐴)‘𝑥))
12	6, 11	eqeq12d 2776	. . . 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 730	. . 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 260	. 2 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝐵)) → (𝐴 +o 𝐵) = ∪ 𝑥 ∈ 𝐵 (𝐴 +o 𝑥))
15	3, 14	sylan2 605	1 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → (𝐴 +o 𝐵) = ∪ 𝑥 ∈ 𝐵 (𝐴 +o 𝑥))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  Vcvv 3450  ∪ ciun 4951   ↦ cmpt 5186  Oncon0 6359  Lim wlim 6360  suc csuc 6361  ‘cfv 6535  (class class class)co 7416  reccrdg 8403   +_o coa 8459

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pr 5398  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6301  df-ord 6362  df-on 6363  df-lim 6364  df-suc 6365  df-iota 6491  df-fun 6537  df-fn 6538  df-f 6539  df-f1 6540  df-fo 6541  df-f1o 6542  df-fv 6543  df-ov 7419  df-oprab 7420  df-mpo 7421  df-2nd 7993  df-frecs 8285  df-wrecs 8316  df-recs 8365  df-rdg 8404  df-oadd 8466

This theorem is used by:  oacl  8529  oa0r  8532  oaordi  8540  oawordri  8544  oawordeulem  8548  oalimcl  8554  oaass  8555  oarec  8556  odi  8573  oeoalem  8591  oaabslem  8642  oaabs2  8644  oa0suclim  44181  naddgeoa  44300  naddwordnexlem4  44307