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Theorem oalim 8551
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 ∧ (𝐵𝐶 ∧ Lim 𝐵)) → (𝐴 +o 𝐵) = 𝑥𝐵 (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.
StepHypRef Expression
1 limelon 6429 . . 3 ((𝐵𝐶 ∧ Lim 𝐵) → 𝐵 ∈ On)
2 simpr 483 . . 3 ((𝐵𝐶 ∧ Lim 𝐵) → Lim 𝐵)
31, 2jca 510 . 2 ((𝐵𝐶 ∧ Lim 𝐵) → (𝐵 ∈ On ∧ Lim 𝐵))
4 rdglim2a 8452 . . . 4 ((𝐵 ∈ On ∧ Lim 𝐵) → (rec((𝑦 ∈ V ↦ suc 𝑦), 𝐴)‘𝐵) = 𝑥𝐵 (rec((𝑦 ∈ V ↦ suc 𝑦), 𝐴)‘𝑥))
54adantl 480 . . 3 ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝐵)) → (rec((𝑦 ∈ V ↦ suc 𝑦), 𝐴)‘𝐵) = 𝑥𝐵 (rec((𝑦 ∈ V ↦ suc 𝑦), 𝐴)‘𝑥))
6 oav 8530 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) = (rec((𝑦 ∈ V ↦ suc 𝑦), 𝐴)‘𝐵))
7 onelon 6390 . . . . . . . 8 ((𝐵 ∈ On ∧ 𝑥𝐵) → 𝑥 ∈ On)
8 oav 8530 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 +o 𝑥) = (rec((𝑦 ∈ V ↦ suc 𝑦), 𝐴)‘𝑥))
97, 8sylan2 591 . . . . . . 7 ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝑥𝐵)) → (𝐴 +o 𝑥) = (rec((𝑦 ∈ V ↦ suc 𝑦), 𝐴)‘𝑥))
109anassrs 466 . . . . . 6 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑥𝐵) → (𝐴 +o 𝑥) = (rec((𝑦 ∈ V ↦ suc 𝑦), 𝐴)‘𝑥))
1110iuneq2dv 5017 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝑥𝐵 (𝐴 +o 𝑥) = 𝑥𝐵 (rec((𝑦 ∈ V ↦ suc 𝑦), 𝐴)‘𝑥))
126, 11eqeq12d 2742 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +o 𝐵) = 𝑥𝐵 (𝐴 +o 𝑥) ↔ (rec((𝑦 ∈ V ↦ suc 𝑦), 𝐴)‘𝐵) = 𝑥𝐵 (rec((𝑦 ∈ V ↦ suc 𝑦), 𝐴)‘𝑥)))
1312adantrr 715 . . 3 ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝐵)) → ((𝐴 +o 𝐵) = 𝑥𝐵 (𝐴 +o 𝑥) ↔ (rec((𝑦 ∈ V ↦ suc 𝑦), 𝐴)‘𝐵) = 𝑥𝐵 (rec((𝑦 ∈ V ↦ suc 𝑦), 𝐴)‘𝑥)))
145, 13mpbird 256 . 2 ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝐵)) → (𝐴 +o 𝐵) = 𝑥𝐵 (𝐴 +o 𝑥))
153, 14sylan2 591 1 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → (𝐴 +o 𝐵) = 𝑥𝐵 (𝐴 +o 𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1534  wcel 2099  Vcvv 3462   ciun 4993  cmpt 5226  Oncon0 6365  Lim wlim 6366  suc csuc 6367  cfv 6543  (class class class)co 7413  reccrdg 8428   +o coa 8482
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pr 5423  ax-un 7735
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3966  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-iun 4995  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6302  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7416  df-oprab 7417  df-mpo 7418  df-2nd 7993  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-oadd 8489
This theorem is referenced by:  oacl  8554  oa0r  8557  oaordi  8565  oawordri  8569  oawordeulem  8573  oalimcl  8579  oaass  8580  oarec  8581  odi  8598  oeoalem  8615  oaabslem  8666  oaabs2  8668  oa0suclim  42975  naddgeoa  43095  naddwordnexlem4  43102
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