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Theorem oalim 7953
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.)
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 6086 . . 3 ((𝐵𝐶 ∧ Lim 𝐵) → 𝐵 ∈ On)
2 simpr 477 . . 3 ((𝐵𝐶 ∧ Lim 𝐵) → Lim 𝐵)
31, 2jca 504 . 2 ((𝐵𝐶 ∧ Lim 𝐵) → (𝐵 ∈ On ∧ Lim 𝐵))
4 rdglim2a 7867 . . . 4 ((𝐵 ∈ On ∧ Lim 𝐵) → (rec((𝑦 ∈ V ↦ suc 𝑦), 𝐴)‘𝐵) = 𝑥𝐵 (rec((𝑦 ∈ V ↦ suc 𝑦), 𝐴)‘𝑥))
54adantl 474 . . 3 ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝐵)) → (rec((𝑦 ∈ V ↦ suc 𝑦), 𝐴)‘𝐵) = 𝑥𝐵 (rec((𝑦 ∈ V ↦ suc 𝑦), 𝐴)‘𝑥))
6 oav 7932 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) = (rec((𝑦 ∈ V ↦ suc 𝑦), 𝐴)‘𝐵))
7 onelon 6048 . . . . . . . 8 ((𝐵 ∈ On ∧ 𝑥𝐵) → 𝑥 ∈ On)
8 oav 7932 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 +o 𝑥) = (rec((𝑦 ∈ V ↦ suc 𝑦), 𝐴)‘𝑥))
97, 8sylan2 583 . . . . . . 7 ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝑥𝐵)) → (𝐴 +o 𝑥) = (rec((𝑦 ∈ V ↦ suc 𝑦), 𝐴)‘𝑥))
109anassrs 460 . . . . . 6 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑥𝐵) → (𝐴 +o 𝑥) = (rec((𝑦 ∈ V ↦ suc 𝑦), 𝐴)‘𝑥))
1110iuneq2dv 4809 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝑥𝐵 (𝐴 +o 𝑥) = 𝑥𝐵 (rec((𝑦 ∈ V ↦ suc 𝑦), 𝐴)‘𝑥))
126, 11eqeq12d 2787 . . . 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 704 . . 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 249 . 2 ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝐵)) → (𝐴 +o 𝐵) = 𝑥𝐵 (𝐴 +o 𝑥))
153, 14sylan2 583 1 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → (𝐴 +o 𝐵) = 𝑥𝐵 (𝐴 +o 𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 387   = wceq 1507  wcel 2050  Vcvv 3409   ciun 4786  cmpt 5002  Oncon0 6023  Lim wlim 6024  suc csuc 6025  cfv 6182  (class class class)co 6970  reccrdg 7843   +o coa 7896
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2744  ax-rep 5043  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180  ax-un 7273
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-ral 3087  df-rex 3088  df-reu 3089  df-rab 3091  df-v 3411  df-sbc 3676  df-csb 3781  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-pss 3839  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-tp 4440  df-op 4442  df-uni 4707  df-iun 4788  df-br 4924  df-opab 4986  df-mpt 5003  df-tr 5025  df-id 5306  df-eprel 5311  df-po 5320  df-so 5321  df-fr 5360  df-we 5362  df-xp 5407  df-rel 5408  df-cnv 5409  df-co 5410  df-dm 5411  df-rn 5412  df-res 5413  df-ima 5414  df-pred 5980  df-ord 6026  df-on 6027  df-lim 6028  df-suc 6029  df-iota 6146  df-fun 6184  df-fn 6185  df-f 6186  df-f1 6187  df-fo 6188  df-f1o 6189  df-fv 6190  df-ov 6973  df-oprab 6974  df-mpo 6975  df-wrecs 7744  df-recs 7806  df-rdg 7844  df-oadd 7903
This theorem is referenced by:  oacl  7956  oa0r  7959  oaordi  7967  oawordri  7971  oawordeulem  7975  oalimcl  7981  oaass  7982  oarec  7983  odi  8000  oeoalem  8017  oaabslem  8064  oaabs2  8066
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