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Theorem oav2 6488
Description: Value of ordinal addition. (Contributed by Mario Carneiro and Jim Kingdon, 12-Aug-2019.)
Assertion
Ref Expression
oav2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) = (𝐴 𝑥𝐵 suc (𝐴 +o 𝑥)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem oav2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 oafnex 6469 . . 3 (𝑦 ∈ V ↦ suc 𝑦) Fn V
2 rdgival 6407 . . 3 (((𝑦 ∈ V ↦ suc 𝑦) Fn V ∧ 𝐴 ∈ On ∧ 𝐵 ∈ On) → (rec((𝑦 ∈ V ↦ suc 𝑦), 𝐴)‘𝐵) = (𝐴 𝑥𝐵 ((𝑦 ∈ V ↦ suc 𝑦)‘(rec((𝑦 ∈ V ↦ suc 𝑦), 𝐴)‘𝑥))))
31, 2mp3an1 1335 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (rec((𝑦 ∈ V ↦ suc 𝑦), 𝐴)‘𝐵) = (𝐴 𝑥𝐵 ((𝑦 ∈ V ↦ suc 𝑦)‘(rec((𝑦 ∈ V ↦ suc 𝑦), 𝐴)‘𝑥))))
4 oav 6479 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) = (rec((𝑦 ∈ V ↦ suc 𝑦), 𝐴)‘𝐵))
5 onelon 4402 . . . . . 6 ((𝐵 ∈ On ∧ 𝑥𝐵) → 𝑥 ∈ On)
6 vex 2755 . . . . . . . . . 10 𝑥 ∈ V
7 oaexg 6473 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝑥 ∈ V) → (𝐴 +o 𝑥) ∈ V)
86, 7mpan2 425 . . . . . . . . 9 (𝐴 ∈ On → (𝐴 +o 𝑥) ∈ V)
9 sucexg 4515 . . . . . . . . . 10 ((𝐴 +o 𝑥) ∈ V → suc (𝐴 +o 𝑥) ∈ V)
108, 9syl 14 . . . . . . . . 9 (𝐴 ∈ On → suc (𝐴 +o 𝑥) ∈ V)
11 suceq 4420 . . . . . . . . . 10 (𝑦 = (𝐴 +o 𝑥) → suc 𝑦 = suc (𝐴 +o 𝑥))
12 eqid 2189 . . . . . . . . . 10 (𝑦 ∈ V ↦ suc 𝑦) = (𝑦 ∈ V ↦ suc 𝑦)
1311, 12fvmptg 5613 . . . . . . . . 9 (((𝐴 +o 𝑥) ∈ V ∧ suc (𝐴 +o 𝑥) ∈ V) → ((𝑦 ∈ V ↦ suc 𝑦)‘(𝐴 +o 𝑥)) = suc (𝐴 +o 𝑥))
148, 10, 13syl2anc 411 . . . . . . . 8 (𝐴 ∈ On → ((𝑦 ∈ V ↦ suc 𝑦)‘(𝐴 +o 𝑥)) = suc (𝐴 +o 𝑥))
1514adantr 276 . . . . . . 7 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → ((𝑦 ∈ V ↦ suc 𝑦)‘(𝐴 +o 𝑥)) = suc (𝐴 +o 𝑥))
16 oav 6479 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 +o 𝑥) = (rec((𝑦 ∈ V ↦ suc 𝑦), 𝐴)‘𝑥))
1716fveq2d 5538 . . . . . . 7 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → ((𝑦 ∈ V ↦ suc 𝑦)‘(𝐴 +o 𝑥)) = ((𝑦 ∈ V ↦ suc 𝑦)‘(rec((𝑦 ∈ V ↦ suc 𝑦), 𝐴)‘𝑥)))
1815, 17eqtr3d 2224 . . . . . 6 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → suc (𝐴 +o 𝑥) = ((𝑦 ∈ V ↦ suc 𝑦)‘(rec((𝑦 ∈ V ↦ suc 𝑦), 𝐴)‘𝑥)))
195, 18sylan2 286 . . . . 5 ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝑥𝐵)) → suc (𝐴 +o 𝑥) = ((𝑦 ∈ V ↦ suc 𝑦)‘(rec((𝑦 ∈ V ↦ suc 𝑦), 𝐴)‘𝑥)))
2019anassrs 400 . . . 4 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑥𝐵) → suc (𝐴 +o 𝑥) = ((𝑦 ∈ V ↦ suc 𝑦)‘(rec((𝑦 ∈ V ↦ suc 𝑦), 𝐴)‘𝑥)))
2120iuneq2dv 3922 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝑥𝐵 suc (𝐴 +o 𝑥) = 𝑥𝐵 ((𝑦 ∈ V ↦ suc 𝑦)‘(rec((𝑦 ∈ V ↦ suc 𝑦), 𝐴)‘𝑥)))
2221uneq2d 3304 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 𝑥𝐵 suc (𝐴 +o 𝑥)) = (𝐴 𝑥𝐵 ((𝑦 ∈ V ↦ suc 𝑦)‘(rec((𝑦 ∈ V ↦ suc 𝑦), 𝐴)‘𝑥))))
233, 4, 223eqtr4d 2232 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) = (𝐴 𝑥𝐵 suc (𝐴 +o 𝑥)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1364  wcel 2160  Vcvv 2752  cun 3142   ciun 3901  cmpt 4079  Oncon0 4381  suc csuc 4383   Fn wfn 5230  cfv 5235  (class class class)co 5896  reccrdg 6394   +o coa 6438
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-iord 4384  df-on 4386  df-suc 4389  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-ov 5899  df-oprab 5900  df-mpo 5901  df-1st 6165  df-2nd 6166  df-recs 6330  df-irdg 6395  df-oadd 6445
This theorem is referenced by:  oasuc  6489
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