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Theorem oav2 6327
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 6308 . . 3 (𝑦 ∈ V ↦ suc 𝑦) Fn V
2 rdgival 6247 . . 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 1287 . 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 6318 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) = (rec((𝑦 ∈ V ↦ suc 𝑦), 𝐴)‘𝐵))
5 onelon 4276 . . . . . 6 ((𝐵 ∈ On ∧ 𝑥𝐵) → 𝑥 ∈ On)
6 vex 2663 . . . . . . . . . 10 𝑥 ∈ V
7 oaexg 6312 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝑥 ∈ V) → (𝐴 +o 𝑥) ∈ V)
86, 7mpan2 421 . . . . . . . . 9 (𝐴 ∈ On → (𝐴 +o 𝑥) ∈ V)
9 sucexg 4384 . . . . . . . . . 10 ((𝐴 +o 𝑥) ∈ V → suc (𝐴 +o 𝑥) ∈ V)
108, 9syl 14 . . . . . . . . 9 (𝐴 ∈ On → suc (𝐴 +o 𝑥) ∈ V)
11 suceq 4294 . . . . . . . . . 10 (𝑦 = (𝐴 +o 𝑥) → suc 𝑦 = suc (𝐴 +o 𝑥))
12 eqid 2117 . . . . . . . . . 10 (𝑦 ∈ V ↦ suc 𝑦) = (𝑦 ∈ V ↦ suc 𝑦)
1311, 12fvmptg 5465 . . . . . . . . 9 (((𝐴 +o 𝑥) ∈ V ∧ suc (𝐴 +o 𝑥) ∈ V) → ((𝑦 ∈ V ↦ suc 𝑦)‘(𝐴 +o 𝑥)) = suc (𝐴 +o 𝑥))
148, 10, 13syl2anc 408 . . . . . . . 8 (𝐴 ∈ On → ((𝑦 ∈ V ↦ suc 𝑦)‘(𝐴 +o 𝑥)) = suc (𝐴 +o 𝑥))
1514adantr 274 . . . . . . 7 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → ((𝑦 ∈ V ↦ suc 𝑦)‘(𝐴 +o 𝑥)) = suc (𝐴 +o 𝑥))
16 oav 6318 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 +o 𝑥) = (rec((𝑦 ∈ V ↦ suc 𝑦), 𝐴)‘𝑥))
1716fveq2d 5393 . . . . . . 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 2152 . . . . . 6 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → suc (𝐴 +o 𝑥) = ((𝑦 ∈ V ↦ suc 𝑦)‘(rec((𝑦 ∈ V ↦ suc 𝑦), 𝐴)‘𝑥)))
195, 18sylan2 284 . . . . 5 ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝑥𝐵)) → suc (𝐴 +o 𝑥) = ((𝑦 ∈ V ↦ suc 𝑦)‘(rec((𝑦 ∈ V ↦ suc 𝑦), 𝐴)‘𝑥)))
2019anassrs 397 . . . 4 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑥𝐵) → suc (𝐴 +o 𝑥) = ((𝑦 ∈ V ↦ suc 𝑦)‘(rec((𝑦 ∈ V ↦ suc 𝑦), 𝐴)‘𝑥)))
2120iuneq2dv 3804 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝑥𝐵 suc (𝐴 +o 𝑥) = 𝑥𝐵 ((𝑦 ∈ V ↦ suc 𝑦)‘(rec((𝑦 ∈ V ↦ suc 𝑦), 𝐴)‘𝑥)))
2221uneq2d 3200 . 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 2160 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 103   = wceq 1316  wcel 1465  Vcvv 2660  cun 3039   ciun 3783  cmpt 3959  Oncon0 4255  suc csuc 4257   Fn wfn 5088  cfv 5093  (class class class)co 5742  reccrdg 6234   +o coa 6278
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-id 4185  df-iord 4258  df-on 4260  df-suc 4263  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-recs 6170  df-irdg 6235  df-oadd 6285
This theorem is referenced by:  oasuc  6328
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