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Theorem oav2 6156
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 6137 . . 3 (𝑦 ∈ V ↦ suc 𝑦) Fn V
2 rdgival 6079 . . 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 1256 . 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 6147 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +𝑜 𝐵) = (rec((𝑦 ∈ V ↦ suc 𝑦), 𝐴)‘𝐵))
5 onelon 4175 . . . . . 6 ((𝐵 ∈ On ∧ 𝑥𝐵) → 𝑥 ∈ On)
6 vex 2615 . . . . . . . . . 10 𝑥 ∈ V
7 oaexg 6141 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝑥 ∈ V) → (𝐴 +𝑜 𝑥) ∈ V)
86, 7mpan2 416 . . . . . . . . 9 (𝐴 ∈ On → (𝐴 +𝑜 𝑥) ∈ V)
9 sucexg 4278 . . . . . . . . . 10 ((𝐴 +𝑜 𝑥) ∈ V → suc (𝐴 +𝑜 𝑥) ∈ V)
108, 9syl 14 . . . . . . . . 9 (𝐴 ∈ On → suc (𝐴 +𝑜 𝑥) ∈ V)
11 suceq 4193 . . . . . . . . . 10 (𝑦 = (𝐴 +𝑜 𝑥) → suc 𝑦 = suc (𝐴 +𝑜 𝑥))
12 eqid 2083 . . . . . . . . . 10 (𝑦 ∈ V ↦ suc 𝑦) = (𝑦 ∈ V ↦ suc 𝑦)
1311, 12fvmptg 5325 . . . . . . . . 9 (((𝐴 +𝑜 𝑥) ∈ V ∧ suc (𝐴 +𝑜 𝑥) ∈ V) → ((𝑦 ∈ V ↦ suc 𝑦)‘(𝐴 +𝑜 𝑥)) = suc (𝐴 +𝑜 𝑥))
148, 10, 13syl2anc 403 . . . . . . . 8 (𝐴 ∈ On → ((𝑦 ∈ V ↦ suc 𝑦)‘(𝐴 +𝑜 𝑥)) = suc (𝐴 +𝑜 𝑥))
1514adantr 270 . . . . . . 7 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → ((𝑦 ∈ V ↦ suc 𝑦)‘(𝐴 +𝑜 𝑥)) = suc (𝐴 +𝑜 𝑥))
16 oav 6147 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 +𝑜 𝑥) = (rec((𝑦 ∈ V ↦ suc 𝑦), 𝐴)‘𝑥))
1716fveq2d 5257 . . . . . . 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 2117 . . . . . 6 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → suc (𝐴 +𝑜 𝑥) = ((𝑦 ∈ V ↦ suc 𝑦)‘(rec((𝑦 ∈ V ↦ suc 𝑦), 𝐴)‘𝑥)))
195, 18sylan2 280 . . . . 5 ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝑥𝐵)) → suc (𝐴 +𝑜 𝑥) = ((𝑦 ∈ V ↦ suc 𝑦)‘(rec((𝑦 ∈ V ↦ suc 𝑦), 𝐴)‘𝑥)))
2019anassrs 392 . . . 4 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑥𝐵) → suc (𝐴 +𝑜 𝑥) = ((𝑦 ∈ V ↦ suc 𝑦)‘(rec((𝑦 ∈ V ↦ suc 𝑦), 𝐴)‘𝑥)))
2120iuneq2dv 3725 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝑥𝐵 suc (𝐴 +𝑜 𝑥) = 𝑥𝐵 ((𝑦 ∈ V ↦ suc 𝑦)‘(rec((𝑦 ∈ V ↦ suc 𝑦), 𝐴)‘𝑥)))
2221uneq2d 3138 . 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 2125 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 102   = wceq 1285  wcel 1434  Vcvv 2612  cun 2982   ciun 3704  cmpt 3865  Oncon0 4154  suc csuc 4156   Fn wfn 4964  cfv 4969  (class class class)co 5591  reccrdg 6066   +𝑜 coa 6110
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3919  ax-sep 3922  ax-pow 3974  ax-pr 4000  ax-un 4224  ax-setind 4316
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2614  df-sbc 2827  df-csb 2920  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-iun 3706  df-br 3812  df-opab 3866  df-mpt 3867  df-tr 3902  df-id 4084  df-iord 4157  df-on 4159  df-suc 4162  df-xp 4407  df-rel 4408  df-cnv 4409  df-co 4410  df-dm 4411  df-rn 4412  df-res 4413  df-ima 4414  df-iota 4934  df-fun 4971  df-fn 4972  df-f 4973  df-f1 4974  df-fo 4975  df-f1o 4976  df-fv 4977  df-ov 5594  df-oprab 5595  df-mpt2 5596  df-1st 5846  df-2nd 5847  df-recs 6002  df-irdg 6067  df-oadd 6117
This theorem is referenced by:  oasuc  6157
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