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Theorem oav2 5982
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 x B 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 5963 . . 3 (y V ↦ suc y) Fn V
2 rdgival 5909 . . 3 (((y V ↦ suc y) Fn V A On B On) → (rec((y V ↦ suc y), A)‘B) = (A x B ((y V ↦ suc y)‘(rec((y V ↦ suc y), A)‘x))))
31, 2mp3an1 1218 . 2 ((A On B On) → (rec((y V ↦ suc y), A)‘B) = (A x B ((y V ↦ suc y)‘(rec((y V ↦ suc y), A)‘x))))
4 oav 5973 . 2 ((A On B On) → (A +𝑜 B) = (rec((y V ↦ suc y), A)‘B))
5 onelon 4087 . . . . . 6 ((B On x B) → x On)
6 vex 2554 . . . . . . . . . 10 x V
7 oaexg 5967 . . . . . . . . . 10 ((A On x V) → (A +𝑜 x) V)
86, 7mpan2 401 . . . . . . . . 9 (A On → (A +𝑜 x) V)
9 sucexg 4190 . . . . . . . . . 10 ((A +𝑜 x) V → suc (A +𝑜 x) V)
108, 9syl 14 . . . . . . . . 9 (A On → suc (A +𝑜 x) V)
11 suceq 4105 . . . . . . . . . 10 (y = (A +𝑜 x) → suc y = suc (A +𝑜 x))
12 eqid 2037 . . . . . . . . . 10 (y V ↦ suc y) = (y V ↦ suc y)
1311, 12fvmptg 5191 . . . . . . . . 9 (((A +𝑜 x) V suc (A +𝑜 x) V) → ((y V ↦ suc y)‘(A +𝑜 x)) = suc (A +𝑜 x))
148, 10, 13syl2anc 391 . . . . . . . 8 (A On → ((y V ↦ suc y)‘(A +𝑜 x)) = suc (A +𝑜 x))
1514adantr 261 . . . . . . 7 ((A On x On) → ((y V ↦ suc y)‘(A +𝑜 x)) = suc (A +𝑜 x))
16 oav 5973 . . . . . . . 8 ((A On x On) → (A +𝑜 x) = (rec((y V ↦ suc y), A)‘x))
1716fveq2d 5125 . . . . . . 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 2071 . . . . . 6 ((A On x On) → suc (A +𝑜 x) = ((y V ↦ suc y)‘(rec((y V ↦ suc y), A)‘x)))
195, 18sylan2 270 . . . . 5 ((A On (B On x B)) → suc (A +𝑜 x) = ((y V ↦ suc y)‘(rec((y V ↦ suc y), A)‘x)))
2019anassrs 380 . . . 4 (((A On B On) x B) → suc (A +𝑜 x) = ((y V ↦ suc y)‘(rec((y V ↦ suc y), A)‘x)))
2120iuneq2dv 3669 . . 3 ((A On B On) → x B suc (A +𝑜 x) = x B ((y V ↦ suc y)‘(rec((y V ↦ suc y), A)‘x)))
2221uneq2d 3091 . 2 ((A On B On) → (A x B suc (A +𝑜 x)) = (A x B ((y V ↦ suc y)‘(rec((y V ↦ suc y), A)‘x))))
233, 4, 223eqtr4d 2079 1 ((A On B On) → (A +𝑜 B) = (A x B suc (A +𝑜 x)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242   wcel 1390  Vcvv 2551  cun 2909   ciun 3648  cmpt 3809  Oncon0 4066  suc csuc 4068   Fn wfn 4840  cfv 4845  (class class class)co 5455  reccrdg 5896   +𝑜 coa 5937
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-id 4021  df-iord 4069  df-on 4071  df-suc 4074  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-oadd 5944
This theorem is referenced by:  oasuc  5983
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