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Theorem omv2 5984
Description: Value of ordinal multiplication. (Contributed by Jim Kingdon, 23-Aug-2019.)
Assertion
Ref Expression
omv2 ((A On B On) → (A ·𝑜 B) = x B ((A ·𝑜 x) +𝑜 A))
Distinct variable groups:   x,A   x,B

Proof of Theorem omv2
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 omfnex 5968 . . . 4 (A On → (y V ↦ (y +𝑜 A)) Fn V)
2 0elon 4095 . . . . 5 On
3 rdgival 5909 . . . . 5 (((y V ↦ (y +𝑜 A)) Fn V On B On) → (rec((y V ↦ (y +𝑜 A)), ∅)‘B) = (∅ ∪ x B ((y V ↦ (y +𝑜 A))‘(rec((y V ↦ (y +𝑜 A)), ∅)‘x))))
42, 3mp3an2 1219 . . . 4 (((y V ↦ (y +𝑜 A)) Fn V B On) → (rec((y V ↦ (y +𝑜 A)), ∅)‘B) = (∅ ∪ x B ((y V ↦ (y +𝑜 A))‘(rec((y V ↦ (y +𝑜 A)), ∅)‘x))))
51, 4sylan 267 . . 3 ((A On B On) → (rec((y V ↦ (y +𝑜 A)), ∅)‘B) = (∅ ∪ x B ((y V ↦ (y +𝑜 A))‘(rec((y V ↦ (y +𝑜 A)), ∅)‘x))))
6 omv 5974 . . 3 ((A On B On) → (A ·𝑜 B) = (rec((y V ↦ (y +𝑜 A)), ∅)‘B))
7 onelon 4087 . . . . . . 7 ((B On x B) → x On)
8 omexg 5970 . . . . . . . . 9 ((A On x On) → (A ·𝑜 x) V)
9 omcl 5980 . . . . . . . . . 10 ((A On x On) → (A ·𝑜 x) On)
10 simpl 102 . . . . . . . . . 10 ((A On x On) → A On)
11 oacl 5979 . . . . . . . . . 10 (((A ·𝑜 x) On A On) → ((A ·𝑜 x) +𝑜 A) On)
129, 10, 11syl2anc 391 . . . . . . . . 9 ((A On x On) → ((A ·𝑜 x) +𝑜 A) On)
13 oveq1 5462 . . . . . . . . . 10 (y = (A ·𝑜 x) → (y +𝑜 A) = ((A ·𝑜 x) +𝑜 A))
14 eqid 2037 . . . . . . . . . 10 (y V ↦ (y +𝑜 A)) = (y V ↦ (y +𝑜 A))
1513, 14fvmptg 5191 . . . . . . . . 9 (((A ·𝑜 x) V ((A ·𝑜 x) +𝑜 A) On) → ((y V ↦ (y +𝑜 A))‘(A ·𝑜 x)) = ((A ·𝑜 x) +𝑜 A))
168, 12, 15syl2anc 391 . . . . . . . 8 ((A On x On) → ((y V ↦ (y +𝑜 A))‘(A ·𝑜 x)) = ((A ·𝑜 x) +𝑜 A))
17 omv 5974 . . . . . . . . 9 ((A On x On) → (A ·𝑜 x) = (rec((y V ↦ (y +𝑜 A)), ∅)‘x))
1817fveq2d 5125 . . . . . . . 8 ((A On x On) → ((y V ↦ (y +𝑜 A))‘(A ·𝑜 x)) = ((y V ↦ (y +𝑜 A))‘(rec((y V ↦ (y +𝑜 A)), ∅)‘x)))
1916, 18eqtr3d 2071 . . . . . . 7 ((A On x On) → ((A ·𝑜 x) +𝑜 A) = ((y V ↦ (y +𝑜 A))‘(rec((y V ↦ (y +𝑜 A)), ∅)‘x)))
207, 19sylan2 270 . . . . . 6 ((A On (B On x B)) → ((A ·𝑜 x) +𝑜 A) = ((y V ↦ (y +𝑜 A))‘(rec((y V ↦ (y +𝑜 A)), ∅)‘x)))
2120anassrs 380 . . . . 5 (((A On B On) x B) → ((A ·𝑜 x) +𝑜 A) = ((y V ↦ (y +𝑜 A))‘(rec((y V ↦ (y +𝑜 A)), ∅)‘x)))
2221iuneq2dv 3669 . . . 4 ((A On B On) → x B ((A ·𝑜 x) +𝑜 A) = x B ((y V ↦ (y +𝑜 A))‘(rec((y V ↦ (y +𝑜 A)), ∅)‘x)))
2322uneq2d 3091 . . 3 ((A On B On) → (∅ ∪ x B ((A ·𝑜 x) +𝑜 A)) = (∅ ∪ x B ((y V ↦ (y +𝑜 A))‘(rec((y V ↦ (y +𝑜 A)), ∅)‘x))))
245, 6, 233eqtr4d 2079 . 2 ((A On B On) → (A ·𝑜 B) = (∅ ∪ x B ((A ·𝑜 x) +𝑜 A)))
25 uncom 3081 . . 3 (∅ ∪ x B ((A ·𝑜 x) +𝑜 A)) = ( x B ((A ·𝑜 x) +𝑜 A) ∪ ∅)
26 un0 3245 . . 3 ( x B ((A ·𝑜 x) +𝑜 A) ∪ ∅) = x B ((A ·𝑜 x) +𝑜 A)
2725, 26eqtri 2057 . 2 (∅ ∪ x B ((A ·𝑜 x) +𝑜 A)) = x B ((A ·𝑜 x) +𝑜 A)
2824, 27syl6eq 2085 1 ((A On B On) → (A ·𝑜 B) = x B ((A ·𝑜 x) +𝑜 A))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242   wcel 1390  Vcvv 2551  cun 2909  c0 3218   ciun 3648  cmpt 3809  Oncon0 4066   Fn wfn 4840  cfv 4845  (class class class)co 5455  reccrdg 5896   +𝑜 coa 5937   ·𝑜 comu 5938
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-nul 3874  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  df-omul 5945
This theorem is referenced by:  omsuc  5990
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