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Theorem omv2 5960
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 5944 . . . 4 (A On → (y V ↦ (y +𝑜 A)) Fn V)
2 0elon 4078 . . . . 5 On
3 rdgival 5889 . . . . 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 1205 . . . 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 5950 . . 3 ((A On B On) → (A ·𝑜 B) = (rec((y V ↦ (y +𝑜 A)), ∅)‘B))
7 onelon 4070 . . . . . . 7 ((B On x B) → x On)
8 omexg 5946 . . . . . . . . 9 ((A On x On) → (A ·𝑜 x) V)
9 omcl 5956 . . . . . . . . . 10 ((A On x On) → (A ·𝑜 x) On)
10 ax-ia1 99 . . . . . . . . . 10 ((A On x On) → A On)
11 oacl 5955 . . . . . . . . . 10 (((A ·𝑜 x) On A On) → ((A ·𝑜 x) +𝑜 A) On)
129, 10, 11syl2anc 393 . . . . . . . . 9 ((A On x On) → ((A ·𝑜 x) +𝑜 A) On)
13 oveq1 5443 . . . . . . . . . 10 (y = (A ·𝑜 x) → (y +𝑜 A) = ((A ·𝑜 x) +𝑜 A))
14 eqid 2022 . . . . . . . . . 10 (y V ↦ (y +𝑜 A)) = (y V ↦ (y +𝑜 A))
1513, 14fvmptg 5173 . . . . . . . . 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 393 . . . . . . . 8 ((A On x On) → ((y V ↦ (y +𝑜 A))‘(A ·𝑜 x)) = ((A ·𝑜 x) +𝑜 A))
17 omv 5950 . . . . . . . . 9 ((A On x On) → (A ·𝑜 x) = (rec((y V ↦ (y +𝑜 A)), ∅)‘x))
1817fveq2d 5107 . . . . . . . 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 2056 . . . . . . 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 382 . . . . 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 3652 . . . 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 3074 . . 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 2064 . 2 ((A On B On) → (A ·𝑜 B) = (∅ ∪ x B ((A ·𝑜 x) +𝑜 A)))
25 uncom 3064 . . 3 (∅ ∪ x B ((A ·𝑜 x) +𝑜 A)) = ( x B ((A ·𝑜 x) +𝑜 A) ∪ ∅)
26 un0 3228 . . 3 ( x B ((A ·𝑜 x) +𝑜 A) ∪ ∅) = x B ((A ·𝑜 x) +𝑜 A)
2725, 26eqtri 2042 . 2 (∅ ∪ x B ((A ·𝑜 x) +𝑜 A)) = x B ((A ·𝑜 x) +𝑜 A)
2824, 27syl6eq 2070 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 1228   wcel 1374  Vcvv 2535  cun 2892  c0 3201   ciun 3631  cmpt 3792  Oncon0 4049   Fn wfn 4824  cfv 4829  (class class class)co 5436  reccrdg 5877   +𝑜 coa 5913   ·𝑜 comu 5914
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 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-coll 3846  ax-sep 3849  ax-nul 3857  ax-pow 3901  ax-pr 3918  ax-un 4120  ax-setind 4204
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-fal 1234  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ne 2188  df-ral 2289  df-rex 2290  df-reu 2291  df-rab 2293  df-v 2537  df-sbc 2742  df-csb 2830  df-dif 2897  df-un 2899  df-in 2901  df-ss 2908  df-nul 3202  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-iun 3633  df-br 3739  df-opab 3793  df-mpt 3794  df-tr 3829  df-id 4004  df-iord 4052  df-on 4054  df-suc 4057  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-rn 4283  df-res 4284  df-ima 4285  df-iota 4794  df-fun 4831  df-fn 4832  df-f 4833  df-f1 4834  df-fo 4835  df-f1o 4836  df-fv 4837  df-ov 5439  df-oprab 5440  df-mpt2 5441  df-1st 5690  df-2nd 5691  df-recs 5842  df-irdg 5878  df-oadd 5920  df-omul 5921
This theorem is referenced by:  omsuc  5966
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