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Theorem omsuc 5966
Description: Multiplication with successor. Definition 8.15 of [TakeutiZaring] p. 62. (Contributed by NM, 17-Sep-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)
Assertion
Ref Expression
omsuc ((A On B On) → (A ·𝑜 suc B) = ((A ·𝑜 B) +𝑜 A))

Proof of Theorem omsuc
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 df-suc 4057 . . . . . . 7 suc B = (B ∪ {B})
2 iuneq1 3644 . . . . . . 7 (suc B = (B ∪ {B}) → x suc B((A ·𝑜 x) +𝑜 A) = x (B ∪ {B})((A ·𝑜 x) +𝑜 A))
31, 2ax-mp 7 . . . . . 6 x suc B((A ·𝑜 x) +𝑜 A) = x (B ∪ {B})((A ·𝑜 x) +𝑜 A)
4 iunxun 3709 . . . . . 6 x (B ∪ {B})((A ·𝑜 x) +𝑜 A) = ( x B ((A ·𝑜 x) +𝑜 A) ∪ x {B} ((A ·𝑜 x) +𝑜 A))
53, 4eqtri 2042 . . . . 5 x suc B((A ·𝑜 x) +𝑜 A) = ( x B ((A ·𝑜 x) +𝑜 A) ∪ x {B} ((A ·𝑜 x) +𝑜 A))
6 oveq2 5444 . . . . . . . 8 (x = B → (A ·𝑜 x) = (A ·𝑜 B))
76oveq1d 5451 . . . . . . 7 (x = B → ((A ·𝑜 x) +𝑜 A) = ((A ·𝑜 B) +𝑜 A))
87iunxsng 3706 . . . . . 6 (B On → x {B} ((A ·𝑜 x) +𝑜 A) = ((A ·𝑜 B) +𝑜 A))
98uneq2d 3074 . . . . 5 (B On → ( x B ((A ·𝑜 x) +𝑜 A) ∪ x {B} ((A ·𝑜 x) +𝑜 A)) = ( x B ((A ·𝑜 x) +𝑜 A) ∪ ((A ·𝑜 B) +𝑜 A)))
105, 9syl5eq 2066 . . . 4 (B On → x suc B((A ·𝑜 x) +𝑜 A) = ( x B ((A ·𝑜 x) +𝑜 A) ∪ ((A ·𝑜 B) +𝑜 A)))
1110adantl 262 . . 3 ((A On B On) → x suc B((A ·𝑜 x) +𝑜 A) = ( x B ((A ·𝑜 x) +𝑜 A) ∪ ((A ·𝑜 B) +𝑜 A)))
12 suceloni 4177 . . . 4 (B On → suc B On)
13 omv2 5960 . . . 4 ((A On suc B On) → (A ·𝑜 suc B) = x suc B((A ·𝑜 x) +𝑜 A))
1412, 13sylan2 270 . . 3 ((A On B On) → (A ·𝑜 suc B) = x suc B((A ·𝑜 x) +𝑜 A))
15 omv2 5960 . . . 4 ((A On B On) → (A ·𝑜 B) = x B ((A ·𝑜 x) +𝑜 A))
1615uneq1d 3073 . . 3 ((A On B On) → ((A ·𝑜 B) ∪ ((A ·𝑜 B) +𝑜 A)) = ( x B ((A ·𝑜 x) +𝑜 A) ∪ ((A ·𝑜 B) +𝑜 A)))
1711, 14, 163eqtr4d 2064 . 2 ((A On B On) → (A ·𝑜 suc B) = ((A ·𝑜 B) ∪ ((A ·𝑜 B) +𝑜 A)))
18 omcl 5956 . . 3 ((A On B On) → (A ·𝑜 B) On)
19 ax-ia1 99 . . 3 ((A On B On) → A On)
20 oaword1 5965 . . . 4 (((A ·𝑜 B) On A On) → (A ·𝑜 B) ⊆ ((A ·𝑜 B) +𝑜 A))
21 ssequn1 3090 . . . 4 ((A ·𝑜 B) ⊆ ((A ·𝑜 B) +𝑜 A) ↔ ((A ·𝑜 B) ∪ ((A ·𝑜 B) +𝑜 A)) = ((A ·𝑜 B) +𝑜 A))
2220, 21sylib 127 . . 3 (((A ·𝑜 B) On A On) → ((A ·𝑜 B) ∪ ((A ·𝑜 B) +𝑜 A)) = ((A ·𝑜 B) +𝑜 A))
2318, 19, 22syl2anc 393 . 2 ((A On B On) → ((A ·𝑜 B) ∪ ((A ·𝑜 B) +𝑜 A)) = ((A ·𝑜 B) +𝑜 A))
2417, 23eqtrd 2054 1 ((A On B On) → (A ·𝑜 suc B) = ((A ·𝑜 B) +𝑜 A))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1228   wcel 1374  cun 2892  wss 2894  {csn 3350   ciun 3631  Oncon0 4049  suc csuc 4051  (class class class)co 5436   +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:  onmsuc  5967
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