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Theorem omsuc 6081
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 4135 . . . . . . 7 suc 𝐵 = (𝐵 ∪ {𝐵})
2 iuneq1 3697 . . . . . . 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 3762 . . . . . 6 𝑥 ∈ (𝐵 ∪ {𝐵})((𝐴 ·𝑜 𝑥) +𝑜 𝐴) = ( 𝑥𝐵 ((𝐴 ·𝑜 𝑥) +𝑜 𝐴) ∪ 𝑥 ∈ {𝐵} ((𝐴 ·𝑜 𝑥) +𝑜 𝐴))
53, 4eqtri 2076 . . . . 5 𝑥 ∈ suc 𝐵((𝐴 ·𝑜 𝑥) +𝑜 𝐴) = ( 𝑥𝐵 ((𝐴 ·𝑜 𝑥) +𝑜 𝐴) ∪ 𝑥 ∈ {𝐵} ((𝐴 ·𝑜 𝑥) +𝑜 𝐴))
6 oveq2 5547 . . . . . . . 8 (𝑥 = 𝐵 → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 𝐵))
76oveq1d 5554 . . . . . . 7 (𝑥 = 𝐵 → ((𝐴 ·𝑜 𝑥) +𝑜 𝐴) = ((𝐴 ·𝑜 𝐵) +𝑜 𝐴))
87iunxsng 3759 . . . . . 6 (𝐵 ∈ On → 𝑥 ∈ {𝐵} ((𝐴 ·𝑜 𝑥) +𝑜 𝐴) = ((𝐴 ·𝑜 𝐵) +𝑜 𝐴))
98uneq2d 3124 . . . . 5 (𝐵 ∈ On → ( 𝑥𝐵 ((𝐴 ·𝑜 𝑥) +𝑜 𝐴) ∪ 𝑥 ∈ {𝐵} ((𝐴 ·𝑜 𝑥) +𝑜 𝐴)) = ( 𝑥𝐵 ((𝐴 ·𝑜 𝑥) +𝑜 𝐴) ∪ ((𝐴 ·𝑜 𝐵) +𝑜 𝐴)))
105, 9syl5eq 2100 . . . 4 (𝐵 ∈ On → 𝑥 ∈ suc 𝐵((𝐴 ·𝑜 𝑥) +𝑜 𝐴) = ( 𝑥𝐵 ((𝐴 ·𝑜 𝑥) +𝑜 𝐴) ∪ ((𝐴 ·𝑜 𝐵) +𝑜 𝐴)))
1110adantl 266 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝑥 ∈ suc 𝐵((𝐴 ·𝑜 𝑥) +𝑜 𝐴) = ( 𝑥𝐵 ((𝐴 ·𝑜 𝑥) +𝑜 𝐴) ∪ ((𝐴 ·𝑜 𝐵) +𝑜 𝐴)))
12 suceloni 4254 . . . 4 (𝐵 ∈ On → suc 𝐵 ∈ On)
13 omv2 6075 . . . 4 ((𝐴 ∈ On ∧ suc 𝐵 ∈ On) → (𝐴 ·𝑜 suc 𝐵) = 𝑥 ∈ suc 𝐵((𝐴 ·𝑜 𝑥) +𝑜 𝐴))
1412, 13sylan2 274 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 suc 𝐵) = 𝑥 ∈ suc 𝐵((𝐴 ·𝑜 𝑥) +𝑜 𝐴))
15 omv2 6075 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 𝐵) = 𝑥𝐵 ((𝐴 ·𝑜 𝑥) +𝑜 𝐴))
1615uneq1d 3123 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·𝑜 𝐵) ∪ ((𝐴 ·𝑜 𝐵) +𝑜 𝐴)) = ( 𝑥𝐵 ((𝐴 ·𝑜 𝑥) +𝑜 𝐴) ∪ ((𝐴 ·𝑜 𝐵) +𝑜 𝐴)))
1711, 14, 163eqtr4d 2098 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 suc 𝐵) = ((𝐴 ·𝑜 𝐵) ∪ ((𝐴 ·𝑜 𝐵) +𝑜 𝐴)))
18 omcl 6071 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 𝐵) ∈ On)
19 simpl 106 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐴 ∈ On)
20 oaword1 6080 . . . 4 (((𝐴 ·𝑜 𝐵) ∈ On ∧ 𝐴 ∈ On) → (𝐴 ·𝑜 𝐵) ⊆ ((𝐴 ·𝑜 𝐵) +𝑜 𝐴))
21 ssequn1 3140 . . . 4 ((𝐴 ·𝑜 𝐵) ⊆ ((𝐴 ·𝑜 𝐵) +𝑜 𝐴) ↔ ((𝐴 ·𝑜 𝐵) ∪ ((𝐴 ·𝑜 𝐵) +𝑜 𝐴)) = ((𝐴 ·𝑜 𝐵) +𝑜 𝐴))
2220, 21sylib 131 . . 3 (((𝐴 ·𝑜 𝐵) ∈ On ∧ 𝐴 ∈ On) → ((𝐴 ·𝑜 𝐵) ∪ ((𝐴 ·𝑜 𝐵) +𝑜 𝐴)) = ((𝐴 ·𝑜 𝐵) +𝑜 𝐴))
2318, 19, 22syl2anc 397 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·𝑜 𝐵) ∪ ((𝐴 ·𝑜 𝐵) +𝑜 𝐴)) = ((𝐴 ·𝑜 𝐵) +𝑜 𝐴))
2417, 23eqtrd 2088 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 101   = wceq 1259  wcel 1409  cun 2942  wss 2944  {csn 3402   ciun 3684  Oncon0 4127  suc csuc 4129  (class class class)co 5539   +𝑜 coa 6028   ·𝑜 comu 6029
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3899  ax-sep 3902  ax-nul 3910  ax-pow 3954  ax-pr 3971  ax-un 4197  ax-setind 4289
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2787  df-csb 2880  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-nul 3252  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-iun 3686  df-br 3792  df-opab 3846  df-mpt 3847  df-tr 3882  df-id 4057  df-iord 4130  df-on 4132  df-suc 4135  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385  df-iota 4894  df-fun 4931  df-fn 4932  df-f 4933  df-f1 4934  df-fo 4935  df-f1o 4936  df-fv 4937  df-ov 5542  df-oprab 5543  df-mpt2 5544  df-1st 5794  df-2nd 5795  df-recs 5950  df-irdg 5987  df-oadd 6035  df-omul 6036
This theorem is referenced by:  onmsuc  6082
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