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Theorem omsuc 6498
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 4389 . . . . . . 7 suc 𝐵 = (𝐵 ∪ {𝐵})
2 iuneq1 3914 . . . . . . 7 (suc 𝐵 = (𝐵 ∪ {𝐵}) → 𝑥 ∈ suc 𝐵((𝐴 ·o 𝑥) +o 𝐴) = 𝑥 ∈ (𝐵 ∪ {𝐵})((𝐴 ·o 𝑥) +o 𝐴))
31, 2ax-mp 5 . . . . . 6 𝑥 ∈ suc 𝐵((𝐴 ·o 𝑥) +o 𝐴) = 𝑥 ∈ (𝐵 ∪ {𝐵})((𝐴 ·o 𝑥) +o 𝐴)
4 iunxun 3981 . . . . . 6 𝑥 ∈ (𝐵 ∪ {𝐵})((𝐴 ·o 𝑥) +o 𝐴) = ( 𝑥𝐵 ((𝐴 ·o 𝑥) +o 𝐴) ∪ 𝑥 ∈ {𝐵} ((𝐴 ·o 𝑥) +o 𝐴))
53, 4eqtri 2210 . . . . 5 𝑥 ∈ suc 𝐵((𝐴 ·o 𝑥) +o 𝐴) = ( 𝑥𝐵 ((𝐴 ·o 𝑥) +o 𝐴) ∪ 𝑥 ∈ {𝐵} ((𝐴 ·o 𝑥) +o 𝐴))
6 oveq2 5905 . . . . . . . 8 (𝑥 = 𝐵 → (𝐴 ·o 𝑥) = (𝐴 ·o 𝐵))
76oveq1d 5912 . . . . . . 7 (𝑥 = 𝐵 → ((𝐴 ·o 𝑥) +o 𝐴) = ((𝐴 ·o 𝐵) +o 𝐴))
87iunxsng 3977 . . . . . 6 (𝐵 ∈ On → 𝑥 ∈ {𝐵} ((𝐴 ·o 𝑥) +o 𝐴) = ((𝐴 ·o 𝐵) +o 𝐴))
98uneq2d 3304 . . . . 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, 9eqtrid 2234 . . . 4 (𝐵 ∈ On → 𝑥 ∈ suc 𝐵((𝐴 ·o 𝑥) +o 𝐴) = ( 𝑥𝐵 ((𝐴 ·o 𝑥) +o 𝐴) ∪ ((𝐴 ·o 𝐵) +o 𝐴)))
1110adantl 277 . . 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 onsuc 4518 . . . 4 (𝐵 ∈ On → suc 𝐵 ∈ On)
13 omv2 6491 . . . 4 ((𝐴 ∈ On ∧ suc 𝐵 ∈ On) → (𝐴 ·o suc 𝐵) = 𝑥 ∈ suc 𝐵((𝐴 ·o 𝑥) +o 𝐴))
1412, 13sylan2 286 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o suc 𝐵) = 𝑥 ∈ suc 𝐵((𝐴 ·o 𝑥) +o 𝐴))
15 omv2 6491 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o 𝐵) = 𝑥𝐵 ((𝐴 ·o 𝑥) +o 𝐴))
1615uneq1d 3303 . . 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 2232 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o suc 𝐵) = ((𝐴 ·o 𝐵) ∪ ((𝐴 ·o 𝐵) +o 𝐴)))
18 omcl 6487 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o 𝐵) ∈ On)
19 simpl 109 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐴 ∈ On)
20 oaword1 6497 . . . 4 (((𝐴 ·o 𝐵) ∈ On ∧ 𝐴 ∈ On) → (𝐴 ·o 𝐵) ⊆ ((𝐴 ·o 𝐵) +o 𝐴))
21 ssequn1 3320 . . . 4 ((𝐴 ·o 𝐵) ⊆ ((𝐴 ·o 𝐵) +o 𝐴) ↔ ((𝐴 ·o 𝐵) ∪ ((𝐴 ·o 𝐵) +o 𝐴)) = ((𝐴 ·o 𝐵) +o 𝐴))
2220, 21sylib 122 . . 3 (((𝐴 ·o 𝐵) ∈ On ∧ 𝐴 ∈ On) → ((𝐴 ·o 𝐵) ∪ ((𝐴 ·o 𝐵) +o 𝐴)) = ((𝐴 ·o 𝐵) +o 𝐴))
2318, 19, 22syl2anc 411 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·o 𝐵) ∪ ((𝐴 ·o 𝐵) +o 𝐴)) = ((𝐴 ·o 𝐵) +o 𝐴))
2417, 23eqtrd 2222 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 104   = wceq 1364  wcel 2160  cun 3142  wss 3144  {csn 3607   ciun 3901  Oncon0 4381  suc csuc 4383  (class class class)co 5897   +o coa 6439   ·o comu 6440
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-iord 4384  df-on 4386  df-suc 4389  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-ov 5900  df-oprab 5901  df-mpo 5902  df-1st 6166  df-2nd 6167  df-recs 6331  df-irdg 6396  df-oadd 6446  df-omul 6447
This theorem is referenced by:  onmsuc  6499
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