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.

Step	Hyp	Ref	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))
3	1, 2	ax-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) = (∪ 𝑥 ∈ 𝐵 ((𝐴 ·o 𝑥) +o 𝐴) ∪ ∪ 𝑥 ∈ {𝐵} ((𝐴 ·o 𝑥) +o 𝐴))
5	3, 4	eqtri 2210	. . . . 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 5905	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → (𝐴 ·o 𝑥) = (𝐴 ·o 𝐵))
7	6	oveq1d 5912	. . . . . . 7 ⊢ (𝑥 = 𝐵 → ((𝐴 ·o 𝑥) +o 𝐴) = ((𝐴 ·o 𝐵) +o 𝐴))
8	7	iunxsng 3977	. . . . . 6 ⊢ (𝐵 ∈ On → ∪ 𝑥 ∈ {𝐵} ((𝐴 ·o 𝑥) +o 𝐴) = ((𝐴 ·o 𝐵) +o 𝐴))
9	8	uneq2d 3304	. . . . 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)))
10	5, 9	eqtrid 2234	. . . 4 ⊢ (𝐵 ∈ On → ∪ 𝑥 ∈ suc 𝐵((𝐴 ·o 𝑥) +o 𝐴) = (∪ 𝑥 ∈ 𝐵 ((𝐴 ·o 𝑥) +o 𝐴) ∪ ((𝐴 ·o 𝐵) +o 𝐴)))
11	10	adantl 277	. . 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		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))
14	12, 13	sylan2 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) = ∪ 𝑥 ∈ 𝐵 ((𝐴 ·o 𝑥) +o 𝐴))
16	15	uneq1d 3303	. . 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)))
17	11, 14, 16	3eqtr4d 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))
22	20, 21	sylib 122	. . 3 ⊢ (((𝐴 ·o 𝐵) ∈ On ∧ 𝐴 ∈ On) → ((𝐴 ·o 𝐵) ∪ ((𝐴 ·o 𝐵) +o 𝐴)) = ((𝐴 ·o 𝐵) +o 𝐴))
23	18, 19, 22	syl2anc 411	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·o 𝐵) ∪ ((𝐴 ·o 𝐵) +o 𝐴)) = ((𝐴 ·o 𝐵) +o 𝐴))
24	17, 23	eqtrd 2222	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o suc 𝐵) = ((𝐴 ·o 𝐵) +o 𝐴))

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