Theorem omsuc 6539

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 4407	. . . . . . 7 ⊢ suc 𝐵 = (𝐵 ∪ {𝐵})
2		iuneq1 3930	. . . . . . 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 3997	. . . . . 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 2217	. . . . 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 5933	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → (𝐴 ·o 𝑥) = (𝐴 ·o 𝐵))
7	6	oveq1d 5940	. . . . . . 7 ⊢ (𝑥 = 𝐵 → ((𝐴 ·o 𝑥) +o 𝐴) = ((𝐴 ·o 𝐵) +o 𝐴))
8	7	iunxsng 3993	. . . . . 6 ⊢ (𝐵 ∈ On → ∪ 𝑥 ∈ {𝐵} ((𝐴 ·o 𝑥) +o 𝐴) = ((𝐴 ·o 𝐵) +o 𝐴))
9	8	uneq2d 3318	. . . . 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 2241	. . . 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 4538	. . . 4 ⊢ (𝐵 ∈ On → suc 𝐵 ∈ On)
13		omv2 6532	. . . 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 6532	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o 𝐵) = ∪ 𝑥 ∈ 𝐵 ((𝐴 ·o 𝑥) +o 𝐴))
16	15	uneq1d 3317	. . 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 2239	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o suc 𝐵) = ((𝐴 ·o 𝐵) ∪ ((𝐴 ·o 𝐵) +o 𝐴)))
18		omcl 6528	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o 𝐵) ∈ On)
19		simpl 109	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐴 ∈ On)
20		oaword1 6538	. . . 4 ⊢ (((𝐴 ·o 𝐵) ∈ On ∧ 𝐴 ∈ On) → (𝐴 ·o 𝐵) ⊆ ((𝐴 ·o 𝐵) +o 𝐴))
21		ssequn1 3334	. . . 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 2229	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o suc 𝐵) = ((𝐴 ·o 𝐵) +o 𝐴))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2167   ∪ cun 3155   ⊆ wss 3157  {csn 3623  ∪ ciun 3917  Oncon0 4399  suc csuc 4401  (class class class)co 5925   +_o coa 6480   ·_o comu 6481

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-iord 4402  df-on 4404  df-suc 4407  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-irdg 6437  df-oadd 6487  df-omul 6488

This theorem is referenced by:  onmsuc  6540