Theorem omsuc 6633

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 4464	. . . . . . 7 ⊢ suc 𝐵 = (𝐵 ∪ {𝐵})
2		iuneq1 3979	. . . . . . 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 4046	. . . . . 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 2250	. . . . 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 6019	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → (𝐴 ·o 𝑥) = (𝐴 ·o 𝐵))
7	6	oveq1d 6026	. . . . . . 7 ⊢ (𝑥 = 𝐵 → ((𝐴 ·o 𝑥) +o 𝐴) = ((𝐴 ·o 𝐵) +o 𝐴))
8	7	iunxsng 4042	. . . . . 6 ⊢ (𝐵 ∈ On → ∪ 𝑥 ∈ {𝐵} ((𝐴 ·o 𝑥) +o 𝐴) = ((𝐴 ·o 𝐵) +o 𝐴))
9	8	uneq2d 3359	. . . . 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 2274	. . . 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 4595	. . . 4 ⊢ (𝐵 ∈ On → suc 𝐵 ∈ On)
13		omv2 6626	. . . 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 6626	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o 𝐵) = ∪ 𝑥 ∈ 𝐵 ((𝐴 ·o 𝑥) +o 𝐴))
16	15	uneq1d 3358	. . 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 2272	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o suc 𝐵) = ((𝐴 ·o 𝐵) ∪ ((𝐴 ·o 𝐵) +o 𝐴)))
18		omcl 6622	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o 𝐵) ∈ On)
19		simpl 109	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐴 ∈ On)
20		oaword1 6632	. . . 4 ⊢ (((𝐴 ·o 𝐵) ∈ On ∧ 𝐴 ∈ On) → (𝐴 ·o 𝐵) ⊆ ((𝐴 ·o 𝐵) +o 𝐴))
21		ssequn1 3375	. . . 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 2262	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o suc 𝐵) = ((𝐴 ·o 𝐵) +o 𝐴))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1395   ∈ wcel 2200   ∪ cun 3196   ⊆ wss 3198  {csn 3667  ∪ ciun 3966  Oncon0 4456  suc csuc 4458  (class class class)co 6011   +_o coa 6572   ·_o comu 6573

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4200  ax-sep 4203  ax-nul 4211  ax-pow 4260  ax-pr 4295  ax-un 4526  ax-setind 4631

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3890  df-iun 3968  df-br 4085  df-opab 4147  df-mpt 4148  df-tr 4184  df-id 4386  df-iord 4459  df-on 4461  df-suc 4464  df-xp 4727  df-rel 4728  df-cnv 4729  df-co 4730  df-dm 4731  df-rn 4732  df-res 4733  df-ima 4734  df-iota 5282  df-fun 5324  df-fn 5325  df-f 5326  df-f1 5327  df-fo 5328  df-f1o 5329  df-fv 5330  df-ov 6014  df-oprab 6015  df-mpo 6016  df-1st 6296  df-2nd 6297  df-recs 6464  df-irdg 6529  df-oadd 6579  df-omul 6580

This theorem is referenced by:  onmsuc  6634