Theorem omsuc 6074

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 4126	. . . . . . 7 ⊢ suc 𝐵 = (𝐵 ∪ {𝐵})
2		iuneq1 3691	. . . . . . 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 7	. . . . . 6 ⊢ ∪ 𝑥 ∈ suc 𝐵((𝐴 ·𝑜 𝑥) +𝑜 𝐴) = ∪ 𝑥 ∈ (𝐵 ∪ {𝐵})((𝐴 ·𝑜 𝑥) +𝑜 𝐴)
4		iunxun 3756	. . . . . 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 2101	. . . . 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 5540	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 𝐵))
7	6	oveq1d 5547	. . . . . . 7 ⊢ (𝑥 = 𝐵 → ((𝐴 ·𝑜 𝑥) +𝑜 𝐴) = ((𝐴 ·𝑜 𝐵) +𝑜 𝐴))
8	7	iunxsng 3753	. . . . . 6 ⊢ (𝐵 ∈ On → ∪ 𝑥 ∈ {𝐵} ((𝐴 ·𝑜 𝑥) +𝑜 𝐴) = ((𝐴 ·𝑜 𝐵) +𝑜 𝐴))
9	8	uneq2d 3126	. . . . 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	syl5eq 2125	. . . 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 271	. . 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		suceloni 4245	. . . 4 ⊢ (𝐵 ∈ On → suc 𝐵 ∈ On)
13		omv2 6068	. . . 4 ⊢ ((𝐴 ∈ On ∧ suc 𝐵 ∈ On) → (𝐴 ·𝑜 suc 𝐵) = ∪ 𝑥 ∈ suc 𝐵((𝐴 ·𝑜 𝑥) +𝑜 𝐴))
14	12, 13	sylan2 280	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 suc 𝐵) = ∪ 𝑥 ∈ suc 𝐵((𝐴 ·𝑜 𝑥) +𝑜 𝐴))
15		omv2 6068	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 𝐵) = ∪ 𝑥 ∈ 𝐵 ((𝐴 ·𝑜 𝑥) +𝑜 𝐴))
16	15	uneq1d 3125	. . 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 2123	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 suc 𝐵) = ((𝐴 ·𝑜 𝐵) ∪ ((𝐴 ·𝑜 𝐵) +𝑜 𝐴)))
18		omcl 6064	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 𝐵) ∈ On)
19		simpl 107	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐴 ∈ On)
20		oaword1 6073	. . . 4 ⊢ (((𝐴 ·𝑜 𝐵) ∈ On ∧ 𝐴 ∈ On) → (𝐴 ·𝑜 𝐵) ⊆ ((𝐴 ·𝑜 𝐵) +𝑜 𝐴))
21		ssequn1 3142	. . . 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 120	. . 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 403	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·𝑜 𝐵) ∪ ((𝐴 ·𝑜 𝐵) +𝑜 𝐴)) = ((𝐴 ·𝑜 𝐵) +𝑜 𝐴))
24	17, 23	eqtrd 2113	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 suc 𝐵) = ((𝐴 ·𝑜 𝐵) +𝑜 𝐴))

Syntax hints:   → wi 4   ∧ wa 102   = wceq 1284   ∈ wcel 1433   ∪ cun 2971   ⊆ wss 2973  {csn 3398  ∪ ciun 3678  Oncon0 4118  suc csuc 4120  (class class class)co 5532   +_𝑜 coa 6021   ·_𝑜 comu 6022

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-id 4048  df-iord 4121  df-on 4123  df-suc 4126  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-irdg 5980  df-oadd 6028  df-omul 6029

This theorem is referenced by:  onmsuc  6075