Theorem omsuc 6051

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 4108	. . . . . . 7 ⊢ suc 𝐵 = (𝐵 ∪ {𝐵})
2		iuneq1 3670	. . . . . . 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 3735	. . . . . 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 2060	. . . . 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 5520	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 𝐵))
7	6	oveq1d 5527	. . . . . . 7 ⊢ (𝑥 = 𝐵 → ((𝐴 ·𝑜 𝑥) +𝑜 𝐴) = ((𝐴 ·𝑜 𝐵) +𝑜 𝐴))
8	7	iunxsng 3732	. . . . . 6 ⊢ (𝐵 ∈ On → ∪ 𝑥 ∈ {𝐵} ((𝐴 ·𝑜 𝑥) +𝑜 𝐴) = ((𝐴 ·𝑜 𝐵) +𝑜 𝐴))
9	8	uneq2d 3097	. . . . 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 2084	. . . 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 262	. . 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 4227	. . . 4 ⊢ (𝐵 ∈ On → suc 𝐵 ∈ On)
13		omv2 6045	. . . 4 ⊢ ((𝐴 ∈ On ∧ suc 𝐵 ∈ On) → (𝐴 ·𝑜 suc 𝐵) = ∪ 𝑥 ∈ suc 𝐵((𝐴 ·𝑜 𝑥) +𝑜 𝐴))
14	12, 13	sylan2 270	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 suc 𝐵) = ∪ 𝑥 ∈ suc 𝐵((𝐴 ·𝑜 𝑥) +𝑜 𝐴))
15		omv2 6045	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 𝐵) = ∪ 𝑥 ∈ 𝐵 ((𝐴 ·𝑜 𝑥) +𝑜 𝐴))
16	15	uneq1d 3096	. . 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 2082	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 suc 𝐵) = ((𝐴 ·𝑜 𝐵) ∪ ((𝐴 ·𝑜 𝐵) +𝑜 𝐴)))
18		omcl 6041	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 𝐵) ∈ On)
19		simpl 102	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐴 ∈ On)
20		oaword1 6050	. . . 4 ⊢ (((𝐴 ·𝑜 𝐵) ∈ On ∧ 𝐴 ∈ On) → (𝐴 ·𝑜 𝐵) ⊆ ((𝐴 ·𝑜 𝐵) +𝑜 𝐴))
21		ssequn1 3113	. . . 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 127	. . 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 391	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·𝑜 𝐵) ∪ ((𝐴 ·𝑜 𝐵) +𝑜 𝐴)) = ((𝐴 ·𝑜 𝐵) +𝑜 𝐴))
24	17, 23	eqtrd 2072	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 suc 𝐵) = ((𝐴 ·𝑜 𝐵) +𝑜 𝐴))

Syntax hints:   → wi 4   ∧ wa 97   = wceq 1243   ∈ wcel 1393   ∪ cun 2915   ⊆ wss 2917  {csn 3375  ∪ ciun 3657  Oncon0 4100  suc csuc 4102  (class class class)co 5512   +_𝑜 coa 5998   ·_𝑜 comu 5999

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-id 4030  df-iord 4103  df-on 4105  df-suc 4108  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-oadd 6005  df-omul 6006

This theorem is referenced by:  onmsuc  6052