Theorem omsuc 6180

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 4170	. . . . . . 7 ⊢ suc 𝐵 = (𝐵 ∪ {𝐵})
2		iuneq1 3725	. . . . . . 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 3790	. . . . . 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 2105	. . . . 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 5614	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 𝐵))
7	6	oveq1d 5621	. . . . . . 7 ⊢ (𝑥 = 𝐵 → ((𝐴 ·𝑜 𝑥) +𝑜 𝐴) = ((𝐴 ·𝑜 𝐵) +𝑜 𝐴))
8	7	iunxsng 3787	. . . . . 6 ⊢ (𝐵 ∈ On → ∪ 𝑥 ∈ {𝐵} ((𝐴 ·𝑜 𝑥) +𝑜 𝐴) = ((𝐴 ·𝑜 𝐵) +𝑜 𝐴))
9	8	uneq2d 3143	. . . . 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 2129	. . . 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 4289	. . . 4 ⊢ (𝐵 ∈ On → suc 𝐵 ∈ On)
13		omv2 6173	. . . 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 6173	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 𝐵) = ∪ 𝑥 ∈ 𝐵 ((𝐴 ·𝑜 𝑥) +𝑜 𝐴))
16	15	uneq1d 3142	. . 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 2127	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 suc 𝐵) = ((𝐴 ·𝑜 𝐵) ∪ ((𝐴 ·𝑜 𝐵) +𝑜 𝐴)))
18		omcl 6169	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 𝐵) ∈ On)
19		simpl 107	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐴 ∈ On)
20		oaword1 6179	. . . 4 ⊢ (((𝐴 ·𝑜 𝐵) ∈ On ∧ 𝐴 ∈ On) → (𝐴 ·𝑜 𝐵) ⊆ ((𝐴 ·𝑜 𝐵) +𝑜 𝐴))
21		ssequn1 3159	. . . 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 2117	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 suc 𝐵) = ((𝐴 ·𝑜 𝐵) +𝑜 𝐴))

Syntax hints:   → wi 4   ∧ wa 102   = wceq 1287   ∈ wcel 1436   ∪ cun 2986   ⊆ wss 2988  {csn 3430  ∪ ciun 3712  Oncon0 4162  suc csuc 4164  (class class class)co 5606   +_𝑜 coa 6125   ·_𝑜 comu 6126

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 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-coll 3927  ax-sep 3930  ax-nul 3938  ax-pow 3982  ax-pr 4008  ax-un 4232  ax-setind 4324

This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-ral 2360  df-rex 2361  df-reu 2362  df-rab 2364  df-v 2617  df-sbc 2830  df-csb 2923  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-nul 3276  df-pw 3416  df-sn 3436  df-pr 3437  df-op 3439  df-uni 3636  df-iun 3714  df-br 3820  df-opab 3874  df-mpt 3875  df-tr 3910  df-id 4092  df-iord 4165  df-on 4167  df-suc 4170  df-xp 4415  df-rel 4416  df-cnv 4417  df-co 4418  df-dm 4419  df-rn 4420  df-res 4421  df-ima 4422  df-iota 4942  df-fun 4979  df-fn 4980  df-f 4981  df-f1 4982  df-fo 4983  df-f1o 4984  df-fv 4985  df-ov 5609  df-oprab 5610  df-mpt2 5611  df-1st 5861  df-2nd 5862  df-recs 6017  df-irdg 6082  df-oadd 6132  df-omul 6133

This theorem is referenced by:  onmsuc  6181