Theorem omv2 6494

Description: Value of ordinal multiplication. (Contributed by Jim Kingdon, 23-Aug-2019.)

Assertion

Ref	Expression
omv2	⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o 𝐵) = ∪ 𝑥 ∈ 𝐵 ((𝐴 ·o 𝑥) +o 𝐴))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem omv2

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		omfnex 6478	. . . 4 ⊢ (𝐴 ∈ On → (𝑦 ∈ V ↦ (𝑦 +o 𝐴)) Fn V)
2		0elon 4413	. . . . 5 ⊢ ∅ ∈ On
3		rdgival 6411	. . . . 5 ⊢ (((𝑦 ∈ V ↦ (𝑦 +o 𝐴)) Fn V ∧ ∅ ∈ On ∧ 𝐵 ∈ On) → (rec((𝑦 ∈ V ↦ (𝑦 +o 𝐴)), ∅)‘𝐵) = (∅ ∪ ∪ 𝑥 ∈ 𝐵 ((𝑦 ∈ V ↦ (𝑦 +o 𝐴))‘(rec((𝑦 ∈ V ↦ (𝑦 +o 𝐴)), ∅)‘𝑥))))
4	2, 3	mp3an2 1336	. . . 4 ⊢ (((𝑦 ∈ V ↦ (𝑦 +o 𝐴)) Fn V ∧ 𝐵 ∈ On) → (rec((𝑦 ∈ V ↦ (𝑦 +o 𝐴)), ∅)‘𝐵) = (∅ ∪ ∪ 𝑥 ∈ 𝐵 ((𝑦 ∈ V ↦ (𝑦 +o 𝐴))‘(rec((𝑦 ∈ V ↦ (𝑦 +o 𝐴)), ∅)‘𝑥))))
5	1, 4	sylan 283	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (rec((𝑦 ∈ V ↦ (𝑦 +o 𝐴)), ∅)‘𝐵) = (∅ ∪ ∪ 𝑥 ∈ 𝐵 ((𝑦 ∈ V ↦ (𝑦 +o 𝐴))‘(rec((𝑦 ∈ V ↦ (𝑦 +o 𝐴)), ∅)‘𝑥))))
6		omv 6484	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o 𝐵) = (rec((𝑦 ∈ V ↦ (𝑦 +o 𝐴)), ∅)‘𝐵))
7		onelon 4405	. . . . . . 7 ⊢ ((𝐵 ∈ On ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ On)
8		omexg 6480	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 ·o 𝑥) ∈ V)
9		omcl 6490	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 ·o 𝑥) ∈ On)
10		simpl 109	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → 𝐴 ∈ On)
11		oacl 6489	. . . . . . . . . 10 ⊢ (((𝐴 ·o 𝑥) ∈ On ∧ 𝐴 ∈ On) → ((𝐴 ·o 𝑥) +o 𝐴) ∈ On)
12	9, 10, 11	syl2anc 411	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → ((𝐴 ·o 𝑥) +o 𝐴) ∈ On)
13		oveq1 5907	. . . . . . . . . 10 ⊢ (𝑦 = (𝐴 ·o 𝑥) → (𝑦 +o 𝐴) = ((𝐴 ·o 𝑥) +o 𝐴))
14		eqid 2189	. . . . . . . . . 10 ⊢ (𝑦 ∈ V ↦ (𝑦 +o 𝐴)) = (𝑦 ∈ V ↦ (𝑦 +o 𝐴))
15	13, 14	fvmptg 5616	. . . . . . . . 9 ⊢ (((𝐴 ·o 𝑥) ∈ V ∧ ((𝐴 ·o 𝑥) +o 𝐴) ∈ On) → ((𝑦 ∈ V ↦ (𝑦 +o 𝐴))‘(𝐴 ·o 𝑥)) = ((𝐴 ·o 𝑥) +o 𝐴))
16	8, 12, 15	syl2anc 411	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → ((𝑦 ∈ V ↦ (𝑦 +o 𝐴))‘(𝐴 ·o 𝑥)) = ((𝐴 ·o 𝑥) +o 𝐴))
17		omv 6484	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 ·o 𝑥) = (rec((𝑦 ∈ V ↦ (𝑦 +o 𝐴)), ∅)‘𝑥))
18	17	fveq2d 5541	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → ((𝑦 ∈ V ↦ (𝑦 +o 𝐴))‘(𝐴 ·o 𝑥)) = ((𝑦 ∈ V ↦ (𝑦 +o 𝐴))‘(rec((𝑦 ∈ V ↦ (𝑦 +o 𝐴)), ∅)‘𝑥)))
19	16, 18	eqtr3d 2224	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → ((𝐴 ·o 𝑥) +o 𝐴) = ((𝑦 ∈ V ↦ (𝑦 +o 𝐴))‘(rec((𝑦 ∈ V ↦ (𝑦 +o 𝐴)), ∅)‘𝑥)))
20	7, 19	sylan2 286	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝑥 ∈ 𝐵)) → ((𝐴 ·o 𝑥) +o 𝐴) = ((𝑦 ∈ V ↦ (𝑦 +o 𝐴))‘(rec((𝑦 ∈ V ↦ (𝑦 +o 𝐴)), ∅)‘𝑥)))
21	20	anassrs 400	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑥 ∈ 𝐵) → ((𝐴 ·o 𝑥) +o 𝐴) = ((𝑦 ∈ V ↦ (𝑦 +o 𝐴))‘(rec((𝑦 ∈ V ↦ (𝑦 +o 𝐴)), ∅)‘𝑥)))
22	21	iuneq2dv 3925	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ∪ 𝑥 ∈ 𝐵 ((𝐴 ·o 𝑥) +o 𝐴) = ∪ 𝑥 ∈ 𝐵 ((𝑦 ∈ V ↦ (𝑦 +o 𝐴))‘(rec((𝑦 ∈ V ↦ (𝑦 +o 𝐴)), ∅)‘𝑥)))
23	22	uneq2d 3304	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∅ ∪ ∪ 𝑥 ∈ 𝐵 ((𝐴 ·o 𝑥) +o 𝐴)) = (∅ ∪ ∪ 𝑥 ∈ 𝐵 ((𝑦 ∈ V ↦ (𝑦 +o 𝐴))‘(rec((𝑦 ∈ V ↦ (𝑦 +o 𝐴)), ∅)‘𝑥))))
24	5, 6, 23	3eqtr4d 2232	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o 𝐵) = (∅ ∪ ∪ 𝑥 ∈ 𝐵 ((𝐴 ·o 𝑥) +o 𝐴)))
25		uncom 3294	. . 3 ⊢ (∅ ∪ ∪ 𝑥 ∈ 𝐵 ((𝐴 ·o 𝑥) +o 𝐴)) = (∪ 𝑥 ∈ 𝐵 ((𝐴 ·o 𝑥) +o 𝐴) ∪ ∅)
26		un0 3471	. . 3 ⊢ (∪ 𝑥 ∈ 𝐵 ((𝐴 ·o 𝑥) +o 𝐴) ∪ ∅) = ∪ 𝑥 ∈ 𝐵 ((𝐴 ·o 𝑥) +o 𝐴)
27	25, 26	eqtri 2210	. 2 ⊢ (∅ ∪ ∪ 𝑥 ∈ 𝐵 ((𝐴 ·o 𝑥) +o 𝐴)) = ∪ 𝑥 ∈ 𝐵 ((𝐴 ·o 𝑥) +o 𝐴)
28	24, 27	eqtrdi 2238	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o 𝐵) = ∪ 𝑥 ∈ 𝐵 ((𝐴 ·o 𝑥) +o 𝐴))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2160  Vcvv 2752   ∪ cun 3142  ∅c0 3437  ∪ ciun 3904   ↦ cmpt 4082  Oncon0 4384   Fn wfn 5233  ‘cfv 5238  (class class class)co 5900  reccrdg 6398   +_o coa 6442   ·_o comu 6443

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4136  ax-sep 4139  ax-nul 4147  ax-pow 4195  ax-pr 4230  ax-un 4454  ax-setind 4557

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3595  df-sn 3616  df-pr 3617  df-op 3619  df-uni 3828  df-iun 3906  df-br 4022  df-opab 4083  df-mpt 4084  df-tr 4120  df-id 4314  df-iord 4387  df-on 4389  df-suc 4392  df-xp 4653  df-rel 4654  df-cnv 4655  df-co 4656  df-dm 4657  df-rn 4658  df-res 4659  df-ima 4660  df-iota 5199  df-fun 5240  df-fn 5241  df-f 5242  df-f1 5243  df-fo 5244  df-f1o 5245  df-fv 5246  df-ov 5903  df-oprab 5904  df-mpo 5905  df-1st 6169  df-2nd 6170  df-recs 6334  df-irdg 6399  df-oadd 6449  df-omul 6450

This theorem is referenced by:  omsuc  6501