Theorem omv2 6581

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 6565	. . . 4 ⊢ (𝐴 ∈ On → (𝑦 ∈ V ↦ (𝑦 +o 𝐴)) Fn V)
2		0elon 4460	. . . . 5 ⊢ ∅ ∈ On
3		rdgival 6498	. . . . 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 1340	. . . 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 6571	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o 𝐵) = (rec((𝑦 ∈ V ↦ (𝑦 +o 𝐴)), ∅)‘𝐵))
7		onelon 4452	. . . . . . 7 ⊢ ((𝐵 ∈ On ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ On)
8		omexg 6567	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 ·o 𝑥) ∈ V)
9		omcl 6577	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 ·o 𝑥) ∈ On)
10		simpl 109	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → 𝐴 ∈ On)
11		oacl 6576	. . . . . . . . . 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 5981	. . . . . . . . . 10 ⊢ (𝑦 = (𝐴 ·o 𝑥) → (𝑦 +o 𝐴) = ((𝐴 ·o 𝑥) +o 𝐴))
14		eqid 2209	. . . . . . . . . 10 ⊢ (𝑦 ∈ V ↦ (𝑦 +o 𝐴)) = (𝑦 ∈ V ↦ (𝑦 +o 𝐴))
15	13, 14	fvmptg 5683	. . . . . . . . 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 6571	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 ·o 𝑥) = (rec((𝑦 ∈ V ↦ (𝑦 +o 𝐴)), ∅)‘𝑥))
18	17	fveq2d 5607	. . . . . . . 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 2244	. . . . . . 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 3965	. . . 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 3338	. . 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 2252	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o 𝐵) = (∅ ∪ ∪ 𝑥 ∈ 𝐵 ((𝐴 ·o 𝑥) +o 𝐴)))
25		uncom 3328	. . 3 ⊢ (∅ ∪ ∪ 𝑥 ∈ 𝐵 ((𝐴 ·o 𝑥) +o 𝐴)) = (∪ 𝑥 ∈ 𝐵 ((𝐴 ·o 𝑥) +o 𝐴) ∪ ∅)
26		un0 3505	. . 3 ⊢ (∪ 𝑥 ∈ 𝐵 ((𝐴 ·o 𝑥) +o 𝐴) ∪ ∅) = ∪ 𝑥 ∈ 𝐵 ((𝐴 ·o 𝑥) +o 𝐴)
27	25, 26	eqtri 2230	. 2 ⊢ (∅ ∪ ∪ 𝑥 ∈ 𝐵 ((𝐴 ·o 𝑥) +o 𝐴)) = ∪ 𝑥 ∈ 𝐵 ((𝐴 ·o 𝑥) +o 𝐴)
28	24, 27	eqtrdi 2258	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o 𝐵) = ∪ 𝑥 ∈ 𝐵 ((𝐴 ·o 𝑥) +o 𝐴))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1375   ∈ wcel 2180  Vcvv 2779   ∪ cun 3175  ∅c0 3471  ∪ ciun 3944   ↦ cmpt 4124  Oncon0 4431   Fn wfn 5289  ‘cfv 5294  (class class class)co 5974  reccrdg 6485   +_o coa 6529   ·_o comu 6530

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 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-13 2182  ax-14 2183  ax-ext 2191  ax-coll 4178  ax-sep 4181  ax-nul 4189  ax-pow 4237  ax-pr 4272  ax-un 4501  ax-setind 4606

This theorem depends on definitions:  df-bi 117  df-3an 985  df-tru 1378  df-fal 1381  df-nf 1487  df-sb 1789  df-eu 2060  df-mo 2061  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ne 2381  df-ral 2493  df-rex 2494  df-reu 2495  df-rab 2497  df-v 2781  df-sbc 3009  df-csb 3105  df-dif 3179  df-un 3181  df-in 3183  df-ss 3190  df-nul 3472  df-pw 3631  df-sn 3652  df-pr 3653  df-op 3655  df-uni 3868  df-iun 3946  df-br 4063  df-opab 4125  df-mpt 4126  df-tr 4162  df-id 4361  df-iord 4434  df-on 4436  df-suc 4439  df-xp 4702  df-rel 4703  df-cnv 4704  df-co 4705  df-dm 4706  df-rn 4707  df-res 4708  df-ima 4709  df-iota 5254  df-fun 5296  df-fn 5297  df-f 5298  df-f1 5299  df-fo 5300  df-f1o 5301  df-fv 5302  df-ov 5977  df-oprab 5978  df-mpo 5979  df-1st 6256  df-2nd 6257  df-recs 6421  df-irdg 6486  df-oadd 6536  df-omul 6537

This theorem is referenced by:  omsuc  6588