Theorem omv 6344

Description: Value of ordinal multiplication. (Contributed by NM, 17-Sep-1995.) (Revised by Mario Carneiro, 23-Aug-2014.)

Assertion

Ref	Expression
omv	⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅)‘𝐵))

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem omv

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0elon 4309	. . 3 ⊢ ∅ ∈ On
2		omfnex 6338	. . . 4 ⊢ (𝐴 ∈ On → (𝑥 ∈ V ↦ (𝑥 +o 𝐴)) Fn V)
3		rdgexggg 6267	. . . 4 ⊢ (((𝑥 ∈ V ↦ (𝑥 +o 𝐴)) Fn V ∧ ∅ ∈ On ∧ 𝐵 ∈ On) → (rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅)‘𝐵) ∈ V)
4	2, 3	syl3an1 1249	. . 3 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ On ∧ 𝐵 ∈ On) → (rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅)‘𝐵) ∈ V)
5	1, 4	mp3an2 1303	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅)‘𝐵) ∈ V)
6		oveq2 5775	. . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑥 +o 𝑦) = (𝑥 +o 𝐴))
7	6	mpteq2dv 4014	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝑥 ∈ V ↦ (𝑥 +o 𝑦)) = (𝑥 ∈ V ↦ (𝑥 +o 𝐴)))
8		rdgeq1 6261	. . . . 5 ⊢ ((𝑥 ∈ V ↦ (𝑥 +o 𝑦)) = (𝑥 ∈ V ↦ (𝑥 +o 𝐴)) → rec((𝑥 ∈ V ↦ (𝑥 +o 𝑦)), ∅) = rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅))
9	7, 8	syl 14	. . . 4 ⊢ (𝑦 = 𝐴 → rec((𝑥 ∈ V ↦ (𝑥 +o 𝑦)), ∅) = rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅))
10	9	fveq1d 5416	. . 3 ⊢ (𝑦 = 𝐴 → (rec((𝑥 ∈ V ↦ (𝑥 +o 𝑦)), ∅)‘𝑧) = (rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅)‘𝑧))
11		fveq2 5414	. . 3 ⊢ (𝑧 = 𝐵 → (rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅)‘𝑧) = (rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅)‘𝐵))
12		df-omul 6311	. . 3 ⊢ ·o = (𝑦 ∈ On, 𝑧 ∈ On ↦ (rec((𝑥 ∈ V ↦ (𝑥 +o 𝑦)), ∅)‘𝑧))
13	10, 11, 12	ovmpog 5898	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ (rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅)‘𝐵) ∈ V) → (𝐴 ·o 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅)‘𝐵))
14	5, 13	mpd3an3 1316	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅)‘𝐵))

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1331   ∈ wcel 1480  Vcvv 2681  ∅c0 3358   ↦ cmpt 3984  Oncon0 4280   Fn wfn 5113  ‘cfv 5118  (class class class)co 5767  reccrdg 6259   +_o coa 6303   ·_o comu 6304

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-id 4210  df-iord 4283  df-on 4285  df-suc 4288  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-recs 6195  df-irdg 6260  df-oadd 6310  df-omul 6311

This theorem is referenced by:  om0  6347  omcl  6350  omv2  6354