Theorem omv 6609

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 4483	. . 3 ⊢ ∅ ∈ On
2		omfnex 6603	. . . 4 ⊢ (𝐴 ∈ On → (𝑥 ∈ V ↦ (𝑥 +o 𝐴)) Fn V)
3		rdgexggg 6529	. . . 4 ⊢ (((𝑥 ∈ V ↦ (𝑥 +o 𝐴)) Fn V ∧ ∅ ∈ On ∧ 𝐵 ∈ On) → (rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅)‘𝐵) ∈ V)
4	2, 3	syl3an1 1304	. . 3 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ On ∧ 𝐵 ∈ On) → (rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅)‘𝐵) ∈ V)
5	1, 4	mp3an2 1359	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅)‘𝐵) ∈ V)
6		oveq2 6015	. . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑥 +o 𝑦) = (𝑥 +o 𝐴))
7	6	mpteq2dv 4175	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝑥 ∈ V ↦ (𝑥 +o 𝑦)) = (𝑥 ∈ V ↦ (𝑥 +o 𝐴)))
8		rdgeq1 6523	. . . . 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 5631	. . 3 ⊢ (𝑦 = 𝐴 → (rec((𝑥 ∈ V ↦ (𝑥 +o 𝑦)), ∅)‘𝑧) = (rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅)‘𝑧))
11		fveq2 5629	. . 3 ⊢ (𝑧 = 𝐵 → (rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅)‘𝑧) = (rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅)‘𝐵))
12		df-omul 6573	. . 3 ⊢ ·o = (𝑦 ∈ On, 𝑧 ∈ On ↦ (rec((𝑥 ∈ V ↦ (𝑥 +o 𝑦)), ∅)‘𝑧))
13	10, 11, 12	ovmpog 6145	. 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 1372	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅)‘𝐵))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1395   ∈ wcel 2200  Vcvv 2799  ∅c0 3491   ↦ cmpt 4145  Oncon0 4454   Fn wfn 5313  ‘cfv 5318  (class class class)co 6007  reccrdg 6521   +_o coa 6565   ·_o comu 6566

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-iord 4457  df-on 4459  df-suc 4462  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-recs 6457  df-irdg 6522  df-oadd 6572  df-omul 6573

This theorem is referenced by:  om0  6612  omcl  6615  omv2  6619