Theorem omv 3120

Description: Value of ordinal multiplication.

Assertion

Ref	Expression
omv	⊢ ((A ∈ On ∧ B ∈ On) → (A ·o B) = (rec({⟨x, y⟩∣y = (x +o A)}, ∅) ‘B))

Distinct variable group(s):   x,y,A

Proof of Theorem omv

Step	Hyp	Ref	Expression
1		fvex 2838	. 2 ⊢ (rec({⟨x, y⟩∣y = (x +o A)}, ∅) ‘B) ∈ V
2		opreq2 3007	. . . . 5 ⊢ (w = A → (x +o w) = (x +o A))
3	2	cleq2d 1112	. . . 4 ⊢ (w = A → (y = (x +o w) ↔ y = (x +o A)))
4	3	biopabdv 2102	. . 3 ⊢ (w = A → {⟨x, y⟩∣y = (x +o w)} = {⟨x, y⟩∣y = (x +o A)})
5		rdgeq1 2972	. . 3 ⊢ ({⟨x, y⟩∣y = (x +o w)} = {⟨x, y⟩∣y = (x +o A)} → rec({⟨x, y⟩∣y = (x +o w)}, ∅) = rec({⟨x, y⟩∣y = (x +o A)}, ∅))
6		fveq1 2831	. . 3 ⊢ (rec({⟨x, y⟩∣y = (x +o w)}, ∅) = rec({⟨x, y⟩∣y = (x +o A)}, ∅) → (rec({⟨x, y⟩∣y = (x +o w)}, ∅) ‘v) = (rec({⟨x, y⟩∣y = (x +o A)}, ∅) ‘v))
7	4, 5, 6	3syl 21	. 2 ⊢ (w = A → (rec({⟨x, y⟩∣y = (x +o w)}, ∅) ‘v) = (rec({⟨x, y⟩∣y = (x +o A)}, ∅) ‘v))
8		fveq2 2832	. 2 ⊢ (v = B → (rec({⟨x, y⟩∣y = (x +o A)}, ∅) ‘v) = (rec({⟨x, y⟩∣y = (x +o A)}, ∅) ‘B))
9		df-omul 3107	. 2 ⊢ ·o = {⟨⟨w, v⟩, z⟩∣((w ∈ On ∧ v ∈ On) ∧ z = (rec({⟨x, y⟩∣y = (x +o w)}, ∅) ‘v))}
10	1, 7, 8, 9	oprabval2 3051	1 ⊢ ((A ∈ On ∧ B ∈ On) → (A ·o B) = (rec({⟨x, y⟩∣y = (x +o A)}, ∅) ‘B))

Syntax hints:   → wi 2   ∧ wa 196   = wceq 1091   ∈ wcel 1092  ∅c0 1707  {copab 2055  Oncon0 2199   ‘cfv 2422  reccrdg 2969  (class class class)co 3001   +_o coa 3101   ·_o comu 3102

This theorem is referenced by:  om0 3125  om0x 3126  omsuc 3133  omlim 3136

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fv 2438  df-rdg 2970  df-opr 3003  df-oprab 3004  df-omul 3107

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