Theorem oesuc 3134

Description: Ordinal exponentiation with a successor exponent. Definition 8.30 of [TakeutiZaring] p. 67.

Assertion

Ref	Expression
oesuc	⊢ ((A ∈ On ∧ B ∈ On) → (A ↑o suc B) = ((A ↑o B) ·o A))

Proof of Theorem oesuc

Step	Hyp	Ref	Expression
1		opreq1 3006	. . . 4 ⊢ (A = ∅ → (A ↑o suc B) = (∅ ↑o suc B))
2		eloni 2209	. . . . . 6 ⊢ (B ∈ On → Ord B)
3		0elsuc 2340	. . . . . 6 ⊢ (Ord B → ∅ ∈ suc B)
4	2, 3	syl 12	. . . . 5 ⊢ (B ∈ On → ∅ ∈ suc B)
5		oe0m1 3129	. . . . . 6 ⊢ ((suc B ∈ On ∧ ∅ ∈ suc B) → (∅ ↑o suc B) = ∅)
6		suceloni 2314	. . . . . 6 ⊢ (B ∈ On → suc B ∈ On)
7	5, 6	sylan 343	. . . . 5 ⊢ ((B ∈ On ∧ ∅ ∈ suc B) → (∅ ↑o suc B) = ∅)
8	4, 7	mpdan 527	. . . 4 ⊢ (B ∈ On → (∅ ↑o suc B) = ∅)
9	1, 8	sylan9eqr 1145	. . 3 ⊢ ((B ∈ On ∧ A = ∅) → (A ↑o suc B) = ∅)
10		opreq1 3006	. . . . 5 ⊢ (A = ∅ → (A ↑o B) = (∅ ↑o B))
11		id 9	. . . . 5 ⊢ (A = ∅ → A = ∅)
12	10, 11	opreq12d 3014	. . . 4 ⊢ (A = ∅ → ((A ↑o B) ·o A) = ((∅ ↑o B) ·o ∅))
13		oe0m0 3128	. . . . . . . . . 10 ⊢ (∅ ↑o ∅) = 1o
14		1o 3109	. . . . . . . . . 10 ⊢ 1o ∈ On
15	13, 14	eqeltr 1159	. . . . . . . . 9 ⊢ (∅ ↑o ∅) ∈ On
16		opreq2 3007	. . . . . . . . . 10 ⊢ (B = ∅ → (∅ ↑o B) = (∅ ↑o ∅))
17	16	eleq1d 1155	. . . . . . . . 9 ⊢ (B = ∅ → ((∅ ↑o B) ∈ On ↔ (∅ ↑o ∅) ∈ On))
18	15, 17	mpbiri 169	. . . . . . . 8 ⊢ (B = ∅ → (∅ ↑o B) ∈ On)
19	18	adantl 305	. . . . . . 7 ⊢ ((B ∈ On ∧ B = ∅) → (∅ ↑o B) ∈ On)
20		0elon 2277	. . . . . . . . 9 ⊢ ∅ ∈ On
21		oe0m1 3129	. . . . . . . . . 10 ⊢ ((B ∈ On ∧ ∅ ∈ B) → (∅ ↑o B) = ∅)
22	21	eleq1d 1155	. . . . . . . . 9 ⊢ ((B ∈ On ∧ ∅ ∈ B) → ((∅ ↑o B) ∈ On ↔ ∅ ∈ On))
23	20, 22	mpbiri 169	. . . . . . . 8 ⊢ ((B ∈ On ∧ ∅ ∈ B) → (∅ ↑o B) ∈ On)
24	23	adantll 309	. . . . . . 7 ⊢ (((B ∈ On ∧ B ∈ On) ∧ ∅ ∈ B) → (∅ ↑o B) ∈ On)
25	19, 24	oe0lem 3121	. . . . . 6 ⊢ ((B ∈ On ∧ B ∈ On) → (∅ ↑o B) ∈ On)
26	25	anidms 332	. . . . 5 ⊢ (B ∈ On → (∅ ↑o B) ∈ On)
27		om0 3125	. . . . 5 ⊢ ((∅ ↑o B) ∈ On → ((∅ ↑o B) ·o ∅) = ∅)
28	26, 27	syl 12	. . . 4 ⊢ (B ∈ On → ((∅ ↑o B) ·o ∅) = ∅)
29	12, 28	sylan9eqr 1145	. . 3 ⊢ ((B ∈ On ∧ A = ∅) → ((A ↑o B) ·o A) = ∅)
30	9, 29	eqtr4d 1131	. 2 ⊢ ((B ∈ On ∧ A = ∅) → (A ↑o suc B) = ((A ↑o B) ·o A))
31		rdgsuct 2983	. . . 4 ⊢ (B ∈ On → (rec({⟨x, y⟩∣y = (x ·o A)}, 1o) ‘suc B) = ({⟨x, y⟩∣y = (x ·o A)} ‘(rec({⟨x, y⟩∣y = (x ·o A)}, 1o) ‘B)))
32	31	ad2antlr 321	. . 3 ⊢ (((A ∈ On ∧ B ∈ On) ∧ ∅ ∈ A) → (rec({⟨x, y⟩∣y = (x ·o A)}, 1o) ‘suc B) = ({⟨x, y⟩∣y = (x ·o A)} ‘(rec({⟨x, y⟩∣y = (x ·o A)}, 1o) ‘B)))
33		oevn0 3123	. . . 4 ⊢ (((A ∈ On ∧ suc B ∈ On) ∧ ∅ ∈ A) → (A ↑o suc B) = (rec({⟨x, y⟩∣y = (x ·o A)}, 1o) ‘suc B))
34	33, 6	sylan12 355	. . 3 ⊢ (((A ∈ On ∧ B ∈ On) ∧ ∅ ∈ A) → (A ↑o suc B) = (rec({⟨x, y⟩∣y = (x ·o A)}, 1o) ‘suc B))
35		oevn0 3123	. . . . 5 ⊢ (((A ∈ On ∧ B ∈ On) ∧ ∅ ∈ A) → (A ↑o B) = (rec({⟨x, y⟩∣y = (x ·o A)}, 1o) ‘B))
36	35	fveq2d 2836	. . . 4 ⊢ (((A ∈ On ∧ B ∈ On) ∧ ∅ ∈ A) → ({⟨x, y⟩∣y = (x ·o A)} ‘(A ↑o B)) = ({⟨x, y⟩∣y = (x ·o A)} ‘(rec({⟨x, y⟩∣y = (x ·o A)}, 1o) ‘B)))
37		oprex 3018	. . . . 5 ⊢ (A ↑o B) ∈ V
38		oprex 3018	. . . . 5 ⊢ ((A ↑o B) ·o A) ∈ V
39		opreq1 3006	. . . . 5 ⊢ (x = (A ↑o B) → (x ·o A) = ((A ↑o B) ·o A))
40	37, 38, 39	fvopab 2877	. . . 4 ⊢ ({⟨x, y⟩∣y = (x ·o A)} ‘(A ↑o B)) = ((A ↑o B) ·o A)
41	36, 40	syl5eqr 1138	. . 3 ⊢ (((A ∈ On ∧ B ∈ On) ∧ ∅ ∈ A) → ((A ↑o B) ·o A) = ({⟨x, y⟩∣y = (x ·o A)} ‘(rec({⟨x, y⟩∣y = (x ·o A)}, 1o) ‘B)))
42	32, 34, 41	3eqtr4d 1134	. 2 ⊢ (((A ∈ On ∧ B ∈ On) ∧ ∅ ∈ A) → (A ↑o suc B) = ((A ↑o B) ·o A))
43	30, 42	oe0lem 3121	1 ⊢ ((A ∈ On ∧ B ∈ On) → (A ↑o suc B) = ((A ↑o B) ·o A))

Syntax hints:   → wi 2   ∧ wa 196   = wceq 1091   ∈ wcel 1092  ∅c0 1707  {copab 2055  Ord word 2198  Oncon0 2199  suc csuc 2201   ‘cfv 2422  reccrdg 2969  (class class class)co 3001  1_oc1o 3099   ·_o comu 3102   ↑_o coe 3103

This theorem is referenced by:  oecl 3140  oe1 3146  oe1m 3147  oen0 3165

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-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  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-fn 2433  df-fv 2438  df-rdg 2970  df-opr 3003  df-oprab 3004  df-1o 3104  df-omul 3107  df-oexp 3108

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