Theorem oeiv 6729

Description: Value of ordinal exponentiation. (Contributed by Jim Kingdon, 9-Jul-2019.)

Assertion

Ref	Expression
oeiv	⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑o 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵))

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem oeiv

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

Step	Hyp	Ref	Expression
1		1on 6694	. . 3 ⊢ 1o ∈ On
2		vex 2824	. . . . . . 7 ⊢ 𝑥 ∈ V
3		omexg 6724	. . . . . . 7 ⊢ ((𝑥 ∈ V ∧ 𝐴 ∈ On) → (𝑥 ·o 𝐴) ∈ V)
4	2, 3	mpan 428	. . . . . 6 ⊢ (𝐴 ∈ On → (𝑥 ·o 𝐴) ∈ V)
5	4	ralrimivw 2624	. . . . 5 ⊢ (𝐴 ∈ On → ∀𝑥 ∈ V (𝑥 ·o 𝐴) ∈ V)
6		eqid 2238	. . . . . 6 ⊢ (𝑥 ∈ V ↦ (𝑥 ·o 𝐴)) = (𝑥 ∈ V ↦ (𝑥 ·o 𝐴))
7	6	fnmpt 5510	. . . . 5 ⊢ (∀𝑥 ∈ V (𝑥 ·o 𝐴) ∈ V → (𝑥 ∈ V ↦ (𝑥 ·o 𝐴)) Fn V)
8	5, 7	syl 14	. . . 4 ⊢ (𝐴 ∈ On → (𝑥 ∈ V ↦ (𝑥 ·o 𝐴)) Fn V)
9		rdgexggg 6648	. . . 4 ⊢ (((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)) Fn V ∧ 1o ∈ On ∧ 𝐵 ∈ On) → (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∈ V)
10	8, 9	syl3an1 1311	. . 3 ⊢ ((𝐴 ∈ On ∧ 1o ∈ On ∧ 𝐵 ∈ On) → (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∈ V)
11	1, 10	mp3an2 1366	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∈ V)
12		oveq2 6093	. . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑥 ·o 𝑦) = (𝑥 ·o 𝐴))
13	12	mpteq2dv 4222	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝑥 ∈ V ↦ (𝑥 ·o 𝑦)) = (𝑥 ∈ V ↦ (𝑥 ·o 𝐴)))
14		rdgeq1 6642	. . . . 5 ⊢ ((𝑥 ∈ V ↦ (𝑥 ·o 𝑦)) = (𝑥 ∈ V ↦ (𝑥 ·o 𝐴)) → rec((𝑥 ∈ V ↦ (𝑥 ·o 𝑦)), 1o) = rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o))
15	13, 14	syl 14	. . . 4 ⊢ (𝑦 = 𝐴 → rec((𝑥 ∈ V ↦ (𝑥 ·o 𝑦)), 1o) = rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o))
16	15	fveq1d 5697	. . 3 ⊢ (𝑦 = 𝐴 → (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝑦)), 1o)‘𝑧) = (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝑧))
17		fveq2 5695	. . 3 ⊢ (𝑧 = 𝐵 → (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝑧) = (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵))
18		df-oexpi 6693	. . 3 ⊢ ↑o = (𝑦 ∈ On, 𝑧 ∈ On ↦ (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝑦)), 1o)‘𝑧))
19	16, 17, 18	ovmpog 6223	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∈ V) → (𝐴 ↑o 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵))
20	11, 19	mpd3an3 1379	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑o 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   = wceq 1402   ∈ wcel 2209  ∀wral 2528  Vcvv 2821   ↦ cmpt 4192  Oncon0 4508   Fn wfn 5372  ‘cfv 5377  (class class class)co 6085  reccrdg 6640  1_oc1o 6680   ·_o comu 6685   ↑_o coei 6686

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-iord 4511  df-on 4513  df-suc 4516  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-irdg 6641  df-1o 6687  df-oadd 6691  df-omul 6692  df-oexpi 6693

This theorem is used by:  oei0  6732  oeicl  6735