Theorem oeiv 6036

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 6008	. . 3 ⊢ 1𝑜 ∈ On
2		vex 2560	. . . . . . 7 ⊢ 𝑥 ∈ V
3		omexg 6031	. . . . . . 7 ⊢ ((𝑥 ∈ V ∧ 𝐴 ∈ On) → (𝑥 ·𝑜 𝐴) ∈ V)
4	2, 3	mpan 400	. . . . . 6 ⊢ (𝐴 ∈ On → (𝑥 ·𝑜 𝐴) ∈ V)
5	4	ralrimivw 2393	. . . . 5 ⊢ (𝐴 ∈ On → ∀𝑥 ∈ V (𝑥 ·𝑜 𝐴) ∈ V)
6		eqid 2040	. . . . . 6 ⊢ (𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)) = (𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴))
7	6	fnmpt 5025	. . . . 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 5964	. . . 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 1168	. . 3 ⊢ ((𝐴 ∈ On ∧ 1𝑜 ∈ On ∧ 𝐵 ∈ On) → (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵) ∈ V)
11	1, 10	mp3an2 1220	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵) ∈ V)
12		oveq2 5520	. . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑥 ·𝑜 𝑦) = (𝑥 ·𝑜 𝐴))
13	12	mpteq2dv 3848	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝑥 ∈ V ↦ (𝑥 ·𝑜 𝑦)) = (𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)))
14		rdgeq1 5958	. . . . 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 5180	. . 3 ⊢ (𝑦 = 𝐴 → (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝑦)), 1𝑜)‘𝑧) = (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝑧))
17		fveq2 5178	. . 3 ⊢ (𝑧 = 𝐵 → (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝑧) = (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵))
18		df-oexpi 6007	. . 3 ⊢ ↑𝑜 = (𝑦 ∈ On, 𝑧 ∈ On ↦ (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝑦)), 1𝑜)‘𝑧))
19	16, 17, 18	ovmpt2g 5635	. 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 1233	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑𝑜 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵))

Syntax hints:   → wi 4   ∧ wa 97   = wceq 1243   ∈ wcel 1393  ∀wral 2306  Vcvv 2557   ↦ cmpt 3818  Oncon0 4100   Fn wfn 4897  ‘cfv 4902  (class class class)co 5512  reccrdg 5956  1_𝑜c1o 5994   ·_𝑜 comu 5999   ↑_𝑜 coei 6000

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-id 4030  df-iord 4103  df-on 4105  df-suc 4108  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-1o 6001  df-oadd 6005  df-omul 6006  df-oexpi 6007

This theorem is referenced by:  oei0  6039  oeicl  6042