Theorem oev2 8487

Description: Alternate value of ordinal exponentiation. Compare oev 8478. (Contributed by NM, 2-Jan-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)

Assertion

Ref	Expression
oev2	⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑o 𝐵) = ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∩ ((V ∖ ∩ 𝐴) ∪ ∩ 𝐵)))

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem oev2

Step	Hyp	Ref	Expression
1		oveq12 7396	. . . . . 6 ⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) → (𝐴 ↑o 𝐵) = (∅ ↑o ∅))
2		oe0m0 8484	. . . . . 6 ⊢ (∅ ↑o ∅) = 1o
3	1, 2	eqtrdi 2780	. . . . 5 ⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) → (𝐴 ↑o 𝐵) = 1o)
4		fveq2 6858	. . . . . . . 8 ⊢ (𝐵 = ∅ → (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘∅))
5		1oex 8444	. . . . . . . . 9 ⊢ 1o ∈ V
6	5	rdg0 8389	. . . . . . . 8 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘∅) = 1o
7	4, 6	eqtrdi 2780	. . . . . . 7 ⊢ (𝐵 = ∅ → (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) = 1o)
8		inteq 4913	. . . . . . . 8 ⊢ (𝐵 = ∅ → ∩ 𝐵 = ∩ ∅)
9		int0 4926	. . . . . . . 8 ⊢ ∩ ∅ = V
10	8, 9	eqtrdi 2780	. . . . . . 7 ⊢ (𝐵 = ∅ → ∩ 𝐵 = V)
11	7, 10	ineq12d 4184	. . . . . 6 ⊢ (𝐵 = ∅ → ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∩ ∩ 𝐵) = (1o ∩ V))
12		inv1 4361	. . . . . . 7 ⊢ (1o ∩ V) = 1o
13	12	a1i 11	. . . . . 6 ⊢ (𝐴 = ∅ → (1o ∩ V) = 1o)
14	11, 13	sylan9eqr 2786	. . . . 5 ⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) → ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∩ ∩ 𝐵) = 1o)
15	3, 14	eqtr4d 2767	. . . 4 ⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) → (𝐴 ↑o 𝐵) = ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∩ ∩ 𝐵))
16		oveq1 7394	. . . . . . 7 ⊢ (𝐴 = ∅ → (𝐴 ↑o 𝐵) = (∅ ↑o 𝐵))
17		oe0m1 8485	. . . . . . . 8 ⊢ (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ (∅ ↑o 𝐵) = ∅))
18	17	biimpa 476	. . . . . . 7 ⊢ ((𝐵 ∈ On ∧ ∅ ∈ 𝐵) → (∅ ↑o 𝐵) = ∅)
19	16, 18	sylan9eqr 2786	. . . . . 6 ⊢ (((𝐵 ∈ On ∧ ∅ ∈ 𝐵) ∧ 𝐴 = ∅) → (𝐴 ↑o 𝐵) = ∅)
20	19	an32s 652	. . . . 5 ⊢ (((𝐵 ∈ On ∧ 𝐴 = ∅) ∧ ∅ ∈ 𝐵) → (𝐴 ↑o 𝐵) = ∅)
21		int0el 4943	. . . . . . . 8 ⊢ (∅ ∈ 𝐵 → ∩ 𝐵 = ∅)
22	21	ineq2d 4183	. . . . . . 7 ⊢ (∅ ∈ 𝐵 → ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∩ ∩ 𝐵) = ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∩ ∅))
23		in0 4358	. . . . . . 7 ⊢ ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∩ ∅) = ∅
24	22, 23	eqtrdi 2780	. . . . . 6 ⊢ (∅ ∈ 𝐵 → ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∩ ∩ 𝐵) = ∅)
25	24	adantl 481	. . . . 5 ⊢ (((𝐵 ∈ On ∧ 𝐴 = ∅) ∧ ∅ ∈ 𝐵) → ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∩ ∩ 𝐵) = ∅)
26	20, 25	eqtr4d 2767	. . . 4 ⊢ (((𝐵 ∈ On ∧ 𝐴 = ∅) ∧ ∅ ∈ 𝐵) → (𝐴 ↑o 𝐵) = ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∩ ∩ 𝐵))
27	15, 26	oe0lem 8477	. . 3 ⊢ ((𝐵 ∈ On ∧ 𝐴 = ∅) → (𝐴 ↑o 𝐵) = ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∩ ∩ 𝐵))
28		inteq 4913	. . . . . . . . . 10 ⊢ (𝐴 = ∅ → ∩ 𝐴 = ∩ ∅)
29	28, 9	eqtrdi 2780	. . . . . . . . 9 ⊢ (𝐴 = ∅ → ∩ 𝐴 = V)
30	29	difeq2d 4089	. . . . . . . 8 ⊢ (𝐴 = ∅ → (V ∖ ∩ 𝐴) = (V ∖ V))
31		difid 4339	. . . . . . . 8 ⊢ (V ∖ V) = ∅
32	30, 31	eqtrdi 2780	. . . . . . 7 ⊢ (𝐴 = ∅ → (V ∖ ∩ 𝐴) = ∅)
33	32	uneq2d 4131	. . . . . 6 ⊢ (𝐴 = ∅ → (∩ 𝐵 ∪ (V ∖ ∩ 𝐴)) = (∩ 𝐵 ∪ ∅))
34		uncom 4121	. . . . . 6 ⊢ (∩ 𝐵 ∪ (V ∖ ∩ 𝐴)) = ((V ∖ ∩ 𝐴) ∪ ∩ 𝐵)
35		un0 4357	. . . . . 6 ⊢ (∩ 𝐵 ∪ ∅) = ∩ 𝐵
36	33, 34, 35	3eqtr3g 2787	. . . . 5 ⊢ (𝐴 = ∅ → ((V ∖ ∩ 𝐴) ∪ ∩ 𝐵) = ∩ 𝐵)
37	36	adantl 481	. . . 4 ⊢ ((𝐵 ∈ On ∧ 𝐴 = ∅) → ((V ∖ ∩ 𝐴) ∪ ∩ 𝐵) = ∩ 𝐵)
38	37	ineq2d 4183	. . 3 ⊢ ((𝐵 ∈ On ∧ 𝐴 = ∅) → ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∩ ((V ∖ ∩ 𝐴) ∪ ∩ 𝐵)) = ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∩ ∩ 𝐵))
39	27, 38	eqtr4d 2767	. 2 ⊢ ((𝐵 ∈ On ∧ 𝐴 = ∅) → (𝐴 ↑o 𝐵) = ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∩ ((V ∖ ∩ 𝐴) ∪ ∩ 𝐵)))
40		oevn0 8479	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐴 ↑o 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵))
41		int0el 4943	. . . . . . . . . 10 ⊢ (∅ ∈ 𝐴 → ∩ 𝐴 = ∅)
42	41	difeq2d 4089	. . . . . . . . 9 ⊢ (∅ ∈ 𝐴 → (V ∖ ∩ 𝐴) = (V ∖ ∅))
43		dif0 4341	. . . . . . . . 9 ⊢ (V ∖ ∅) = V
44	42, 43	eqtrdi 2780	. . . . . . . 8 ⊢ (∅ ∈ 𝐴 → (V ∖ ∩ 𝐴) = V)
45	44	uneq2d 4131	. . . . . . 7 ⊢ (∅ ∈ 𝐴 → (∩ 𝐵 ∪ (V ∖ ∩ 𝐴)) = (∩ 𝐵 ∪ V))
46		unv 4362	. . . . . . 7 ⊢ (∩ 𝐵 ∪ V) = V
47	45, 34, 46	3eqtr3g 2787	. . . . . 6 ⊢ (∅ ∈ 𝐴 → ((V ∖ ∩ 𝐴) ∪ ∩ 𝐵) = V)
48	47	adantl 481	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → ((V ∖ ∩ 𝐴) ∪ ∩ 𝐵) = V)
49	48	ineq2d 4183	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∩ ((V ∖ ∩ 𝐴) ∪ ∩ 𝐵)) = ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∩ V))
50		inv1 4361	. . . 4 ⊢ ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∩ V) = (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵)
51	49, 50	eqtr2di 2781	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) = ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∩ ((V ∖ ∩ 𝐴) ∪ ∩ 𝐵)))
52	40, 51	eqtrd 2764	. 2 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐴 ↑o 𝐵) = ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∩ ((V ∖ ∩ 𝐴) ∪ ∩ 𝐵)))
53	39, 52	oe0lem 8477	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑o 𝐵) = ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∩ ((V ∖ ∩ 𝐴) ∪ ∩ 𝐵)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3447   ∖ cdif 3911   ∪ cun 3912   ∩ cin 3913  ∅c0 4296  ∩ cint 4910   ↦ cmpt 5188  Oncon0 6332  ‘cfv 6511  (class class class)co 7387  reccrdg 8377  1_oc1o 8427   ·_o comu 8432   ↑_o coe 8433

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-oexp 8440

This theorem is referenced by: (None)