Theorem oev2 8514

Description: Alternate value of ordinal exponentiation. Compare oev 8505. (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 7428	. . . . . 6 ⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) → (𝐴 ↑o 𝐵) = (∅ ↑o ∅))
2		oe0m0 8511	. . . . . 6 ⊢ (∅ ↑o ∅) = 1o
3	1, 2	eqtrdi 2816	. . . . 5 ⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) → (𝐴 ↑o 𝐵) = 1o)
4		fveq2 6885	. . . . . . . 8 ⊢ (𝐵 = ∅ → (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘∅))
5		1oex 8469	. . . . . . . . 9 ⊢ 1o ∈ V
6	5	rdg0 8414	. . . . . . . 8 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘∅) = 1o
7	4, 6	eqtrdi 2816	. . . . . . 7 ⊢ (𝐵 = ∅ → (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) = 1o)
8		inteq 4917	. . . . . . . 8 ⊢ (𝐵 = ∅ → ∩ 𝐵 = ∩ ∅)
9		int0 4929	. . . . . . . 8 ⊢ ∩ ∅ = V
10	8, 9	eqtrdi 2816	. . . . . . 7 ⊢ (𝐵 = ∅ → ∩ 𝐵 = V)
11	7, 10	ineq12d 4174	. . . . . 6 ⊢ (𝐵 = ∅ → ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∩ ∩ 𝐵) = (1o ∩ V))
12		inv1 4355	. . . . . . 7 ⊢ (1o ∩ V) = 1o
13	12	a1i 11	. . . . . 6 ⊢ (𝐴 = ∅ → (1o ∩ V) = 1o)
14	11, 13	sylan9eqr 2822	. . . . 5 ⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) → ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∩ ∩ 𝐵) = 1o)
15	3, 14	eqtr4d 2803	. . . 4 ⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) → (𝐴 ↑o 𝐵) = ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∩ ∩ 𝐵))
16		oveq1 7426	. . . . . . 7 ⊢ (𝐴 = ∅ → (𝐴 ↑o 𝐵) = (∅ ↑o 𝐵))
17		oe0m1 8512	. . . . . . . 8 ⊢ (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ (∅ ↑o 𝐵) = ∅))
18	17	biimpa 482	. . . . . . 7 ⊢ ((𝐵 ∈ On ∧ ∅ ∈ 𝐵) → (∅ ↑o 𝐵) = ∅)
19	16, 18	sylan9eqr 2822	. . . . . 6 ⊢ (((𝐵 ∈ On ∧ ∅ ∈ 𝐵) ∧ 𝐴 = ∅) → (𝐴 ↑o 𝐵) = ∅)
20	19	an32s 665	. . . . 5 ⊢ (((𝐵 ∈ On ∧ 𝐴 = ∅) ∧ ∅ ∈ 𝐵) → (𝐴 ↑o 𝐵) = ∅)
21		int0el 4946	. . . . . . . 8 ⊢ (∅ ∈ 𝐵 → ∩ 𝐵 = ∅)
22	21	ineq2d 4173	. . . . . . 7 ⊢ (∅ ∈ 𝐵 → ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∩ ∩ 𝐵) = ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∩ ∅))
23		in0 4352	. . . . . . 7 ⊢ ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∩ ∅) = ∅
24	22, 23	eqtrdi 2816	. . . . . 6 ⊢ (∅ ∈ 𝐵 → ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∩ ∩ 𝐵) = ∅)
25	24	adantl 487	. . . . 5 ⊢ (((𝐵 ∈ On ∧ 𝐴 = ∅) ∧ ∅ ∈ 𝐵) → ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∩ ∩ 𝐵) = ∅)
26	20, 25	eqtr4d 2803	. . . 4 ⊢ (((𝐵 ∈ On ∧ 𝐴 = ∅) ∧ ∅ ∈ 𝐵) → (𝐴 ↑o 𝐵) = ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∩ ∩ 𝐵))
27	15, 26	oe0lem 8504	. . 3 ⊢ ((𝐵 ∈ On ∧ 𝐴 = ∅) → (𝐴 ↑o 𝐵) = ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∩ ∩ 𝐵))
28		inteq 4917	. . . . . . . . . 10 ⊢ (𝐴 = ∅ → ∩ 𝐴 = ∩ ∅)
29	28, 9	eqtrdi 2816	. . . . . . . . 9 ⊢ (𝐴 = ∅ → ∩ 𝐴 = V)
30	29	difeq2d 4081	. . . . . . . 8 ⊢ (𝐴 = ∅ → (V ∖ ∩ 𝐴) = (V ∖ V))
31		difid 4332	. . . . . . . 8 ⊢ (V ∖ V) = ∅
32	30, 31	eqtrdi 2816	. . . . . . 7 ⊢ (𝐴 = ∅ → (V ∖ ∩ 𝐴) = ∅)
33	32	uneq2d 4122	. . . . . 6 ⊢ (𝐴 = ∅ → (∩ 𝐵 ∪ (V ∖ ∩ 𝐴)) = (∩ 𝐵 ∪ ∅))
34		uncom 4112	. . . . . 6 ⊢ (∩ 𝐵 ∪ (V ∖ ∩ 𝐴)) = ((V ∖ ∩ 𝐴) ∪ ∩ 𝐵)
35		un0 4351	. . . . . 6 ⊢ (∩ 𝐵 ∪ ∅) = ∩ 𝐵
36	33, 34, 35	3eqtr3g 2823	. . . . 5 ⊢ (𝐴 = ∅ → ((V ∖ ∩ 𝐴) ∪ ∩ 𝐵) = ∩ 𝐵)
37	36	adantl 487	. . . 4 ⊢ ((𝐵 ∈ On ∧ 𝐴 = ∅) → ((V ∖ ∩ 𝐴) ∪ ∩ 𝐵) = ∩ 𝐵)
38	37	ineq2d 4173	. . 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 2803	. 2 ⊢ ((𝐵 ∈ On ∧ 𝐴 = ∅) → (𝐴 ↑o 𝐵) = ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∩ ((V ∖ ∩ 𝐴) ∪ ∩ 𝐵)))
40		oevn0 8506	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐴 ↑o 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵))
41		int0el 4946	. . . . . . . . . 10 ⊢ (∅ ∈ 𝐴 → ∩ 𝐴 = ∅)
42	41	difeq2d 4081	. . . . . . . . 9 ⊢ (∅ ∈ 𝐴 → (V ∖ ∩ 𝐴) = (V ∖ ∅))
43		dif0 4334	. . . . . . . . 9 ⊢ (V ∖ ∅) = V
44	42, 43	eqtrdi 2816	. . . . . . . 8 ⊢ (∅ ∈ 𝐴 → (V ∖ ∩ 𝐴) = V)
45	44	uneq2d 4122	. . . . . . 7 ⊢ (∅ ∈ 𝐴 → (∩ 𝐵 ∪ (V ∖ ∩ 𝐴)) = (∩ 𝐵 ∪ V))
46		unv 4356	. . . . . . 7 ⊢ (∩ 𝐵 ∪ V) = V
47	45, 34, 46	3eqtr3g 2823	. . . . . 6 ⊢ (∅ ∈ 𝐴 → ((V ∖ ∩ 𝐴) ∪ ∩ 𝐵) = V)
48	47	adantl 487	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → ((V ∖ ∩ 𝐴) ∪ ∩ 𝐵) = V)
49	48	ineq2d 4173	. . . 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 4355	. . . 4 ⊢ ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∩ V) = (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵)
51	49, 50	eqtr2di 2817	. . 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 2800	. 2 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐴 ↑o 𝐵) = ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∩ ((V ∖ ∩ 𝐴) ∪ ∩ 𝐵)))
53	39, 52	oe0lem 8504	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑o 𝐵) = ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∩ ((V ∖ ∩ 𝐴) ∪ ∩ 𝐵)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146  Vcvv 3457   ∖ cdif 3903   ∪ cun 3904   ∩ cin 3905  ∅c0 4286  ∩ cint 4914   ↦ cmpt 5194  Oncon0 6364  ‘cfv 6540  (class class class)co 7419  reccrdg 8402  1_oc1o 8452   ·_o comu 8457   ↑_o coe 8458

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-int 4915  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7869  df-2nd 7993  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-1o 8459  df-oexp 8465

This theorem is used by: (None)