Theorem oev2 7603

Description: Alternate value of ordinal exponentiation. Compare oev 7594. (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 6659	. . . . . 6 ⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) → (𝐴 ↑𝑜 𝐵) = (∅ ↑𝑜 ∅))
2		oe0m0 7600	. . . . . 6 ⊢ (∅ ↑𝑜 ∅) = 1𝑜
3	1, 2	syl6eq 2672	. . . . 5 ⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) → (𝐴 ↑𝑜 𝐵) = 1𝑜)
4		fveq2 6191	. . . . . . . 8 ⊢ (𝐵 = ∅ → (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘∅))
5		1on 7567	. . . . . . . . . 10 ⊢ 1𝑜 ∈ On
6	5	elexi 3213	. . . . . . . . 9 ⊢ 1𝑜 ∈ V
7	6	rdg0 7517	. . . . . . . 8 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘∅) = 1𝑜
8	4, 7	syl6eq 2672	. . . . . . 7 ⊢ (𝐵 = ∅ → (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵) = 1𝑜)
9		inteq 4478	. . . . . . . 8 ⊢ (𝐵 = ∅ → ∩ 𝐵 = ∩ ∅)
10		int0 4490	. . . . . . . 8 ⊢ ∩ ∅ = V
11	9, 10	syl6eq 2672	. . . . . . 7 ⊢ (𝐵 = ∅ → ∩ 𝐵 = V)
12	8, 11	ineq12d 3815	. . . . . 6 ⊢ (𝐵 = ∅ → ((rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵) ∩ ∩ 𝐵) = (1𝑜 ∩ V))
13		inv1 3970	. . . . . . 7 ⊢ (1𝑜 ∩ V) = 1𝑜
14	13	a1i 11	. . . . . 6 ⊢ (𝐴 = ∅ → (1𝑜 ∩ V) = 1𝑜)
15	12, 14	sylan9eqr 2678	. . . . 5 ⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) → ((rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵) ∩ ∩ 𝐵) = 1𝑜)
16	3, 15	eqtr4d 2659	. . . 4 ⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) → (𝐴 ↑𝑜 𝐵) = ((rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵) ∩ ∩ 𝐵))
17		oveq1 6657	. . . . . . 7 ⊢ (𝐴 = ∅ → (𝐴 ↑𝑜 𝐵) = (∅ ↑𝑜 𝐵))
18		oe0m1 7601	. . . . . . . 8 ⊢ (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ (∅ ↑𝑜 𝐵) = ∅))
19	18	biimpa 501	. . . . . . 7 ⊢ ((𝐵 ∈ On ∧ ∅ ∈ 𝐵) → (∅ ↑𝑜 𝐵) = ∅)
20	17, 19	sylan9eqr 2678	. . . . . 6 ⊢ (((𝐵 ∈ On ∧ ∅ ∈ 𝐵) ∧ 𝐴 = ∅) → (𝐴 ↑𝑜 𝐵) = ∅)
21	20	an32s 846	. . . . 5 ⊢ (((𝐵 ∈ On ∧ 𝐴 = ∅) ∧ ∅ ∈ 𝐵) → (𝐴 ↑𝑜 𝐵) = ∅)
22		int0el 4508	. . . . . . . 8 ⊢ (∅ ∈ 𝐵 → ∩ 𝐵 = ∅)
23	22	ineq2d 3814	. . . . . . 7 ⊢ (∅ ∈ 𝐵 → ((rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵) ∩ ∩ 𝐵) = ((rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵) ∩ ∅))
24		in0 3968	. . . . . . 7 ⊢ ((rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵) ∩ ∅) = ∅
25	23, 24	syl6eq 2672	. . . . . 6 ⊢ (∅ ∈ 𝐵 → ((rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵) ∩ ∩ 𝐵) = ∅)
26	25	adantl 482	. . . . 5 ⊢ (((𝐵 ∈ On ∧ 𝐴 = ∅) ∧ ∅ ∈ 𝐵) → ((rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵) ∩ ∩ 𝐵) = ∅)
27	21, 26	eqtr4d 2659	. . . 4 ⊢ (((𝐵 ∈ On ∧ 𝐴 = ∅) ∧ ∅ ∈ 𝐵) → (𝐴 ↑𝑜 𝐵) = ((rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵) ∩ ∩ 𝐵))
28	16, 27	oe0lem 7593	. . 3 ⊢ ((𝐵 ∈ On ∧ 𝐴 = ∅) → (𝐴 ↑𝑜 𝐵) = ((rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵) ∩ ∩ 𝐵))
29		inteq 4478	. . . . . . . . . 10 ⊢ (𝐴 = ∅ → ∩ 𝐴 = ∩ ∅)
30	29, 10	syl6eq 2672	. . . . . . . . 9 ⊢ (𝐴 = ∅ → ∩ 𝐴 = V)
31	30	difeq2d 3728	. . . . . . . 8 ⊢ (𝐴 = ∅ → (V ∖ ∩ 𝐴) = (V ∖ V))
32		difid 3948	. . . . . . . 8 ⊢ (V ∖ V) = ∅
33	31, 32	syl6eq 2672	. . . . . . 7 ⊢ (𝐴 = ∅ → (V ∖ ∩ 𝐴) = ∅)
34	33	uneq2d 3767	. . . . . 6 ⊢ (𝐴 = ∅ → (∩ 𝐵 ∪ (V ∖ ∩ 𝐴)) = (∩ 𝐵 ∪ ∅))
35		uncom 3757	. . . . . 6 ⊢ (∩ 𝐵 ∪ (V ∖ ∩ 𝐴)) = ((V ∖ ∩ 𝐴) ∪ ∩ 𝐵)
36		un0 3967	. . . . . 6 ⊢ (∩ 𝐵 ∪ ∅) = ∩ 𝐵
37	34, 35, 36	3eqtr3g 2679	. . . . 5 ⊢ (𝐴 = ∅ → ((V ∖ ∩ 𝐴) ∪ ∩ 𝐵) = ∩ 𝐵)
38	37	adantl 482	. . . 4 ⊢ ((𝐵 ∈ On ∧ 𝐴 = ∅) → ((V ∖ ∩ 𝐴) ∪ ∩ 𝐵) = ∩ 𝐵)
39	38	ineq2d 3814	. . 3 ⊢ ((𝐵 ∈ On ∧ 𝐴 = ∅) → ((rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵) ∩ ((V ∖ ∩ 𝐴) ∪ ∩ 𝐵)) = ((rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵) ∩ ∩ 𝐵))
40	28, 39	eqtr4d 2659	. 2 ⊢ ((𝐵 ∈ On ∧ 𝐴 = ∅) → (𝐴 ↑𝑜 𝐵) = ((rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵) ∩ ((V ∖ ∩ 𝐴) ∪ ∩ 𝐵)))
41		oevn0 7595	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐴 ↑𝑜 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵))
42		int0el 4508	. . . . . . . . . 10 ⊢ (∅ ∈ 𝐴 → ∩ 𝐴 = ∅)
43	42	difeq2d 3728	. . . . . . . . 9 ⊢ (∅ ∈ 𝐴 → (V ∖ ∩ 𝐴) = (V ∖ ∅))
44		dif0 3950	. . . . . . . . 9 ⊢ (V ∖ ∅) = V
45	43, 44	syl6eq 2672	. . . . . . . 8 ⊢ (∅ ∈ 𝐴 → (V ∖ ∩ 𝐴) = V)
46	45	uneq2d 3767	. . . . . . 7 ⊢ (∅ ∈ 𝐴 → (∩ 𝐵 ∪ (V ∖ ∩ 𝐴)) = (∩ 𝐵 ∪ V))
47		unv 3971	. . . . . . 7 ⊢ (∩ 𝐵 ∪ V) = V
48	46, 35, 47	3eqtr3g 2679	. . . . . 6 ⊢ (∅ ∈ 𝐴 → ((V ∖ ∩ 𝐴) ∪ ∩ 𝐵) = V)
49	48	adantl 482	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → ((V ∖ ∩ 𝐴) ∪ ∩ 𝐵) = V)
50	49	ineq2d 3814	. . . 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))
51		inv1 3970	. . . 4 ⊢ ((rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵) ∩ V) = (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵)
52	50, 51	syl6req 2673	. . 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)))
53	41, 52	eqtrd 2656	. 2 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐴 ↑𝑜 𝐵) = ((rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵) ∩ ((V ∖ ∩ 𝐴) ∪ ∩ 𝐵)))
54	40, 53	oe0lem 7593	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑𝑜 𝐵) = ((rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵) ∩ ((V ∖ ∩ 𝐴) ∪ ∩ 𝐵)))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  Vcvv 3200   ∖ cdif 3571   ∪ cun 3572   ∩ cin 3573  ∅c0 3915  ∩ cint 4475   ↦ cmpt 4729  Oncon0 5723  ‘cfv 5888  (class class class)co 6650  reccrdg 7505  1_𝑜c1o 7553   ·_𝑜 comu 7558   ↑_𝑜 coe 7559

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oexp 7566

This theorem is referenced by: (None)