Theorem oev2 7490

Description: Alternate value of ordinal exponentiation. Compare oev 7481. (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 6558	. . . . . 6 ⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) → (𝐴 ↑𝑜 𝐵) = (∅ ↑𝑜 ∅))
2		oe0m0 7487	. . . . . 6 ⊢ (∅ ↑𝑜 ∅) = 1𝑜
3	1, 2	syl6eq 2660	. . . . 5 ⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) → (𝐴 ↑𝑜 𝐵) = 1𝑜)
4		fveq2 6103	. . . . . . . 8 ⊢ (𝐵 = ∅ → (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘∅))
5		1on 7454	. . . . . . . . . 10 ⊢ 1𝑜 ∈ On
6	5	elexi 3186	. . . . . . . . 9 ⊢ 1𝑜 ∈ V
7	6	rdg0 7404	. . . . . . . 8 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘∅) = 1𝑜
8	4, 7	syl6eq 2660	. . . . . . 7 ⊢ (𝐵 = ∅ → (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵) = 1𝑜)
9		inteq 4413	. . . . . . . 8 ⊢ (𝐵 = ∅ → ∩ 𝐵 = ∩ ∅)
10		int0 4425	. . . . . . . 8 ⊢ ∩ ∅ = V
11	9, 10	syl6eq 2660	. . . . . . 7 ⊢ (𝐵 = ∅ → ∩ 𝐵 = V)
12	8, 11	ineq12d 3777	. . . . . 6 ⊢ (𝐵 = ∅ → ((rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵) ∩ ∩ 𝐵) = (1𝑜 ∩ V))
13		inv1 3922	. . . . . . 7 ⊢ (1𝑜 ∩ V) = 1𝑜
14	13	a1i 11	. . . . . 6 ⊢ (𝐴 = ∅ → (1𝑜 ∩ V) = 1𝑜)
15	12, 14	sylan9eqr 2666	. . . . 5 ⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) → ((rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵) ∩ ∩ 𝐵) = 1𝑜)
16	3, 15	eqtr4d 2647	. . . 4 ⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) → (𝐴 ↑𝑜 𝐵) = ((rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵) ∩ ∩ 𝐵))
17		oveq1 6556	. . . . . . 7 ⊢ (𝐴 = ∅ → (𝐴 ↑𝑜 𝐵) = (∅ ↑𝑜 𝐵))
18		oe0m1 7488	. . . . . . . 8 ⊢ (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ (∅ ↑𝑜 𝐵) = ∅))
19	18	biimpa 500	. . . . . . 7 ⊢ ((𝐵 ∈ On ∧ ∅ ∈ 𝐵) → (∅ ↑𝑜 𝐵) = ∅)
20	17, 19	sylan9eqr 2666	. . . . . 6 ⊢ (((𝐵 ∈ On ∧ ∅ ∈ 𝐵) ∧ 𝐴 = ∅) → (𝐴 ↑𝑜 𝐵) = ∅)
21	20	an32s 842	. . . . 5 ⊢ (((𝐵 ∈ On ∧ 𝐴 = ∅) ∧ ∅ ∈ 𝐵) → (𝐴 ↑𝑜 𝐵) = ∅)
22		int0el 4443	. . . . . . . 8 ⊢ (∅ ∈ 𝐵 → ∩ 𝐵 = ∅)
23	22	ineq2d 3776	. . . . . . 7 ⊢ (∅ ∈ 𝐵 → ((rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵) ∩ ∩ 𝐵) = ((rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵) ∩ ∅))
24		in0 3920	. . . . . . 7 ⊢ ((rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵) ∩ ∅) = ∅
25	23, 24	syl6eq 2660	. . . . . 6 ⊢ (∅ ∈ 𝐵 → ((rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵) ∩ ∩ 𝐵) = ∅)
26	25	adantl 481	. . . . 5 ⊢ (((𝐵 ∈ On ∧ 𝐴 = ∅) ∧ ∅ ∈ 𝐵) → ((rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵) ∩ ∩ 𝐵) = ∅)
27	21, 26	eqtr4d 2647	. . . 4 ⊢ (((𝐵 ∈ On ∧ 𝐴 = ∅) ∧ ∅ ∈ 𝐵) → (𝐴 ↑𝑜 𝐵) = ((rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵) ∩ ∩ 𝐵))
28	16, 27	oe0lem 7480	. . 3 ⊢ ((𝐵 ∈ On ∧ 𝐴 = ∅) → (𝐴 ↑𝑜 𝐵) = ((rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵) ∩ ∩ 𝐵))
29		inteq 4413	. . . . . . . . . 10 ⊢ (𝐴 = ∅ → ∩ 𝐴 = ∩ ∅)
30	29, 10	syl6eq 2660	. . . . . . . . 9 ⊢ (𝐴 = ∅ → ∩ 𝐴 = V)
31	30	difeq2d 3690	. . . . . . . 8 ⊢ (𝐴 = ∅ → (V ∖ ∩ 𝐴) = (V ∖ V))
32		difid 3902	. . . . . . . 8 ⊢ (V ∖ V) = ∅
33	31, 32	syl6eq 2660	. . . . . . 7 ⊢ (𝐴 = ∅ → (V ∖ ∩ 𝐴) = ∅)
34	33	uneq2d 3729	. . . . . 6 ⊢ (𝐴 = ∅ → (∩ 𝐵 ∪ (V ∖ ∩ 𝐴)) = (∩ 𝐵 ∪ ∅))
35		uncom 3719	. . . . . 6 ⊢ (∩ 𝐵 ∪ (V ∖ ∩ 𝐴)) = ((V ∖ ∩ 𝐴) ∪ ∩ 𝐵)
36		un0 3919	. . . . . 6 ⊢ (∩ 𝐵 ∪ ∅) = ∩ 𝐵
37	34, 35, 36	3eqtr3g 2667	. . . . 5 ⊢ (𝐴 = ∅ → ((V ∖ ∩ 𝐴) ∪ ∩ 𝐵) = ∩ 𝐵)
38	37	adantl 481	. . . 4 ⊢ ((𝐵 ∈ On ∧ 𝐴 = ∅) → ((V ∖ ∩ 𝐴) ∪ ∩ 𝐵) = ∩ 𝐵)
39	38	ineq2d 3776	. . 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 2647	. 2 ⊢ ((𝐵 ∈ On ∧ 𝐴 = ∅) → (𝐴 ↑𝑜 𝐵) = ((rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵) ∩ ((V ∖ ∩ 𝐴) ∪ ∩ 𝐵)))
41		oevn0 7482	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐴 ↑𝑜 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵))
42		int0el 4443	. . . . . . . . . 10 ⊢ (∅ ∈ 𝐴 → ∩ 𝐴 = ∅)
43	42	difeq2d 3690	. . . . . . . . 9 ⊢ (∅ ∈ 𝐴 → (V ∖ ∩ 𝐴) = (V ∖ ∅))
44		dif0 3904	. . . . . . . . 9 ⊢ (V ∖ ∅) = V
45	43, 44	syl6eq 2660	. . . . . . . 8 ⊢ (∅ ∈ 𝐴 → (V ∖ ∩ 𝐴) = V)
46	45	uneq2d 3729	. . . . . . 7 ⊢ (∅ ∈ 𝐴 → (∩ 𝐵 ∪ (V ∖ ∩ 𝐴)) = (∩ 𝐵 ∪ V))
47		unv 3923	. . . . . . 7 ⊢ (∩ 𝐵 ∪ V) = V
48	46, 35, 47	3eqtr3g 2667	. . . . . 6 ⊢ (∅ ∈ 𝐴 → ((V ∖ ∩ 𝐴) ∪ ∩ 𝐵) = V)
49	48	adantl 481	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → ((V ∖ ∩ 𝐴) ∪ ∩ 𝐵) = V)
50	49	ineq2d 3776	. . . 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 3922	. . . 4 ⊢ ((rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵) ∩ V) = (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵)
52	50, 51	syl6req 2661	. . 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 2644	. 2 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐴 ↑𝑜 𝐵) = ((rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵) ∩ ((V ∖ ∩ 𝐴) ∪ ∩ 𝐵)))
54	40, 53	oe0lem 7480	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑𝑜 𝐵) = ((rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵) ∩ ((V ∖ ∩ 𝐴) ∪ ∩ 𝐵)))

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ∖ cdif 3537   ∪ cun 3538   ∩ cin 3539  ∅c0 3874  ∩ cint 4410   ↦ cmpt 4643  Oncon0 5640  ‘cfv 5804  (class class class)co 6549  reccrdg 7392  1_𝑜c1o 7440   ·_𝑜 comu 7445   ↑_𝑜 coe 7446

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oexp 7453

This theorem is referenced by: (None)