Theorem oev2 8140

Description: Alternate value of ordinal exponentiation. Compare oev 8131. (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 7157	. . . . . 6 ⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) → (𝐴 ↑o 𝐵) = (∅ ↑o ∅))
2		oe0m0 8137	. . . . . 6 ⊢ (∅ ↑o ∅) = 1o
3	1, 2	syl6eq 2870	. . . . 5 ⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) → (𝐴 ↑o 𝐵) = 1o)
4		fveq2 6663	. . . . . . . 8 ⊢ (𝐵 = ∅ → (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘∅))
5		1oex 8102	. . . . . . . . 9 ⊢ 1o ∈ V
6	5	rdg0 8049	. . . . . . . 8 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘∅) = 1o
7	4, 6	syl6eq 2870	. . . . . . 7 ⊢ (𝐵 = ∅ → (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) = 1o)
8		inteq 4870	. . . . . . . 8 ⊢ (𝐵 = ∅ → ∩ 𝐵 = ∩ ∅)
9		int0 4881	. . . . . . . 8 ⊢ ∩ ∅ = V
10	8, 9	syl6eq 2870	. . . . . . 7 ⊢ (𝐵 = ∅ → ∩ 𝐵 = V)
11	7, 10	ineq12d 4188	. . . . . 6 ⊢ (𝐵 = ∅ → ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∩ ∩ 𝐵) = (1o ∩ V))
12		inv1 4346	. . . . . . 7 ⊢ (1o ∩ V) = 1o
13	12	a1i 11	. . . . . 6 ⊢ (𝐴 = ∅ → (1o ∩ V) = 1o)
14	11, 13	sylan9eqr 2876	. . . . 5 ⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) → ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∩ ∩ 𝐵) = 1o)
15	3, 14	eqtr4d 2857	. . . 4 ⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) → (𝐴 ↑o 𝐵) = ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∩ ∩ 𝐵))
16		oveq1 7155	. . . . . . 7 ⊢ (𝐴 = ∅ → (𝐴 ↑o 𝐵) = (∅ ↑o 𝐵))
17		oe0m1 8138	. . . . . . . 8 ⊢ (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ (∅ ↑o 𝐵) = ∅))
18	17	biimpa 479	. . . . . . 7 ⊢ ((𝐵 ∈ On ∧ ∅ ∈ 𝐵) → (∅ ↑o 𝐵) = ∅)
19	16, 18	sylan9eqr 2876	. . . . . 6 ⊢ (((𝐵 ∈ On ∧ ∅ ∈ 𝐵) ∧ 𝐴 = ∅) → (𝐴 ↑o 𝐵) = ∅)
20	19	an32s 650	. . . . 5 ⊢ (((𝐵 ∈ On ∧ 𝐴 = ∅) ∧ ∅ ∈ 𝐵) → (𝐴 ↑o 𝐵) = ∅)
21		int0el 4898	. . . . . . . 8 ⊢ (∅ ∈ 𝐵 → ∩ 𝐵 = ∅)
22	21	ineq2d 4187	. . . . . . 7 ⊢ (∅ ∈ 𝐵 → ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∩ ∩ 𝐵) = ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∩ ∅))
23		in0 4343	. . . . . . 7 ⊢ ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∩ ∅) = ∅
24	22, 23	syl6eq 2870	. . . . . 6 ⊢ (∅ ∈ 𝐵 → ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∩ ∩ 𝐵) = ∅)
25	24	adantl 484	. . . . 5 ⊢ (((𝐵 ∈ On ∧ 𝐴 = ∅) ∧ ∅ ∈ 𝐵) → ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∩ ∩ 𝐵) = ∅)
26	20, 25	eqtr4d 2857	. . . 4 ⊢ (((𝐵 ∈ On ∧ 𝐴 = ∅) ∧ ∅ ∈ 𝐵) → (𝐴 ↑o 𝐵) = ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∩ ∩ 𝐵))
27	15, 26	oe0lem 8130	. . 3 ⊢ ((𝐵 ∈ On ∧ 𝐴 = ∅) → (𝐴 ↑o 𝐵) = ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∩ ∩ 𝐵))
28		inteq 4870	. . . . . . . . . 10 ⊢ (𝐴 = ∅ → ∩ 𝐴 = ∩ ∅)
29	28, 9	syl6eq 2870	. . . . . . . . 9 ⊢ (𝐴 = ∅ → ∩ 𝐴 = V)
30	29	difeq2d 4097	. . . . . . . 8 ⊢ (𝐴 = ∅ → (V ∖ ∩ 𝐴) = (V ∖ V))
31		difid 4328	. . . . . . . 8 ⊢ (V ∖ V) = ∅
32	30, 31	syl6eq 2870	. . . . . . 7 ⊢ (𝐴 = ∅ → (V ∖ ∩ 𝐴) = ∅)
33	32	uneq2d 4137	. . . . . 6 ⊢ (𝐴 = ∅ → (∩ 𝐵 ∪ (V ∖ ∩ 𝐴)) = (∩ 𝐵 ∪ ∅))
34		uncom 4127	. . . . . 6 ⊢ (∩ 𝐵 ∪ (V ∖ ∩ 𝐴)) = ((V ∖ ∩ 𝐴) ∪ ∩ 𝐵)
35		un0 4342	. . . . . 6 ⊢ (∩ 𝐵 ∪ ∅) = ∩ 𝐵
36	33, 34, 35	3eqtr3g 2877	. . . . 5 ⊢ (𝐴 = ∅ → ((V ∖ ∩ 𝐴) ∪ ∩ 𝐵) = ∩ 𝐵)
37	36	adantl 484	. . . 4 ⊢ ((𝐵 ∈ On ∧ 𝐴 = ∅) → ((V ∖ ∩ 𝐴) ∪ ∩ 𝐵) = ∩ 𝐵)
38	37	ineq2d 4187	. . 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 2857	. 2 ⊢ ((𝐵 ∈ On ∧ 𝐴 = ∅) → (𝐴 ↑o 𝐵) = ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∩ ((V ∖ ∩ 𝐴) ∪ ∩ 𝐵)))
40		oevn0 8132	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐴 ↑o 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵))
41		int0el 4898	. . . . . . . . . 10 ⊢ (∅ ∈ 𝐴 → ∩ 𝐴 = ∅)
42	41	difeq2d 4097	. . . . . . . . 9 ⊢ (∅ ∈ 𝐴 → (V ∖ ∩ 𝐴) = (V ∖ ∅))
43		dif0 4330	. . . . . . . . 9 ⊢ (V ∖ ∅) = V
44	42, 43	syl6eq 2870	. . . . . . . 8 ⊢ (∅ ∈ 𝐴 → (V ∖ ∩ 𝐴) = V)
45	44	uneq2d 4137	. . . . . . 7 ⊢ (∅ ∈ 𝐴 → (∩ 𝐵 ∪ (V ∖ ∩ 𝐴)) = (∩ 𝐵 ∪ V))
46		unv 4347	. . . . . . 7 ⊢ (∩ 𝐵 ∪ V) = V
47	45, 34, 46	3eqtr3g 2877	. . . . . 6 ⊢ (∅ ∈ 𝐴 → ((V ∖ ∩ 𝐴) ∪ ∩ 𝐵) = V)
48	47	adantl 484	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → ((V ∖ ∩ 𝐴) ∪ ∩ 𝐵) = V)
49	48	ineq2d 4187	. . . 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 4346	. . . 4 ⊢ ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∩ V) = (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵)
51	49, 50	syl6req 2871	. . 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 2854	. 2 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐴 ↑o 𝐵) = ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∩ ((V ∖ ∩ 𝐴) ∪ ∩ 𝐵)))
53	39, 52	oe0lem 8130	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑o 𝐵) = ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∩ ((V ∖ ∩ 𝐴) ∪ ∩ 𝐵)))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1531   ∈ wcel 2108  Vcvv 3493   ∖ cdif 3931   ∪ cun 3932   ∩ cin 3933  ∅c0 4289  ∩ cint 4867   ↦ cmpt 5137  Oncon0 6184  ‘cfv 6348  (class class class)co 7148  reccrdg 8037  1_oc1o 8087   ·_o comu 8092   ↑_o coe 8093

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1083  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7573  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-1o 8094  df-oexp 8100

This theorem is referenced by: (None)