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 e. On /\ B e. On ) -> ( A ^o B ) = ( ( rec ( ( x e. _V |-> ( x .o A ) ) , 1o ) ` B ) i^i ( ( _V \ |^| A ) u. |^| 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 e. _V |-> ( x .o A ) ) , 1o ) ` B ) = ( rec ( ( x e. _V |-> ( x .o A ) ) , 1o ) ` (/) ) )
5		1on 7567	. . . . . . . . . 10 |- 1o e. On
6	5	elexi 3213	. . . . . . . . 9 |- 1o e. _V
7	6	rdg0 7517	. . . . . . . 8 |- ( rec ( ( x e. _V |-> ( x .o A ) ) , 1o ) ` (/) ) = 1o
8	4, 7	syl6eq 2672	. . . . . . 7 |- ( B = (/) -> ( rec ( ( x e. _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 e. _V |-> ( x .o A ) ) , 1o ) ` B ) i^i |^| B ) = ( 1o i^i _V ) )
13		inv1 3970	. . . . . . 7 |- ( 1o i^i _V ) = 1o
14	13	a1i 11	. . . . . 6 |- ( A = (/) -> ( 1o i^i _V ) = 1o )
15	12, 14	sylan9eqr 2678	. . . . 5 |- ( ( A = (/) /\ B = (/) ) -> ( ( rec ( ( x e. _V |-> ( x .o A ) ) , 1o ) ` B ) i^i |^| B ) = 1o )
16	3, 15	eqtr4d 2659	. . . 4 |- ( ( A = (/) /\ B = (/) ) -> ( A ^o B ) = ( ( rec ( ( x e. _V |-> ( x .o A ) ) , 1o ) ` B ) i^i |^| B ) )
17		oveq1 6657	. . . . . . 7 |- ( A = (/) -> ( A ^o B ) = ( (/) ^o B ) )
18		oe0m1 7601	. . . . . . . 8 |- ( B e. On -> ( (/) e. B <-> ( (/) ^o B ) = (/) ) )
19	18	biimpa 501	. . . . . . 7 |- ( ( B e. On /\ (/) e. B ) -> ( (/) ^o B ) = (/) )
20	17, 19	sylan9eqr 2678	. . . . . 6 |- ( ( ( B e. On /\ (/) e. B ) /\ A = (/) ) -> ( A ^o B ) = (/) )
21	20	an32s 846	. . . . 5 |- ( ( ( B e. On /\ A = (/) ) /\ (/) e. B ) -> ( A ^o B ) = (/) )
22		int0el 4508	. . . . . . . 8 |- ( (/) e. B -> |^| B = (/) )
23	22	ineq2d 3814	. . . . . . 7 |- ( (/) e. B -> ( ( rec ( ( x e. _V |-> ( x .o A ) ) , 1o ) ` B ) i^i |^| B ) = ( ( rec ( ( x e. _V |-> ( x .o A ) ) , 1o ) ` B ) i^i (/) ) )
24		in0 3968	. . . . . . 7 |- ( ( rec ( ( x e. _V |-> ( x .o A ) ) , 1o ) ` B ) i^i (/) ) = (/)
25	23, 24	syl6eq 2672	. . . . . 6 |- ( (/) e. B -> ( ( rec ( ( x e. _V |-> ( x .o A ) ) , 1o ) ` B ) i^i |^| B ) = (/) )
26	25	adantl 482	. . . . 5 |- ( ( ( B e. On /\ A = (/) ) /\ (/) e. B ) -> ( ( rec ( ( x e. _V |-> ( x .o A ) ) , 1o ) ` B ) i^i |^| B ) = (/) )
27	21, 26	eqtr4d 2659	. . . 4 |- ( ( ( B e. On /\ A = (/) ) /\ (/) e. B ) -> ( A ^o B ) = ( ( rec ( ( x e. _V |-> ( x .o A ) ) , 1o ) ` B ) i^i |^| B ) )
28	16, 27	oe0lem 7593	. . 3 |- ( ( B e. On /\ A = (/) ) -> ( A ^o B ) = ( ( rec ( ( x e. _V |-> ( x .o A ) ) , 1o ) ` B ) i^i |^| 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 u. ( _V \ |^| A ) ) = ( |^| B u. (/) ) )
35		uncom 3757	. . . . . 6 |- ( |^| B u. ( _V \ |^| A ) ) = ( ( _V \ |^| A ) u. |^| B )
36		un0 3967	. . . . . 6 |- ( |^| B u. (/) ) = |^| B
37	34, 35, 36	3eqtr3g 2679	. . . . 5 |- ( A = (/) -> ( ( _V \ |^| A ) u. |^| B ) = |^| B )
38	37	adantl 482	. . . 4 |- ( ( B e. On /\ A = (/) ) -> ( ( _V \ |^| A ) u. |^| B ) = |^| B )
39	38	ineq2d 3814	. . 3 |- ( ( B e. On /\ A = (/) ) -> ( ( rec ( ( x e. _V |-> ( x .o A ) ) , 1o ) ` B ) i^i ( ( _V \ |^| A ) u. |^| B ) ) = ( ( rec ( ( x e. _V |-> ( x .o A ) ) , 1o ) ` B ) i^i |^| B ) )
40	28, 39	eqtr4d 2659	. 2 |- ( ( B e. On /\ A = (/) ) -> ( A ^o B ) = ( ( rec ( ( x e. _V |-> ( x .o A ) ) , 1o ) ` B ) i^i ( ( _V \ |^| A ) u. |^| B ) ) )
41		oevn0 7595	. . 3 |- ( ( ( A e. On /\ B e. On ) /\ (/) e. A ) -> ( A ^o B ) = ( rec ( ( x e. _V |-> ( x .o A ) ) , 1o ) ` B ) )
42		int0el 4508	. . . . . . . . . 10 |- ( (/) e. A -> |^| A = (/) )
43	42	difeq2d 3728	. . . . . . . . 9 |- ( (/) e. A -> ( _V \ |^| A ) = ( _V \ (/) ) )
44		dif0 3950	. . . . . . . . 9 |- ( _V \ (/) ) = _V
45	43, 44	syl6eq 2672	. . . . . . . 8 |- ( (/) e. A -> ( _V \ |^| A ) = _V )
46	45	uneq2d 3767	. . . . . . 7 |- ( (/) e. A -> ( |^| B u. ( _V \ |^| A ) ) = ( |^| B u. _V ) )
47		unv 3971	. . . . . . 7 |- ( |^| B u. _V ) = _V
48	46, 35, 47	3eqtr3g 2679	. . . . . 6 |- ( (/) e. A -> ( ( _V \ |^| A ) u. |^| B ) = _V )
49	48	adantl 482	. . . . 5 |- ( ( ( A e. On /\ B e. On ) /\ (/) e. A ) -> ( ( _V \ |^| A ) u. |^| B ) = _V )
50	49	ineq2d 3814	. . . 4 |- ( ( ( A e. On /\ B e. On ) /\ (/) e. A ) -> ( ( rec ( ( x e. _V |-> ( x .o A ) ) , 1o ) ` B ) i^i ( ( _V \ |^| A ) u. |^| B ) ) = ( ( rec ( ( x e. _V |-> ( x .o A ) ) , 1o ) ` B ) i^i _V ) )
51		inv1 3970	. . . 4 |- ( ( rec ( ( x e. _V |-> ( x .o A ) ) , 1o ) ` B ) i^i _V ) = ( rec ( ( x e. _V |-> ( x .o A ) ) , 1o ) ` B )
52	50, 51	syl6req 2673	. . 3 |- ( ( ( A e. On /\ B e. On ) /\ (/) e. A ) -> ( rec ( ( x e. _V |-> ( x .o A ) ) , 1o ) ` B ) = ( ( rec ( ( x e. _V |-> ( x .o A ) ) , 1o ) ` B ) i^i ( ( _V \ |^| A ) u. |^| B ) ) )
53	41, 52	eqtrd 2656	. 2 |- ( ( ( A e. On /\ B e. On ) /\ (/) e. A ) -> ( A ^o B ) = ( ( rec ( ( x e. _V |-> ( x .o A ) ) , 1o ) ` B ) i^i ( ( _V \ |^| A ) u. |^| B ) ) )
54	40, 53	oe0lem 7593	1 |- ( ( A e. On /\ B e. On ) -> ( A ^o B ) = ( ( rec ( ( x e. _V |-> ( x .o A ) ) , 1o ) ` B ) i^i ( ( _V \ |^| A ) u. |^| B ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   \ cdif 3571   u. cun 3572   i^i cin 3573   (/)c0 3915   |^|cint 4475   |-> cmpt 4729   Oncon0 5723   ` cfv 5888  (class class class)co 6650   reccrdg 7505   1oc1o 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)