Theorem oecl 7617

Description: Closure law for ordinal exponentiation. (Contributed by NM, 1-Jan-2005.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Assertion

Ref	Expression
oecl	|- ( ( A e. On /\ B e. On ) -> ( A ^o B ) e. On )

Proof of Theorem oecl

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 6658	. . . . . . . 8 |- ( B = (/) -> ( (/) ^o B ) = ( (/) ^o (/) ) )
2		oe0m0 7600	. . . . . . . . 9 |- ( (/) ^o (/) ) = 1o
3		1on 7567	. . . . . . . . 9 |- 1o e. On
4	2, 3	eqeltri 2697	. . . . . . . 8 |- ( (/) ^o (/) ) e. On
5	1, 4	syl6eqel 2709	. . . . . . 7 |- ( B = (/) -> ( (/) ^o B ) e. On )
6	5	adantl 482	. . . . . 6 |- ( ( B e. On /\ B = (/) ) -> ( (/) ^o B ) e. On )
7		oe0m1 7601	. . . . . . . . 9 |- ( B e. On -> ( (/) e. B <-> ( (/) ^o B ) = (/) ) )
8	7	biimpa 501	. . . . . . . 8 |- ( ( B e. On /\ (/) e. B ) -> ( (/) ^o B ) = (/) )
9		0elon 5778	. . . . . . . 8 |- (/) e. On
10	8, 9	syl6eqel 2709	. . . . . . 7 |- ( ( B e. On /\ (/) e. B ) -> ( (/) ^o B ) e. On )
11	10	adantll 750	. . . . . 6 |- ( ( ( B e. On /\ B e. On ) /\ (/) e. B ) -> ( (/) ^o B ) e. On )
12	6, 11	oe0lem 7593	. . . . 5 |- ( ( B e. On /\ B e. On ) -> ( (/) ^o B ) e. On )
13	12	anidms 677	. . . 4 |- ( B e. On -> ( (/) ^o B ) e. On )
14		oveq1 6657	. . . . 5 |- ( A = (/) -> ( A ^o B ) = ( (/) ^o B ) )
15	14	eleq1d 2686	. . . 4 |- ( A = (/) -> ( ( A ^o B ) e. On <-> ( (/) ^o B ) e. On ) )
16	13, 15	syl5ibr 236	. . 3 |- ( A = (/) -> ( B e. On -> ( A ^o B ) e. On ) )
17	16	impcom 446	. 2 |- ( ( B e. On /\ A = (/) ) -> ( A ^o B ) e. On )
18		oveq2 6658	. . . . . . 7 |- ( x = (/) -> ( A ^o x ) = ( A ^o (/) ) )
19	18	eleq1d 2686	. . . . . 6 |- ( x = (/) -> ( ( A ^o x ) e. On <-> ( A ^o (/) ) e. On ) )
20		oveq2 6658	. . . . . . 7 |- ( x = y -> ( A ^o x ) = ( A ^o y ) )
21	20	eleq1d 2686	. . . . . 6 |- ( x = y -> ( ( A ^o x ) e. On <-> ( A ^o y ) e. On ) )
22		oveq2 6658	. . . . . . 7 |- ( x = suc y -> ( A ^o x ) = ( A ^o suc y ) )
23	22	eleq1d 2686	. . . . . 6 |- ( x = suc y -> ( ( A ^o x ) e. On <-> ( A ^o suc y ) e. On ) )
24		oveq2 6658	. . . . . . 7 |- ( x = B -> ( A ^o x ) = ( A ^o B ) )
25	24	eleq1d 2686	. . . . . 6 |- ( x = B -> ( ( A ^o x ) e. On <-> ( A ^o B ) e. On ) )
26		oe0 7602	. . . . . . . 8 |- ( A e. On -> ( A ^o (/) ) = 1o )
27	26, 3	syl6eqel 2709	. . . . . . 7 |- ( A e. On -> ( A ^o (/) ) e. On )
28	27	adantr 481	. . . . . 6 |- ( ( A e. On /\ (/) e. A ) -> ( A ^o (/) ) e. On )
29		omcl 7616	. . . . . . . . . . 11 |- ( ( ( A ^o y ) e. On /\ A e. On ) -> ( ( A ^o y ) .o A ) e. On )
30	29	expcom 451	. . . . . . . . . 10 |- ( A e. On -> ( ( A ^o y ) e. On -> ( ( A ^o y ) .o A ) e. On ) )
31	30	adantr 481	. . . . . . . . 9 |- ( ( A e. On /\ y e. On ) -> ( ( A ^o y ) e. On -> ( ( A ^o y ) .o A ) e. On ) )
32		oesuc 7607	. . . . . . . . . 10 |- ( ( A e. On /\ y e. On ) -> ( A ^o suc y ) = ( ( A ^o y ) .o A ) )
33	32	eleq1d 2686	. . . . . . . . 9 |- ( ( A e. On /\ y e. On ) -> ( ( A ^o suc y ) e. On <-> ( ( A ^o y ) .o A ) e. On ) )
34	31, 33	sylibrd 249	. . . . . . . 8 |- ( ( A e. On /\ y e. On ) -> ( ( A ^o y ) e. On -> ( A ^o suc y ) e. On ) )
35	34	expcom 451	. . . . . . 7 |- ( y e. On -> ( A e. On -> ( ( A ^o y ) e. On -> ( A ^o suc y ) e. On ) ) )
36	35	adantrd 484	. . . . . 6 |- ( y e. On -> ( ( A e. On /\ (/) e. A ) -> ( ( A ^o y ) e. On -> ( A ^o suc y ) e. On ) ) )
37		vex 3203	. . . . . . . . 9 |- x e. _V
38		iunon 7436	. . . . . . . . 9 |- ( ( x e. _V /\ A. y e. x ( A ^o y ) e. On ) -> U_ y e. x ( A ^o y ) e. On )
39	37, 38	mpan 706	. . . . . . . 8 |- ( A. y e. x ( A ^o y ) e. On -> U_ y e. x ( A ^o y ) e. On )
40		oelim 7614	. . . . . . . . . . . 12 |- ( ( ( A e. On /\ ( x e. _V /\ Lim x ) ) /\ (/) e. A ) -> ( A ^o x ) = U_ y e. x ( A ^o y ) )
41	37, 40	mpanlr1 722	. . . . . . . . . . 11 |- ( ( ( A e. On /\ Lim x ) /\ (/) e. A ) -> ( A ^o x ) = U_ y e. x ( A ^o y ) )
42	41	anasss 679	. . . . . . . . . 10 |- ( ( A e. On /\ ( Lim x /\ (/) e. A ) ) -> ( A ^o x ) = U_ y e. x ( A ^o y ) )
43	42	an12s 843	. . . . . . . . 9 |- ( ( Lim x /\ ( A e. On /\ (/) e. A ) ) -> ( A ^o x ) = U_ y e. x ( A ^o y ) )
44	43	eleq1d 2686	. . . . . . . 8 |- ( ( Lim x /\ ( A e. On /\ (/) e. A ) ) -> ( ( A ^o x ) e. On <-> U_ y e. x ( A ^o y ) e. On ) )
45	39, 44	syl5ibr 236	. . . . . . 7 |- ( ( Lim x /\ ( A e. On /\ (/) e. A ) ) -> ( A. y e. x ( A ^o y ) e. On -> ( A ^o x ) e. On ) )
46	45	ex 450	. . . . . 6 |- ( Lim x -> ( ( A e. On /\ (/) e. A ) -> ( A. y e. x ( A ^o y ) e. On -> ( A ^o x ) e. On ) ) )
47	19, 21, 23, 25, 28, 36, 46	tfinds3 7064	. . . . 5 |- ( B e. On -> ( ( A e. On /\ (/) e. A ) -> ( A ^o B ) e. On ) )
48	47	expd 452	. . . 4 |- ( B e. On -> ( A e. On -> ( (/) e. A -> ( A ^o B ) e. On ) ) )
49	48	com12 32	. . 3 |- ( A e. On -> ( B e. On -> ( (/) e. A -> ( A ^o B ) e. On ) ) )
50	49	imp31 448	. 2 |- ( ( ( A e. On /\ B e. On ) /\ (/) e. A ) -> ( A ^o B ) e. On )
51	17, 50	oe0lem 7593	1 |- ( ( A e. On /\ B e. On ) -> ( A ^o B ) e. On )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   (/)c0 3915   U_ciun 4520   Oncon0 5723   Lim wlim 5724   suc csuc 5725  (class class class)co 6650   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-rep 4771  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-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-oadd 7564  df-omul 7565  df-oexp 7566

This theorem is referenced by:  oen0  7666  oeordi  7667  oeord  7668  oecan  7669  oeword  7670  oewordri  7672  oeworde  7673  oeordsuc  7674  oeoalem  7676  oeoa  7677  oeoelem  7678  oeoe  7679  oelimcl  7680  oeeulem  7681  oeeui  7682  oaabs2  7725  omabs  7727  cantnfle  8568  cantnflt  8569  cantnfp1  8578  cantnflem1d  8585  cantnflem1  8586  cantnflem2  8587  cantnflem3  8588  cantnflem4  8589  cantnf  8590  oemapwe  8591  cantnffval2  8592  cnfcomlem  8596  cnfcom  8597  cnfcom3lem  8600  cnfcom3  8601  infxpenc  8841