Theorem oevn0 7060

Description: Value of ordinal exponentiation at a nonzero mantissa. (Contributed by NM, 31-Dec-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)

Assertion

Ref	Expression
oevn0	|- ( ( ( A e. On /\ B e. On ) /\ (/) e. A ) -> ( A ^o B ) = ( rec ( ( x e. _V |-> ( x .o A ) ) , 1o ) ` B ) )

Distinct variable group:   x, A

Allowed substitution hint:   B( x)

Proof of Theorem oevn0

Step	Hyp	Ref	Expression
1		on0eln0 4877	. . . . 5 |- ( A e. On -> ( (/) e. A <-> A =/= (/) ) )
2		df-ne 2647	. . . . 5 |- ( A =/= (/) <-> -. A = (/) )
3	1, 2	syl6bb 261	. . . 4 |- ( A e. On -> ( (/) e. A <-> -. A = (/) ) )
4	3	adantr 465	. . 3 |- ( ( A e. On /\ B e. On ) -> ( (/) e. A <-> -. A = (/) ) )
5		oev 7059	. . . . 5 |- ( ( A e. On /\ B e. On ) -> ( A ^o B ) = if ( A = (/) , ( 1o \ B ) , ( rec ( ( x e. _V |-> ( x .o A ) ) , 1o ) ` B ) ) )
6		iffalse 3902	. . . . 5 |- ( -. A = (/) -> if ( A = (/) , ( 1o \ B ) , ( rec ( ( x e. _V |-> ( x .o A ) ) , 1o ) ` B ) ) = ( rec ( ( x e. _V |-> ( x .o A ) ) , 1o ) ` B ) )
7	5, 6	sylan9eq 2513	. . . 4 |- ( ( ( A e. On /\ B e. On ) /\ -. A = (/) ) -> ( A ^o B ) = ( rec ( ( x e. _V |-> ( x .o A ) ) , 1o ) ` B ) )
8	7	ex 434	. . 3 |- ( ( A e. On /\ B e. On ) -> ( -. A = (/) -> ( A ^o B ) = ( rec ( ( x e. _V |-> ( x .o A ) ) , 1o ) ` B ) ) )
9	4, 8	sylbid 215	. 2 |- ( ( A e. On /\ B e. On ) -> ( (/) e. A -> ( A ^o B ) = ( rec ( ( x e. _V |-> ( x .o A ) ) , 1o ) ` B ) ) )
10	9	imp 429	1 |- ( ( ( A e. On /\ B e. On ) /\ (/) e. A ) -> ( A ^o B ) = ( rec ( ( x e. _V |-> ( x .o A ) ) , 1o ) ` B ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1370   e. wcel 1758   =/= wne 2645   _Vcvv 3072   \ cdif 3428   (/)c0 3740   ifcif 3894   |-> cmpt 4453   Oncon0 4822   ` cfv 5521  (class class class)co 6195   reccrdg 6970   1oc1o 7018   .o comu 7023   ^o coe 7024

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pr 4634  ax-un 6477

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-sbc 3289  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-ord 4825  df-on 4826  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-iota 5484  df-fun 5523  df-fv 5529  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-recs 6937  df-rdg 6971  df-1o 7025  df-oexp 7031

This theorem is referenced by:  oe0  7067  oev2  7068  oesuclem  7070  oelim  7079