Theorem oevn0 7157

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 4922	. . . . 5 |- ( A e. On -> ( (/) e. A <-> A =/= (/) ) )
2		df-ne 2651	. . . . 5 |- ( A =/= (/) <-> -. A = (/) )
3	1, 2	syl6bb 261	. . . 4 |- ( A e. On -> ( (/) e. A <-> -. A = (/) ) )
4	3	adantr 463	. . 3 |- ( ( A e. On /\ B e. On ) -> ( (/) e. A <-> -. A = (/) ) )
5		oev 7156	. . . . 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 3938	. . . . 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 2515	. . . 4 |- ( ( ( A e. On /\ B e. On ) /\ -. A = (/) ) -> ( A ^o B ) = ( rec ( ( x e. _V |-> ( x .o A ) ) , 1o ) ` B ) )
8	7	ex 432	. . 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 427	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 367   = wceq 1398   e. wcel 1823   =/= wne 2649   _Vcvv 3106   \ cdif 3458   (/)c0 3783   ifcif 3929   |-> cmpt 4497   Oncon0 4867   ` cfv 5570  (class class class)co 6270   reccrdg 7067   1oc1o 7115   .o comu 7120   ^o coe 7121

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676  ax-un 6565

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-iota 5534  df-fun 5572  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-recs 7034  df-rdg 7068  df-1o 7122  df-oexp 7128

This theorem is referenced by:  oe0  7164  oev2  7165  oesuclem  7167  oelim  7176