Theorem oe0m 7598

Description: Ordinal exponentiation with zero mantissa. (Contributed by NM, 31-Dec-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)

Assertion

Ref	Expression
oe0m	|- ( A e. On -> ( (/) ^o A ) = ( 1o \ A ) )

Proof of Theorem oe0m

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		0elon 5778	. . 3 |- (/) e. On
2		oev 7594	. . 3 |- ( ( (/) e. On /\ A e. On ) -> ( (/) ^o A ) = if ( (/) = (/) , ( 1o \ A ) , ( rec ( ( x e. _V |-> ( x .o (/) ) ) , 1o ) ` A ) ) )
3	1, 2	mpan 706	. 2 |- ( A e. On -> ( (/) ^o A ) = if ( (/) = (/) , ( 1o \ A ) , ( rec ( ( x e. _V |-> ( x .o (/) ) ) , 1o ) ` A ) ) )
4		eqid 2622	. . 3 |- (/) = (/)
5	4	iftruei 4093	. 2 |- if ( (/) = (/) , ( 1o \ A ) , ( rec ( ( x e. _V |-> ( x .o (/) ) ) , 1o ) ` A ) ) = ( 1o \ A )
6	3, 5	syl6eq 2672	1 |- ( A e. On -> ( (/) ^o A ) = ( 1o \ A ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   _Vcvv 3200   \ cdif 3571   (/)c0 3915   ifcif 4086   |-> 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-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-rab 2921  df-v 3202  df-sbc 3436  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-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-suc 5729  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oexp 7566

This theorem is referenced by:  oe0m0  7600  oe0m1  7601  cantnflem2  8587