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Theorem oe0 7229
Description: Ordinal exponentiation with zero exponent. Definition 8.30 of [TakeutiZaring] p. 67. (Contributed by NM, 31-Dec-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
Assertion
Ref Expression
oe0  |-  ( A  e.  On  ->  ( A  ^o  (/) )  =  1o )

Proof of Theorem oe0
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 oveq1 6302 . . . . 5  |-  ( A  =  (/)  ->  ( A  ^o  (/) )  =  (
(/)  ^o  (/) ) )
2 oe0m0 7227 . . . . 5  |-  ( (/)  ^o  (/) )  =  1o
31, 2syl6eq 2503 . . . 4  |-  ( A  =  (/)  ->  ( A  ^o  (/) )  =  1o )
43adantl 468 . . 3  |-  ( ( A  e.  On  /\  A  =  (/) )  -> 
( A  ^o  (/) )  =  1o )
5 0elon 5479 . . . . . 6  |-  (/)  e.  On
6 oevn0 7222 . . . . . 6  |-  ( ( ( A  e.  On  /\  (/)  e.  On )  /\  (/) 
e.  A )  -> 
( A  ^o  (/) )  =  ( rec ( ( x  e.  _V  |->  ( x  .o  A ) ) ,  1o ) `
 (/) ) )
75, 6mpanl2 688 . . . . 5  |-  ( ( A  e.  On  /\  (/) 
e.  A )  -> 
( A  ^o  (/) )  =  ( rec ( ( x  e.  _V  |->  ( x  .o  A ) ) ,  1o ) `
 (/) ) )
8 1on 7194 . . . . . . 7  |-  1o  e.  On
98elexi 3057 . . . . . 6  |-  1o  e.  _V
109rdg0 7144 . . . . 5  |-  ( rec ( ( x  e. 
_V  |->  ( x  .o  A ) ) ,  1o ) `  (/) )  =  1o
117, 10syl6eq 2503 . . . 4  |-  ( ( A  e.  On  /\  (/) 
e.  A )  -> 
( A  ^o  (/) )  =  1o )
1211adantll 721 . . 3  |-  ( ( ( A  e.  On  /\  A  e.  On )  /\  (/)  e.  A )  ->  ( A  ^o  (/) )  =  1o )
134, 12oe0lem 7220 . 2  |-  ( ( A  e.  On  /\  A  e.  On )  ->  ( A  ^o  (/) )  =  1o )
1413anidms 651 1  |-  ( A  e.  On  ->  ( A  ^o  (/) )  =  1o )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    = wceq 1446    e. wcel 1889   _Vcvv 3047   (/)c0 3733    |-> cmpt 4464   Oncon0 5426   ` cfv 5585  (class class class)co 6295   reccrdg 7132   1oc1o 7180    .o comu 7185    ^o coe 7186
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-reu 2746  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6698  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-oexp 7193
This theorem is referenced by:  oecl  7244  oe1  7250  oe1m  7251  oen0  7292  oewordri  7298  oeoalem  7302  oeoelem  7304  oeoe  7305  oeeulem  7307  nnecl  7319  oaabs2  7351  cantnff  8184
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