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Theorem oe0 6608
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 5952 . . . . 5  |-  ( A  =  (/)  ->  ( A  ^o  (/) )  =  (
(/)  ^o  (/) ) )
2 oe0m0 6606 . . . . 5  |-  ( (/)  ^o  (/) )  =  1o
31, 2syl6eq 2406 . . . 4  |-  ( A  =  (/)  ->  ( A  ^o  (/) )  =  1o )
43adantl 452 . . 3  |-  ( ( A  e.  On  /\  A  =  (/) )  -> 
( A  ^o  (/) )  =  1o )
5 0elon 4527 . . . . . 6  |-  (/)  e.  On
6 oevn0 6601 . . . . . 6  |-  ( ( ( A  e.  On  /\  (/)  e.  On )  /\  (/) 
e.  A )  -> 
( A  ^o  (/) )  =  ( rec ( ( x  e.  _V  |->  ( x  .o  A ) ) ,  1o ) `
 (/) ) )
75, 6mpanl2 662 . . . . 5  |-  ( ( A  e.  On  /\  (/) 
e.  A )  -> 
( A  ^o  (/) )  =  ( rec ( ( x  e.  _V  |->  ( x  .o  A ) ) ,  1o ) `
 (/) ) )
8 1on 6573 . . . . . . 7  |-  1o  e.  On
98elexi 2873 . . . . . 6  |-  1o  e.  _V
109rdg0 6521 . . . . 5  |-  ( rec ( ( x  e. 
_V  |->  ( x  .o  A ) ) ,  1o ) `  (/) )  =  1o
117, 10syl6eq 2406 . . . 4  |-  ( ( A  e.  On  /\  (/) 
e.  A )  -> 
( A  ^o  (/) )  =  1o )
1211adantll 694 . . 3  |-  ( ( ( A  e.  On  /\  A  e.  On )  /\  (/)  e.  A )  ->  ( A  ^o  (/) )  =  1o )
134, 12oe0lem 6599 . 2  |-  ( ( A  e.  On  /\  A  e.  On )  ->  ( A  ^o  (/) )  =  1o )
1413anidms 626 1  |-  ( A  e.  On  ->  ( A  ^o  (/) )  =  1o )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1642    e. wcel 1710   _Vcvv 2864   (/)c0 3531    e. cmpt 4158   Oncon0 4474   ` cfv 5337  (class class class)co 5945   reccrdg 6509   1oc1o 6559    .o comu 6564    ^o coe 6565
This theorem is referenced by:  oecl  6623  oe1  6629  oe1m  6630  oen0  6671  oewordri  6677  oeoalem  6681  oeoelem  6683  oeoe  6684  oeeulem  6686  nnecl  6698  oaabs2  6730  cantnff  7465
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-recs 6475  df-rdg 6510  df-1o 6566  df-oexp 6572
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