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Theorem oe0 6407
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
StepHypRef Expression
1 oveq1 5717 . . . . 5  |-  ( A  =  (/)  ->  ( A  ^o  (/) )  =  (
(/)  ^o  (/) ) )
2 oe0m0 6405 . . . . 5  |-  ( (/)  ^o  (/) )  =  1o
31, 2syl6eq 2301 . . . 4  |-  ( A  =  (/)  ->  ( A  ^o  (/) )  =  1o )
43adantl 454 . . 3  |-  ( ( A  e.  On  /\  A  =  (/) )  -> 
( A  ^o  (/) )  =  1o )
5 0elon 4338 . . . . . 6  |-  (/)  e.  On
6 oevn0 6400 . . . . . 6  |-  ( ( ( A  e.  On  /\  (/)  e.  On )  /\  (/) 
e.  A )  -> 
( A  ^o  (/) )  =  ( rec ( ( x  e.  _V  |->  ( x  .o  A ) ) ,  1o ) `
 (/) ) )
75, 6mpanl2 665 . . . . 5  |-  ( ( A  e.  On  /\  (/) 
e.  A )  -> 
( A  ^o  (/) )  =  ( rec ( ( x  e.  _V  |->  ( x  .o  A ) ) ,  1o ) `
 (/) ) )
8 1on 6372 . . . . . . 7  |-  1o  e.  On
98elexi 2736 . . . . . 6  |-  1o  e.  _V
109rdg0 6320 . . . . 5  |-  ( rec ( ( x  e. 
_V  |->  ( x  .o  A ) ) ,  1o ) `  (/) )  =  1o
117, 10syl6eq 2301 . . . 4  |-  ( ( A  e.  On  /\  (/) 
e.  A )  -> 
( A  ^o  (/) )  =  1o )
1211adantll 697 . . 3  |-  ( ( ( A  e.  On  /\  A  e.  On )  /\  (/)  e.  A )  ->  ( A  ^o  (/) )  =  1o )
134, 12oe0lem 6398 . 2  |-  ( ( A  e.  On  /\  A  e.  On )  ->  ( A  ^o  (/) )  =  1o )
1413anidms 629 1  |-  ( A  e.  On  ->  ( A  ^o  (/) )  =  1o )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    = wceq 1619    e. wcel 1621   _Vcvv 2727   (/)c0 3362    e. cmpt 3974   Oncon0 4285   ` cfv 4592  (class class class)co 5710   reccrdg 6308   1oc1o 6358    .o comu 6363    ^o coe 6364
This theorem is referenced by:  oecl  6422  oe1  6428  oe1m  6429  oen0  6470  oewordri  6476  oeoalem  6480  oeoelem  6482  oeoe  6483  oeeulem  6485  nnecl  6497  oaabs2  6529  cantnff  7259
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-recs 6274  df-rdg 6309  df-1o 6365  df-oexp 6371
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