MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  oe0 Structured version   Unicode version

Theorem oe0 7190
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 6303 . . . . 5  |-  ( A  =  (/)  ->  ( A  ^o  (/) )  =  (
(/)  ^o  (/) ) )
2 oe0m0 7188 . . . . 5  |-  ( (/)  ^o  (/) )  =  1o
31, 2syl6eq 2514 . . . 4  |-  ( A  =  (/)  ->  ( A  ^o  (/) )  =  1o )
43adantl 466 . . 3  |-  ( ( A  e.  On  /\  A  =  (/) )  -> 
( A  ^o  (/) )  =  1o )
5 0elon 4940 . . . . . 6  |-  (/)  e.  On
6 oevn0 7183 . . . . . 6  |-  ( ( ( A  e.  On  /\  (/)  e.  On )  /\  (/) 
e.  A )  -> 
( A  ^o  (/) )  =  ( rec ( ( x  e.  _V  |->  ( x  .o  A ) ) ,  1o ) `
 (/) ) )
75, 6mpanl2 681 . . . . 5  |-  ( ( A  e.  On  /\  (/) 
e.  A )  -> 
( A  ^o  (/) )  =  ( rec ( ( x  e.  _V  |->  ( x  .o  A ) ) ,  1o ) `
 (/) ) )
8 1on 7155 . . . . . . 7  |-  1o  e.  On
98elexi 3119 . . . . . 6  |-  1o  e.  _V
109rdg0 7105 . . . . 5  |-  ( rec ( ( x  e. 
_V  |->  ( x  .o  A ) ) ,  1o ) `  (/) )  =  1o
117, 10syl6eq 2514 . . . 4  |-  ( ( A  e.  On  /\  (/) 
e.  A )  -> 
( A  ^o  (/) )  =  1o )
1211adantll 713 . . 3  |-  ( ( ( A  e.  On  /\  A  e.  On )  /\  (/)  e.  A )  ->  ( A  ^o  (/) )  =  1o )
134, 12oe0lem 7181 . 2  |-  ( ( A  e.  On  /\  A  e.  On )  ->  ( A  ^o  (/) )  =  1o )
1413anidms 645 1  |-  ( A  e.  On  ->  ( A  ^o  (/) )  =  1o )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819   _Vcvv 3109   (/)c0 3793    |-> cmpt 4515   Oncon0 4887   ` cfv 5594  (class class class)co 6296   reccrdg 7093   1oc1o 7141    .o comu 7146    ^o coe 7147
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-recs 7060  df-rdg 7094  df-1o 7148  df-oexp 7154
This theorem is referenced by:  oecl  7205  oe1  7211  oe1m  7212  oen0  7253  oewordri  7259  oeoalem  7263  oeoelem  7265  oeoe  7266  oeeulem  7268  nnecl  7280  oaabs2  7312  cantnff  8110
  Copyright terms: Public domain W3C validator