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Theorem oa0 7184
Description: Addition with zero. Proposition 8.3 of [TakeutiZaring] p. 57. (Contributed by NM, 3-May-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)
Assertion
Ref Expression
oa0  |-  ( A  e.  On  ->  ( A  +o  (/) )  =  A )

Proof of Theorem oa0
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 0elon 4940 . . 3  |-  (/)  e.  On
2 oav 7179 . . 3  |-  ( ( A  e.  On  /\  (/) 
e.  On )  -> 
( A  +o  (/) )  =  ( rec ( ( x  e.  _V  |->  suc  x ) ,  A
) `  (/) ) )
31, 2mpan2 671 . 2  |-  ( A  e.  On  ->  ( A  +o  (/) )  =  ( rec ( ( x  e.  _V  |->  suc  x
) ,  A ) `
 (/) ) )
4 rdg0g 7111 . 2  |-  ( A  e.  On  ->  ( rec ( ( x  e. 
_V  |->  suc  x ) ,  A ) `  (/) )  =  A )
53, 4eqtrd 2498 1  |-  ( A  e.  On  ->  ( A  +o  (/) )  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1395    e. wcel 1819   _Vcvv 3109   (/)c0 3793    |-> cmpt 4515   Oncon0 4887   suc csuc 4889   ` cfv 5594  (class class class)co 6296   reccrdg 7093    +o coa 7145
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-oadd 7152
This theorem is referenced by:  oa1suc  7199  oacl  7203  oa0r  7206  om0r  7207  oawordri  7217  oaord1  7218  oaword1  7219  oawordeulem  7221  oa00  7226  oaass  7228  oarec  7229  odi  7246  oeoalem  7263  nna0  7271  nna0r  7276  nnm0r  7277  nnawordi  7288  cantnflt  8108  cantnfltOLD  8138
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