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Theorem add1p1 8663
Description: Adding two times 1 to a number. (Contributed by AV, 22-Sep-2018.)
Assertion
Ref Expression
add1p1  |-  ( N  e.  CC  ->  (
( N  +  1 )  +  1 )  =  ( N  + 
2 ) )

Proof of Theorem add1p1
StepHypRef Expression
1 id 19 . . 3  |-  ( N  e.  CC  ->  N  e.  CC )
2 1cnd 7502 . . 3  |-  ( N  e.  CC  ->  1  e.  CC )
31, 2, 2addassd 7508 . 2  |-  ( N  e.  CC  ->  (
( N  +  1 )  +  1 )  =  ( N  +  ( 1  +  1 ) ) )
4 1p1e2 8537 . . . 4  |-  ( 1  +  1 )  =  2
54a1i 9 . . 3  |-  ( N  e.  CC  ->  (
1  +  1 )  =  2 )
65oveq2d 5668 . 2  |-  ( N  e.  CC  ->  ( N  +  ( 1  +  1 ) )  =  ( N  + 
2 ) )
73, 6eqtrd 2120 1  |-  ( N  e.  CC  ->  (
( N  +  1 )  +  1 )  =  ( N  + 
2 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1289    e. wcel 1438  (class class class)co 5652   CCcc 7346   1c1 7349    + caddc 7351   2c2 8471
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-1cn 7436  ax-addass 7445
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rex 2365  df-v 2621  df-un 3003  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-iota 4980  df-fv 5023  df-ov 5655  df-2 8479
This theorem is referenced by:  nneoor  8846
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