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Theorem add1p1 9199
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 8004 . . 3  |-  ( N  e.  CC  ->  1  e.  CC )
31, 2, 2addassd 8011 . 2  |-  ( N  e.  CC  ->  (
( N  +  1 )  +  1 )  =  ( N  +  ( 1  +  1 ) ) )
4 1p1e2 9067 . . . 4  |-  ( 1  +  1 )  =  2
54a1i 9 . . 3  |-  ( N  e.  CC  ->  (
1  +  1 )  =  2 )
65oveq2d 5913 . 2  |-  ( N  e.  CC  ->  ( N  +  ( 1  +  1 ) )  =  ( N  + 
2 ) )
73, 6eqtrd 2222 1  |-  ( N  e.  CC  ->  (
( N  +  1 )  +  1 )  =  ( N  + 
2 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1364    e. wcel 2160  (class class class)co 5897   CCcc 7840   1c1 7843    + caddc 7845   2c2 9001
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171  ax-1cn 7935  ax-addass 7944
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rex 2474  df-v 2754  df-un 3148  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-iota 5196  df-fv 5243  df-ov 5900  df-2 9009
This theorem is referenced by:  nneoor  9386
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