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Theorem add1p1 9164
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 7970 . . 3  |-  ( N  e.  CC  ->  1  e.  CC )
31, 2, 2addassd 7976 . 2  |-  ( N  e.  CC  ->  (
( N  +  1 )  +  1 )  =  ( N  +  ( 1  +  1 ) ) )
4 1p1e2 9032 . . . 4  |-  ( 1  +  1 )  =  2
54a1i 9 . . 3  |-  ( N  e.  CC  ->  (
1  +  1 )  =  2 )
65oveq2d 5888 . 2  |-  ( N  e.  CC  ->  ( N  +  ( 1  +  1 ) )  =  ( N  + 
2 ) )
73, 6eqtrd 2210 1  |-  ( N  e.  CC  ->  (
( N  +  1 )  +  1 )  =  ( N  + 
2 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1353    e. wcel 2148  (class class class)co 5872   CCcc 7806   1c1 7809    + caddc 7811   2c2 8966
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159  ax-1cn 7901  ax-addass 7910
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-v 2739  df-un 3133  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4003  df-iota 5177  df-fv 5223  df-ov 5875  df-2 8974
This theorem is referenced by:  nneoor  9351
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