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Theorem add1p1 8399
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 7249 . . 3  |-  ( N  e.  CC  ->  1  e.  CC )
31, 2, 2addassd 7255 . 2  |-  ( N  e.  CC  ->  (
( N  +  1 )  +  1 )  =  ( N  +  ( 1  +  1 ) ) )
4 1p1e2 8274 . . . 4  |-  ( 1  +  1 )  =  2
54a1i 9 . . 3  |-  ( N  e.  CC  ->  (
1  +  1 )  =  2 )
65oveq2d 5579 . 2  |-  ( N  e.  CC  ->  ( N  +  ( 1  +  1 ) )  =  ( N  + 
2 ) )
73, 6eqtrd 2115 1  |-  ( N  e.  CC  ->  (
( N  +  1 )  +  1 )  =  ( N  + 
2 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1285    e. wcel 1434  (class class class)co 5563   CCcc 7093   1c1 7096    + caddc 7098   2c2 8208
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-1cn 7183  ax-addass 7192
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-rex 2359  df-v 2612  df-un 2986  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-br 3806  df-iota 4917  df-fv 4960  df-ov 5566  df-2 8217
This theorem is referenced by:  nneoor  8582
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