ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  3p3e6 Unicode version

Theorem 3p3e6 8970
Description: 3 + 3 = 6. (Contributed by NM, 11-May-2004.)
Assertion
Ref Expression
3p3e6  |-  ( 3  +  3 )  =  6

Proof of Theorem 3p3e6
StepHypRef Expression
1 df-3 8888 . . . 4  |-  3  =  ( 2  +  1 )
21oveq2i 5832 . . 3  |-  ( 3  +  3 )  =  ( 3  +  ( 2  +  1 ) )
3 3cn 8903 . . . 4  |-  3  e.  CC
4 2cn 8899 . . . 4  |-  2  e.  CC
5 ax-1cn 7820 . . . 4  |-  1  e.  CC
63, 4, 5addassi 7881 . . 3  |-  ( ( 3  +  2 )  +  1 )  =  ( 3  +  ( 2  +  1 ) )
72, 6eqtr4i 2181 . 2  |-  ( 3  +  3 )  =  ( ( 3  +  2 )  +  1 )
8 df-6 8891 . . 3  |-  6  =  ( 5  +  1 )
9 3p2e5 8969 . . . 4  |-  ( 3  +  2 )  =  5
109oveq1i 5831 . . 3  |-  ( ( 3  +  2 )  +  1 )  =  ( 5  +  1 )
118, 10eqtr4i 2181 . 2  |-  6  =  ( ( 3  +  2 )  +  1 )
127, 11eqtr4i 2181 1  |-  ( 3  +  3 )  =  6
Colors of variables: wff set class
Syntax hints:    = wceq 1335  (class class class)co 5821   1c1 7728    + caddc 7730   2c2 8879   3c3 8880   5c5 8882   6c6 8883
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139  ax-resscn 7819  ax-1cn 7820  ax-1re 7821  ax-addrcl 7824  ax-addass 7829
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-iota 5134  df-fv 5177  df-ov 5824  df-2 8887  df-3 8888  df-4 8889  df-5 8890  df-6 8891
This theorem is referenced by:  3t2e6  8984  binom4  13283  ex-dvds  13293  ex-gcd  13294
  Copyright terms: Public domain W3C validator