ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  6p3e9 Unicode version

Theorem 6p3e9 8983
Description: 6 + 3 = 9. (Contributed by NM, 11-May-2004.)
Assertion
Ref Expression
6p3e9  |-  ( 6  +  3 )  =  9

Proof of Theorem 6p3e9
StepHypRef Expression
1 df-3 8893 . . . 4  |-  3  =  ( 2  +  1 )
21oveq2i 5835 . . 3  |-  ( 6  +  3 )  =  ( 6  +  ( 2  +  1 ) )
3 6cn 8915 . . . 4  |-  6  e.  CC
4 2cn 8904 . . . 4  |-  2  e.  CC
5 ax-1cn 7825 . . . 4  |-  1  e.  CC
63, 4, 5addassi 7886 . . 3  |-  ( ( 6  +  2 )  +  1 )  =  ( 6  +  ( 2  +  1 ) )
72, 6eqtr4i 2181 . 2  |-  ( 6  +  3 )  =  ( ( 6  +  2 )  +  1 )
8 df-9 8899 . . 3  |-  9  =  ( 8  +  1 )
9 6p2e8 8982 . . . 4  |-  ( 6  +  2 )  =  8
109oveq1i 5834 . . 3  |-  ( ( 6  +  2 )  +  1 )  =  ( 8  +  1 )
118, 10eqtr4i 2181 . 2  |-  9  =  ( ( 6  +  2 )  +  1 )
127, 11eqtr4i 2181 1  |-  ( 6  +  3 )  =  9
Colors of variables: wff set class
Syntax hints:    = wceq 1335  (class class class)co 5824   1c1 7733    + caddc 7735   2c2 8884   3c3 8885   6c6 8888   8c8 8890   9c9 8891
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 7824  ax-1cn 7825  ax-1re 7826  ax-addrcl 7829  ax-addass 7834
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 5135  df-fv 5178  df-ov 5827  df-2 8892  df-3 8893  df-4 8894  df-5 8895  df-6 8896  df-7 8897  df-8 8898  df-9 8899
This theorem is referenced by:  3t3e9  8990  6p4e10  9366  ex-gcd  13307
  Copyright terms: Public domain W3C validator