MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  5p4e9 Unicode version

Theorem 5p4e9 9862
Description: 5 + 4 = 9. (Contributed by NM, 11-May-2004.)
Assertion
Ref Expression
5p4e9  |-  ( 5  +  4 )  =  9

Proof of Theorem 5p4e9
StepHypRef Expression
1 df-4 9806 . . . 4  |-  4  =  ( 3  +  1 )
21oveq2i 5869 . . 3  |-  ( 5  +  4 )  =  ( 5  +  ( 3  +  1 ) )
3 5re 9821 . . . . 5  |-  5  e.  RR
43recni 8849 . . . 4  |-  5  e.  CC
5 3cn 9818 . . . 4  |-  3  e.  CC
6 ax-1cn 8795 . . . 4  |-  1  e.  CC
74, 5, 6addassi 8845 . . 3  |-  ( ( 5  +  3 )  +  1 )  =  ( 5  +  ( 3  +  1 ) )
82, 7eqtr4i 2306 . 2  |-  ( 5  +  4 )  =  ( ( 5  +  3 )  +  1 )
9 df-9 9811 . . 3  |-  9  =  ( 8  +  1 )
10 5p3e8 9861 . . . 4  |-  ( 5  +  3 )  =  8
1110oveq1i 5868 . . 3  |-  ( ( 5  +  3 )  +  1 )  =  ( 8  +  1 )
129, 11eqtr4i 2306 . 2  |-  9  =  ( ( 5  +  3 )  +  1 )
138, 12eqtr4i 2306 1  |-  ( 5  +  4 )  =  9
Colors of variables: wff set class
Syntax hints:    = wceq 1623  (class class class)co 5858   1c1 8738    + caddc 8740   3c3 9796   4c4 9797   5c5 9798   8c8 9801   9c9 9802
This theorem is referenced by:  5p5e10  9863  139prm  13125  1259lem4  13132  2503lem2  13136  4001lem1  13139  4001lem2  13140
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-addass 8802  ax-i2m1 8805  ax-1ne0 8806  ax-rrecex 8809  ax-cnre 8810
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-ov 5861  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811
  Copyright terms: Public domain W3C validator