MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  6p3e9 Structured version   Visualization version   GIF version

Theorem 6p3e9 12402
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 12306 . . . 4 3 = (2 + 1)
21oveq2i 7431 . . 3 (6 + 3) = (6 + (2 + 1))
3 6cn 12333 . . . 4 6 ∈ ℂ
4 2cn 12317 . . . 4 2 ∈ ℂ
5 ax-1cn 11196 . . . 4 1 ∈ ℂ
63, 4, 5addassi 11254 . . 3 ((6 + 2) + 1) = (6 + (2 + 1))
72, 6eqtr4i 2759 . 2 (6 + 3) = ((6 + 2) + 1)
8 df-9 12312 . . 3 9 = (8 + 1)
9 6p2e8 12401 . . . 4 (6 + 2) = 8
109oveq1i 7430 . . 3 ((6 + 2) + 1) = (8 + 1)
118, 10eqtr4i 2759 . 2 9 = ((6 + 2) + 1)
127, 11eqtr4i 2759 1 (6 + 3) = 9
Colors of variables: wff setvar class
Syntax hints:   = wceq 1534  (class class class)co 7420  1c1 11139   + caddc 11141  2c2 12297  3c3 12298  6c6 12301  8c8 12303  9c9 12304
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-1cn 11196  ax-addcl 11198  ax-addass 11203
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-iota 6500  df-fv 6556  df-ov 7423  df-2 12305  df-3 12306  df-4 12307  df-5 12308  df-6 12309  df-7 12310  df-8 12311  df-9 12312
This theorem is referenced by:  3t3e9  12409  6p4e10  12779  2exp8  17057  139prm  17092  2503lem2  17106  4001lem1  17109  4001lem2  17110  4001lem4  17112  log2ublem3  26879  ex-gcd  30266  hgt750lem2  34284  kur14lem8  34823  problem5  35273  fmtno5lem1  46893  139prmALT  46936  gboge9  47104  gbpart9  47109  nnsum4primeseven  47140
  Copyright terms: Public domain W3C validator