MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  2p2e4 Structured version   Visualization version   GIF version

Theorem 2p2e4 11350
Description: Two plus two equals four. For more information, see "2+2=4 Trivia" on the Metamath Proof Explorer Home Page: mmset.html#trivia. This proof is simple, but it depends on many other proof steps because 2 and 4 are complex numbers and thus it depends on our construction of complex numbers. The proof o2p2e4 7778 is similar but proves 2 + 2 = 4 using ordinal natural numbers (finite integers starting at 0), so that proof depends on fewer intermediate steps. (Contributed by NM, 27-May-1999.)
Assertion
Ref Expression
2p2e4 (2 + 2) = 4

Proof of Theorem 2p2e4
StepHypRef Expression
1 df-2 11284 . . 3 2 = (1 + 1)
21oveq2i 6806 . 2 (2 + 2) = (2 + (1 + 1))
3 df-4 11286 . . 3 4 = (3 + 1)
4 df-3 11285 . . . 4 3 = (2 + 1)
54oveq1i 6805 . . 3 (3 + 1) = ((2 + 1) + 1)
6 2cn 11296 . . . 4 2 ∈ ℂ
7 ax-1cn 10199 . . . 4 1 ∈ ℂ
86, 7, 7addassi 10253 . . 3 ((2 + 1) + 1) = (2 + (1 + 1))
93, 5, 83eqtri 2797 . 2 4 = (2 + (1 + 1))
102, 9eqtr4i 2796 1 (2 + 2) = 4
Colors of variables: wff setvar class
Syntax hints:   = wceq 1631  (class class class)co 6795  1c1 10142   + caddc 10144  2c2 11275  3c3 11276  4c4 11277
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-resscn 10198  ax-1cn 10199  ax-icn 10200  ax-addcl 10201  ax-addrcl 10202  ax-mulcl 10203  ax-mulrcl 10204  ax-addass 10206  ax-i2m1 10209  ax-1ne0 10210  ax-rrecex 10213  ax-cnre 10214
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-iota 5993  df-fv 6038  df-ov 6798  df-2 11284  df-3 11285  df-4 11286
This theorem is referenced by:  2t2e4  11383  i4  13173  4bc2eq6  13319  bpoly4  14995  fsumcube  14996  ef01bndlem  15119  6gcd4e2  15462  pythagtriplem1  15727  prmlem2  16033  43prm  16035  1259lem4  16047  2503lem1  16050  2503lem2  16051  2503lem3  16052  4001lem1  16054  4001lem4  16057  cphipval2  23258  quart1lem  24802  log2ub  24896  hgt750lem2  31069  wallispi2lem1  40802  stirlinglem8  40812  sqwvfourb  40960  fmtnorec4  41986  m11nprm  42043  3exp4mod41  42058  gbowgt5  42175  gbpart7  42180  sbgoldbaltlem1  42192  sbgoldbalt  42194  sgoldbeven3prm  42196  mogoldbb  42198  nnsum3primes4  42201  2t6m3t4e0  42651  2p2ne5  43072
  Copyright terms: Public domain W3C validator