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Theorem 2p2e4 11760
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 8155 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 11688 . . 3 2 = (1 + 1)
21oveq2i 7156 . 2 (2 + 2) = (2 + (1 + 1))
3 df-4 11690 . . 3 4 = (3 + 1)
4 df-3 11689 . . . 4 3 = (2 + 1)
54oveq1i 7155 . . 3 (3 + 1) = ((2 + 1) + 1)
6 2cn 11700 . . . 4 2 ∈ ℂ
7 ax-1cn 10583 . . . 4 1 ∈ ℂ
86, 7, 7addassi 10639 . . 3 ((2 + 1) + 1) = (2 + (1 + 1))
93, 5, 83eqtri 2845 . 2 4 = (2 + (1 + 1))
102, 9eqtr4i 2844 1 (2 + 2) = 4
Colors of variables: wff setvar class
Syntax hints:   = wceq 1528  (class class class)co 7145  1c1 10526   + caddc 10528  2c2 11680  3c3 11681  4c4 11682
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-1cn 10583  ax-addcl 10585  ax-addass 10590
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-iota 6307  df-fv 6356  df-ov 7148  df-2 11688  df-3 11689  df-4 11690
This theorem is referenced by:  2t2e4  11789  i4  13555  4bc2eq6  13677  bpoly4  15401  fsumcube  15402  ef01bndlem  15525  6gcd4e2  15874  pythagtriplem1  16141  prmlem2  16441  43prm  16443  1259lem4  16455  2503lem1  16458  2503lem2  16459  2503lem3  16460  4001lem1  16462  4001lem4  16465  cphipval2  23771  quart1lem  25360  log2ub  25454  hgt750lem2  31822  3cubeslem3l  39161  3cubeslem3r  39162  wallispi2lem1  42233  stirlinglem8  42243  sqwvfourb  42391  fmtnorec4  43588  m11nprm  43643  3exp4mod41  43658  gbowgt5  43804  gbpart7  43809  sbgoldbaltlem1  43821  sbgoldbalt  43823  sgoldbeven3prm  43825  mogoldbb  43827  nnsum3primes4  43830  2t6m3t4e0  44324  2p2ne5  44827
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