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Theorem 2p2e4 11761
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 8157 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 11689 . . 3 2 = (1 + 1)
21oveq2i 7156 . 2 (2 + 2) = (2 + (1 + 1))
3 df-4 11691 . . 3 4 = (3 + 1)
4 df-3 11690 . . . 4 3 = (2 + 1)
54oveq1i 7155 . . 3 (3 + 1) = ((2 + 1) + 1)
6 2cn 11701 . . . 4 2 ∈ ℂ
7 ax-1cn 10584 . . . 4 1 ∈ ℂ
86, 7, 7addassi 10640 . . 3 ((2 + 1) + 1) = (2 + (1 + 1))
93, 5, 83eqtri 2848 . 2 4 = (2 + (1 + 1))
102, 9eqtr4i 2847 1 (2 + 2) = 4
Colors of variables: wff setvar class
Syntax hints:   = wceq 1528  (class class class)co 7145  1c1 10527   + caddc 10529  2c2 11681  3c3 11682  4c4 11683
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 2793  ax-1cn 10584  ax-addcl 10586  ax-addass 10591
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 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rex 3144  df-rab 3147  df-v 3497  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4833  df-br 5059  df-iota 6308  df-fv 6357  df-ov 7148  df-2 11689  df-3 11690  df-4 11691
This theorem is referenced by:  2t2e4  11790  i4  13557  4bc2eq6  13679  bpoly4  15403  fsumcube  15404  ef01bndlem  15527  6gcd4e2  15876  pythagtriplem1  16143  prmlem2  16443  43prm  16445  1259lem4  16457  2503lem1  16460  2503lem2  16461  2503lem3  16462  4001lem1  16464  4001lem4  16467  cphipval2  23773  quart1lem  25360  log2ub  25455  hgt750lem2  31823  3cubeslem3l  39163  3cubeslem3r  39164  wallispi2lem1  42237  stirlinglem8  42247  sqwvfourb  42395  fmtnorec4  43558  m11nprm  43613  3exp4mod41  43628  gbowgt5  43774  gbpart7  43779  sbgoldbaltlem1  43791  sbgoldbalt  43793  sgoldbeven3prm  43795  mogoldbb  43797  nnsum3primes4  43800  2t6m3t4e0  44294  2p2ne5  44797
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