Theorem 2p2e4 12302

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 8466 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

Step	Hyp	Ref	Expression
1		df-2 12235	. . 3 ⊢ 2 = (1 + 1)
2	1	oveq2i 7367	. 2 ⊢ (2 + 2) = (2 + (1 + 1))
3		df-4 12237	. . 3 ⊢ 4 = (3 + 1)
4		df-3 12236	. . . 4 ⊢ 3 = (2 + 1)
5	4	oveq1i 7366	. . 3 ⊢ (3 + 1) = ((2 + 1) + 1)
6		2cn 12247	. . . 4 ⊢ 2 ∈ ℂ
7		ax-1cn 11087	. . . 4 ⊢ 1 ∈ ℂ
8	6, 7, 7	addassi 11146	. . 3 ⊢ ((2 + 1) + 1) = (2 + (1 + 1))
9	3, 5, 8	3eqtri 2766	. 2 ⊢ 4 = (2 + (1 + 1))
10	2, 9	eqtr4i 2765	1 ⊢ (2 + 2) = 4

Syntax hints:   = wceq 1547  (class class class)co 7356  1c1 11030   + caddc 11032  2c2 12227  3c3 12228  4c4 12229

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-1cn 11087  ax-addcl 11089  ax-addass 11094

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-iota 6441  df-fv 6493  df-ov 7359  df-2 12235  df-3 12236  df-4 12237

This theorem is referenced by:  2t2e4  12331  i4  14157  4bc2eq6  14282  bpoly4  16015  fsumcube  16016  ef01bndlem  16142  6gcd4e2  16498  pythagtriplem1  16778  prmlem2  17081  43prm  17083  1259lem4  17095  2503lem1  17098  2503lem2  17099  2503lem3  17100  4001lem1  17102  4001lem4  17105  cphipval2  25226  quart1lem  26837  log2ub  26931  hgt750lem2  34836  3lexlogpow5ineq1  42539  3lexlogpow5ineq5  42545  3cubeslem3l  43135  3cubeslem3r  43136  wallispi2lem1  46514  stirlinglem8  46524  sqwvfourb  46672  sin3t  47334  cos3t  47335  fmtnorec4  48027  m11nprm  48079  3exp4mod41  48094  gbowgt5  48253  gbpart7  48258  sbgoldbaltlem1  48270  sbgoldbalt  48272  sgoldbeven3prm  48274  mogoldbb  48276  nnsum3primes4  48279  pgnbgreunbgrlem2lem3  48607  2t6m3t4e0  48839  ackval1012  49181  2p2ne5  50288