Theorem 2p2e4 12273

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 12206	. . 3 ⊢ 2 = (1 + 1)
2	1	oveq2i 7367	. 2 ⊢ (2 + 2) = (2 + (1 + 1))
3		df-4 12208	. . 3 ⊢ 4 = (3 + 1)
4		df-3 12207	. . . 4 ⊢ 3 = (2 + 1)
5	4	oveq1i 7366	. . 3 ⊢ (3 + 1) = ((2 + 1) + 1)
6		2cn 12218	. . . 4 ⊢ 2 ∈ ℂ
7		ax-1cn 11082	. . . 4 ⊢ 1 ∈ ℂ
8	6, 7, 7	addassi 11140	. . 3 ⊢ ((2 + 1) + 1) = (2 + (1 + 1))
9	3, 5, 8	3eqtri 2761	. 2 ⊢ 4 = (2 + (1 + 1))
10	2, 9	eqtr4i 2760	1 ⊢ (2 + 2) = 4

Syntax hints:   = wceq 1541  (class class class)co 7356  1c1 11025   + caddc 11027  2c2 12198  3c3 12199  4c4 12200

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2706  ax-1cn 11082  ax-addcl 11084  ax-addass 11089

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4284  df-if 4478  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-br 5097  df-iota 6446  df-fv 6498  df-ov 7359  df-2 12206  df-3 12207  df-4 12208

This theorem is referenced by:  2t2e4  12302  i4  14125  4bc2eq6  14250  bpoly4  15980  fsumcube  15981  ef01bndlem  16107  6gcd4e2  16463  pythagtriplem1  16742  prmlem2  17045  43prm  17047  1259lem4  17059  2503lem1  17062  2503lem2  17063  2503lem3  17064  4001lem1  17066  4001lem4  17069  cphipval2  25195  quart1lem  26819  log2ub  26913  hgt750lem2  34758  3lexlogpow5ineq1  42247  3lexlogpow5ineq5  42253  3cubeslem3l  42870  3cubeslem3r  42871  wallispi2lem1  46257  stirlinglem8  46267  sqwvfourb  46415  fmtnorec4  47737  m11nprm  47789  3exp4mod41  47804  gbowgt5  47950  gbpart7  47955  sbgoldbaltlem1  47967  sbgoldbalt  47969  sgoldbeven3prm  47971  mogoldbb  47973  nnsum3primes4  47976  pgnbgreunbgrlem2lem3  48304  2t6m3t4e0  48536  ackval1012  48878  2p2ne5  49985