Theorem 2p2e4 12390

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 8532 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 12318	. . 3 ⊢ 2 = (1 + 1)
2	1	oveq2i 7430	. 2 ⊢ (2 + 2) = (2 + (1 + 1))
3		df-4 12320	. . 3 ⊢ 4 = (3 + 1)
4		df-3 12319	. . . 4 ⊢ 3 = (2 + 1)
5	4	oveq1i 7429	. . 3 ⊢ (3 + 1) = ((2 + 1) + 1)
6		2cn 12331	. . . 4 ⊢ 2 ∈ ℂ
7		ax-1cn 11173	. . . 4 ⊢ 1 ∈ ℂ
8	6, 7, 7	addassi 11234	. . 3 ⊢ ((2 + 1) + 1) = (2 + (1 + 1))
9	3, 5, 8	3eqtri 2792	. 2 ⊢ 4 = (2 + (1 + 1))
10	2, 9	eqtr4i 2791	1 ⊢ (2 + 2) = 4

This proof depends on syntax axioms:   = wceq 1570  (class class class)co 7419  1c1 11116   + caddc 11118  2c2 12310  3c3 12311  4c4 12312

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-1cn 11173  ax-addcl 11175  ax-addass 11180

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-iota 6496  df-fv 6548  df-ov 7422  df-2 12318  df-3 12319  df-4 12320

This theorem is used by:  2t2e4  12419  i4  14258  4bc2eq6  14383  bpoly4  16135  fsumcube  16136  ef01bndlem  16262  6gcd4e2  16618  pythagtriplem1  16898  prmlem2  17202  43prm  17204  1259lem4  17216  2503lem1  17219  2503lem2  17220  2503lem3  17221  4001lem1  17223  4001lem4  17226  cphipval2  25451  quart1lem  27071  log2ub  27165  hgt750lem2  35104  3lexlogpow5ineq1  42879  3lexlogpow5ineq5  42885  3cubeslem3l  43475  3cubeslem3r  43476  wallispi2lem1  46843  stirlinglem8  46853  sqwvfourb  47001  sin3t  47666  cos3t  47667  fmtnorec4  48359  m11nprm  48411  3exp4mod41  48426  gbowgt5  48585  gbpart7  48590  sbgoldbaltlem1  48602  sbgoldbalt  48604  sgoldbeven3prm  48606  mogoldbb  48608  nnsum3primes4  48611  pgnbgreunbgrlem2lem3  48939  2t6m3t4e0  49185  ackval1012  49527  2p2ne5  50675