Theorem 2p2e4 12258

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 8459 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 12191	. . 3 ⊢ 2 = (1 + 1)
2	1	oveq2i 7360	. 2 ⊢ (2 + 2) = (2 + (1 + 1))
3		df-4 12193	. . 3 ⊢ 4 = (3 + 1)
4		df-3 12192	. . . 4 ⊢ 3 = (2 + 1)
5	4	oveq1i 7359	. . 3 ⊢ (3 + 1) = ((2 + 1) + 1)
6		2cn 12203	. . . 4 ⊢ 2 ∈ ℂ
7		ax-1cn 11067	. . . 4 ⊢ 1 ∈ ℂ
8	6, 7, 7	addassi 11125	. . 3 ⊢ ((2 + 1) + 1) = (2 + (1 + 1))
9	3, 5, 8	3eqtri 2756	. 2 ⊢ 4 = (2 + (1 + 1))
10	2, 9	eqtr4i 2755	1 ⊢ (2 + 2) = 4

Syntax hints:   = wceq 1540  (class class class)co 7349  1c1 11010   + caddc 11012  2c2 12183  3c3 12184  4c4 12185

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-1cn 11067  ax-addcl 11069  ax-addass 11074

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-iota 6438  df-fv 6490  df-ov 7352  df-2 12191  df-3 12192  df-4 12193

This theorem is referenced by:  2t2e4  12287  i4  14111  4bc2eq6  14236  bpoly4  15966  fsumcube  15967  ef01bndlem  16093  6gcd4e2  16449  pythagtriplem1  16728  prmlem2  17031  43prm  17033  1259lem4  17045  2503lem1  17048  2503lem2  17049  2503lem3  17050  4001lem1  17052  4001lem4  17055  cphipval2  25139  quart1lem  26763  log2ub  26857  hgt750lem2  34620  3lexlogpow5ineq1  42027  3lexlogpow5ineq5  42033  3cubeslem3l  42659  3cubeslem3r  42660  wallispi2lem1  46052  stirlinglem8  46062  sqwvfourb  46210  fmtnorec4  47533  m11nprm  47585  3exp4mod41  47600  gbowgt5  47746  gbpart7  47751  sbgoldbaltlem1  47763  sbgoldbalt  47765  sgoldbeven3prm  47767  mogoldbb  47769  nnsum3primes4  47772  pgnbgreunbgrlem2lem3  48100  2t6m3t4e0  48332  ackval1012  48675  2p2ne5  49783