Theorem 2p2e4 12275

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 8468 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 12208	. . 3 ⊢ 2 = (1 + 1)
2	1	oveq2i 7369	. 2 ⊢ (2 + 2) = (2 + (1 + 1))
3		df-4 12210	. . 3 ⊢ 4 = (3 + 1)
4		df-3 12209	. . . 4 ⊢ 3 = (2 + 1)
5	4	oveq1i 7368	. . 3 ⊢ (3 + 1) = ((2 + 1) + 1)
6		2cn 12220	. . . 4 ⊢ 2 ∈ ℂ
7		ax-1cn 11084	. . . 4 ⊢ 1 ∈ ℂ
8	6, 7, 7	addassi 11142	. . 3 ⊢ ((2 + 1) + 1) = (2 + (1 + 1))
9	3, 5, 8	3eqtri 2763	. 2 ⊢ 4 = (2 + (1 + 1))
10	2, 9	eqtr4i 2762	1 ⊢ (2 + 2) = 4

Syntax hints:   = wceq 1541  (class class class)co 7358  1c1 11027   + caddc 11029  2c2 12200  3c3 12201  4c4 12202

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 2708  ax-1cn 11084  ax-addcl 11086  ax-addass 11091

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 2715  df-cleq 2728  df-clel 2811  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-iota 6448  df-fv 6500  df-ov 7361  df-2 12208  df-3 12209  df-4 12210

This theorem is referenced by:  2t2e4  12304  i4  14127  4bc2eq6  14252  bpoly4  15982  fsumcube  15983  ef01bndlem  16109  6gcd4e2  16465  pythagtriplem1  16744  prmlem2  17047  43prm  17049  1259lem4  17061  2503lem1  17064  2503lem2  17065  2503lem3  17066  4001lem1  17068  4001lem4  17071  cphipval2  25197  quart1lem  26821  log2ub  26915  hgt750lem2  34809  3lexlogpow5ineq1  42308  3lexlogpow5ineq5  42314  3cubeslem3l  42928  3cubeslem3r  42929  wallispi2lem1  46315  stirlinglem8  46325  sqwvfourb  46473  fmtnorec4  47795  m11nprm  47847  3exp4mod41  47862  gbowgt5  48008  gbpart7  48013  sbgoldbaltlem1  48025  sbgoldbalt  48027  sgoldbeven3prm  48029  mogoldbb  48031  nnsum3primes4  48034  pgnbgreunbgrlem2lem3  48362  2t6m3t4e0  48594  ackval1012  48936  2p2ne5  50043