Theorem 2p2e4 12422

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 8591 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 12350	. . 3 ⊢ 2 = (1 + 1)
2	1	oveq2i 7454	. 2 ⊢ (2 + 2) = (2 + (1 + 1))
3		df-4 12352	. . 3 ⊢ 4 = (3 + 1)
4		df-3 12351	. . . 4 ⊢ 3 = (2 + 1)
5	4	oveq1i 7453	. . 3 ⊢ (3 + 1) = ((2 + 1) + 1)
6		2cn 12362	. . . 4 ⊢ 2 ∈ ℂ
7		ax-1cn 11236	. . . 4 ⊢ 1 ∈ ℂ
8	6, 7, 7	addassi 11294	. . 3 ⊢ ((2 + 1) + 1) = (2 + (1 + 1))
9	3, 5, 8	3eqtri 2772	. 2 ⊢ 4 = (2 + (1 + 1))
10	2, 9	eqtr4i 2771	1 ⊢ (2 + 2) = 4

Syntax hints:   = wceq 1537  (class class class)co 7443  1c1 11179   + caddc 11181  2c2 12342  3c3 12343  4c4 12344

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-1cn 11236  ax-addcl 11238  ax-addass 11243

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-iota 6520  df-fv 6576  df-ov 7446  df-2 12350  df-3 12351  df-4 12352

This theorem is referenced by:  2t2e4  12451  i4  14247  4bc2eq6  14372  bpoly4  16101  fsumcube  16102  ef01bndlem  16226  6gcd4e2  16579  pythagtriplem1  16857  prmlem2  17161  43prm  17163  1259lem4  17175  2503lem1  17178  2503lem2  17179  2503lem3  17180  4001lem1  17182  4001lem4  17185  cphipval2  25286  quart1lem  26908  log2ub  27002  hgt750lem2  34621  3lexlogpow5ineq1  42003  3lexlogpow5ineq5  42009  3cubeslem3l  42634  3cubeslem3r  42635  wallispi2lem1  45981  stirlinglem8  45991  sqwvfourb  46139  fmtnorec4  47412  m11nprm  47464  3exp4mod41  47479  gbowgt5  47625  gbpart7  47630  sbgoldbaltlem1  47642  sbgoldbalt  47644  sgoldbeven3prm  47646  mogoldbb  47648  nnsum3primes4  47651  2t6m3t4e0  48062  ackval1012  48413  2p2ne5  48881