Theorem 2p2e4 11760

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 8149 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 11688	. . 3 ⊢ 2 = (1 + 1)
2	1	oveq2i 7146	. 2 ⊢ (2 + 2) = (2 + (1 + 1))
3		df-4 11690	. . 3 ⊢ 4 = (3 + 1)
4		df-3 11689	. . . 4 ⊢ 3 = (2 + 1)
5	4	oveq1i 7145	. . 3 ⊢ (3 + 1) = ((2 + 1) + 1)
6		2cn 11700	. . . 4 ⊢ 2 ∈ ℂ
7		ax-1cn 10584	. . . 4 ⊢ 1 ∈ ℂ
8	6, 7, 7	addassi 10640	. . 3 ⊢ ((2 + 1) + 1) = (2 + (1 + 1))
9	3, 5, 8	3eqtri 2825	. 2 ⊢ 4 = (2 + (1 + 1))
10	2, 9	eqtr4i 2824	1 ⊢ (2 + 2) = 4

Syntax hints:   = wceq 1538  (class class class)co 7135  1c1 10527   + caddc 10529  2c2 11680  3c3 11681  4c4 11682

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770  ax-1cn 10584  ax-addcl 10586  ax-addass 10591

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443  df-un 3886  df-in 3888  df-ss 3898  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-iota 6283  df-fv 6332  df-ov 7138  df-2 11688  df-3 11689  df-4 11690

This theorem is referenced by:  2t2e4  11789  i4  13563  4bc2eq6  13685  bpoly4  15405  fsumcube  15406  ef01bndlem  15529  6gcd4e2  15876  pythagtriplem1  16143  prmlem2  16445  43prm  16447  1259lem4  16459  2503lem1  16462  2503lem2  16463  2503lem3  16464  4001lem1  16466  4001lem4  16469  cphipval2  23845  quart1lem  25441  log2ub  25535  hgt750lem2  32033  3cubeslem3l  39627  3cubeslem3r  39628  wallispi2lem1  42713  stirlinglem8  42723  sqwvfourb  42871  fmtnorec4  44066  m11nprm  44119  3exp4mod41  44134  gbowgt5  44280  gbpart7  44285  sbgoldbaltlem1  44297  sbgoldbalt  44299  sgoldbeven3prm  44301  mogoldbb  44303  nnsum3primes4  44306  2t6m3t4e0  44750  ackval1012  45104  2p2ne5  45326