Theorem 2p2e4 12399

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 8528 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 12327	. . 3 ⊢ 2 = (1 + 1)
2	1	oveq2i 7424	. 2 ⊢ (2 + 2) = (2 + (1 + 1))
3		df-4 12329	. . 3 ⊢ 4 = (3 + 1)
4		df-3 12328	. . . 4 ⊢ 3 = (2 + 1)
5	4	oveq1i 7423	. . 3 ⊢ (3 + 1) = ((2 + 1) + 1)
6		2cn 12340	. . . 4 ⊢ 2 ∈ ℂ
7		ax-1cn 11182	. . . 4 ⊢ 1 ∈ ℂ
8	6, 7, 7	addassi 11243	. . 3 ⊢ ((2 + 1) + 1) = (2 + (1 + 1))
9	3, 5, 8	3eqtri 2787	. 2 ⊢ 4 = (2 + (1 + 1))
10	2, 9	eqtr4i 2786	1 ⊢ (2 + 2) = 4

This proof depends on syntax axioms:   = wceq 1570  (class class class)co 7413  1c1 11125   + caddc 11127  2c2 12319  3c3 12320  4c4 12321

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 2147  ax-9 2155  ax-ext 2732  ax-1cn 11182  ax-addcl 11184  ax-addass 11189

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 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-iota 6489  df-fv 6541  df-ov 7416  df-2 12327  df-3 12328  df-4 12329

This theorem is used by:  2t2e4  12428  i4  14268  4bc2eq6  14393  bpoly4  16145  fsumcube  16146  ef01bndlem  16272  6gcd4e2  16628  pythagtriplem1  16908  prmlem2  17212  43prm  17214  1259lem4  17226  2503lem1  17229  2503lem2  17230  2503lem3  17231  4001lem1  17233  4001lem4  17236  cphipval2  25469  quart1lem  27092  log2ub  27186  hgt750lem2  35160  3lexlogpow5ineq1  42920  3lexlogpow5ineq5  42926  4p4e8ALT  43125  2p4e6  43133  2p6e8  43135  3p4e7  43137  3cubeslem3l  43531  3cubeslem3r  43532  wallispi2lem1  46899  stirlinglem8  46909  sqwvfourb  47057  sin3t  47735  cos3t  47736  goldpolyfactor  47745  fmtnorec4  48452  m11nprm  48504  3exp4mod41  48519  gbowgt5  48678  gbpart7  48683  sbgoldbaltlem1  48695  sbgoldbalt  48697  sgoldbeven3prm  48699  mogoldbb  48701  nnsum3primes4  48704  pgnbgreunbgrlem2lem3  49032  2t6m3t4e0  49278  ackval1012  49620  2p2ne5  50769