Theorem 2p2e4 12370

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 8522 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 12298	. . 3 ⊢ 2 = (1 + 1)
2	1	oveq2i 7421	. 2 ⊢ (2 + 2) = (2 + (1 + 1))
3		df-4 12300	. . 3 ⊢ 4 = (3 + 1)
4		df-3 12299	. . . 4 ⊢ 3 = (2 + 1)
5	4	oveq1i 7420	. . 3 ⊢ (3 + 1) = ((2 + 1) + 1)
6		2cn 12311	. . . 4 ⊢ 2 ∈ ℂ
7		ax-1cn 11153	. . . 4 ⊢ 1 ∈ ℂ
8	6, 7, 7	addassi 11214	. . 3 ⊢ ((2 + 1) + 1) = (2 + (1 + 1))
9	3, 5, 8	3eqtri 2790	. 2 ⊢ 4 = (2 + (1 + 1))
10	2, 9	eqtr4i 2789	1 ⊢ (2 + 2) = 4

Syntax hints:   = wceq 1570  (class class class)co 7410  1c1 11096   + caddc 11098  2c2 12290  3c3 12291  4c4 12292

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-1cn 11153  ax-addcl 11155  ax-addass 11160

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-iota 6492  df-fv 6544  df-ov 7413  df-2 12298  df-3 12299  df-4 12300

This theorem is referenced by:  2t2e4  12399  i4  14236  4bc2eq6  14361  bpoly4  16108  fsumcube  16109  ef01bndlem  16235  6gcd4e2  16591  pythagtriplem1  16871  prmlem2  17175  43prm  17177  1259lem4  17189  2503lem1  17192  2503lem2  17193  2503lem3  17194  4001lem1  17196  4001lem4  17199  cphipval2  25400  quart1lem  27020  log2ub  27114  hgt750lem2  35039  3lexlogpow5ineq1  42821  3lexlogpow5ineq5  42827  3cubeslem3l  43417  3cubeslem3r  43418  wallispi2lem1  46785  stirlinglem8  46795  sqwvfourb  46943  sin3t  47608  cos3t  47609  fmtnorec4  48301  m11nprm  48353  3exp4mod41  48368  gbowgt5  48527  gbpart7  48532  sbgoldbaltlem1  48544  sbgoldbalt  48546  sgoldbeven3prm  48548  mogoldbb  48550  nnsum3primes4  48553  pgnbgreunbgrlem2lem3  48881  2t6m3t4e0  49128  ackval1012  49470  2p2ne5  50618