Theorem 2p2e4 12255

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 8456 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 12188	. . 3 ⊢ 2 = (1 + 1)
2	1	oveq2i 7357	. 2 ⊢ (2 + 2) = (2 + (1 + 1))
3		df-4 12190	. . 3 ⊢ 4 = (3 + 1)
4		df-3 12189	. . . 4 ⊢ 3 = (2 + 1)
5	4	oveq1i 7356	. . 3 ⊢ (3 + 1) = ((2 + 1) + 1)
6		2cn 12200	. . . 4 ⊢ 2 ∈ ℂ
7		ax-1cn 11064	. . . 4 ⊢ 1 ∈ ℂ
8	6, 7, 7	addassi 11122	. . 3 ⊢ ((2 + 1) + 1) = (2 + (1 + 1))
9	3, 5, 8	3eqtri 2758	. 2 ⊢ 4 = (2 + (1 + 1))
10	2, 9	eqtr4i 2757	1 ⊢ (2 + 2) = 4

Syntax hints:   = wceq 1541  (class class class)co 7346  1c1 11007   + caddc 11009  2c2 12180  3c3 12181  4c4 12182

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 2113  ax-9 2121  ax-ext 2703  ax-1cn 11064  ax-addcl 11066  ax-addass 11071

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 2710  df-cleq 2723  df-clel 2806  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-iota 6437  df-fv 6489  df-ov 7349  df-2 12188  df-3 12189  df-4 12190

This theorem is referenced by:  2t2e4  12284  i4  14111  4bc2eq6  14236  bpoly4  15966  fsumcube  15967  ef01bndlem  16093  6gcd4e2  16449  pythagtriplem1  16728  prmlem2  17031  43prm  17033  1259lem4  17045  2503lem1  17048  2503lem2  17049  2503lem3  17050  4001lem1  17052  4001lem4  17055  cphipval2  25168  quart1lem  26792  log2ub  26886  hgt750lem2  34665  3lexlogpow5ineq1  42095  3lexlogpow5ineq5  42101  3cubeslem3l  42727  3cubeslem3r  42728  wallispi2lem1  46117  stirlinglem8  46127  sqwvfourb  46275  fmtnorec4  47588  m11nprm  47640  3exp4mod41  47655  gbowgt5  47801  gbpart7  47806  sbgoldbaltlem1  47818  sbgoldbalt  47820  sgoldbeven3prm  47822  mogoldbb  47824  nnsum3primes4  47827  pgnbgreunbgrlem2lem3  48155  2t6m3t4e0  48387  ackval1012  48730  2p2ne5  49838