Theorem 2p2e4 11021

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 7508 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 10956	. . 3 ⊢ 2 = (1 + 1)
2	1	oveq2i 6560	. 2 ⊢ (2 + 2) = (2 + (1 + 1))
3		df-4 10958	. . 3 ⊢ 4 = (3 + 1)
4		df-3 10957	. . . 4 ⊢ 3 = (2 + 1)
5	4	oveq1i 6559	. . 3 ⊢ (3 + 1) = ((2 + 1) + 1)
6		2cn 10968	. . . 4 ⊢ 2 ∈ ℂ
7		ax-1cn 9873	. . . 4 ⊢ 1 ∈ ℂ
8	6, 7, 7	addassi 9927	. . 3 ⊢ ((2 + 1) + 1) = (2 + (1 + 1))
9	3, 5, 8	3eqtri 2636	. 2 ⊢ 4 = (2 + (1 + 1))
10	2, 9	eqtr4i 2635	1 ⊢ (2 + 2) = 4

Syntax hints:   = wceq 1475  (class class class)co 6549  1c1 9816   + caddc 9818  2c2 10947  3c3 10948  4c4 10949

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-addass 9880  ax-i2m1 9883  ax-1ne0 9884  ax-rrecex 9887  ax-cnre 9888

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-ov 6552  df-2 10956  df-3 10957  df-4 10958

This theorem is referenced by:  2t2e4  11054  i4  12829  4bc2eq6  12978  bpoly4  14629  fsumcube  14630  ef01bndlem  14753  6gcd4e2  15093  pythagtriplem1  15359  prmlem2  15665  43prm  15667  1259lem4  15679  2503lem1  15682  2503lem2  15683  2503lem3  15684  4001lem1  15686  4001lem4  15689  cphipval2  22848  quart1lem  24382  log2ub  24476  wallispi2lem1  38964  stirlinglem8  38974  sqwvfourb  39122  fmtnorec4  39999  m11nprm  40056  3exp4mod41  40071  gbogt5  40184  gbpart7  40189  sgoldbaltlem1  40201  sgoldbalt  40203  nnsum3primes4  40204  2t6m3t4e0  41919  2p2ne5  42353