Theorem 2p2e4 11144

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 7621 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 11079	. . 3 ⊢ 2 = (1 + 1)
2	1	oveq2i 6661	. 2 ⊢ (2 + 2) = (2 + (1 + 1))
3		df-4 11081	. . 3 ⊢ 4 = (3 + 1)
4		df-3 11080	. . . 4 ⊢ 3 = (2 + 1)
5	4	oveq1i 6660	. . 3 ⊢ (3 + 1) = ((2 + 1) + 1)
6		2cn 11091	. . . 4 ⊢ 2 ∈ ℂ
7		ax-1cn 9994	. . . 4 ⊢ 1 ∈ ℂ
8	6, 7, 7	addassi 10048	. . 3 ⊢ ((2 + 1) + 1) = (2 + (1 + 1))
9	3, 5, 8	3eqtri 2648	. 2 ⊢ 4 = (2 + (1 + 1))
10	2, 9	eqtr4i 2647	1 ⊢ (2 + 2) = 4

Syntax hints:   = wceq 1483  (class class class)co 6650  1c1 9937   + caddc 9939  2c2 11070  3c3 11071  4c4 11072

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-addass 10001  ax-i2m1 10004  ax-1ne0 10005  ax-rrecex 10008  ax-cnre 10009

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653  df-2 11079  df-3 11080  df-4 11081

This theorem is referenced by:  2t2e4  11177  i4  12967  4bc2eq6  13116  bpoly4  14790  fsumcube  14791  ef01bndlem  14914  6gcd4e2  15255  pythagtriplem1  15521  prmlem2  15827  43prm  15829  1259lem4  15841  2503lem1  15844  2503lem2  15845  2503lem3  15846  4001lem1  15848  4001lem4  15851  cphipval2  23040  quart1lem  24582  log2ub  24676  hgt750lem2  30730  wallispi2lem1  40288  stirlinglem8  40298  sqwvfourb  40446  fmtnorec4  41461  m11nprm  41518  3exp4mod41  41533  gbowgt5  41650  gbpart7  41655  sbgoldbaltlem1  41667  sbgoldbalt  41669  sgoldbeven3prm  41671  mogoldbb  41673  nnsum3primes4  41676  2t6m3t4e0  42126  2p2ne5  42544