Theorem 2p2e4 11088

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 7566 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 11023	. . 3 ⊢ 2 = (1 + 1)
2	1	oveq2i 6615	. 2 ⊢ (2 + 2) = (2 + (1 + 1))
3		df-4 11025	. . 3 ⊢ 4 = (3 + 1)
4		df-3 11024	. . . 4 ⊢ 3 = (2 + 1)
5	4	oveq1i 6614	. . 3 ⊢ (3 + 1) = ((2 + 1) + 1)
6		2cn 11035	. . . 4 ⊢ 2 ∈ ℂ
7		ax-1cn 9938	. . . 4 ⊢ 1 ∈ ℂ
8	6, 7, 7	addassi 9992	. . 3 ⊢ ((2 + 1) + 1) = (2 + (1 + 1))
9	3, 5, 8	3eqtri 2647	. 2 ⊢ 4 = (2 + (1 + 1))
10	2, 9	eqtr4i 2646	1 ⊢ (2 + 2) = 4

Syntax hints:   = wceq 1480  (class class class)co 6604  1c1 9881   + caddc 9883  2c2 11014  3c3 11015  4c4 11016

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-addass 9945  ax-i2m1 9948  ax-1ne0 9949  ax-rrecex 9952  ax-cnre 9953

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-iota 5810  df-fv 5855  df-ov 6607  df-2 11023  df-3 11024  df-4 11025

This theorem is referenced by:  2t2e4  11121  i4  12907  4bc2eq6  13056  bpoly4  14715  fsumcube  14716  ef01bndlem  14839  6gcd4e2  15179  pythagtriplem1  15445  prmlem2  15751  43prm  15753  1259lem4  15765  2503lem1  15768  2503lem2  15769  2503lem3  15770  4001lem1  15772  4001lem4  15775  cphipval2  22948  quart1lem  24482  log2ub  24576  wallispi2lem1  39592  stirlinglem8  39602  sqwvfourb  39750  fmtnorec4  40757  m11nprm  40814  3exp4mod41  40829  gbogt5  40942  gbpart7  40947  sgoldbaltlem1  40959  sgoldbalt  40961  nnsum3primes4  40962  2t6m3t4e0  41411  2p2ne5  41844