Theorem 2p2e4 9364

Description: Two plus two equals four. For more information, see "2+2=4 Trivia" on the Metamath Proof Explorer Home Page: https://us.metamath.org/mpeuni/mmset.html#trivia. (Contributed by NM, 27-May-1999.)

Assertion

Ref	Expression
2p2e4	⊢ (2 + 2) = 4

Proof of Theorem 2p2e4

Step	Hyp	Ref	Expression
1		df-2 9296	. . 3 ⊢ 2 = (1 + 1)
2	1	oveq2i 6061	. 2 ⊢ (2 + 2) = (2 + (1 + 1))
3		df-4 9298	. . 3 ⊢ 4 = (3 + 1)
4		df-3 9297	. . . 4 ⊢ 3 = (2 + 1)
5	4	oveq1i 6060	. . 3 ⊢ (3 + 1) = ((2 + 1) + 1)
6		2cn 9308	. . . 4 ⊢ 2 ∈ ℂ
7		ax-1cn 8220	. . . 4 ⊢ 1 ∈ ℂ
8	6, 7, 7	addassi 8282	. . 3 ⊢ ((2 + 1) + 1) = (2 + (1 + 1))
9	3, 5, 8	3eqtri 2257	. 2 ⊢ 4 = (2 + (1 + 1))
10	2, 9	eqtr4i 2256	1 ⊢ (2 + 2) = 4

Syntax hints:   = wceq 1398  (class class class)co 6050  1c1 8128   + caddc 8130  2c2 9288  3c3 9289  4c4 9290

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-addrcl 8224  ax-addass 8229

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-rex 2526  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-iota 5312  df-fv 5360  df-ov 6053  df-2 9296  df-3 9297  df-4 9298

This theorem is referenced by:  2t2e4  9392  i4  11004  4bc2eq6  11137  resqrexlemover  11695  resqrexlemcalc1  11699  ef01bndlem  12442  6gcd4e2  12691  pythagtriplem1  12963