Theorem 2p2e4 9312

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 9244	. . 3 ⊢ 2 = (1 + 1)
2	1	oveq2i 6039	. 2 ⊢ (2 + 2) = (2 + (1 + 1))
3		df-4 9246	. . 3 ⊢ 4 = (3 + 1)
4		df-3 9245	. . . 4 ⊢ 3 = (2 + 1)
5	4	oveq1i 6038	. . 3 ⊢ (3 + 1) = ((2 + 1) + 1)
6		2cn 9256	. . . 4 ⊢ 2 ∈ ℂ
7		ax-1cn 8168	. . . 4 ⊢ 1 ∈ ℂ
8	6, 7, 7	addassi 8230	. . 3 ⊢ ((2 + 1) + 1) = (2 + (1 + 1))
9	3, 5, 8	3eqtri 2256	. 2 ⊢ 4 = (2 + (1 + 1))
10	2, 9	eqtr4i 2255	1 ⊢ (2 + 2) = 4

Syntax hints:   = wceq 1398  (class class class)co 6028  1c1 8076   + caddc 8078  2c2 9236  3c3 9237  4c4 9238

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 2213  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-addrcl 8172  ax-addass 8177

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-rex 2517  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-iota 5293  df-fv 5341  df-ov 6031  df-2 9244  df-3 9245  df-4 9246

This theorem is referenced by:  2t2e4  9340  i4  10950  4bc2eq6  11082  resqrexlemover  11633  resqrexlemcalc1  11637  ef01bndlem  12380  6gcd4e2  12629  pythagtriplem1  12901