Theorem 2p2e4 9233

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 9165	. . 3 ⊢ 2 = (1 + 1)
2	1	oveq2i 6011	. 2 ⊢ (2 + 2) = (2 + (1 + 1))
3		df-4 9167	. . 3 ⊢ 4 = (3 + 1)
4		df-3 9166	. . . 4 ⊢ 3 = (2 + 1)
5	4	oveq1i 6010	. . 3 ⊢ (3 + 1) = ((2 + 1) + 1)
6		2cn 9177	. . . 4 ⊢ 2 ∈ ℂ
7		ax-1cn 8088	. . . 4 ⊢ 1 ∈ ℂ
8	6, 7, 7	addassi 8150	. . 3 ⊢ ((2 + 1) + 1) = (2 + (1 + 1))
9	3, 5, 8	3eqtri 2254	. 2 ⊢ 4 = (2 + (1 + 1))
10	2, 9	eqtr4i 2253	1 ⊢ (2 + 2) = 4

Syntax hints:   = wceq 1395  (class class class)co 6000  1c1 7996   + caddc 7998  2c2 9157  3c3 9158  4c4 9159

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-addrcl 8092  ax-addass 8097

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-br 4083  df-iota 5277  df-fv 5325  df-ov 6003  df-2 9165  df-3 9166  df-4 9167

This theorem is referenced by:  2t2e4  9261  i4  10859  4bc2eq6  10991  resqrexlemover  11516  resqrexlemcalc1  11520  ef01bndlem  12262  6gcd4e2  12511  pythagtriplem1  12783