Theorem 2p2e4 9269

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 9201	. . 3 ⊢ 2 = (1 + 1)
2	1	oveq2i 6028	. 2 ⊢ (2 + 2) = (2 + (1 + 1))
3		df-4 9203	. . 3 ⊢ 4 = (3 + 1)
4		df-3 9202	. . . 4 ⊢ 3 = (2 + 1)
5	4	oveq1i 6027	. . 3 ⊢ (3 + 1) = ((2 + 1) + 1)
6		2cn 9213	. . . 4 ⊢ 2 ∈ ℂ
7		ax-1cn 8124	. . . 4 ⊢ 1 ∈ ℂ
8	6, 7, 7	addassi 8186	. . 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 1397  (class class class)co 6017  1c1 8032   + caddc 8034  2c2 9193  3c3 9194  4c4 9195

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-addrcl 8128  ax-addass 8133

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-iota 5286  df-fv 5334  df-ov 6020  df-2 9201  df-3 9202  df-4 9203

This theorem is referenced by:  2t2e4  9297  i4  10903  4bc2eq6  11035  resqrexlemover  11570  resqrexlemcalc1  11574  ef01bndlem  12316  6gcd4e2  12565  pythagtriplem1  12837