Theorem 2p2e4 8226

Description: Two plus two equals four. For more information, see "2+2=4 Trivia" on the Metamath Proof Explorer Home Page: http://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 8165	. . 3 |- 2 = ( 1 + 1 )
2	1	oveq2i 5554	. 2 |- ( 2 + 2 ) = ( 2 + ( 1 + 1 ) )
3		df-4 8167	. . 3 |- 4 = ( 3 + 1 )
4		df-3 8166	. . . 4 |- 3 = ( 2 + 1 )
5	4	oveq1i 5553	. . 3 |- ( 3 + 1 ) = ( ( 2 + 1 ) + 1 )
6		2cn 8177	. . . 4 |- 2 e. CC
7		ax-1cn 7131	. . . 4 |- 1 e. CC
8	6, 7, 7	addassi 7189	. . 3 |- ( ( 2 + 1 ) + 1 ) = ( 2 + ( 1 + 1 ) )
9	3, 5, 8	3eqtri 2106	. 2 |- 4 = ( 2 + ( 1 + 1 ) )
10	2, 9	eqtr4i 2105	1 |- ( 2 + 2 ) = 4

Syntax hints:   = wceq 1285  (class class class)co 5543   1c1 7044   + caddc 7046   2c2 8156   3c3 8157   4c4 8158

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-resscn 7130  ax-1cn 7131  ax-1re 7132  ax-addrcl 7135  ax-addass 7140

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-iota 4897  df-fv 4940  df-ov 5546  df-2 8165  df-3 8166  df-4 8167

This theorem is referenced by:  2t2e4  8253  i4  9674  4bc2eq6  9798  resqrexlemover  10034  resqrexlemcalc1  10038  6gcd4e2  10528