Theorem 2p2e4 9248

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 9180	. . 3 |- 2 = ( 1 + 1 )
2	1	oveq2i 6018	. 2 |- ( 2 + 2 ) = ( 2 + ( 1 + 1 ) )
3		df-4 9182	. . 3 |- 4 = ( 3 + 1 )
4		df-3 9181	. . . 4 |- 3 = ( 2 + 1 )
5	4	oveq1i 6017	. . 3 |- ( 3 + 1 ) = ( ( 2 + 1 ) + 1 )
6		2cn 9192	. . . 4 |- 2 e. CC
7		ax-1cn 8103	. . . 4 |- 1 e. CC
8	6, 7, 7	addassi 8165	. . 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 6007   1c1 8011   + caddc 8013   2c2 9172   3c3 9173   4c4 9174

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 8102  ax-1cn 8103  ax-1re 8104  ax-addrcl 8107  ax-addass 8112

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 3889  df-br 4084  df-iota 5278  df-fv 5326  df-ov 6010  df-2 9180  df-3 9181  df-4 9182

This theorem is referenced by:  2t2e4  9276  i4  10876  4bc2eq6  11008  resqrexlemover  11537  resqrexlemcalc1  11541  ef01bndlem  12283  6gcd4e2  12532  pythagtriplem1  12804