Theorem 2p2e4 9329

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 9261	. . 3 |- 2 = ( 1 + 1 )
2	1	oveq2i 6039	. 2 |- ( 2 + 2 ) = ( 2 + ( 1 + 1 ) )
3		df-4 9263	. . 3 |- 4 = ( 3 + 1 )
4		df-3 9262	. . . 4 |- 3 = ( 2 + 1 )
5	4	oveq1i 6038	. . 3 |- ( 3 + 1 ) = ( ( 2 + 1 ) + 1 )
6		2cn 9273	. . . 4 |- 2 e. CC
7		ax-1cn 8185	. . . 4 |- 1 e. CC
8	6, 7, 7	addassi 8247	. . 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 8093   + caddc 8095   2c2 9253   3c3 9254   4c4 9255

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 8184  ax-1cn 8185  ax-1re 8186  ax-addrcl 8189  ax-addass 8194

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 9261  df-3 9262  df-4 9263

This theorem is referenced by:  2t2e4  9357  i4  10967  4bc2eq6  11099  resqrexlemover  11650  resqrexlemcalc1  11654  ef01bndlem  12397  6gcd4e2  12646  pythagtriplem1  12918