Theorem 2p2e4 8037

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 7973	. . 3 |- 2 = ( 1 + 1 )
2	1	oveq2i 5523	. 2 |- ( 2 + 2 ) = ( 2 + ( 1 + 1 ) )
3		df-4 7975	. . 3 |- 4 = ( 3 + 1 )
4		df-3 7974	. . . 4 |- 3 = ( 2 + 1 )
5	4	oveq1i 5522	. . 3 |- ( 3 + 1 ) = ( ( 2 + 1 ) + 1 )
6		2cn 7986	. . . 4 |- 2 e. CC
7		ax-1cn 6977	. . . 4 |- 1 e. CC
8	6, 7, 7	addassi 7035	. . 3 |- ( ( 2 + 1 ) + 1 ) = ( 2 + ( 1 + 1 ) )
9	3, 5, 8	3eqtri 2064	. 2 |- 4 = ( 2 + ( 1 + 1 ) )
10	2, 9	eqtr4i 2063	1 |- ( 2 + 2 ) = 4

Syntax hints:   = wceq 1243  (class class class)co 5512   1c1 6890   + caddc 6892   2c2 7964   3c3 7965   4c4 7966

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-resscn 6976  ax-1cn 6977  ax-1re 6978  ax-addrcl 6981  ax-addass 6986

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-iota 4867  df-fv 4910  df-ov 5515  df-2 7973  df-3 7974  df-4 7975

This theorem is referenced by:  2t2e4  8069  i4  9355  resqrexlemover  9608  resqrexlemcalc1  9612