Theorem 2p2e4 8699

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 8637	. . 3 |- 2 = ( 1 + 1 )
2	1	oveq2i 5717	. 2 |- ( 2 + 2 ) = ( 2 + ( 1 + 1 ) )
3		df-4 8639	. . 3 |- 4 = ( 3 + 1 )
4		df-3 8638	. . . 4 |- 3 = ( 2 + 1 )
5	4	oveq1i 5716	. . 3 |- ( 3 + 1 ) = ( ( 2 + 1 ) + 1 )
6		2cn 8649	. . . 4 |- 2 e. CC
7		ax-1cn 7588	. . . 4 |- 1 e. CC
8	6, 7, 7	addassi 7646	. . 3 |- ( ( 2 + 1 ) + 1 ) = ( 2 + ( 1 + 1 ) )
9	3, 5, 8	3eqtri 2124	. 2 |- 4 = ( 2 + ( 1 + 1 ) )
10	2, 9	eqtr4i 2123	1 |- ( 2 + 2 ) = 4

Syntax hints:   = wceq 1299  (class class class)co 5706   1c1 7501   + caddc 7503   2c2 8629   3c3 8630   4c4 8631

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-resscn 7587  ax-1cn 7588  ax-1re 7589  ax-addrcl 7592  ax-addass 7597

This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-rex 2381  df-v 2643  df-un 3025  df-in 3027  df-ss 3034  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-br 3876  df-iota 5024  df-fv 5067  df-ov 5709  df-2 8637  df-3 8638  df-4 8639

This theorem is referenced by:  2t2e4  8726  i4  10236  4bc2eq6  10361  resqrexlemover  10622  resqrexlemcalc1  10626  ef01bndlem  11261  6gcd4e2  11476