Theorem 2p2e4 9117

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 9049	. . 3 |- 2 = ( 1 + 1 )
2	1	oveq2i 5933	. 2 |- ( 2 + 2 ) = ( 2 + ( 1 + 1 ) )
3		df-4 9051	. . 3 |- 4 = ( 3 + 1 )
4		df-3 9050	. . . 4 |- 3 = ( 2 + 1 )
5	4	oveq1i 5932	. . 3 |- ( 3 + 1 ) = ( ( 2 + 1 ) + 1 )
6		2cn 9061	. . . 4 |- 2 e. CC
7		ax-1cn 7972	. . . 4 |- 1 e. CC
8	6, 7, 7	addassi 8034	. . 3 |- ( ( 2 + 1 ) + 1 ) = ( 2 + ( 1 + 1 ) )
9	3, 5, 8	3eqtri 2221	. 2 |- 4 = ( 2 + ( 1 + 1 ) )
10	2, 9	eqtr4i 2220	1 |- ( 2 + 2 ) = 4

Syntax hints:   = wceq 1364  (class class class)co 5922   1c1 7880   + caddc 7882   2c2 9041   3c3 9042   4c4 9043

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-addrcl 7976  ax-addass 7981

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-iota 5219  df-fv 5266  df-ov 5925  df-2 9049  df-3 9050  df-4 9051

This theorem is referenced by:  2t2e4  9145  i4  10734  4bc2eq6  10866  resqrexlemover  11175  resqrexlemcalc1  11179  ef01bndlem  11921  6gcd4e2  12162  pythagtriplem1  12434