Theorem 2p2e4 9134

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 9066	. . 3 |- 2 = ( 1 + 1 )
2	1	oveq2i 5936	. 2 |- ( 2 + 2 ) = ( 2 + ( 1 + 1 ) )
3		df-4 9068	. . 3 |- 4 = ( 3 + 1 )
4		df-3 9067	. . . 4 |- 3 = ( 2 + 1 )
5	4	oveq1i 5935	. . 3 |- ( 3 + 1 ) = ( ( 2 + 1 ) + 1 )
6		2cn 9078	. . . 4 |- 2 e. CC
7		ax-1cn 7989	. . . 4 |- 1 e. CC
8	6, 7, 7	addassi 8051	. . 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 5925   1c1 7897   + caddc 7899   2c2 9058   3c3 9059   4c4 9060

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 7988  ax-1cn 7989  ax-1re 7990  ax-addrcl 7993  ax-addass 7998

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 3629  df-pr 3630  df-op 3632  df-uni 3841  df-br 4035  df-iota 5220  df-fv 5267  df-ov 5928  df-2 9066  df-3 9067  df-4 9068

This theorem is referenced by:  2t2e4  9162  i4  10751  4bc2eq6  10883  resqrexlemover  11192  resqrexlemcalc1  11196  ef01bndlem  11938  6gcd4e2  12187  pythagtriplem1  12459