Theorem 2p2e4 9111

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 9043	. . 3 |- 2 = ( 1 + 1 )
2	1	oveq2i 5930	. 2 |- ( 2 + 2 ) = ( 2 + ( 1 + 1 ) )
3		df-4 9045	. . 3 |- 4 = ( 3 + 1 )
4		df-3 9044	. . . 4 |- 3 = ( 2 + 1 )
5	4	oveq1i 5929	. . 3 |- ( 3 + 1 ) = ( ( 2 + 1 ) + 1 )
6		2cn 9055	. . . 4 |- 2 e. CC
7		ax-1cn 7967	. . . 4 |- 1 e. CC
8	6, 7, 7	addassi 8029	. . 3 |- ( ( 2 + 1 ) + 1 ) = ( 2 + ( 1 + 1 ) )
9	3, 5, 8	3eqtri 2218	. 2 |- 4 = ( 2 + ( 1 + 1 ) )
10	2, 9	eqtr4i 2217	1 |- ( 2 + 2 ) = 4

Syntax hints:   = wceq 1364  (class class class)co 5919   1c1 7875   + caddc 7877   2c2 9035   3c3 9036   4c4 9037

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-addrcl 7971  ax-addass 7976

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rex 2478  df-v 2762  df-un 3158  df-in 3160  df-ss 3167  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-iota 5216  df-fv 5263  df-ov 5922  df-2 9043  df-3 9044  df-4 9045

This theorem is referenced by:  2t2e4  9139  i4  10716  4bc2eq6  10848  resqrexlemover  11157  resqrexlemcalc1  11161  ef01bndlem  11902  6gcd4e2  12135  pythagtriplem1  12406