Theorem 2p2e4 9163

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 9095	. . 3 |- 2 = ( 1 + 1 )
2	1	oveq2i 5955	. 2 |- ( 2 + 2 ) = ( 2 + ( 1 + 1 ) )
3		df-4 9097	. . 3 |- 4 = ( 3 + 1 )
4		df-3 9096	. . . 4 |- 3 = ( 2 + 1 )
5	4	oveq1i 5954	. . 3 |- ( 3 + 1 ) = ( ( 2 + 1 ) + 1 )
6		2cn 9107	. . . 4 |- 2 e. CC
7		ax-1cn 8018	. . . 4 |- 1 e. CC
8	6, 7, 7	addassi 8080	. . 3 |- ( ( 2 + 1 ) + 1 ) = ( 2 + ( 1 + 1 ) )
9	3, 5, 8	3eqtri 2230	. 2 |- 4 = ( 2 + ( 1 + 1 ) )
10	2, 9	eqtr4i 2229	1 |- ( 2 + 2 ) = 4

Syntax hints:   = wceq 1373  (class class class)co 5944   1c1 7926   + caddc 7928   2c2 9087   3c3 9088   4c4 9089

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187  ax-resscn 8017  ax-1cn 8018  ax-1re 8019  ax-addrcl 8022  ax-addass 8027

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-rex 2490  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-iota 5232  df-fv 5279  df-ov 5947  df-2 9095  df-3 9096  df-4 9097

This theorem is referenced by:  2t2e4  9191  i4  10787  4bc2eq6  10919  resqrexlemover  11321  resqrexlemcalc1  11325  ef01bndlem  12067  6gcd4e2  12316  pythagtriplem1  12588