Theorem 2p2e4 9060

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 8992	. . 3 |- 2 = ( 1 + 1 )
2	1	oveq2i 5899	. 2 |- ( 2 + 2 ) = ( 2 + ( 1 + 1 ) )
3		df-4 8994	. . 3 |- 4 = ( 3 + 1 )
4		df-3 8993	. . . 4 |- 3 = ( 2 + 1 )
5	4	oveq1i 5898	. . 3 |- ( 3 + 1 ) = ( ( 2 + 1 ) + 1 )
6		2cn 9004	. . . 4 |- 2 e. CC
7		ax-1cn 7918	. . . 4 |- 1 e. CC
8	6, 7, 7	addassi 7979	. . 3 |- ( ( 2 + 1 ) + 1 ) = ( 2 + ( 1 + 1 ) )
9	3, 5, 8	3eqtri 2212	. 2 |- 4 = ( 2 + ( 1 + 1 ) )
10	2, 9	eqtr4i 2211	1 |- ( 2 + 2 ) = 4

Syntax hints:   = wceq 1363  (class class class)co 5888   1c1 7826   + caddc 7828   2c2 8984   3c3 8985   4c4 8986

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169  ax-resscn 7917  ax-1cn 7918  ax-1re 7919  ax-addrcl 7922  ax-addass 7927

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-rex 2471  df-v 2751  df-un 3145  df-in 3147  df-ss 3154  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-iota 5190  df-fv 5236  df-ov 5891  df-2 8992  df-3 8993  df-4 8994

This theorem is referenced by:  2t2e4  9087  i4  10637  4bc2eq6  10768  resqrexlemover  11033  resqrexlemcalc1  11037  ef01bndlem  11778  6gcd4e2  12010  pythagtriplem1  12279