Theorem 2p2e4 9414

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 9346	. . 3 |- 2 = ( 1 + 1 )
2	1	oveq2i 6090	. 2 |- ( 2 + 2 ) = ( 2 + ( 1 + 1 ) )
3		df-4 9348	. . 3 |- 4 = ( 3 + 1 )
4		df-3 9347	. . . 4 |- 3 = ( 2 + 1 )
5	4	oveq1i 6089	. . 3 |- ( 3 + 1 ) = ( ( 2 + 1 ) + 1 )
6		2cn 9358	. . . 4 |- 2 e. CC
7		ax-1cn 8266	. . . 4 |- 1 e. CC
8	6, 7, 7	addassi 8328	. . 3 |- ( ( 2 + 1 ) + 1 ) = ( 2 + ( 1 + 1 ) )
9	3, 5, 8	3eqtri 2263	. 2 |- 4 = ( 2 + ( 1 + 1 ) )
10	2, 9	eqtr4i 2262	1 |- ( 2 + 2 ) = 4

This proof depends on syntax axioms:   = wceq 1402  (class class class)co 6079   1c1 8174   + caddc 8176   2c2 9338   3c3 9339   4c4 9340

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-addrcl 8270  ax-addass 8275

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-br 4129  df-iota 5335  df-fv 5383  df-ov 6082  df-2 9346  df-3 9347  df-4 9348

This theorem is used by:  2t2e4  9442  i4  11062  4bc2eq6  11196  resqrexlemover  11759  resqrexlemcalc1  11763  ef01bndlem  12506  6gcd4e2  12755  pythagtriplem1  13027  log2ublog2  16069