Theorem 2p2e4 9431

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 9363	. . 3 ⊢ 2 = (1 + 1)
2	1	oveq2i 6096	. 2 ⊢ (2 + 2) = (2 + (1 + 1))
3		df-4 9365	. . 3 ⊢ 4 = (3 + 1)
4		df-3 9364	. . . 4 ⊢ 3 = (2 + 1)
5	4	oveq1i 6095	. . 3 ⊢ (3 + 1) = ((2 + 1) + 1)
6		2cn 9375	. . . 4 ⊢ 2 ∈ ℂ
7		ax-1cn 8272	. . . 4 ⊢ 1 ∈ ℂ
8	6, 7, 7	addassi 8334	. . 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 6085  1c1 8180   + caddc 8182  2c2 9355  3c3 9356  4c4 9357

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 8271  ax-1cn 8272  ax-1re 8273  ax-addrcl 8276  ax-addass 8281

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 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-iota 5337  df-fv 5385  df-ov 6088  df-2 9363  df-3 9364  df-4 9365

This theorem is used by:  2t2e4  9459  i4  11079  4bc2eq6  11213  resqrexlemover  11776  resqrexlemcalc1  11780  ef01bndlem  12523  6gcd4e2  12772  pythagtriplem1  13044  log2ublog2  16086