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Theorem 2p2e4 8226
Description: Two plus two equals four. For more information, see "2+2=4 Trivia" on the Metamath Proof Explorer Home Page: http://us.metamath.org/mpeuni/mmset.html#trivia. (Contributed by NM, 27-May-1999.)
Assertion
Ref Expression
2p2e4  |-  ( 2  +  2 )  =  4

Proof of Theorem 2p2e4
StepHypRef Expression
1 df-2 8165 . . 3  |-  2  =  ( 1  +  1 )
21oveq2i 5554 . 2  |-  ( 2  +  2 )  =  ( 2  +  ( 1  +  1 ) )
3 df-4 8167 . . 3  |-  4  =  ( 3  +  1 )
4 df-3 8166 . . . 4  |-  3  =  ( 2  +  1 )
54oveq1i 5553 . . 3  |-  ( 3  +  1 )  =  ( ( 2  +  1 )  +  1 )
6 2cn 8177 . . . 4  |-  2  e.  CC
7 ax-1cn 7131 . . . 4  |-  1  e.  CC
86, 7, 7addassi 7189 . . 3  |-  ( ( 2  +  1 )  +  1 )  =  ( 2  +  ( 1  +  1 ) )
93, 5, 83eqtri 2106 . 2  |-  4  =  ( 2  +  ( 1  +  1 ) )
102, 9eqtr4i 2105 1  |-  ( 2  +  2 )  =  4
Colors of variables: wff set class
Syntax hints:    = wceq 1285  (class class class)co 5543   1c1 7044    + caddc 7046   2c2 8156   3c3 8157   4c4 8158
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-resscn 7130  ax-1cn 7131  ax-1re 7132  ax-addrcl 7135  ax-addass 7140
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-iota 4897  df-fv 4940  df-ov 5546  df-2 8165  df-3 8166  df-4 8167
This theorem is referenced by:  2t2e4  8253  i4  9674  4bc2eq6  9798  resqrexlemover  10034  resqrexlemcalc1  10038  6gcd4e2  10528
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