Theorem 4p3e7 9381

Description: 4 + 3 = 7. (Contributed by NM, 11-May-2004.)

Assertion

Ref	Expression
4p3e7	|- ( 4 + 3 ) = 7

Proof of Theorem 4p3e7

Step	Hyp	Ref	Expression
1		df-3 9296	. . . 4 |- 3 = ( 2 + 1 )
2	1	oveq2i 6060	. . 3 |- ( 4 + 3 ) = ( 4 + ( 2 + 1 ) )
3		4cn 9314	. . . 4 |- 4 e. CC
4		2cn 9307	. . . 4 |- 2 e. CC
5		ax-1cn 8219	. . . 4 |- 1 e. CC
6	3, 4, 5	addassi 8281	. . 3 |- ( ( 4 + 2 ) + 1 ) = ( 4 + ( 2 + 1 ) )
7	2, 6	eqtr4i 2256	. 2 |- ( 4 + 3 ) = ( ( 4 + 2 ) + 1 )
8		df-7 9300	. . 3 |- 7 = ( 6 + 1 )
9		4p2e6 9380	. . . 4 |- ( 4 + 2 ) = 6
10	9	oveq1i 6059	. . 3 |- ( ( 4 + 2 ) + 1 ) = ( 6 + 1 )
11	8, 10	eqtr4i 2256	. 2 |- 7 = ( ( 4 + 2 ) + 1 )
12	7, 11	eqtr4i 2256	1 |- ( 4 + 3 ) = 7

Syntax hints:   = wceq 1398  (class class class)co 6049   1c1 8127   + caddc 8129   2c2 9287   3c3 9288   4c4 9289   6c6 9291   7c7 9292

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214  ax-resscn 8218  ax-1cn 8219  ax-1re 8220  ax-addrcl 8223  ax-addass 8228

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-rex 2526  df-v 2814  df-un 3214  df-in 3216  df-ss 3223  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-br 4109  df-iota 5311  df-fv 5359  df-ov 6052  df-2 9295  df-3 9296  df-4 9297  df-5 9298  df-6 9299  df-7 9300

This theorem is referenced by:  4p4e8  9382  2lgslem3d  15961  2lgsoddprmlem3d  15975