Theorem 4p4e8 9400

Description: 4 + 4 = 8. (Contributed by NM, 11-May-2004.)

Assertion

Ref	Expression
4p4e8	|- ( 4 + 4 ) = 8

Proof of Theorem 4p4e8

Step	Hyp	Ref	Expression
1		df-4 9315	. . . 4 |- 4 = ( 3 + 1 )
2	1	oveq2i 6069	. . 3 |- ( 4 + 4 ) = ( 4 + ( 3 + 1 ) )
3		4cn 9332	. . . 4 |- 4 e. CC
4		3cn 9329	. . . 4 |- 3 e. CC
5		ax-1cn 8236	. . . 4 |- 1 e. CC
6	3, 4, 5	addassi 8298	. . 3 |- ( ( 4 + 3 ) + 1 ) = ( 4 + ( 3 + 1 ) )
7	2, 6	eqtr4i 2258	. 2 |- ( 4 + 4 ) = ( ( 4 + 3 ) + 1 )
8		df-8 9319	. . 3 |- 8 = ( 7 + 1 )
9		4p3e7 9399	. . . 4 |- ( 4 + 3 ) = 7
10	9	oveq1i 6068	. . 3 |- ( ( 4 + 3 ) + 1 ) = ( 7 + 1 )
11	8, 10	eqtr4i 2258	. 2 |- 8 = ( ( 4 + 3 ) + 1 )
12	7, 11	eqtr4i 2258	1 |- ( 4 + 4 ) = 8

Syntax hints:   = wceq 1398  (class class class)co 6058   1c1 8144   + caddc 8146   3c3 9306   4c4 9307   7c7 9310   8c8 9311

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 2216  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-addrcl 8240  ax-addass 8245

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-rex 2528  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-iota 5317  df-fv 5365  df-ov 6061  df-2 9313  df-3 9314  df-4 9315  df-5 9316  df-6 9317  df-7 9318  df-8 9319

This theorem is referenced by:  4t2e8  9413