Theorem 4p4e8 9288

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 9203	. . . 4 |- 4 = ( 3 + 1 )
2	1	oveq2i 6028	. . 3 |- ( 4 + 4 ) = ( 4 + ( 3 + 1 ) )
3		4cn 9220	. . . 4 |- 4 e. CC
4		3cn 9217	. . . 4 |- 3 e. CC
5		ax-1cn 8124	. . . 4 |- 1 e. CC
6	3, 4, 5	addassi 8186	. . 3 |- ( ( 4 + 3 ) + 1 ) = ( 4 + ( 3 + 1 ) )
7	2, 6	eqtr4i 2255	. 2 |- ( 4 + 4 ) = ( ( 4 + 3 ) + 1 )
8		df-8 9207	. . 3 |- 8 = ( 7 + 1 )
9		4p3e7 9287	. . . 4 |- ( 4 + 3 ) = 7
10	9	oveq1i 6027	. . 3 |- ( ( 4 + 3 ) + 1 ) = ( 7 + 1 )
11	8, 10	eqtr4i 2255	. 2 |- 8 = ( ( 4 + 3 ) + 1 )
12	7, 11	eqtr4i 2255	1 |- ( 4 + 4 ) = 8

Syntax hints:   = wceq 1397  (class class class)co 6017   1c1 8032   + caddc 8034   3c3 9194   4c4 9195   7c7 9198   8c8 9199

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-addrcl 8128  ax-addass 8133

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-iota 5286  df-fv 5334  df-ov 6020  df-2 9201  df-3 9202  df-4 9203  df-5 9204  df-6 9205  df-7 9206  df-8 9207

This theorem is referenced by:  4t2e8  9301