Theorem 4p4e8 9429

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 9344	. . . 4 |- 4 = ( 3 + 1 )
2	1	oveq2i 6086	. . 3 |- ( 4 + 4 ) = ( 4 + ( 3 + 1 ) )
3		4cn 9361	. . . 4 |- 4 e. CC
4		3cn 9358	. . . 4 |- 3 e. CC
5		ax-1cn 8262	. . . 4 |- 1 e. CC
6	3, 4, 5	addassi 8324	. . 3 |- ( ( 4 + 3 ) + 1 ) = ( 4 + ( 3 + 1 ) )
7	2, 6	eqtr4i 2262	. 2 |- ( 4 + 4 ) = ( ( 4 + 3 ) + 1 )
8		df-8 9348	. . 3 |- 8 = ( 7 + 1 )
9		4p3e7 9428	. . . 4 |- ( 4 + 3 ) = 7
10	9	oveq1i 6085	. . 3 |- ( ( 4 + 3 ) + 1 ) = ( 7 + 1 )
11	8, 10	eqtr4i 2262	. 2 |- 8 = ( ( 4 + 3 ) + 1 )
12	7, 11	eqtr4i 2262	1 |- ( 4 + 4 ) = 8

Syntax hints:   = wceq 1402  (class class class)co 6075   1c1 8170   + caddc 8172   3c3 9335   4c4 9336   7c7 9339   8c8 9340

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 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 8261  ax-1cn 8262  ax-1re 8263  ax-addrcl 8266  ax-addass 8271

This theorem 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 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-iota 5332  df-fv 5380  df-ov 6078  df-2 9342  df-3 9343  df-4 9344  df-5 9345  df-6 9346  df-7 9347  df-8 9348

This theorem is referenced by:  4t2e8  9442