Theorem 4p3e7 9428

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 9343	. . . 4 |- 3 = ( 2 + 1 )
2	1	oveq2i 6086	. . 3 |- ( 4 + 3 ) = ( 4 + ( 2 + 1 ) )
3		4cn 9361	. . . 4 |- 4 e. CC
4		2cn 9354	. . . 4 |- 2 e. CC
5		ax-1cn 8262	. . . 4 |- 1 e. CC
6	3, 4, 5	addassi 8324	. . 3 |- ( ( 4 + 2 ) + 1 ) = ( 4 + ( 2 + 1 ) )
7	2, 6	eqtr4i 2262	. 2 |- ( 4 + 3 ) = ( ( 4 + 2 ) + 1 )
8		df-7 9347	. . 3 |- 7 = ( 6 + 1 )
9		4p2e6 9427	. . . 4 |- ( 4 + 2 ) = 6
10	9	oveq1i 6085	. . 3 |- ( ( 4 + 2 ) + 1 ) = ( 6 + 1 )
11	8, 10	eqtr4i 2262	. 2 |- 7 = ( ( 4 + 2 ) + 1 )
12	7, 11	eqtr4i 2262	1 |- ( 4 + 3 ) = 7

Syntax hints:   = wceq 1402  (class class class)co 6075   1c1 8170   + caddc 8172   2c2 9334   3c3 9335   4c4 9336   6c6 9338   7c7 9339

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

This theorem is referenced by:  4p4e8  9429  2lgslem3d  16129  2lgsoddprmlem3d  16143