Theorem 7p2e9 9435

Description: 7 + 2 = 9. (Contributed by NM, 11-May-2004.)

Assertion

Ref	Expression
7p2e9	|- ( 7 + 2 ) = 9

Proof of Theorem 7p2e9

Step	Hyp	Ref	Expression
1		df-2 9342	. . . . 5 |- 2 = ( 1 + 1 )
2	1	oveq2i 6086	. . . 4 |- ( 7 + 2 ) = ( 7 + ( 1 + 1 ) )
3		7cn 9367	. . . . 5 |- 7 e. CC
4		ax-1cn 8262	. . . . 5 |- 1 e. CC
5	3, 4, 4	addassi 8324	. . . 4 |- ( ( 7 + 1 ) + 1 ) = ( 7 + ( 1 + 1 ) )
6	2, 5	eqtr4i 2262	. . 3 |- ( 7 + 2 ) = ( ( 7 + 1 ) + 1 )
7		df-8 9348	. . . 4 |- 8 = ( 7 + 1 )
8	7	oveq1i 6085	. . 3 |- ( 8 + 1 ) = ( ( 7 + 1 ) + 1 )
9	6, 8	eqtr4i 2262	. 2 |- ( 7 + 2 ) = ( 8 + 1 )
10		df-9 9349	. 2 |- 9 = ( 8 + 1 )
11	9, 10	eqtr4i 2262	1 |- ( 7 + 2 ) = 9

Syntax hints:   = wceq 1402  (class class class)co 6075   1c1 8170   + caddc 8172   2c2 9334   7c7 9339   8c8 9340   9c9 9341

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  df-9 9349

This theorem is referenced by:  7p3e10  9830  7t7e49  9869  cos2bnd  12505