Theorem 5p4e9 9435

Description: 5 + 4 = 9. (Contributed by NM, 11-May-2004.)

Assertion

Ref	Expression
5p4e9	|- ( 5 + 4 ) = 9

Proof of Theorem 5p4e9

Step	Hyp	Ref	Expression
1		df-4 9347	. . . 4 |- 4 = ( 3 + 1 )
2	1	oveq2i 6089	. . 3 |- ( 5 + 4 ) = ( 5 + ( 3 + 1 ) )
3		5cn 9366	. . . 4 |- 5 e. CC
4		3cn 9361	. . . 4 |- 3 e. CC
5		ax-1cn 8265	. . . 4 |- 1 e. CC
6	3, 4, 5	addassi 8327	. . 3 |- ( ( 5 + 3 ) + 1 ) = ( 5 + ( 3 + 1 ) )
7	2, 6	eqtr4i 2262	. 2 |- ( 5 + 4 ) = ( ( 5 + 3 ) + 1 )
8		df-9 9352	. . 3 |- 9 = ( 8 + 1 )
9		5p3e8 9434	. . . 4 |- ( 5 + 3 ) = 8
10	9	oveq1i 6088	. . 3 |- ( ( 5 + 3 ) + 1 ) = ( 8 + 1 )
11	8, 10	eqtr4i 2262	. 2 |- 9 = ( ( 5 + 3 ) + 1 )
12	7, 11	eqtr4i 2262	1 |- ( 5 + 4 ) = 9

Syntax hints:   = wceq 1402  (class class class)co 6078   1c1 8173   + caddc 8175   3c3 9338   4c4 9339   5c5 9340   8c8 9343   9c9 9344

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 8264  ax-1cn 8265  ax-1re 8266  ax-addrcl 8269  ax-addass 8274

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 3714  df-pr 3715  df-op 3717  df-uni 3934  df-br 4129  df-iota 5335  df-fv 5383  df-ov 6081  df-2 9345  df-3 9346  df-4 9347  df-5 9348  df-6 9349  df-7 9350  df-8 9351  df-9 9352

This theorem is referenced by:  5p5e10  9829