Theorem 7p2e9 9406

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 9313	. . . . 5 ⊢ 2 = (1 + 1)
2	1	oveq2i 6069	. . . 4 ⊢ (7 + 2) = (7 + (1 + 1))
3		7cn 9338	. . . . 5 ⊢ 7 ∈ ℂ
4		ax-1cn 8236	. . . . 5 ⊢ 1 ∈ ℂ
5	3, 4, 4	addassi 8298	. . . 4 ⊢ ((7 + 1) + 1) = (7 + (1 + 1))
6	2, 5	eqtr4i 2258	. . 3 ⊢ (7 + 2) = ((7 + 1) + 1)
7		df-8 9319	. . . 4 ⊢ 8 = (7 + 1)
8	7	oveq1i 6068	. . 3 ⊢ (8 + 1) = ((7 + 1) + 1)
9	6, 8	eqtr4i 2258	. 2 ⊢ (7 + 2) = (8 + 1)
10		df-9 9320	. 2 ⊢ 9 = (8 + 1)
11	9, 10	eqtr4i 2258	1 ⊢ (7 + 2) = 9

Syntax hints:   = wceq 1398  (class class class)co 6058  1c1 8144   + caddc 8146  2c2 9305  7c7 9310  8c8 9311  9c9 9312

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-addrcl 8240  ax-addass 8245

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-rex 2528  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-iota 5317  df-fv 5365  df-ov 6061  df-2 9313  df-3 9314  df-4 9315  df-5 9316  df-6 9317  df-7 9318  df-8 9319  df-9 9320

This theorem is referenced by:  7p3e10  9801  7t7e49  9840  cos2bnd  12471