Theorem 2exp11

Description: Two to the eleventh power is 2048. (Contributed by AV, 16-Aug-2021)

	Ref	Expression
Assertion	2exp11	⊢ ( 2 ↑ ; 1 1 ) = ; ; ; 2 0 4 8

Proof

Step	Hyp	Ref	Expression
1		8p3e11	⊢ ( 8 + 3 ) = ; 1 1
2	1	eqcomi	⊢ ; 1 1 = ( 8 + 3 )
3	2	oveq2i	⊢ ( 2 ↑ ; 1 1 ) = ( 2 ↑ ( 8 + 3 ) )
4		2cn	⊢ 2 ∈ ℂ
5		8nn0	⊢ 8 ∈ ℕ0
6		3nn0	⊢ 3 ∈ ℕ0
7		expadd	⊢ ( ( 2 ∈ ℂ ∧ 8 ∈ ℕ0 ∧ 3 ∈ ℕ0 ) → ( 2 ↑ ( 8 + 3 ) ) = ( ( 2 ↑ 8 ) · ( 2 ↑ 3 ) ) )
8	4 5 6 7	mp3an	⊢ ( 2 ↑ ( 8 + 3 ) ) = ( ( 2 ↑ 8 ) · ( 2 ↑ 3 ) )
9	3 8	eqtri	⊢ ( 2 ↑ ; 1 1 ) = ( ( 2 ↑ 8 ) · ( 2 ↑ 3 ) )
10		2exp8	⊢ ( 2 ↑ 8 ) = ; ; 2 5 6
11		cu2	⊢ ( 2 ↑ 3 ) = 8
12	10 11	oveq12i	⊢ ( ( 2 ↑ 8 ) · ( 2 ↑ 3 ) ) = ( ; ; 2 5 6 · 8 )
13		2nn0	⊢ 2 ∈ ℕ0
14		5nn0	⊢ 5 ∈ ℕ0
15	13 14	deccl	⊢ ; 2 5 ∈ ℕ0
16		6nn0	⊢ 6 ∈ ℕ0
17		eqid	⊢ ; ; 2 5 6 = ; ; 2 5 6
18		4nn0	⊢ 4 ∈ ℕ0
19		0nn0	⊢ 0 ∈ ℕ0
20	13 19	deccl	⊢ ; 2 0 ∈ ℕ0
21		eqid	⊢ ; 2 5 = ; 2 5
22		1nn0	⊢ 1 ∈ ℕ0
23		8cn	⊢ 8 ∈ ℂ
24		8t2e16	⊢ ( 8 · 2 ) = ; 1 6
25	23 4 24	mulcomli	⊢ ( 2 · 8 ) = ; 1 6
26		1p1e2	⊢ ( 1 + 1 ) = 2
27		6p4e10	⊢ ( 6 + 4 ) = ; 1 0
28	22 16 18 25 26 19 27	decaddci	⊢ ( ( 2 · 8 ) + 4 ) = ; 2 0
29		5cn	⊢ 5 ∈ ℂ
30		8t5e40	⊢ ( 8 · 5 ) = ; 4 0
31	23 29 30	mulcomli	⊢ ( 5 · 8 ) = ; 4 0
32	5 13 14 21 19 18 28 31	decmul1c	⊢ ( ; 2 5 · 8 ) = ; ; 2 0 0
33		4cn	⊢ 4 ∈ ℂ
34	33	addid2i	⊢ ( 0 + 4 ) = 4
35	20 19 18 32 34	decaddi	⊢ ( ( ; 2 5 · 8 ) + 4 ) = ; ; 2 0 4
36		6cn	⊢ 6 ∈ ℂ
37		8t6e48	⊢ ( 8 · 6 ) = ; 4 8
38	23 36 37	mulcomli	⊢ ( 6 · 8 ) = ; 4 8
39	5 15 16 17 5 18 35 38	decmul1c	⊢ ( ; ; 2 5 6 · 8 ) = ; ; ; 2 0 4 8
40	12 39	eqtri	⊢ ( ( 2 ↑ 8 ) · ( 2 ↑ 3 ) ) = ; ; ; 2 0 4 8
41	9 40	eqtri	⊢ ( 2 ↑ ; 1 1 ) = ; ; ; 2 0 4 8