2exp16 - Metamath Proof Explorer

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Theorem 2exp16 17168

Description: Two to the sixteenth power is 65536. (Contributed by Mario Carneiro, 20-Apr-2015.)

**Assertion**
Ref	Expression
2exp16	⊢ (2↑;16) = ;;;;65536

**Proof of Theorem 2exp16**
Step	Hyp	Ref	Expression
1		2nn0 12532	. 2 ⊢ 2 ∈ ℕ₀
2		8nn0 12538	. 2 ⊢ 8 ∈ ℕ₀
3		8cn 12349	. . 3 ⊢ 8 ∈ ℂ
4		2cn 12327	. . 3 ⊢ 2 ∈ ℂ
5		8t2e16 12843	. . 3 ⊢ (8 · 2) = ;16
6	3, 4, 5	mulcomli 11229	. 2 ⊢ (2 · 8) = ;16
7		2exp8 17166	. 2 ⊢ (2↑8) = ;;256
8		5nn0 12535	. . . . 5 ⊢ 5 ∈ ℕ₀
9	1, 8	deccl 12738	. . . 4 ⊢ ;25 ∈ ℕ₀
10		6nn0 12536	. . . 4 ⊢ 6 ∈ ℕ₀
11	9, 10	deccl 12738	. . 3 ⊢ ;;256 ∈ ℕ₀
12		eqid 2765	. . 3 ⊢ ;;256 = ;;256
13		1nn0 12531	. . . . 5 ⊢ 1 ∈ ℕ₀
14	13, 8	deccl 12738	. . . 4 ⊢ ;15 ∈ ℕ₀
15		3nn0 12533	. . . 4 ⊢ 3 ∈ ℕ₀
16	14, 15	deccl 12738	. . 3 ⊢ ;;153 ∈ ℕ₀
17		eqid 2765	. . . 4 ⊢ ;25 = ;25
18		eqid 2765	. . . 4 ⊢ ;;153 = ;;153
19	13, 1	deccl 12738	. . . . 5 ⊢ ;12 ∈ ℕ₀
20	19, 2	deccl 12738	. . . 4 ⊢ ;;128 ∈ ℕ₀
21		4nn0 12534	. . . . . 6 ⊢ 4 ∈ ℕ₀
22	13, 21	deccl 12738	. . . . 5 ⊢ ;14 ∈ ℕ₀
23		eqid 2765	. . . . . 6 ⊢ ;15 = ;15
24		eqid 2765	. . . . . 6 ⊢ ;;128 = ;;128
25		0nn0 12530	. . . . . . . 8 ⊢ 0 ∈ ℕ₀
26	13	dec0h 12750	. . . . . . . 8 ⊢ 1 = ;01
27		eqid 2765	. . . . . . . 8 ⊢ ;12 = ;12
28		0p1e1 12372	. . . . . . . 8 ⊢ (0 + 1) = 1
29		1p2e3 12394	. . . . . . . 8 ⊢ (1 + 2) = 3
30	25, 13, 13, 1, 26, 27, 28, 29	decadd 12782	. . . . . . 7 ⊢ (1 + ;12) = ;13
31		3p1e4 12396	. . . . . . 7 ⊢ (3 + 1) = 4
32	13, 15, 13, 30, 31	decaddi 12788	. . . . . 6 ⊢ ((1 + ;12) + 1) = ;14
33		5cn 12340	. . . . . . 7 ⊢ 5 ∈ ℂ
34		8p5e13 12811	. . . . . . 7 ⊢ (8 + 5) = ;13
35	3, 33, 34	addcomli 11413	. . . . . 6 ⊢ (5 + 8) = ;13
36	13, 8, 19, 2, 23, 24, 32, 15, 35	decaddc 12783	. . . . 5 ⊢ (;15 + ;;128) = ;;143
37		eqid 2765	. . . . . . 7 ⊢ ;14 = ;14
38		4p1e5 12397	. . . . . . 7 ⊢ (4 + 1) = 5
39	13, 21, 13, 37, 38	decaddi 12788	. . . . . 6 ⊢ (;14 + 1) = ;15
40		2t2e4 12415	. . . . . . . 8 ⊢ (2 · 2) = 4
41		1p1e2 12375	. . . . . . . 8 ⊢ (1 + 1) = 2
42	40, 41	oveq12i 7428	. . . . . . 7 ⊢ ((2 · 2) + (1 + 1)) = (4 + 2)
43		4p2e6 12404	. . . . . . 7 ⊢ (4 + 2) = 6
44	42, 43	eqtri 2788	. . . . . 6 ⊢ ((2 · 2) + (1 + 1)) = 6
45		5t2e10 12828	. . . . . . 7 ⊢ (5 · 2) = ;10
46	33	addlidi 11409	. . . . . . 7 ⊢ (0 + 5) = 5
47	13, 25, 8, 45, 46	decaddi 12788	. . . . . 6 ⊢ ((5 · 2) + 5) = ;15
48	1, 8, 13, 8, 17, 39, 1, 8, 13, 44, 47	decmac 12780	. . . . 5 ⊢ ((;25 · 2) + (;14 + 1)) = ;65
49		6t2e12 12832	. . . . . 6 ⊢ (6 · 2) = ;12
50		3cn 12333	. . . . . . 7 ⊢ 3 ∈ ℂ
51		3p2e5 12402	. . . . . . 7 ⊢ (3 + 2) = 5
52	50, 4, 51	addcomli 11413	. . . . . 6 ⊢ (2 + 3) = 5
53	13, 1, 15, 49, 52	decaddi 12788	. . . . 5 ⊢ ((6 · 2) + 3) = ;15
54	9, 10, 22, 15, 12, 36, 1, 8, 13, 48, 53	decmac 12780	. . . 4 ⊢ ((;;256 · 2) + (;15 + ;;128)) = ;;655
55	15	dec0h 12750	. . . . 5 ⊢ 3 = ;03
56	50	addlidi 11409	. . . . . . 7 ⊢ (0 + 3) = 3
57	56, 55	eqtri 2788	. . . . . 6 ⊢ (0 + 3) = ;03
58	4	addlidi 11409	. . . . . . . 8 ⊢ (0 + 2) = 2
59	58	oveq2i 7427	. . . . . . 7 ⊢ ((2 · 5) + (0 + 2)) = ((2 · 5) + 2)
60	33, 4, 45	mulcomli 11229	. . . . . . . 8 ⊢ (2 · 5) = ;10
61	13, 25, 1, 60, 58	decaddi 12788	. . . . . . 7 ⊢ ((2 · 5) + 2) = ;12
62	59, 61	eqtri 2788	. . . . . 6 ⊢ ((2 · 5) + (0 + 2)) = ;12
63		5t5e25 12831	. . . . . . 7 ⊢ (5 · 5) = ;25
64		5p3e8 12408	. . . . . . 7 ⊢ (5 + 3) = 8
65	1, 8, 15, 63, 64	decaddi 12788	. . . . . 6 ⊢ ((5 · 5) + 3) = ;28
66	1, 8, 25, 15, 17, 57, 8, 2, 1, 62, 65	decmac 12780	. . . . 5 ⊢ ((;25 · 5) + (0 + 3)) = ;;128
67		6t5e30 12835	. . . . . 6 ⊢ (6 · 5) = ;30
68	15, 25, 15, 67, 56	decaddi 12788	. . . . 5 ⊢ ((6 · 5) + 3) = ;33
69	9, 10, 25, 15, 12, 55, 8, 15, 15, 66, 68	decmac 12780	. . . 4 ⊢ ((;;256 · 5) + 3) = ;;;1283
70	1, 8, 14, 15, 17, 18, 11, 15, 20, 54, 69	decma2c 12781	. . 3 ⊢ ((;;256 · ;25) + ;;153) = ;;;6553
71		6cn 12343	. . . . . . 7 ⊢ 6 ∈ ℂ
72	71, 4, 49	mulcomli 11229	. . . . . 6 ⊢ (2 · 6) = ;12
73	13, 1, 15, 72, 52	decaddi 12788	. . . . 5 ⊢ ((2 · 6) + 3) = ;15
74	71, 33, 67	mulcomli 11229	. . . . . 6 ⊢ (5 · 6) = ;30
75	15, 25, 15, 74, 56	decaddi 12788	. . . . 5 ⊢ ((5 · 6) + 3) = ;33
76	1, 8, 15, 17, 10, 15, 15, 73, 75	decrmac 12786	. . . 4 ⊢ ((;25 · 6) + 3) = ;;153
77		6t6e36 12836	. . . 4 ⊢ (6 · 6) = ;36
78	10, 9, 10, 12, 10, 15, 76, 77	decmul1c 12793	. . 3 ⊢ (;;256 · 6) = ;;;1536
79	11, 9, 10, 12, 10, 16, 70, 78	decmul2c 12794	. 2 ⊢ (;;256 · ;;256) = ;;;;65536
80	1, 2, 6, 7, 79	numexp2x 17156	1 ⊢ (2↑;16) = ;;;;65536

Colors of variables: wff setvar class

This proof depends on syntax axioms: = wceq 1570 (class class class)co 7416 0cc0 11111 1c1 11112 + caddc 11114 · cmul 11116 2c2 12306 3c3 12307 4c4 12308 5c5 12309 6c6 12310 8c8 12312 cdc 12723 ↑cexp 14111

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7738 ax-cnex 11167 ax-resscn 11168 ax-1cn 11169 ax-icn 11170 ax-addcl 11171 ax-addrcl 11172 ax-mulcl 11173 ax-mulrcl 11174 ax-mulcom 11175 ax-addass 11176 ax-mulass 11177 ax-distr 11178 ax-i2m1 11179 ax-1ne0 11180 ax-1rid 11181 ax-rnegex 11182 ax-rrecex 11183 ax-cnre 11184 ax-pre-lttri 11185 ax-pre-lttrn 11186 ax-pre-ltadd 11187 ax-pre-mulgt0 11188

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4875 df-iun 4960 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7865 df-2nd 7989 df-frecs 8280 df-wrecs 8311 df-recs 8360 df-rdg 8399 df-er 8696 df-en 8946 df-dom 8947 df-sdom 8948 df-pnf 11256 df-mnf 11257 df-xr 11258 df-ltxr 11259 df-le 11260 df-sub 11454 df-neg 11455 df-nn 12245 df-2 12314 df-3 12315 df-4 12316 df-5 12317 df-6 12318 df-7 12319 df-8 12320 df-9 12321 df-n0 12516 df-z 12603 df-dec 12724 df-uz 12875 df-seq 14052 df-exp 14112

This theorem is used by: 1259lem1 17209 fmtno4 48337 ackval41 49508 ackval42 49509 ackval42a 49510

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