2exp16 - Intuitionistic Logic Explorer

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Theorem 2exp16 13218

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 9582	. 2 ⊢ 2 ∈ ℕ₀
2		8nn0 9588	. 2 ⊢ 8 ∈ ℕ₀
3		8cn 9391	. . 3 ⊢ 8 ∈ ℂ
4		2cn 9376	. . 3 ⊢ 2 ∈ ℂ
5		8t2e16 9893	. . 3 ⊢ (8 · 2) = ;16
6	3, 4, 5	mulcomli 8333	. 2 ⊢ (2 · 8) = ;16
7		2exp8 13216	. 2 ⊢ (2↑8) = ;;256
8		5nn0 9585	. . . . 5 ⊢ 5 ∈ ℕ₀
9	1, 8	deccl 9793	. . . 4 ⊢ ;25 ∈ ℕ₀
10		6nn0 9586	. . . 4 ⊢ 6 ∈ ℕ₀
11	9, 10	deccl 9793	. . 3 ⊢ ;;256 ∈ ℕ₀
12		eqid 2238	. . 3 ⊢ ;;256 = ;;256
13		1nn0 9581	. . . . 5 ⊢ 1 ∈ ℕ₀
14	13, 8	deccl 9793	. . . 4 ⊢ ;15 ∈ ℕ₀
15		3nn0 9583	. . . 4 ⊢ 3 ∈ ℕ₀
16	14, 15	deccl 9793	. . 3 ⊢ ;;153 ∈ ℕ₀
17		eqid 2238	. . . 4 ⊢ ;25 = ;25
18		eqid 2238	. . . 4 ⊢ ;;153 = ;;153
19	13, 1	deccl 9793	. . . . 5 ⊢ ;12 ∈ ℕ₀
20	19, 2	deccl 9793	. . . 4 ⊢ ;;128 ∈ ℕ₀
21		4nn0 9584	. . . . . 6 ⊢ 4 ∈ ℕ₀
22	13, 21	deccl 9793	. . . . 5 ⊢ ;14 ∈ ℕ₀
23		eqid 2238	. . . . . 6 ⊢ ;15 = ;15
24		eqid 2238	. . . . . 6 ⊢ ;;128 = ;;128
25		0nn0 9580	. . . . . . . 8 ⊢ 0 ∈ ℕ₀
26	13	dec0h 9800	. . . . . . . 8 ⊢ 1 = ;01
27		eqid 2238	. . . . . . . 8 ⊢ ;12 = ;12
28		0p1e1 9419	. . . . . . . 8 ⊢ (0 + 1) = 1
29		1p2e3 9440	. . . . . . . 8 ⊢ (1 + 2) = 3
30	25, 13, 13, 1, 26, 27, 28, 29	decadd 9832	. . . . . . 7 ⊢ (1 + ;12) = ;13
31		3p1e4 9441	. . . . . . 7 ⊢ (3 + 1) = 4
32	13, 15, 13, 30, 31	decaddi 9838	. . . . . 6 ⊢ ((1 + ;12) + 1) = ;14
33		5cn 9385	. . . . . . 7 ⊢ 5 ∈ ℂ
34		8p5e13 9861	. . . . . . 7 ⊢ (8 + 5) = ;13
35	3, 33, 34	addcomli 8471	. . . . . 6 ⊢ (5 + 8) = ;13
36	13, 8, 19, 2, 23, 24, 32, 15, 35	decaddc 9833	. . . . 5 ⊢ (;15 + ;;128) = ;;143
37		eqid 2238	. . . . . . 7 ⊢ ;14 = ;14
38		4p1e5 9442	. . . . . . 7 ⊢ (4 + 1) = 5
39	13, 21, 13, 37, 38	decaddi 9838	. . . . . 6 ⊢ (;14 + 1) = ;15
40		2t2e4 9460	. . . . . . . 8 ⊢ (2 · 2) = 4
41		1p1e2 9422	. . . . . . . 8 ⊢ (1 + 1) = 2
42	40, 41	oveq12i 6097	. . . . . . 7 ⊢ ((2 · 2) + (1 + 1)) = (4 + 2)
43		4p2e6 9449	. . . . . . 7 ⊢ (4 + 2) = 6
44	42, 43	eqtri 2259	. . . . . 6 ⊢ ((2 · 2) + (1 + 1)) = 6
45		5t2e10 9878	. . . . . . 7 ⊢ (5 · 2) = ;10
46	33	addlidi 8469	. . . . . . 7 ⊢ (0 + 5) = 5
47	13, 25, 8, 45, 46	decaddi 9838	. . . . . 6 ⊢ ((5 · 2) + 5) = ;15
48	1, 8, 13, 8, 17, 39, 1, 8, 13, 44, 47	decmac 9830	. . . . 5 ⊢ ((;25 · 2) + (;14 + 1)) = ;65
49		6t2e12 9882	. . . . . 6 ⊢ (6 · 2) = ;12
50		3cn 9380	. . . . . . 7 ⊢ 3 ∈ ℂ
51		3p2e5 9447	. . . . . . 7 ⊢ (3 + 2) = 5
52	50, 4, 51	addcomli 8471	. . . . . 6 ⊢ (2 + 3) = 5
53	13, 1, 15, 49, 52	decaddi 9838	. . . . 5 ⊢ ((6 · 2) + 3) = ;15
54	9, 10, 22, 15, 12, 36, 1, 8, 13, 48, 53	decmac 9830	. . . 4 ⊢ ((;;256 · 2) + (;15 + ;;128)) = ;;655
55	15	dec0h 9800	. . . . 5 ⊢ 3 = ;03
56	50	addlidi 8469	. . . . . . 7 ⊢ (0 + 3) = 3
57	56, 55	eqtri 2259	. . . . . 6 ⊢ (0 + 3) = ;03
58	4	addlidi 8469	. . . . . . . 8 ⊢ (0 + 2) = 2
59	58	oveq2i 6096	. . . . . . 7 ⊢ ((2 · 5) + (0 + 2)) = ((2 · 5) + 2)
60	33, 4, 45	mulcomli 8333	. . . . . . . 8 ⊢ (2 · 5) = ;10
61	13, 25, 1, 60, 58	decaddi 9838	. . . . . . 7 ⊢ ((2 · 5) + 2) = ;12
62	59, 61	eqtri 2259	. . . . . 6 ⊢ ((2 · 5) + (0 + 2)) = ;12
63		5t5e25 9881	. . . . . . 7 ⊢ (5 · 5) = ;25
64		5p3e8 9453	. . . . . . 7 ⊢ (5 + 3) = 8
65	1, 8, 15, 63, 64	decaddi 9838	. . . . . 6 ⊢ ((5 · 5) + 3) = ;28
66	1, 8, 25, 15, 17, 57, 8, 2, 1, 62, 65	decmac 9830	. . . . 5 ⊢ ((;25 · 5) + (0 + 3)) = ;;128
67		6t5e30 9885	. . . . . 6 ⊢ (6 · 5) = ;30
68	15, 25, 15, 67, 56	decaddi 9838	. . . . 5 ⊢ ((6 · 5) + 3) = ;33
69	9, 10, 25, 15, 12, 55, 8, 15, 15, 66, 68	decmac 9830	. . . 4 ⊢ ((;;256 · 5) + 3) = ;;;1283
70	1, 8, 14, 15, 17, 18, 11, 15, 20, 54, 69	decma2c 9831	. . 3 ⊢ ((;;256 · ;25) + ;;153) = ;;;6553
71		6cn 9387	. . . . . . 7 ⊢ 6 ∈ ℂ
72	71, 4, 49	mulcomli 8333	. . . . . 6 ⊢ (2 · 6) = ;12
73	13, 1, 15, 72, 52	decaddi 9838	. . . . 5 ⊢ ((2 · 6) + 3) = ;15
74	71, 33, 67	mulcomli 8333	. . . . . 6 ⊢ (5 · 6) = ;30
75	15, 25, 15, 74, 56	decaddi 9838	. . . . 5 ⊢ ((5 · 6) + 3) = ;33
76	1, 8, 15, 17, 10, 15, 15, 73, 75	decrmac 9836	. . . 4 ⊢ ((;25 · 6) + 3) = ;;153
77		6t6e36 9886	. . . 4 ⊢ (6 · 6) = ;36
78	10, 9, 10, 12, 10, 15, 76, 77	decmul1c 9843	. . 3 ⊢ (;;256 · 6) = ;;;1536
79	11, 9, 10, 12, 10, 16, 70, 78	decmul2c 9844	. 2 ⊢ (;;256 · ;;256) = ;;;;65536
80	1, 2, 6, 7, 79	numexp2x 13206	1 ⊢ (2↑;16) = ;;;;65536

Colors of variables: wff set class

This proof depends on syntax axioms: = wceq 1402 (class class class)co 6085 0cc0 8179 1c1 8180 + caddc 8182 · cmul 8184 2c2 9356 3c3 9357 4c4 9358 5c5 9359 6c6 9360 8c8 9362 cdc 9779 ↑cexp 10977

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 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-14 2212 ax-ext 2220 ax-coll 4246 ax-sep 4249 ax-nul 4259 ax-pow 4311 ax-pr 4346 ax-un 4578 ax-setind 4684 ax-iinf 4735 ax-cnex 8270 ax-resscn 8271 ax-1cn 8272 ax-1re 8273 ax-icn 8274 ax-addcl 8275 ax-addrcl 8276 ax-mulcl 8277 ax-mulrcl 8278 ax-addcom 8279 ax-mulcom 8280 ax-addass 8281 ax-mulass 8282 ax-distr 8283 ax-i2m1 8284 ax-0lt1 8285 ax-1rid 8286 ax-0id 8287 ax-rnegex 8288 ax-precex 8289 ax-cnre 8290 ax-pre-ltirr 8291 ax-pre-ltwlin 8292 ax-pre-lttrn 8293 ax-pre-apti 8294 ax-pre-ltadd 8295 ax-pre-mulgt0 8296 ax-pre-mulext 8297

This proof depends on definitions: df-bi 117 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rmo 2536 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-if 3639 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-int 3971 df-iun 4014 df-br 4131 df-opab 4193 df-mpt 4194 df-tr 4230 df-id 4438 df-po 4441 df-iso 4442 df-iord 4511 df-on 4513 df-ilim 4514 df-suc 4516 df-iom 4738 df-xp 4780 df-rel 4781 df-cnv 4782 df-co 4783 df-dm 4784 df-rn 4785 df-res 4786 df-ima 4787 df-iota 5337 df-fun 5379 df-fn 5380 df-f 5381 df-f1 5382 df-fo 5383 df-f1o 5384 df-fv 5385 df-riota 6038 df-ov 6088 df-oprab 6089 df-mpo 6090 df-1st 6374 df-2nd 6375 df-recs 6576 df-frec 6662 df-pnf 8362 df-mnf 8363 df-xr 8364 df-ltxr 8365 df-le 8366 df-sub 8499 df-neg 8500 df-reap 8904 df-ap 8911 df-div 9004 df-inn 9306 df-2 9364 df-3 9365 df-4 9366 df-5 9367 df-6 9368 df-7 9369 df-8 9370 df-9 9371 df-n0 9566 df-z 9647 df-dec 9780 df-uz 9924 df-seqfrec 10887 df-exp 10978

This theorem is used by: (None)

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