3exp7 - Mathbox for metakunt

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Theorem 3exp7 42538

Description: 3 to the power of 7 equals 2187. (Contributed by metakunt, 21-Aug-2024.)

**Assertion**
Ref	Expression
3exp7	⊢ (3↑7) = ;;;2187

**Proof of Theorem 3exp7**
Step	Hyp	Ref	Expression
1		3nn0 12446	. 2 ⊢ 3 ∈ ℕ₀
2		6nn0 12449	. 2 ⊢ 6 ∈ ℕ₀
3		6p1e7 12315	. 2 ⊢ (6 + 1) = 7
4		7nn0 12450	. . . 4 ⊢ 7 ∈ ℕ₀
5		2nn0 12445	. . . 4 ⊢ 2 ∈ ℕ₀
6	4, 5	deccl 12650	. . 3 ⊢ ;72 ∈ ℕ₀
7		9nn0 12452	. . 3 ⊢ 9 ∈ ℕ₀
8		3cn 12253	. . . . 5 ⊢ 3 ∈ ℂ
9		2cn 12247	. . . . 5 ⊢ 2 ∈ ℂ
10		3t2e6 12333	. . . . 5 ⊢ (3 · 2) = 6
11	8, 9, 10	mulcomli 11145	. . . 4 ⊢ (2 · 3) = 6
12		3exp3 17053	. . . 4 ⊢ (3↑3) = ;27
13	5, 4	deccl 12650	. . . . 5 ⊢ ;27 ∈ ℕ₀
14		eqid 2739	. . . . 5 ⊢ ;27 = ;27
15		1nn0 12444	. . . . . 6 ⊢ 1 ∈ ℕ₀
16		8nn0 12451	. . . . . 6 ⊢ 8 ∈ ℕ₀
17	15, 16	deccl 12650	. . . . 5 ⊢ ;18 ∈ ℕ₀
18		0nn0 12443	. . . . . 6 ⊢ 0 ∈ ℕ₀
19	5	dec0h 12657	. . . . . 6 ⊢ 2 = ;02
20		eqid 2739	. . . . . 6 ⊢ ;18 = ;18
21	13	nn0cni 12440	. . . . . . . . 9 ⊢ ;27 ∈ ℂ
22	21	mul02i 11326	. . . . . . . 8 ⊢ (0 · ;27) = 0
23		6cn 12263	. . . . . . . . 9 ⊢ 6 ∈ ℂ
24		ax-1cn 11087	. . . . . . . . 9 ⊢ 1 ∈ ℂ
25	23, 24, 3	addcomli 11329	. . . . . . . 8 ⊢ (1 + 6) = 7
26	22, 25	oveq12i 7368	. . . . . . 7 ⊢ ((0 · ;27) + (1 + 6)) = (0 + 7)
27		7cn 12266	. . . . . . . 8 ⊢ 7 ∈ ℂ
28	27	addlidi 11325	. . . . . . 7 ⊢ (0 + 7) = 7
29	26, 28	eqtri 2762	. . . . . 6 ⊢ ((0 · ;27) + (1 + 6)) = 7
30	16	dec0h 12657	. . . . . . 7 ⊢ 8 = ;08
31		2t2e4 12331	. . . . . . . . 9 ⊢ (2 · 2) = 4
32	9	addlidi 11325	. . . . . . . . 9 ⊢ (0 + 2) = 2
33	31, 32	oveq12i 7368	. . . . . . . 8 ⊢ ((2 · 2) + (0 + 2)) = (4 + 2)
34		4p2e6 12320	. . . . . . . 8 ⊢ (4 + 2) = 6
35	33, 34	eqtri 2762	. . . . . . 7 ⊢ ((2 · 2) + (0 + 2)) = 6
36		4nn0 12447	. . . . . . . 8 ⊢ 4 ∈ ℕ₀
37		7t2e14 12744	. . . . . . . . 9 ⊢ (7 · 2) = ;14
38	27, 9, 37	mulcomli 11145	. . . . . . . 8 ⊢ (2 · 7) = ;14
39		1p1e2 12292	. . . . . . . 8 ⊢ (1 + 1) = 2
40		8cn 12269	. . . . . . . . 9 ⊢ 8 ∈ ℂ
41		4cn 12257	. . . . . . . . 9 ⊢ 4 ∈ ℂ
42		8p4e12 12717	. . . . . . . . 9 ⊢ (8 + 4) = ;12
43	40, 41, 42	addcomli 11329	. . . . . . . 8 ⊢ (4 + 8) = ;12
44	15, 36, 16, 38, 39, 5, 43	decaddci 12696	. . . . . . 7 ⊢ ((2 · 7) + 8) = ;22
45	5, 4, 18, 16, 14, 30, 5, 5, 5, 35, 44	decma2c 12688	. . . . . 6 ⊢ ((2 · ;27) + 8) = ;62
46	18, 5, 15, 16, 19, 20, 13, 5, 2, 29, 45	decmac 12687	. . . . 5 ⊢ ((2 · ;27) + ;18) = ;72
47		4p4e8 12322	. . . . . . 7 ⊢ (4 + 4) = 8
48	15, 36, 36, 37, 47	decaddi 12695	. . . . . 6 ⊢ ((7 · 2) + 4) = ;18
49		7t7e49 12749	. . . . . 6 ⊢ (7 · 7) = ;49
50	4, 5, 4, 14, 7, 36, 48, 49	decmul2c 12701	. . . . 5 ⊢ (7 · ;27) = ;;189
51	13, 5, 4, 14, 7, 17, 46, 50	decmul1c 12700	. . . 4 ⊢ (;27 · ;27) = ;;729
52	1, 1, 11, 12, 51	numexp2x 17040	. . 3 ⊢ (3↑6) = ;;729
53		eqid 2739	. . . 4 ⊢ ;72 = ;72
54		7t3e21 12745	. . . . 5 ⊢ (7 · 3) = ;21
55		1p0e1 12291	. . . . 5 ⊢ (1 + 0) = 1
56	5, 15, 18, 54, 55	decaddi 12695	. . . 4 ⊢ ((7 · 3) + 0) = ;21
57	11	oveq1i 7366	. . . . 5 ⊢ ((2 · 3) + 2) = (6 + 2)
58		6p2e8 12326	. . . . 5 ⊢ (6 + 2) = 8
59	57, 58	eqtri 2762	. . . 4 ⊢ ((2 · 3) + 2) = 8
60	4, 5, 18, 5, 53, 19, 1, 56, 59	decma 12686	. . 3 ⊢ ((;72 · 3) + 2) = ;;218
61		9t3e27 12758	. . 3 ⊢ (9 · 3) = ;27
62	1, 6, 7, 52, 4, 5, 60, 61	decmul1c 12700	. 2 ⊢ ((3↑6) · 3) = ;;;2187
63	1, 2, 3, 62	numexpp1 17039	1 ⊢ (3↑7) = ;;;2187

Colors of variables: wff setvar class

Syntax hints: = wceq 1547 (class class class)co 7356 0cc0 11029 1c1 11030 + caddc 11032 · cmul 11034 2c2 12227 3c3 12228 4c4 12229 6c6 12231 7c7 12232 8c8 12233 9c9 12234 cdc 12635 ↑cexp 14014

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-op 4562 df-uni 4839 df-iun 4923 df-br 5073 df-opab 5135 df-mpt 5154 df-tr 5180 df-id 5513 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5571 df-we 5573 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-pred 6252 df-ord 6313 df-on 6314 df-lim 6315 df-suc 6316 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7807 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-er 8633 df-en 8884 df-dom 8885 df-sdom 8886 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-5 12238 df-6 12239 df-7 12240 df-8 12241 df-9 12242 df-n0 12429 df-z 12516 df-dec 12636 df-uz 12780 df-seq 13955 df-exp 14015

This theorem is referenced by: 3lexlogpow5ineq2 42540

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