ex-ind-dvds - Metamath Proof Explorer

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Theorem ex-ind-dvds 30546

Description: Example of a proof by induction (divisibility result). (Contributed by Stanislas Polu, 9-Mar-2020.) (Revised by BJ, 24-Mar-2020.)

**Assertion**
Ref	Expression
ex-ind-dvds	⊢ (𝑁 ∈ ℕ₀ → 3 ∥ ((4↑𝑁) + 2))

Proof of Theorem ex-ind-dvds Dummy variables 𝑘 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 7368	. . . 4 ⊢ (𝑘 = 0 → (4↑𝑘) = (4↑0))
2	1	oveq1d 7375	. . 3 ⊢ (𝑘 = 0 → ((4↑𝑘) + 2) = ((4↑0) + 2))
3	2	breq2d 5098	. 2 ⊢ (𝑘 = 0 → (3 ∥ ((4↑𝑘) + 2) ↔ 3 ∥ ((4↑0) + 2)))
4		oveq2 7368	. . . 4 ⊢ (𝑘 = 𝑛 → (4↑𝑘) = (4↑𝑛))
5	4	oveq1d 7375	. . 3 ⊢ (𝑘 = 𝑛 → ((4↑𝑘) + 2) = ((4↑𝑛) + 2))
6	5	breq2d 5098	. 2 ⊢ (𝑘 = 𝑛 → (3 ∥ ((4↑𝑘) + 2) ↔ 3 ∥ ((4↑𝑛) + 2)))
7		oveq2 7368	. . . 4 ⊢ (𝑘 = (𝑛 + 1) → (4↑𝑘) = (4↑(𝑛 + 1)))
8	7	oveq1d 7375	. . 3 ⊢ (𝑘 = (𝑛 + 1) → ((4↑𝑘) + 2) = ((4↑(𝑛 + 1)) + 2))
9	8	breq2d 5098	. 2 ⊢ (𝑘 = (𝑛 + 1) → (3 ∥ ((4↑𝑘) + 2) ↔ 3 ∥ ((4↑(𝑛 + 1)) + 2)))
10		oveq2 7368	. . . 4 ⊢ (𝑘 = 𝑁 → (4↑𝑘) = (4↑𝑁))
11	10	oveq1d 7375	. . 3 ⊢ (𝑘 = 𝑁 → ((4↑𝑘) + 2) = ((4↑𝑁) + 2))
12	11	breq2d 5098	. 2 ⊢ (𝑘 = 𝑁 → (3 ∥ ((4↑𝑘) + 2) ↔ 3 ∥ ((4↑𝑁) + 2)))
13		3z 12551	. . . 4 ⊢ 3 ∈ ℤ
14		iddvds 16229	. . . 4 ⊢ (3 ∈ ℤ → 3 ∥ 3)
15	13, 14	ax-mp 5	. . 3 ⊢ 3 ∥ 3
16		4nn0 12447	. . . . . 6 ⊢ 4 ∈ ℕ₀
17	16	numexp0 17037	. . . . 5 ⊢ (4↑0) = 1
18	17	oveq1i 7370	. . . 4 ⊢ ((4↑0) + 2) = (1 + 2)
19		1p2e3 12310	. . . 4 ⊢ (1 + 2) = 3
20	18, 19	eqtri 2760	. . 3 ⊢ ((4↑0) + 2) = 3
21	15, 20	breqtrri 5113	. 2 ⊢ 3 ∥ ((4↑0) + 2)
22	13	a1i 11	. . . . 5 ⊢ ((𝑛 ∈ ℕ₀ ∧ 3 ∥ ((4↑𝑛) + 2)) → 3 ∈ ℤ)
23	16	a1i 11	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ₀ ∧ 3 ∥ ((4↑𝑛) + 2)) → 4 ∈ ℕ₀)
24		simpl 482	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ₀ ∧ 3 ∥ ((4↑𝑛) + 2)) → 𝑛 ∈ ℕ₀)
25	23, 24	nn0expcld 14199	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ₀ ∧ 3 ∥ ((4↑𝑛) + 2)) → (4↑𝑛) ∈ ℕ₀)
26	25	nn0zd 12540	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ₀ ∧ 3 ∥ ((4↑𝑛) + 2)) → (4↑𝑛) ∈ ℤ)
27		2z 12550	. . . . . . . 8 ⊢ 2 ∈ ℤ
28	27	a1i 11	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ₀ ∧ 3 ∥ ((4↑𝑛) + 2)) → 2 ∈ ℤ)
29	26, 28	zaddcld 12628	. . . . . 6 ⊢ ((𝑛 ∈ ℕ₀ ∧ 3 ∥ ((4↑𝑛) + 2)) → ((4↑𝑛) + 2) ∈ ℤ)
30		4z 12552	. . . . . . 7 ⊢ 4 ∈ ℤ
31	30	a1i 11	. . . . . 6 ⊢ ((𝑛 ∈ ℕ₀ ∧ 3 ∥ ((4↑𝑛) + 2)) → 4 ∈ ℤ)
32	29, 31	zmulcld 12630	. . . . 5 ⊢ ((𝑛 ∈ ℕ₀ ∧ 3 ∥ ((4↑𝑛) + 2)) → (((4↑𝑛) + 2) · 4) ∈ ℤ)
33	22, 28	zmulcld 12630	. . . . 5 ⊢ ((𝑛 ∈ ℕ₀ ∧ 3 ∥ ((4↑𝑛) + 2)) → (3 · 2) ∈ ℤ)
34	16	a1i 11	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ₀ → 4 ∈ ℕ₀)
35		id 22	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ₀ → 𝑛 ∈ ℕ₀)
36	34, 35	nn0expcld 14199	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ₀ → (4↑𝑛) ∈ ℕ₀)
37	36	nn0zd 12540	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ₀ → (4↑𝑛) ∈ ℤ)
38	37	adantr 480	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ₀ ∧ 3 ∥ ((4↑𝑛) + 2)) → (4↑𝑛) ∈ ℤ)
39	38, 28	zaddcld 12628	. . . . . 6 ⊢ ((𝑛 ∈ ℕ₀ ∧ 3 ∥ ((4↑𝑛) + 2)) → ((4↑𝑛) + 2) ∈ ℤ)
40		simpr 484	. . . . . 6 ⊢ ((𝑛 ∈ ℕ₀ ∧ 3 ∥ ((4↑𝑛) + 2)) → 3 ∥ ((4↑𝑛) + 2))
41	22, 39, 31, 40	dvdsmultr1d 16257	. . . . 5 ⊢ ((𝑛 ∈ ℕ₀ ∧ 3 ∥ ((4↑𝑛) + 2)) → 3 ∥ (((4↑𝑛) + 2) · 4))
42		dvdsmul1 16237	. . . . . . 7 ⊢ ((3 ∈ ℤ ∧ 2 ∈ ℤ) → 3 ∥ (3 · 2))
43	13, 27, 42	mp2an 693	. . . . . 6 ⊢ 3 ∥ (3 · 2)
44	43	a1i 11	. . . . 5 ⊢ ((𝑛 ∈ ℕ₀ ∧ 3 ∥ ((4↑𝑛) + 2)) → 3 ∥ (3 · 2))
45	22, 32, 33, 41, 44	dvds2subd 16253	. . . 4 ⊢ ((𝑛 ∈ ℕ₀ ∧ 3 ∥ ((4↑𝑛) + 2)) → 3 ∥ ((((4↑𝑛) + 2) · 4) − (3 · 2)))
46	36	nn0cnd 12491	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ₀ → (4↑𝑛) ∈ ℂ)
47		2cnd 12250	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ₀ → 2 ∈ ℂ)
48		4cn 12257	. . . . . . . . 9 ⊢ 4 ∈ ℂ
49	48	a1i 11	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ₀ → 4 ∈ ℂ)
50	46, 47, 49	adddird 11161	. . . . . . 7 ⊢ (𝑛 ∈ ℕ₀ → (((4↑𝑛) + 2) · 4) = (((4↑𝑛) · 4) + (2 · 4)))
51	50	oveq1d 7375	. . . . . 6 ⊢ (𝑛 ∈ ℕ₀ → ((((4↑𝑛) + 2) · 4) − (2 · 3)) = ((((4↑𝑛) · 4) + (2 · 4)) − (2 · 3)))
52		3cn 12253	. . . . . . . . 9 ⊢ 3 ∈ ℂ
53		2cn 12247	. . . . . . . . 9 ⊢ 2 ∈ ℂ
54	52, 53	mulcomi 11144	. . . . . . . 8 ⊢ (3 · 2) = (2 · 3)
55	54	a1i 11	. . . . . . 7 ⊢ (𝑛 ∈ ℕ₀ → (3 · 2) = (2 · 3))
56	55	oveq2d 7376	. . . . . 6 ⊢ (𝑛 ∈ ℕ₀ → ((((4↑𝑛) + 2) · 4) − (3 · 2)) = ((((4↑𝑛) + 2) · 4) − (2 · 3)))
57	49, 35	expp1d 14100	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ₀ → (4↑(𝑛 + 1)) = ((4↑𝑛) · 4))
58		ax-1cn 11087	. . . . . . . . . . . 12 ⊢ 1 ∈ ℂ
59		3p1e4 12312	. . . . . . . . . . . . . 14 ⊢ (3 + 1) = 4
60	52, 58, 59	addcomli 11329	. . . . . . . . . . . . 13 ⊢ (1 + 3) = 4
61	60	eqcomi 2746	. . . . . . . . . . . 12 ⊢ 4 = (1 + 3)
62	58, 52, 61	mvrraddi 11401	. . . . . . . . . . 11 ⊢ (4 − 3) = 1
63	62	oveq2i 7371	. . . . . . . . . 10 ⊢ (2 · (4 − 3)) = (2 · 1)
64	53, 48, 52	subdii 11590	. . . . . . . . . 10 ⊢ (2 · (4 − 3)) = ((2 · 4) − (2 · 3))
65		2t1e2 12330	. . . . . . . . . 10 ⊢ (2 · 1) = 2
66	63, 64, 65	3eqtr3ri 2769	. . . . . . . . 9 ⊢ 2 = ((2 · 4) − (2 · 3))
67	66	a1i 11	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ₀ → 2 = ((2 · 4) − (2 · 3)))
68	57, 67	oveq12d 7378	. . . . . . 7 ⊢ (𝑛 ∈ ℕ₀ → ((4↑(𝑛 + 1)) + 2) = (((4↑𝑛) · 4) + ((2 · 4) − (2 · 3))))
69	46, 49	mulcld 11156	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ₀ → ((4↑𝑛) · 4) ∈ ℂ)
70	47, 49	mulcld 11156	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ₀ → (2 · 4) ∈ ℂ)
71	52	a1i 11	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ₀ → 3 ∈ ℂ)
72	47, 71	mulcld 11156	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ₀ → (2 · 3) ∈ ℂ)
73	69, 70, 72	addsubassd 11516	. . . . . . 7 ⊢ (𝑛 ∈ ℕ₀ → ((((4↑𝑛) · 4) + (2 · 4)) − (2 · 3)) = (((4↑𝑛) · 4) + ((2 · 4) − (2 · 3))))
74	68, 73	eqtr4d 2775	. . . . . 6 ⊢ (𝑛 ∈ ℕ₀ → ((4↑(𝑛 + 1)) + 2) = ((((4↑𝑛) · 4) + (2 · 4)) − (2 · 3)))
75	51, 56, 74	3eqtr4rd 2783	. . . . 5 ⊢ (𝑛 ∈ ℕ₀ → ((4↑(𝑛 + 1)) + 2) = ((((4↑𝑛) + 2) · 4) − (3 · 2)))
76	75	adantr 480	. . . 4 ⊢ ((𝑛 ∈ ℕ₀ ∧ 3 ∥ ((4↑𝑛) + 2)) → ((4↑(𝑛 + 1)) + 2) = ((((4↑𝑛) + 2) · 4) − (3 · 2)))
77	45, 76	breqtrrd 5114	. . 3 ⊢ ((𝑛 ∈ ℕ₀ ∧ 3 ∥ ((4↑𝑛) + 2)) → 3 ∥ ((4↑(𝑛 + 1)) + 2))
78	77	ex 412	. 2 ⊢ (𝑛 ∈ ℕ₀ → (3 ∥ ((4↑𝑛) + 2) → 3 ∥ ((4↑(𝑛 + 1)) + 2)))
79	3, 6, 9, 12, 21, 78	nn0ind 12615	1 ⊢ (𝑁 ∈ ℕ₀ → 3 ∥ ((4↑𝑁) + 2))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 class class class wbr 5086 (class class class)co 7360 ℂcc 11027 0cc0 11029 1c1 11030 + caddc 11032 · cmul 11034 − cmin 11368 2c2 12227 3c3 12228 4c4 12229 ℕ₀cn0 12428 ℤcz 12515 ↑cexp 14014 ∥ cdvds 16212

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5231 ax-nul 5241 ax-pow 5302 ax-pr 5370 ax-un 7682 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 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-om 7811 df-2nd 7936 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-rdg 8342 df-er 8636 df-en 8887 df-dom 8888 df-sdom 8889 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-n0 12429 df-z 12516 df-uz 12780 df-seq 13955 df-exp 14015 df-dvds 16213

This theorem is referenced by: (None)

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