ex-ind-dvds - Metamath Proof Explorer

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Theorem ex-ind-dvds 30721

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 7408	. . . 4 ⊢ (𝑘 = 0 → (4↑𝑘) = (4↑0))
2	1	oveq1d 7415	. . 3 ⊢ (𝑘 = 0 → ((4↑𝑘) + 2) = ((4↑0) + 2))
3	2	breq2d 5117	. 2 ⊢ (𝑘 = 0 → (3 ∥ ((4↑𝑘) + 2) ↔ 3 ∥ ((4↑0) + 2)))
4		oveq2 7408	. . . 4 ⊢ (𝑘 = 𝑛 → (4↑𝑘) = (4↑𝑛))
5	4	oveq1d 7415	. . 3 ⊢ (𝑘 = 𝑛 → ((4↑𝑘) + 2) = ((4↑𝑛) + 2))
6	5	breq2d 5117	. 2 ⊢ (𝑘 = 𝑛 → (3 ∥ ((4↑𝑘) + 2) ↔ 3 ∥ ((4↑𝑛) + 2)))
7		oveq2 7408	. . . 4 ⊢ (𝑘 = (𝑛 + 1) → (4↑𝑘) = (4↑(𝑛 + 1)))
8	7	oveq1d 7415	. . 3 ⊢ (𝑘 = (𝑛 + 1) → ((4↑𝑘) + 2) = ((4↑(𝑛 + 1)) + 2))
9	8	breq2d 5117	. 2 ⊢ (𝑘 = (𝑛 + 1) → (3 ∥ ((4↑𝑘) + 2) ↔ 3 ∥ ((4↑(𝑛 + 1)) + 2)))
10		oveq2 7408	. . . 4 ⊢ (𝑘 = 𝑁 → (4↑𝑘) = (4↑𝑁))
11	10	oveq1d 7415	. . 3 ⊢ (𝑘 = 𝑁 → ((4↑𝑘) + 2) = ((4↑𝑁) + 2))
12	11	breq2d 5117	. 2 ⊢ (𝑘 = 𝑁 → (3 ∥ ((4↑𝑘) + 2) ↔ 3 ∥ ((4↑𝑁) + 2)))
13		3z 12618	. . . 4 ⊢ 3 ∈ ℤ
14		iddvds 16317	. . . 4 ⊢ (3 ∈ ℤ → 3 ∥ 3)
15	13, 14	ax-mp 5	. . 3 ⊢ 3 ∥ 3
16		4nn0 12514	. . . . . 6 ⊢ 4 ∈ ℕ₀
17	16	numexp0 17125	. . . . 5 ⊢ (4↑0) = 1
18	17	oveq1i 7410	. . . 4 ⊢ ((4↑0) + 2) = (1 + 2)
19		1p2e3 12374	. . . 4 ⊢ (1 + 2) = 3
20	18, 19	eqtri 2788	. . 3 ⊢ ((4↑0) + 2) = 3
21	15, 20	breqtrri 5132	. 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 487	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ₀ ∧ 3 ∥ ((4↑𝑛) + 2)) → 𝑛 ∈ ℕ₀)
25	23, 24	nn0expcld 14273	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ₀ ∧ 3 ∥ ((4↑𝑛) + 2)) → (4↑𝑛) ∈ ℕ₀)
26	25	nn0zd 12607	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ₀ ∧ 3 ∥ ((4↑𝑛) + 2)) → (4↑𝑛) ∈ ℤ)
27		2z 12617	. . . . . . . 8 ⊢ 2 ∈ ℤ
28	27	a1i 11	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ₀ ∧ 3 ∥ ((4↑𝑛) + 2)) → 2 ∈ ℤ)
29	26, 28	zaddcld 12695	. . . . . 6 ⊢ ((𝑛 ∈ ℕ₀ ∧ 3 ∥ ((4↑𝑛) + 2)) → ((4↑𝑛) + 2) ∈ ℤ)
30		4z 12619	. . . . . . 7 ⊢ 4 ∈ ℤ
31	30	a1i 11	. . . . . 6 ⊢ ((𝑛 ∈ ℕ₀ ∧ 3 ∥ ((4↑𝑛) + 2)) → 4 ∈ ℤ)
32	29, 31	zmulcld 12697	. . . . 5 ⊢ ((𝑛 ∈ ℕ₀ ∧ 3 ∥ ((4↑𝑛) + 2)) → (((4↑𝑛) + 2) · 4) ∈ ℤ)
33	22, 28	zmulcld 12697	. . . . 5 ⊢ ((𝑛 ∈ ℕ₀ ∧ 3 ∥ ((4↑𝑛) + 2)) → (3 · 2) ∈ ℤ)
34	16	a1i 11	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ₀ → 4 ∈ ℕ₀)
35		id 23	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ₀ → 𝑛 ∈ ℕ₀)
36	34, 35	nn0expcld 14273	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ₀ → (4↑𝑛) ∈ ℕ₀)
37	36	nn0zd 12607	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ₀ → (4↑𝑛) ∈ ℤ)
38	37	adantr 485	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ₀ ∧ 3 ∥ ((4↑𝑛) + 2)) → (4↑𝑛) ∈ ℤ)
39	38, 28	zaddcld 12695	. . . . . 6 ⊢ ((𝑛 ∈ ℕ₀ ∧ 3 ∥ ((4↑𝑛) + 2)) → ((4↑𝑛) + 2) ∈ ℤ)
40		simpr 489	. . . . . 6 ⊢ ((𝑛 ∈ ℕ₀ ∧ 3 ∥ ((4↑𝑛) + 2)) → 3 ∥ ((4↑𝑛) + 2))
41	22, 39, 31, 40	dvdsmultr1d 16345	. . . . 5 ⊢ ((𝑛 ∈ ℕ₀ ∧ 3 ∥ ((4↑𝑛) + 2)) → 3 ∥ (((4↑𝑛) + 2) · 4))
42		dvdsmul1 16325	. . . . . . 7 ⊢ ((3 ∈ ℤ ∧ 2 ∈ ℤ) → 3 ∥ (3 · 2))
43	13, 27, 42	mp2an 704	. . . . . 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 16341	. . . 4 ⊢ ((𝑛 ∈ ℕ₀ ∧ 3 ∥ ((4↑𝑛) + 2)) → 3 ∥ ((((4↑𝑛) + 2) · 4) − (3 · 2)))
46	36	nn0cnd 12558	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ₀ → (4↑𝑛) ∈ ℂ)
47		2cnd 12310	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ₀ → 2 ∈ ℂ)
48		4cn 12317	. . . . . . . . 9 ⊢ 4 ∈ ℂ
49	48	a1i 11	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ₀ → 4 ∈ ℂ)
50	46, 47, 49	adddird 11222	. . . . . . 7 ⊢ (𝑛 ∈ ℕ₀ → (((4↑𝑛) + 2) · 4) = (((4↑𝑛) · 4) + (2 · 4)))
51	50	oveq1d 7415	. . . . . 6 ⊢ (𝑛 ∈ ℕ₀ → ((((4↑𝑛) + 2) · 4) − (2 · 3)) = ((((4↑𝑛) · 4) + (2 · 4)) − (2 · 3)))
52		3cn 12313	. . . . . . . . 9 ⊢ 3 ∈ ℂ
53		2cn 12307	. . . . . . . . 9 ⊢ 2 ∈ ℂ
54	52, 53	mulcomi 11205	. . . . . . . 8 ⊢ (3 · 2) = (2 · 3)
55	54	a1i 11	. . . . . . 7 ⊢ (𝑛 ∈ ℕ₀ → (3 · 2) = (2 · 3))
56	55	oveq2d 7416	. . . . . 6 ⊢ (𝑛 ∈ ℕ₀ → ((((4↑𝑛) + 2) · 4) − (3 · 2)) = ((((4↑𝑛) + 2) · 4) − (2 · 3)))
57	49, 35	expp1d 14174	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ₀ → (4↑(𝑛 + 1)) = ((4↑𝑛) · 4))
58		ax-1cn 11146	. . . . . . . . . . . 12 ⊢ 1 ∈ ℂ
59		3p1e4 12376	. . . . . . . . . . . . . 14 ⊢ (3 + 1) = 4
60	52, 58, 59	addcomli 11390	. . . . . . . . . . . . 13 ⊢ (1 + 3) = 4
61	60	eqcomi 2774	. . . . . . . . . . . 12 ⊢ 4 = (1 + 3)
62	58, 52, 61	mvrraddi 11462	. . . . . . . . . . 11 ⊢ (4 − 3) = 1
63	62	oveq2i 7411	. . . . . . . . . 10 ⊢ (2 · (4 − 3)) = (2 · 1)
64	53, 48, 52	subdii 11651	. . . . . . . . . 10 ⊢ (2 · (4 − 3)) = ((2 · 4) − (2 · 3))
65		2t1e2 12394	. . . . . . . . . 10 ⊢ (2 · 1) = 2
66	63, 64, 65	3eqtr3ri 2797	. . . . . . . . 9 ⊢ 2 = ((2 · 4) − (2 · 3))
67	66	a1i 11	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ₀ → 2 = ((2 · 4) − (2 · 3)))
68	57, 67	oveq12d 7418	. . . . . . 7 ⊢ (𝑛 ∈ ℕ₀ → ((4↑(𝑛 + 1)) + 2) = (((4↑𝑛) · 4) + ((2 · 4) − (2 · 3))))
69	46, 49	mulcld 11217	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ₀ → ((4↑𝑛) · 4) ∈ ℂ)
70	47, 49	mulcld 11217	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ₀ → (2 · 4) ∈ ℂ)
71	52	a1i 11	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ₀ → 3 ∈ ℂ)
72	47, 71	mulcld 11217	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ₀ → (2 · 3) ∈ ℂ)
73	69, 70, 72	addsubassd 11577	. . . . . . 7 ⊢ (𝑛 ∈ ℕ₀ → ((((4↑𝑛) · 4) + (2 · 4)) − (2 · 3)) = (((4↑𝑛) · 4) + ((2 · 4) − (2 · 3))))
74	68, 73	eqtr4d 2803	. . . . . 6 ⊢ (𝑛 ∈ ℕ₀ → ((4↑(𝑛 + 1)) + 2) = ((((4↑𝑛) · 4) + (2 · 4)) − (2 · 3)))
75	51, 56, 74	3eqtr4rd 2811	. . . . 5 ⊢ (𝑛 ∈ ℕ₀ → ((4↑(𝑛 + 1)) + 2) = ((((4↑𝑛) + 2) · 4) − (3 · 2)))
76	75	adantr 485	. . . 4 ⊢ ((𝑛 ∈ ℕ₀ ∧ 3 ∥ ((4↑𝑛) + 2)) → ((4↑(𝑛 + 1)) + 2) = ((((4↑𝑛) + 2) · 4) − (3 · 2)))
77	45, 76	breqtrrd 5133	. . 3 ⊢ ((𝑛 ∈ ℕ₀ ∧ 3 ∥ ((4↑𝑛) + 2)) → 3 ∥ ((4↑(𝑛 + 1)) + 2))
78	77	ex 417	. 2 ⊢ (𝑛 ∈ ℕ₀ → (3 ∥ ((4↑𝑛) + 2) → 3 ∥ ((4↑(𝑛 + 1)) + 2)))
79	3, 6, 9, 12, 21, 78	nn0ind 12682	1 ⊢ (𝑁 ∈ ℕ₀ → 3 ∥ ((4↑𝑁) + 2))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 400 = wceq 1563 ∈ wcel 2145 class class class wbr 5105 (class class class)co 7400 ℂcc 11086 0cc0 11088 1c1 11089 + caddc 11091 · cmul 11093 − cmin 11429 2c2 12286 3c3 12287 4c4 12288 ℕ₀cn0 12495 ℤcz 12582 ↑cexp 14088 ∥ cdvds 16300

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1818 ax-4 1832 ax-5 1933 ax-6 1990 ax-7 2031 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2737 ax-sep 5251 ax-nul 5261 ax-pow 5327 ax-pr 5395 ax-un 7722 ax-cnex 11144 ax-resscn 11145 ax-1cn 11146 ax-icn 11147 ax-addcl 11148 ax-addrcl 11149 ax-mulcl 11150 ax-mulrcl 11151 ax-mulcom 11152 ax-addass 11153 ax-mulass 11154 ax-distr 11155 ax-i2m1 11156 ax-1ne0 11157 ax-1rid 11158 ax-rnegex 11159 ax-rrecex 11160 ax-cnre 11161 ax-pre-lttri 11162 ax-pre-lttrn 11163 ax-pre-ltadd 11164 ax-pre-mulgt0 11165

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1566 df-fal 1576 df-ex 1803 df-nf 1807 df-sb 2094 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3065 df-ral 3080 df-rex 3090 df-reu 3371 df-rab 3418 df-v 3459 df-sbc 3748 df-csb 3856 df-dif 3910 df-un 3912 df-in 3914 df-ss 3924 df-pss 3927 df-nul 4289 df-if 4484 df-pw 4560 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4869 df-iun 4954 df-br 5106 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5547 df-eprel 5552 df-po 5560 df-so 5561 df-fr 5605 df-we 5607 df-xp 5658 df-rel 5659 df-cnv 5660 df-co 5661 df-dm 5662 df-rn 5663 df-res 5664 df-ima 5665 df-pred 6292 df-ord 6353 df-on 6354 df-lim 6355 df-suc 6356 df-iota 6481 df-fun 6527 df-fn 6528 df-f 6529 df-f1 6530 df-fo 6531 df-f1o 6532 df-fv 6533 df-riota 7357 df-ov 7403 df-oprab 7404 df-mpo 7405 df-om 7851 df-2nd 7975 df-frecs 8266 df-wrecs 8297 df-recs 8346 df-rdg 8385 df-er 8682 df-en 8932 df-dom 8933 df-sdom 8934 df-pnf 11233 df-mnf 11234 df-xr 11235 df-ltxr 11236 df-le 11237 df-sub 11431 df-neg 11432 df-nn 12225 df-2 12294 df-3 12295 df-4 12296 df-n0 12496 df-z 12583 df-uz 12854 df-seq 14029 df-exp 14089 df-dvds 16301

This theorem is referenced by: (None)

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