ex-ind-dvds - Metamath Proof Explorer

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Theorem ex-ind-dvds 28240

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 7164	. . . 4 ⊢ (𝑘 = 0 → (4↑𝑘) = (4↑0))
2	1	oveq1d 7171	. . 3 ⊢ (𝑘 = 0 → ((4↑𝑘) + 2) = ((4↑0) + 2))
3	2	breq2d 5078	. 2 ⊢ (𝑘 = 0 → (3 ∥ ((4↑𝑘) + 2) ↔ 3 ∥ ((4↑0) + 2)))
4		oveq2 7164	. . . 4 ⊢ (𝑘 = 𝑛 → (4↑𝑘) = (4↑𝑛))
5	4	oveq1d 7171	. . 3 ⊢ (𝑘 = 𝑛 → ((4↑𝑘) + 2) = ((4↑𝑛) + 2))
6	5	breq2d 5078	. 2 ⊢ (𝑘 = 𝑛 → (3 ∥ ((4↑𝑘) + 2) ↔ 3 ∥ ((4↑𝑛) + 2)))
7		oveq2 7164	. . . 4 ⊢ (𝑘 = (𝑛 + 1) → (4↑𝑘) = (4↑(𝑛 + 1)))
8	7	oveq1d 7171	. . 3 ⊢ (𝑘 = (𝑛 + 1) → ((4↑𝑘) + 2) = ((4↑(𝑛 + 1)) + 2))
9	8	breq2d 5078	. 2 ⊢ (𝑘 = (𝑛 + 1) → (3 ∥ ((4↑𝑘) + 2) ↔ 3 ∥ ((4↑(𝑛 + 1)) + 2)))
10		oveq2 7164	. . . 4 ⊢ (𝑘 = 𝑁 → (4↑𝑘) = (4↑𝑁))
11	10	oveq1d 7171	. . 3 ⊢ (𝑘 = 𝑁 → ((4↑𝑘) + 2) = ((4↑𝑁) + 2))
12	11	breq2d 5078	. 2 ⊢ (𝑘 = 𝑁 → (3 ∥ ((4↑𝑘) + 2) ↔ 3 ∥ ((4↑𝑁) + 2)))
13		3z 12016	. . . 4 ⊢ 3 ∈ ℤ
14		iddvds 15623	. . . 4 ⊢ (3 ∈ ℤ → 3 ∥ 3)
15	13, 14	ax-mp 5	. . 3 ⊢ 3 ∥ 3
16		4nn0 11917	. . . . . 6 ⊢ 4 ∈ ℕ₀
17	16	numexp0 16412	. . . . 5 ⊢ (4↑0) = 1
18	17	oveq1i 7166	. . . 4 ⊢ ((4↑0) + 2) = (1 + 2)
19		1p2e3 11781	. . . 4 ⊢ (1 + 2) = 3
20	18, 19	eqtri 2844	. . 3 ⊢ ((4↑0) + 2) = 3
21	15, 20	breqtrri 5093	. 2 ⊢ 3 ∥ ((4↑0) + 2)
22	13	a1i 11	. . . . 5 ⊢ ((𝑛 ∈ ℕ₀ ∧ 3 ∥ ((4↑𝑛) + 2)) → 3 ∈ ℤ)
23	16	a1i 11	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ₀ → 4 ∈ ℕ₀)
24		id 22	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ₀ → 𝑛 ∈ ℕ₀)
25	23, 24	nn0expcld 13608	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ₀ → (4↑𝑛) ∈ ℕ₀)
26	25	nn0zd 12086	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ₀ → (4↑𝑛) ∈ ℤ)
27	26	adantr 483	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ₀ ∧ 3 ∥ ((4↑𝑛) + 2)) → (4↑𝑛) ∈ ℤ)
28		2z 12015	. . . . . . . 8 ⊢ 2 ∈ ℤ
29	28	a1i 11	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ₀ ∧ 3 ∥ ((4↑𝑛) + 2)) → 2 ∈ ℤ)
30	27, 29	zaddcld 12092	. . . . . 6 ⊢ ((𝑛 ∈ ℕ₀ ∧ 3 ∥ ((4↑𝑛) + 2)) → ((4↑𝑛) + 2) ∈ ℤ)
31		4z 12017	. . . . . . 7 ⊢ 4 ∈ ℤ
32	31	a1i 11	. . . . . 6 ⊢ ((𝑛 ∈ ℕ₀ ∧ 3 ∥ ((4↑𝑛) + 2)) → 4 ∈ ℤ)
33		simpr 487	. . . . . 6 ⊢ ((𝑛 ∈ ℕ₀ ∧ 3 ∥ ((4↑𝑛) + 2)) → 3 ∥ ((4↑𝑛) + 2))
34	22, 30, 32, 33	dvdsmultr1d 15648	. . . . 5 ⊢ ((𝑛 ∈ ℕ₀ ∧ 3 ∥ ((4↑𝑛) + 2)) → 3 ∥ (((4↑𝑛) + 2) · 4))
35		dvdsmul1 15631	. . . . . . 7 ⊢ ((3 ∈ ℤ ∧ 2 ∈ ℤ) → 3 ∥ (3 · 2))
36	13, 28, 35	mp2an 690	. . . . . 6 ⊢ 3 ∥ (3 · 2)
37	36	a1i 11	. . . . 5 ⊢ ((𝑛 ∈ ℕ₀ ∧ 3 ∥ ((4↑𝑛) + 2)) → 3 ∥ (3 · 2))
38	16	a1i 11	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ₀ ∧ 3 ∥ ((4↑𝑛) + 2)) → 4 ∈ ℕ₀)
39		simpl 485	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ₀ ∧ 3 ∥ ((4↑𝑛) + 2)) → 𝑛 ∈ ℕ₀)
40	38, 39	nn0expcld 13608	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ₀ ∧ 3 ∥ ((4↑𝑛) + 2)) → (4↑𝑛) ∈ ℕ₀)
41	40	nn0zd 12086	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ₀ ∧ 3 ∥ ((4↑𝑛) + 2)) → (4↑𝑛) ∈ ℤ)
42	41, 29	zaddcld 12092	. . . . . 6 ⊢ ((𝑛 ∈ ℕ₀ ∧ 3 ∥ ((4↑𝑛) + 2)) → ((4↑𝑛) + 2) ∈ ℤ)
43	42, 32	zmulcld 12094	. . . . 5 ⊢ ((𝑛 ∈ ℕ₀ ∧ 3 ∥ ((4↑𝑛) + 2)) → (((4↑𝑛) + 2) · 4) ∈ ℤ)
44	22, 29	zmulcld 12094	. . . . 5 ⊢ ((𝑛 ∈ ℕ₀ ∧ 3 ∥ ((4↑𝑛) + 2)) → (3 · 2) ∈ ℤ)
45	22, 34, 37, 43, 44	dvds2subd 15645	. . . 4 ⊢ ((𝑛 ∈ ℕ₀ ∧ 3 ∥ ((4↑𝑛) + 2)) → 3 ∥ ((((4↑𝑛) + 2) · 4) − (3 · 2)))
46	25	nn0cnd 11958	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ₀ → (4↑𝑛) ∈ ℂ)
47		2cnd 11716	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ₀ → 2 ∈ ℂ)
48		4cn 11723	. . . . . . . . 9 ⊢ 4 ∈ ℂ
49	48	a1i 11	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ₀ → 4 ∈ ℂ)
50	46, 47, 49	adddird 10666	. . . . . . 7 ⊢ (𝑛 ∈ ℕ₀ → (((4↑𝑛) + 2) · 4) = (((4↑𝑛) · 4) + (2 · 4)))
51	50	oveq1d 7171	. . . . . 6 ⊢ (𝑛 ∈ ℕ₀ → ((((4↑𝑛) + 2) · 4) − (2 · 3)) = ((((4↑𝑛) · 4) + (2 · 4)) − (2 · 3)))
52		3cn 11719	. . . . . . . . 9 ⊢ 3 ∈ ℂ
53		2cn 11713	. . . . . . . . 9 ⊢ 2 ∈ ℂ
54	52, 53	mulcomi 10649	. . . . . . . 8 ⊢ (3 · 2) = (2 · 3)
55	54	a1i 11	. . . . . . 7 ⊢ (𝑛 ∈ ℕ₀ → (3 · 2) = (2 · 3))
56	55	oveq2d 7172	. . . . . 6 ⊢ (𝑛 ∈ ℕ₀ → ((((4↑𝑛) + 2) · 4) − (3 · 2)) = ((((4↑𝑛) + 2) · 4) − (2 · 3)))
57	49, 24	expp1d 13512	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ₀ → (4↑(𝑛 + 1)) = ((4↑𝑛) · 4))
58		ax-1cn 10595	. . . . . . . . . . . 12 ⊢ 1 ∈ ℂ
59		3p1e4 11783	. . . . . . . . . . . . . 14 ⊢ (3 + 1) = 4
60	52, 58, 59	addcomli 10832	. . . . . . . . . . . . 13 ⊢ (1 + 3) = 4
61	60	eqcomi 2830	. . . . . . . . . . . 12 ⊢ 4 = (1 + 3)
62	58, 52, 61	mvrraddi 10903	. . . . . . . . . . 11 ⊢ (4 − 3) = 1
63	62	oveq2i 7167	. . . . . . . . . 10 ⊢ (2 · (4 − 3)) = (2 · 1)
64	53, 48, 52	subdii 11089	. . . . . . . . . 10 ⊢ (2 · (4 − 3)) = ((2 · 4) − (2 · 3))
65		2t1e2 11801	. . . . . . . . . 10 ⊢ (2 · 1) = 2
66	63, 64, 65	3eqtr3ri 2853	. . . . . . . . 9 ⊢ 2 = ((2 · 4) − (2 · 3))
67	66	a1i 11	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ₀ → 2 = ((2 · 4) − (2 · 3)))
68	57, 67	oveq12d 7174	. . . . . . 7 ⊢ (𝑛 ∈ ℕ₀ → ((4↑(𝑛 + 1)) + 2) = (((4↑𝑛) · 4) + ((2 · 4) − (2 · 3))))
69	46, 49	mulcld 10661	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ₀ → ((4↑𝑛) · 4) ∈ ℂ)
70	47, 49	mulcld 10661	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ₀ → (2 · 4) ∈ ℂ)
71	52	a1i 11	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ₀ → 3 ∈ ℂ)
72	47, 71	mulcld 10661	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ₀ → (2 · 3) ∈ ℂ)
73	69, 70, 72	addsubassd 11017	. . . . . . 7 ⊢ (𝑛 ∈ ℕ₀ → ((((4↑𝑛) · 4) + (2 · 4)) − (2 · 3)) = (((4↑𝑛) · 4) + ((2 · 4) − (2 · 3))))
74	68, 73	eqtr4d 2859	. . . . . 6 ⊢ (𝑛 ∈ ℕ₀ → ((4↑(𝑛 + 1)) + 2) = ((((4↑𝑛) · 4) + (2 · 4)) − (2 · 3)))
75	51, 56, 74	3eqtr4rd 2867	. . . . 5 ⊢ (𝑛 ∈ ℕ₀ → ((4↑(𝑛 + 1)) + 2) = ((((4↑𝑛) + 2) · 4) − (3 · 2)))
76	75	adantr 483	. . . 4 ⊢ ((𝑛 ∈ ℕ₀ ∧ 3 ∥ ((4↑𝑛) + 2)) → ((4↑(𝑛 + 1)) + 2) = ((((4↑𝑛) + 2) · 4) − (3 · 2)))
77	45, 76	breqtrrd 5094	. . 3 ⊢ ((𝑛 ∈ ℕ₀ ∧ 3 ∥ ((4↑𝑛) + 2)) → 3 ∥ ((4↑(𝑛 + 1)) + 2))
78	77	ex 415	. 2 ⊢ (𝑛 ∈ ℕ₀ → (3 ∥ ((4↑𝑛) + 2) → 3 ∥ ((4↑(𝑛 + 1)) + 2)))
79	3, 6, 9, 12, 21, 78	nn0ind 12078	1 ⊢ (𝑁 ∈ ℕ₀ → 3 ∥ ((4↑𝑁) + 2))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 class class class wbr 5066 (class class class)co 7156 ℂcc 10535 0cc0 10537 1c1 10538 + caddc 10540 · cmul 10542 − cmin 10870 2c2 11693 3c3 11694 4c4 11695 ℕ₀cn0 11898 ℤcz 11982 ↑cexp 13430 ∥ cdvds 15607

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-n0 11899 df-z 11983 df-uz 12245 df-seq 13371 df-exp 13431 df-dvds 15608

This theorem is referenced by: (None)

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