ex-ind-dvds - Metamath Proof Explorer

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Theorem ex-ind-dvds 30849

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 7431	. . . 4 ⊢ (𝑘 = 0 → (4↑𝑘) = (4↑0))
2	1	oveq1d 7438	. . 3 ⊢ (𝑘 = 0 → ((4↑𝑘) + 2) = ((4↑0) + 2))
3	2	breq2d 5126	. 2 ⊢ (𝑘 = 0 → (3 ∥ ((4↑𝑘) + 2) ↔ 3 ∥ ((4↑0) + 2)))
4		oveq2 7431	. . . 4 ⊢ (𝑘 = 𝑛 → (4↑𝑘) = (4↑𝑛))
5	4	oveq1d 7438	. . 3 ⊢ (𝑘 = 𝑛 → ((4↑𝑘) + 2) = ((4↑𝑛) + 2))
6	5	breq2d 5126	. 2 ⊢ (𝑘 = 𝑛 → (3 ∥ ((4↑𝑘) + 2) ↔ 3 ∥ ((4↑𝑛) + 2)))
7		oveq2 7431	. . . 4 ⊢ (𝑘 = (𝑛 + 1) → (4↑𝑘) = (4↑(𝑛 + 1)))
8	7	oveq1d 7438	. . 3 ⊢ (𝑘 = (𝑛 + 1) → ((4↑𝑘) + 2) = ((4↑(𝑛 + 1)) + 2))
9	8	breq2d 5126	. 2 ⊢ (𝑘 = (𝑛 + 1) → (3 ∥ ((4↑𝑘) + 2) ↔ 3 ∥ ((4↑(𝑛 + 1)) + 2)))
10		oveq2 7431	. . . 4 ⊢ (𝑘 = 𝑁 → (4↑𝑘) = (4↑𝑁))
11	10	oveq1d 7438	. . 3 ⊢ (𝑘 = 𝑁 → ((4↑𝑘) + 2) = ((4↑𝑁) + 2))
12	11	breq2d 5126	. 2 ⊢ (𝑘 = 𝑁 → (3 ∥ ((4↑𝑘) + 2) ↔ 3 ∥ ((4↑𝑁) + 2)))
13		3z 12645	. . . 4 ⊢ 3 ∈ ℤ
14		iddvds 16352	. . . 4 ⊢ (3 ∈ ℤ → 3 ∥ 3)
15	13, 14	ax-mp 5	. . 3 ⊢ 3 ∥ 3
16		4nn0 12541	. . . . . 6 ⊢ 4 ∈ ℕ₀
17	16	numexp0 17160	. . . . 5 ⊢ (4↑0) = 1
18	17	oveq1i 7433	. . . 4 ⊢ ((4↑0) + 2) = (1 + 2)
19		1p2e3 12401	. . . 4 ⊢ (1 + 2) = 3
20	18, 19	eqtri 2789	. . 3 ⊢ ((4↑0) + 2) = 3
21	15, 20	breqtrri 5143	. 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 488	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ₀ ∧ 3 ∥ ((4↑𝑛) + 2)) → 𝑛 ∈ ℕ₀)
25	23, 24	nn0expcld 14302	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ₀ ∧ 3 ∥ ((4↑𝑛) + 2)) → (4↑𝑛) ∈ ℕ₀)
26	25	nn0zd 12634	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ₀ ∧ 3 ∥ ((4↑𝑛) + 2)) → (4↑𝑛) ∈ ℤ)
27		2z 12644	. . . . . . . 8 ⊢ 2 ∈ ℤ
28	27	a1i 11	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ₀ ∧ 3 ∥ ((4↑𝑛) + 2)) → 2 ∈ ℤ)
29	26, 28	zaddcld 12722	. . . . . 6 ⊢ ((𝑛 ∈ ℕ₀ ∧ 3 ∥ ((4↑𝑛) + 2)) → ((4↑𝑛) + 2) ∈ ℤ)
30		4z 12646	. . . . . . 7 ⊢ 4 ∈ ℤ
31	30	a1i 11	. . . . . 6 ⊢ ((𝑛 ∈ ℕ₀ ∧ 3 ∥ ((4↑𝑛) + 2)) → 4 ∈ ℤ)
32	29, 31	zmulcld 12724	. . . . 5 ⊢ ((𝑛 ∈ ℕ₀ ∧ 3 ∥ ((4↑𝑛) + 2)) → (((4↑𝑛) + 2) · 4) ∈ ℤ)
33	22, 28	zmulcld 12724	. . . . 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 14302	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ₀ → (4↑𝑛) ∈ ℕ₀)
37	36	nn0zd 12634	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ₀ → (4↑𝑛) ∈ ℤ)
38	37	adantr 486	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ₀ ∧ 3 ∥ ((4↑𝑛) + 2)) → (4↑𝑛) ∈ ℤ)
39	38, 28	zaddcld 12722	. . . . . 6 ⊢ ((𝑛 ∈ ℕ₀ ∧ 3 ∥ ((4↑𝑛) + 2)) → ((4↑𝑛) + 2) ∈ ℤ)
40		simpr 490	. . . . . 6 ⊢ ((𝑛 ∈ ℕ₀ ∧ 3 ∥ ((4↑𝑛) + 2)) → 3 ∥ ((4↑𝑛) + 2))
41	22, 39, 31, 40	dvdsmultr1d 16380	. . . . 5 ⊢ ((𝑛 ∈ ℕ₀ ∧ 3 ∥ ((4↑𝑛) + 2)) → 3 ∥ (((4↑𝑛) + 2) · 4))
42		dvdsmul1 16360	. . . . . . 7 ⊢ ((3 ∈ ℤ ∧ 2 ∈ ℤ) → 3 ∥ (3 · 2))
43	13, 27, 42	mp2an 705	. . . . . 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 16376	. . . 4 ⊢ ((𝑛 ∈ ℕ₀ ∧ 3 ∥ ((4↑𝑛) + 2)) → 3 ∥ ((((4↑𝑛) + 2) · 4) − (3 · 2)))
46	36	nn0cnd 12585	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ₀ → (4↑𝑛) ∈ ℂ)
47		2cnd 12337	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ₀ → 2 ∈ ℂ)
48		4cn 12344	. . . . . . . . 9 ⊢ 4 ∈ ℂ
49	48	a1i 11	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ₀ → 4 ∈ ℂ)
50	46, 47, 49	adddird 11252	. . . . . . 7 ⊢ (𝑛 ∈ ℕ₀ → (((4↑𝑛) + 2) · 4) = (((4↑𝑛) · 4) + (2 · 4)))
51	50	oveq1d 7438	. . . . . 6 ⊢ (𝑛 ∈ ℕ₀ → ((((4↑𝑛) + 2) · 4) − (2 · 3)) = ((((4↑𝑛) · 4) + (2 · 4)) − (2 · 3)))
52		3cn 12340	. . . . . . . . 9 ⊢ 3 ∈ ℂ
53		2cn 12334	. . . . . . . . 9 ⊢ 2 ∈ ℂ
54	52, 53	mulcomi 11235	. . . . . . . 8 ⊢ (3 · 2) = (2 · 3)
55	54	a1i 11	. . . . . . 7 ⊢ (𝑛 ∈ ℕ₀ → (3 · 2) = (2 · 3))
56	55	oveq2d 7439	. . . . . 6 ⊢ (𝑛 ∈ ℕ₀ → ((((4↑𝑛) + 2) · 4) − (3 · 2)) = ((((4↑𝑛) + 2) · 4) − (2 · 3)))
57	49, 35	expp1d 14203	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ₀ → (4↑(𝑛 + 1)) = ((4↑𝑛) · 4))
58		ax-1cn 11176	. . . . . . . . . . . 12 ⊢ 1 ∈ ℂ
59		3p1e4 12403	. . . . . . . . . . . . . 14 ⊢ (3 + 1) = 4
60	52, 58, 59	addcomli 11420	. . . . . . . . . . . . 13 ⊢ (1 + 3) = 4
61	60	eqcomi 2775	. . . . . . . . . . . 12 ⊢ 4 = (1 + 3)
62	58, 52, 61	mvrraddi 11492	. . . . . . . . . . 11 ⊢ (4 − 3) = 1
63	62	oveq2i 7434	. . . . . . . . . 10 ⊢ (2 · (4 − 3)) = (2 · 1)
64	53, 48, 52	subdii 11681	. . . . . . . . . 10 ⊢ (2 · (4 − 3)) = ((2 · 4) − (2 · 3))
65		2t1e2 12421	. . . . . . . . . 10 ⊢ (2 · 1) = 2
66	63, 64, 65	3eqtr3ri 2798	. . . . . . . . 9 ⊢ 2 = ((2 · 4) − (2 · 3))
67	66	a1i 11	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ₀ → 2 = ((2 · 4) − (2 · 3)))
68	57, 67	oveq12d 7441	. . . . . . 7 ⊢ (𝑛 ∈ ℕ₀ → ((4↑(𝑛 + 1)) + 2) = (((4↑𝑛) · 4) + ((2 · 4) − (2 · 3))))
69	46, 49	mulcld 11247	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ₀ → ((4↑𝑛) · 4) ∈ ℂ)
70	47, 49	mulcld 11247	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ₀ → (2 · 4) ∈ ℂ)
71	52	a1i 11	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ₀ → 3 ∈ ℂ)
72	47, 71	mulcld 11247	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ₀ → (2 · 3) ∈ ℂ)
73	69, 70, 72	addsubassd 11607	. . . . . . 7 ⊢ (𝑛 ∈ ℕ₀ → ((((4↑𝑛) · 4) + (2 · 4)) − (2 · 3)) = (((4↑𝑛) · 4) + ((2 · 4) − (2 · 3))))
74	68, 73	eqtr4d 2804	. . . . . 6 ⊢ (𝑛 ∈ ℕ₀ → ((4↑(𝑛 + 1)) + 2) = ((((4↑𝑛) · 4) + (2 · 4)) − (2 · 3)))
75	51, 56, 74	3eqtr4rd 2812	. . . . 5 ⊢ (𝑛 ∈ ℕ₀ → ((4↑(𝑛 + 1)) + 2) = ((((4↑𝑛) + 2) · 4) − (3 · 2)))
76	75	adantr 486	. . . 4 ⊢ ((𝑛 ∈ ℕ₀ ∧ 3 ∥ ((4↑𝑛) + 2)) → ((4↑(𝑛 + 1)) + 2) = ((((4↑𝑛) + 2) · 4) − (3 · 2)))
77	45, 76	breqtrrd 5144	. . 3 ⊢ ((𝑛 ∈ ℕ₀ ∧ 3 ∥ ((4↑𝑛) + 2)) → 3 ∥ ((4↑(𝑛 + 1)) + 2))
78	77	ex 418	. 2 ⊢ (𝑛 ∈ ℕ₀ → (3 ∥ ((4↑𝑛) + 2) → 3 ∥ ((4↑(𝑛 + 1)) + 2)))
79	3, 6, 9, 12, 21, 78	nn0ind 12709	1 ⊢ (𝑁 ∈ ℕ₀ → 3 ∥ ((4↑𝑁) + 2))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2146 class class class wbr 5114 (class class class)co 7423 ℂcc 11116 0cc0 11118 1c1 11119 + caddc 11121 · cmul 11123 − cmin 11459 2c2 12313 3c3 12314 4c4 12315 ℕ₀cn0 12522 ℤcz 12609 ↑cexp 14117 ∥ cdvds 16335

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2738 ax-sep 5262 ax-nul 5274 ax-pow 5341 ax-pr 5409 ax-un 7745 ax-cnex 11174 ax-resscn 11175 ax-1cn 11176 ax-icn 11177 ax-addcl 11178 ax-addrcl 11179 ax-mulcl 11180 ax-mulrcl 11181 ax-mulcom 11182 ax-addass 11183 ax-mulass 11184 ax-distr 11185 ax-i2m1 11186 ax-1ne0 11187 ax-1rid 11188 ax-rnegex 11189 ax-rrecex 11190 ax-cnre 11191 ax-pre-lttri 11192 ax-pre-lttrn 11193 ax-pre-ltadd 11194 ax-pre-mulgt0 11195

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2570 df-eu 2600 df-clab 2745 df-cleq 2758 df-clel 2841 df-nfc 2915 df-ne 2962 df-nel 3068 df-ral 3083 df-rex 3093 df-reu 3373 df-rab 3420 df-v 3460 df-sbc 3748 df-csb 3857 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3928 df-nul 4290 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4878 df-iun 4963 df-br 5115 df-opab 5179 df-mpt 5198 df-tr 5224 df-id 5561 df-eprel 5566 df-po 5574 df-so 5575 df-fr 5619 df-we 5621 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6309 df-ord 6370 df-on 6371 df-lim 6372 df-suc 6373 df-iota 6499 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7380 df-ov 7426 df-oprab 7427 df-mpo 7428 df-om 7872 df-2nd 7996 df-frecs 8287 df-wrecs 8318 df-recs 8367 df-rdg 8406 df-er 8703 df-en 8953 df-dom 8954 df-sdom 8955 df-pnf 11263 df-mnf 11264 df-xr 11265 df-ltxr 11266 df-le 11267 df-sub 11461 df-neg 11462 df-nn 12252 df-2 12321 df-3 12322 df-4 12323 df-n0 12523 df-z 12610 df-uz 12881 df-seq 14058 df-exp 14118 df-dvds 16336

This theorem is used by: (None)

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