ex-ind-dvds - Metamath Proof Explorer

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Theorem ex-ind-dvds 30146

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 7420	. . . 4 ⊢ (𝑘 = 0 → (4↑𝑘) = (4↑0))
2	1	oveq1d 7427	. . 3 ⊢ (𝑘 = 0 → ((4↑𝑘) + 2) = ((4↑0) + 2))
3	2	breq2d 5160	. 2 ⊢ (𝑘 = 0 → (3 ∥ ((4↑𝑘) + 2) ↔ 3 ∥ ((4↑0) + 2)))
4		oveq2 7420	. . . 4 ⊢ (𝑘 = 𝑛 → (4↑𝑘) = (4↑𝑛))
5	4	oveq1d 7427	. . 3 ⊢ (𝑘 = 𝑛 → ((4↑𝑘) + 2) = ((4↑𝑛) + 2))
6	5	breq2d 5160	. 2 ⊢ (𝑘 = 𝑛 → (3 ∥ ((4↑𝑘) + 2) ↔ 3 ∥ ((4↑𝑛) + 2)))
7		oveq2 7420	. . . 4 ⊢ (𝑘 = (𝑛 + 1) → (4↑𝑘) = (4↑(𝑛 + 1)))
8	7	oveq1d 7427	. . 3 ⊢ (𝑘 = (𝑛 + 1) → ((4↑𝑘) + 2) = ((4↑(𝑛 + 1)) + 2))
9	8	breq2d 5160	. 2 ⊢ (𝑘 = (𝑛 + 1) → (3 ∥ ((4↑𝑘) + 2) ↔ 3 ∥ ((4↑(𝑛 + 1)) + 2)))
10		oveq2 7420	. . . 4 ⊢ (𝑘 = 𝑁 → (4↑𝑘) = (4↑𝑁))
11	10	oveq1d 7427	. . 3 ⊢ (𝑘 = 𝑁 → ((4↑𝑘) + 2) = ((4↑𝑁) + 2))
12	11	breq2d 5160	. 2 ⊢ (𝑘 = 𝑁 → (3 ∥ ((4↑𝑘) + 2) ↔ 3 ∥ ((4↑𝑁) + 2)))
13		3z 12602	. . . 4 ⊢ 3 ∈ ℤ
14		iddvds 16220	. . . 4 ⊢ (3 ∈ ℤ → 3 ∥ 3)
15	13, 14	ax-mp 5	. . 3 ⊢ 3 ∥ 3
16		4nn0 12498	. . . . . 6 ⊢ 4 ∈ ℕ₀
17	16	numexp0 17016	. . . . 5 ⊢ (4↑0) = 1
18	17	oveq1i 7422	. . . 4 ⊢ ((4↑0) + 2) = (1 + 2)
19		1p2e3 12362	. . . 4 ⊢ (1 + 2) = 3
20	18, 19	eqtri 2759	. . 3 ⊢ ((4↑0) + 2) = 3
21	15, 20	breqtrri 5175	. 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 14216	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ₀ ∧ 3 ∥ ((4↑𝑛) + 2)) → (4↑𝑛) ∈ ℕ₀)
26	25	nn0zd 12591	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ₀ ∧ 3 ∥ ((4↑𝑛) + 2)) → (4↑𝑛) ∈ ℤ)
27		2z 12601	. . . . . . . 8 ⊢ 2 ∈ ℤ
28	27	a1i 11	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ₀ ∧ 3 ∥ ((4↑𝑛) + 2)) → 2 ∈ ℤ)
29	26, 28	zaddcld 12677	. . . . . 6 ⊢ ((𝑛 ∈ ℕ₀ ∧ 3 ∥ ((4↑𝑛) + 2)) → ((4↑𝑛) + 2) ∈ ℤ)
30		4z 12603	. . . . . . 7 ⊢ 4 ∈ ℤ
31	30	a1i 11	. . . . . 6 ⊢ ((𝑛 ∈ ℕ₀ ∧ 3 ∥ ((4↑𝑛) + 2)) → 4 ∈ ℤ)
32	29, 31	zmulcld 12679	. . . . 5 ⊢ ((𝑛 ∈ ℕ₀ ∧ 3 ∥ ((4↑𝑛) + 2)) → (((4↑𝑛) + 2) · 4) ∈ ℤ)
33	22, 28	zmulcld 12679	. . . . 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 14216	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ₀ → (4↑𝑛) ∈ ℕ₀)
37	36	nn0zd 12591	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ₀ → (4↑𝑛) ∈ ℤ)
38	37	adantr 480	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ₀ ∧ 3 ∥ ((4↑𝑛) + 2)) → (4↑𝑛) ∈ ℤ)
39	38, 28	zaddcld 12677	. . . . . 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 16247	. . . . 5 ⊢ ((𝑛 ∈ ℕ₀ ∧ 3 ∥ ((4↑𝑛) + 2)) → 3 ∥ (((4↑𝑛) + 2) · 4))
42		dvdsmul1 16228	. . . . . . 7 ⊢ ((3 ∈ ℤ ∧ 2 ∈ ℤ) → 3 ∥ (3 · 2))
43	13, 27, 42	mp2an 689	. . . . . 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 16243	. . . 4 ⊢ ((𝑛 ∈ ℕ₀ ∧ 3 ∥ ((4↑𝑛) + 2)) → 3 ∥ ((((4↑𝑛) + 2) · 4) − (3 · 2)))
46	36	nn0cnd 12541	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ₀ → (4↑𝑛) ∈ ℂ)
47		2cnd 12297	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ₀ → 2 ∈ ℂ)
48		4cn 12304	. . . . . . . . 9 ⊢ 4 ∈ ℂ
49	48	a1i 11	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ₀ → 4 ∈ ℂ)
50	46, 47, 49	adddird 11246	. . . . . . 7 ⊢ (𝑛 ∈ ℕ₀ → (((4↑𝑛) + 2) · 4) = (((4↑𝑛) · 4) + (2 · 4)))
51	50	oveq1d 7427	. . . . . 6 ⊢ (𝑛 ∈ ℕ₀ → ((((4↑𝑛) + 2) · 4) − (2 · 3)) = ((((4↑𝑛) · 4) + (2 · 4)) − (2 · 3)))
52		3cn 12300	. . . . . . . . 9 ⊢ 3 ∈ ℂ
53		2cn 12294	. . . . . . . . 9 ⊢ 2 ∈ ℂ
54	52, 53	mulcomi 11229	. . . . . . . 8 ⊢ (3 · 2) = (2 · 3)
55	54	a1i 11	. . . . . . 7 ⊢ (𝑛 ∈ ℕ₀ → (3 · 2) = (2 · 3))
56	55	oveq2d 7428	. . . . . 6 ⊢ (𝑛 ∈ ℕ₀ → ((((4↑𝑛) + 2) · 4) − (3 · 2)) = ((((4↑𝑛) + 2) · 4) − (2 · 3)))
57	49, 35	expp1d 14119	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ₀ → (4↑(𝑛 + 1)) = ((4↑𝑛) · 4))
58		ax-1cn 11174	. . . . . . . . . . . 12 ⊢ 1 ∈ ℂ
59		3p1e4 12364	. . . . . . . . . . . . . 14 ⊢ (3 + 1) = 4
60	52, 58, 59	addcomli 11413	. . . . . . . . . . . . 13 ⊢ (1 + 3) = 4
61	60	eqcomi 2740	. . . . . . . . . . . 12 ⊢ 4 = (1 + 3)
62	58, 52, 61	mvrraddi 11484	. . . . . . . . . . 11 ⊢ (4 − 3) = 1
63	62	oveq2i 7423	. . . . . . . . . 10 ⊢ (2 · (4 − 3)) = (2 · 1)
64	53, 48, 52	subdii 11670	. . . . . . . . . 10 ⊢ (2 · (4 − 3)) = ((2 · 4) − (2 · 3))
65		2t1e2 12382	. . . . . . . . . 10 ⊢ (2 · 1) = 2
66	63, 64, 65	3eqtr3ri 2768	. . . . . . . . 9 ⊢ 2 = ((2 · 4) − (2 · 3))
67	66	a1i 11	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ₀ → 2 = ((2 · 4) − (2 · 3)))
68	57, 67	oveq12d 7430	. . . . . . 7 ⊢ (𝑛 ∈ ℕ₀ → ((4↑(𝑛 + 1)) + 2) = (((4↑𝑛) · 4) + ((2 · 4) − (2 · 3))))
69	46, 49	mulcld 11241	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ₀ → ((4↑𝑛) · 4) ∈ ℂ)
70	47, 49	mulcld 11241	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ₀ → (2 · 4) ∈ ℂ)
71	52	a1i 11	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ₀ → 3 ∈ ℂ)
72	47, 71	mulcld 11241	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ₀ → (2 · 3) ∈ ℂ)
73	69, 70, 72	addsubassd 11598	. . . . . . 7 ⊢ (𝑛 ∈ ℕ₀ → ((((4↑𝑛) · 4) + (2 · 4)) − (2 · 3)) = (((4↑𝑛) · 4) + ((2 · 4) − (2 · 3))))
74	68, 73	eqtr4d 2774	. . . . . 6 ⊢ (𝑛 ∈ ℕ₀ → ((4↑(𝑛 + 1)) + 2) = ((((4↑𝑛) · 4) + (2 · 4)) − (2 · 3)))
75	51, 56, 74	3eqtr4rd 2782	. . . . 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 5176	. . 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 12664	1 ⊢ (𝑁 ∈ ℕ₀ → 3 ∥ ((4↑𝑁) + 2))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2105 class class class wbr 5148 (class class class)co 7412 ℂcc 11114 0cc0 11116 1c1 11117 + caddc 11119 · cmul 11121 − cmin 11451 2c2 12274 3c3 12275 4c4 12276 ℕ₀cn0 12479 ℤcz 12565 ↑cexp 14034 ∥ cdvds 16204

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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-er 8709 df-en 8946 df-dom 8947 df-sdom 8948 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 df-n0 12480 df-z 12566 df-uz 12830 df-seq 13974 df-exp 14035 df-dvds 16205

This theorem is referenced by: (None)

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