inductionexd - Mathbox for Stanislas Polu

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Theorem inductionexd 44861

Description: Simple induction example. (Contributed by Stanislas Polu, 9-Mar-2020.)

**Assertion**
Ref	Expression
inductionexd	⊢ (𝑁 ∈ ℕ → 3 ∥ ((4↑𝑁) + 5))

Proof of Theorem inductionexd Dummy variables 𝑘 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 7420	. . . 4 ⊢ (𝑘 = 1 → (4↑𝑘) = (4↑1))
2	1	oveq1d 7427	. . 3 ⊢ (𝑘 = 1 → ((4↑𝑘) + 5) = ((4↑1) + 5))
3	2	breq2d 5122	. 2 ⊢ (𝑘 = 1 → (3 ∥ ((4↑𝑘) + 5) ↔ 3 ∥ ((4↑1) + 5)))
4		oveq2 7420	. . . 4 ⊢ (𝑘 = 𝑛 → (4↑𝑘) = (4↑𝑛))
5	4	oveq1d 7427	. . 3 ⊢ (𝑘 = 𝑛 → ((4↑𝑘) + 5) = ((4↑𝑛) + 5))
6	5	breq2d 5122	. 2 ⊢ (𝑘 = 𝑛 → (3 ∥ ((4↑𝑘) + 5) ↔ 3 ∥ ((4↑𝑛) + 5)))
7		oveq2 7420	. . . 4 ⊢ (𝑘 = (𝑛 + 1) → (4↑𝑘) = (4↑(𝑛 + 1)))
8	7	oveq1d 7427	. . 3 ⊢ (𝑘 = (𝑛 + 1) → ((4↑𝑘) + 5) = ((4↑(𝑛 + 1)) + 5))
9	8	breq2d 5122	. 2 ⊢ (𝑘 = (𝑛 + 1) → (3 ∥ ((4↑𝑘) + 5) ↔ 3 ∥ ((4↑(𝑛 + 1)) + 5)))
10		oveq2 7420	. . . 4 ⊢ (𝑘 = 𝑁 → (4↑𝑘) = (4↑𝑁))
11	10	oveq1d 7427	. . 3 ⊢ (𝑘 = 𝑁 → ((4↑𝑘) + 5) = ((4↑𝑁) + 5))
12	11	breq2d 5122	. 2 ⊢ (𝑘 = 𝑁 → (3 ∥ ((4↑𝑘) + 5) ↔ 3 ∥ ((4↑𝑁) + 5)))
13		3z 12628	. . . 4 ⊢ 3 ∈ ℤ
14		4z 12629	. . . . . 6 ⊢ 4 ∈ ℤ
15		1nn0 12521	. . . . . 6 ⊢ 1 ∈ ℕ₀
16		zexpcl 14114	. . . . . 6 ⊢ ((4 ∈ ℤ ∧ 1 ∈ ℕ₀) → (4↑1) ∈ ℤ)
17	14, 15, 16	mp2an 704	. . . . 5 ⊢ (4↑1) ∈ ℤ
18		5nn 12328	. . . . . 6 ⊢ 5 ∈ ℕ
19	18	nnzi 12619	. . . . 5 ⊢ 5 ∈ ℤ
20		zaddcl 12635	. . . . 5 ⊢ (((4↑1) ∈ ℤ ∧ 5 ∈ ℤ) → ((4↑1) + 5) ∈ ℤ)
21	17, 19, 20	mp2an 704	. . . 4 ⊢ ((4↑1) + 5) ∈ ℤ
22	13, 13, 21	3pm3.2i 1358	. . 3 ⊢ (3 ∈ ℤ ∧ 3 ∈ ℤ ∧ ((4↑1) + 5) ∈ ℤ)
23		3t3e9 12409	. . . 4 ⊢ (3 · 3) = 9
24		4nn0 12524	. . . . . . 7 ⊢ 4 ∈ ℕ₀
25	24	numexp1 17137	. . . . . 6 ⊢ (4↑1) = 4
26	25	oveq1i 7422	. . . . 5 ⊢ ((4↑1) + 5) = (4 + 5)
27		5cn 12330	. . . . . 6 ⊢ 5 ∈ ℂ
28		4cn 12327	. . . . . 6 ⊢ 4 ∈ ℂ
29		5p4e9 12399	. . . . . 6 ⊢ (5 + 4) = 9
30	27, 28, 29	addcomli 11403	. . . . 5 ⊢ (4 + 5) = 9
31	26, 30	eqtri 2786	. . . 4 ⊢ ((4↑1) + 5) = 9
32	23, 31	eqtr4i 2789	. . 3 ⊢ (3 · 3) = ((4↑1) + 5)
33		dvds0lem 16325	. . 3 ⊢ (((3 ∈ ℤ ∧ 3 ∈ ℤ ∧ ((4↑1) + 5) ∈ ℤ) ∧ (3 · 3) = ((4↑1) + 5)) → 3 ∥ ((4↑1) + 5))
34	22, 32, 33	mp2an 704	. 2 ⊢ 3 ∥ ((4↑1) + 5)
35	13	a1i 11	. . . . 5 ⊢ ((𝑛 ∈ ℕ ∧ 3 ∥ ((4↑𝑛) + 5)) → 3 ∈ ℤ)
36		4nn 12325	. . . . . . . . . . 11 ⊢ 4 ∈ ℕ
37	36	a1i 11	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → 4 ∈ ℕ)
38		nnnn0 12512	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ₀)
39	37, 38	nnexpcld 14283	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → (4↑𝑛) ∈ ℕ)
40	39	nnzd 12618	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → (4↑𝑛) ∈ ℤ)
41	40	adantr 485	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ ∧ 3 ∥ ((4↑𝑛) + 5)) → (4↑𝑛) ∈ ℤ)
42	19	a1i 11	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ ∧ 3 ∥ ((4↑𝑛) + 5)) → 5 ∈ ℤ)
43	41, 42	zaddcld 12705	. . . . . 6 ⊢ ((𝑛 ∈ ℕ ∧ 3 ∥ ((4↑𝑛) + 5)) → ((4↑𝑛) + 5) ∈ ℤ)
44	14	a1i 11	. . . . . 6 ⊢ ((𝑛 ∈ ℕ ∧ 3 ∥ ((4↑𝑛) + 5)) → 4 ∈ ℤ)
45	43, 44	zmulcld 12707	. . . . 5 ⊢ ((𝑛 ∈ ℕ ∧ 3 ∥ ((4↑𝑛) + 5)) → (((4↑𝑛) + 5) · 4) ∈ ℤ)
46	35, 42	zmulcld 12707	. . . . 5 ⊢ ((𝑛 ∈ ℕ ∧ 3 ∥ ((4↑𝑛) + 5)) → (3 · 5) ∈ ℤ)
47		simpr 489	. . . . . 6 ⊢ ((𝑛 ∈ ℕ ∧ 3 ∥ ((4↑𝑛) + 5)) → 3 ∥ ((4↑𝑛) + 5))
48	35, 43, 44, 47	dvdsmultr1d 16356	. . . . 5 ⊢ ((𝑛 ∈ ℕ ∧ 3 ∥ ((4↑𝑛) + 5)) → 3 ∥ (((4↑𝑛) + 5) · 4))
49		dvdsmul1 16336	. . . . . . 7 ⊢ ((3 ∈ ℤ ∧ 5 ∈ ℤ) → 3 ∥ (3 · 5))
50	13, 19, 49	mp2an 704	. . . . . 6 ⊢ 3 ∥ (3 · 5)
51	50	a1i 11	. . . . 5 ⊢ ((𝑛 ∈ ℕ ∧ 3 ∥ ((4↑𝑛) + 5)) → 3 ∥ (3 · 5))
52	35, 45, 46, 48, 51	dvds2subd 16352	. . . 4 ⊢ ((𝑛 ∈ ℕ ∧ 3 ∥ ((4↑𝑛) + 5)) → 3 ∥ ((((4↑𝑛) + 5) · 4) − (3 · 5)))
53	39	nncnd 12250	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → (4↑𝑛) ∈ ℂ)
54	27	a1i 11	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → 5 ∈ ℂ)
55	28	a1i 11	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → 4 ∈ ℂ)
56	53, 54, 55	adddird 11235	. . . . . . 7 ⊢ (𝑛 ∈ ℕ → (((4↑𝑛) + 5) · 4) = (((4↑𝑛) · 4) + (5 · 4)))
57	56	oveq1d 7427	. . . . . 6 ⊢ (𝑛 ∈ ℕ → ((((4↑𝑛) + 5) · 4) − ;15) = ((((4↑𝑛) · 4) + (5 · 4)) − ;15))
58		3cn 12323	. . . . . . . . 9 ⊢ 3 ∈ ℂ
59		5t3e15 12818	. . . . . . . . 9 ⊢ (5 · 3) = ;15
60	27, 58, 59	mulcomli 11219	. . . . . . . 8 ⊢ (3 · 5) = ;15
61	60	a1i 11	. . . . . . 7 ⊢ (𝑛 ∈ ℕ → (3 · 5) = ;15)
62	61	oveq2d 7428	. . . . . 6 ⊢ (𝑛 ∈ ℕ → ((((4↑𝑛) + 5) · 4) − (3 · 5)) = ((((4↑𝑛) + 5) · 4) − ;15))
63	55, 38	expp1d 14185	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → (4↑(𝑛 + 1)) = ((4↑𝑛) · 4))
64		ax-1cn 11159	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℂ
65		3p1e4 12386	. . . . . . . . . . . . . . . 16 ⊢ (3 + 1) = 4
66	58, 64, 65	addcomli 11403	. . . . . . . . . . . . . . 15 ⊢ (1 + 3) = 4
67	66	eqcomi 2772	. . . . . . . . . . . . . 14 ⊢ 4 = (1 + 3)
68	67	oveq1i 7422	. . . . . . . . . . . . 13 ⊢ (4 − 3) = ((1 + 3) − 3)
69	64, 58	pncan3oi 11474	. . . . . . . . . . . . 13 ⊢ ((1 + 3) − 3) = 1
70	68, 69	eqtri 2786	. . . . . . . . . . . 12 ⊢ (4 − 3) = 1
71	70	oveq2i 7423	. . . . . . . . . . 11 ⊢ (5 · (4 − 3)) = (5 · 1)
72	27, 28, 58	subdii 11664	. . . . . . . . . . 11 ⊢ (5 · (4 − 3)) = ((5 · 4) − (5 · 3))
73	27	mulridi 11214	. . . . . . . . . . 11 ⊢ (5 · 1) = 5
74	71, 72, 73	3eqtr3ri 2795	. . . . . . . . . 10 ⊢ 5 = ((5 · 4) − (5 · 3))
75	59	eqcomi 2772	. . . . . . . . . . 11 ⊢ ;15 = (5 · 3)
76	75	oveq2i 7423	. . . . . . . . . 10 ⊢ ((5 · 4) − ;15) = ((5 · 4) − (5 · 3))
77	74, 76	eqtr4i 2789	. . . . . . . . 9 ⊢ 5 = ((5 · 4) − ;15)
78	77	a1i 11	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → 5 = ((5 · 4) − ;15))
79	63, 78	oveq12d 7430	. . . . . . 7 ⊢ (𝑛 ∈ ℕ → ((4↑(𝑛 + 1)) + 5) = (((4↑𝑛) · 4) + ((5 · 4) − ;15)))
80	53, 55	mulcld 11230	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → ((4↑𝑛) · 4) ∈ ℂ)
81	54, 55	mulcld 11230	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → (5 · 4) ∈ ℂ)
82		5nn0 12525	. . . . . . . . . . 11 ⊢ 5 ∈ ℕ₀
83	15, 82	deccl 12727	. . . . . . . . . 10 ⊢ ;15 ∈ ℕ₀
84	83	nn0cni 12517	. . . . . . . . 9 ⊢ ;15 ∈ ℂ
85	84	a1i 11	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → ;15 ∈ ℂ)
86	80, 81, 85	addsubassd 11590	. . . . . . 7 ⊢ (𝑛 ∈ ℕ → ((((4↑𝑛) · 4) + (5 · 4)) − ;15) = (((4↑𝑛) · 4) + ((5 · 4) − ;15)))
87	79, 86	eqtr4d 2801	. . . . . 6 ⊢ (𝑛 ∈ ℕ → ((4↑(𝑛 + 1)) + 5) = ((((4↑𝑛) · 4) + (5 · 4)) − ;15))
88	57, 62, 87	3eqtr4rd 2809	. . . . 5 ⊢ (𝑛 ∈ ℕ → ((4↑(𝑛 + 1)) + 5) = ((((4↑𝑛) + 5) · 4) − (3 · 5)))
89	88	adantr 485	. . . 4 ⊢ ((𝑛 ∈ ℕ ∧ 3 ∥ ((4↑𝑛) + 5)) → ((4↑(𝑛 + 1)) + 5) = ((((4↑𝑛) + 5) · 4) − (3 · 5)))
90	52, 89	breqtrrd 5140	. . 3 ⊢ ((𝑛 ∈ ℕ ∧ 3 ∥ ((4↑𝑛) + 5)) → 3 ∥ ((4↑(𝑛 + 1)) + 5))
91	90	ex 417	. 2 ⊢ (𝑛 ∈ ℕ → (3 ∥ ((4↑𝑛) + 5) → 3 ∥ ((4↑(𝑛 + 1)) + 5)))
92	3, 6, 9, 12, 34, 91	nnind 12252	1 ⊢ (𝑁 ∈ ℕ → 3 ∥ ((4↑𝑁) + 5))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 400 ∧ w3a 1103 = wceq 1570 ∈ wcel 2143 class class class wbr 5110 (class class class)co 7412 ℂcc 11099 1c1 11102 + caddc 11104 · cmul 11106 − cmin 11442 ℕcn 12234 3c3 12297 4c4 12298 5c5 12299 9c9 12303 ℕ₀cn0 12505 ℤcz 12592 cdc 12712 ↑cexp 14099 ∥ cdvds 16311

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-sep 5258 ax-nul 5270 ax-pow 5338 ax-pr 5406 ax-un 7734 ax-cnex 11157 ax-resscn 11158 ax-1cn 11159 ax-icn 11160 ax-addcl 11161 ax-addrcl 11162 ax-mulcl 11163 ax-mulrcl 11164 ax-mulcom 11165 ax-addass 11166 ax-mulass 11167 ax-distr 11168 ax-i2m1 11169 ax-1ne0 11170 ax-1rid 11171 ax-rnegex 11172 ax-rrecex 11173 ax-cnre 11174 ax-pre-lttri 11175 ax-pre-lttrn 11176 ax-pre-ltadd 11177 ax-pre-mulgt0 11178

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3746 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4288 df-if 4489 df-pw 4565 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4874 df-iun 4959 df-br 5111 df-opab 5175 df-mpt 5194 df-tr 5220 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7864 df-2nd 7988 df-frecs 8279 df-wrecs 8310 df-recs 8359 df-rdg 8398 df-er 8695 df-en 8945 df-dom 8946 df-sdom 8947 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11444 df-neg 11445 df-nn 12235 df-2 12304 df-3 12305 df-4 12306 df-5 12307 df-6 12308 df-7 12309 df-8 12310 df-9 12311 df-n0 12506 df-z 12593 df-dec 12713 df-uz 12864 df-seq 14040 df-exp 14100 df-dvds 16312

This theorem is referenced by: (None)

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