inductionexd - Mathbox for Stanislas Polu

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Theorem inductionexd 43498

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 7422	. . . 4 ⊢ (𝑘 = 1 → (4↑𝑘) = (4↑1))
2	1	oveq1d 7429	. . 3 ⊢ (𝑘 = 1 → ((4↑𝑘) + 5) = ((4↑1) + 5))
3	2	breq2d 5154	. 2 ⊢ (𝑘 = 1 → (3 ∥ ((4↑𝑘) + 5) ↔ 3 ∥ ((4↑1) + 5)))
4		oveq2 7422	. . . 4 ⊢ (𝑘 = 𝑛 → (4↑𝑘) = (4↑𝑛))
5	4	oveq1d 7429	. . 3 ⊢ (𝑘 = 𝑛 → ((4↑𝑘) + 5) = ((4↑𝑛) + 5))
6	5	breq2d 5154	. 2 ⊢ (𝑘 = 𝑛 → (3 ∥ ((4↑𝑘) + 5) ↔ 3 ∥ ((4↑𝑛) + 5)))
7		oveq2 7422	. . . 4 ⊢ (𝑘 = (𝑛 + 1) → (4↑𝑘) = (4↑(𝑛 + 1)))
8	7	oveq1d 7429	. . 3 ⊢ (𝑘 = (𝑛 + 1) → ((4↑𝑘) + 5) = ((4↑(𝑛 + 1)) + 5))
9	8	breq2d 5154	. 2 ⊢ (𝑘 = (𝑛 + 1) → (3 ∥ ((4↑𝑘) + 5) ↔ 3 ∥ ((4↑(𝑛 + 1)) + 5)))
10		oveq2 7422	. . . 4 ⊢ (𝑘 = 𝑁 → (4↑𝑘) = (4↑𝑁))
11	10	oveq1d 7429	. . 3 ⊢ (𝑘 = 𝑁 → ((4↑𝑘) + 5) = ((4↑𝑁) + 5))
12	11	breq2d 5154	. 2 ⊢ (𝑘 = 𝑁 → (3 ∥ ((4↑𝑘) + 5) ↔ 3 ∥ ((4↑𝑁) + 5)))
13		3z 12611	. . . 4 ⊢ 3 ∈ ℤ
14		4z 12612	. . . . . 6 ⊢ 4 ∈ ℤ
15		1nn0 12504	. . . . . 6 ⊢ 1 ∈ ℕ₀
16		zexpcl 14059	. . . . . 6 ⊢ ((4 ∈ ℤ ∧ 1 ∈ ℕ₀) → (4↑1) ∈ ℤ)
17	14, 15, 16	mp2an 691	. . . . 5 ⊢ (4↑1) ∈ ℤ
18		5nn 12314	. . . . . 6 ⊢ 5 ∈ ℕ
19	18	nnzi 12602	. . . . 5 ⊢ 5 ∈ ℤ
20		zaddcl 12618	. . . . 5 ⊢ (((4↑1) ∈ ℤ ∧ 5 ∈ ℤ) → ((4↑1) + 5) ∈ ℤ)
21	17, 19, 20	mp2an 691	. . . 4 ⊢ ((4↑1) + 5) ∈ ℤ
22	13, 13, 21	3pm3.2i 1337	. . 3 ⊢ (3 ∈ ℤ ∧ 3 ∈ ℤ ∧ ((4↑1) + 5) ∈ ℤ)
23		3t3e9 12395	. . . 4 ⊢ (3 · 3) = 9
24		4nn0 12507	. . . . . . 7 ⊢ 4 ∈ ℕ₀
25	24	numexp1 17031	. . . . . 6 ⊢ (4↑1) = 4
26	25	oveq1i 7424	. . . . 5 ⊢ ((4↑1) + 5) = (4 + 5)
27		5cn 12316	. . . . . 6 ⊢ 5 ∈ ℂ
28		4cn 12313	. . . . . 6 ⊢ 4 ∈ ℂ
29		5p4e9 12386	. . . . . 6 ⊢ (5 + 4) = 9
30	27, 28, 29	addcomli 11422	. . . . 5 ⊢ (4 + 5) = 9
31	26, 30	eqtri 2755	. . . 4 ⊢ ((4↑1) + 5) = 9
32	23, 31	eqtr4i 2758	. . 3 ⊢ (3 · 3) = ((4↑1) + 5)
33		dvds0lem 16229	. . 3 ⊢ (((3 ∈ ℤ ∧ 3 ∈ ℤ ∧ ((4↑1) + 5) ∈ ℤ) ∧ (3 · 3) = ((4↑1) + 5)) → 3 ∥ ((4↑1) + 5))
34	22, 32, 33	mp2an 691	. 2 ⊢ 3 ∥ ((4↑1) + 5)
35	13	a1i 11	. . . . 5 ⊢ ((𝑛 ∈ ℕ ∧ 3 ∥ ((4↑𝑛) + 5)) → 3 ∈ ℤ)
36		4nn 12311	. . . . . . . . . . 11 ⊢ 4 ∈ ℕ
37	36	a1i 11	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → 4 ∈ ℕ)
38		nnnn0 12495	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ₀)
39	37, 38	nnexpcld 14225	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → (4↑𝑛) ∈ ℕ)
40	39	nnzd 12601	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → (4↑𝑛) ∈ ℤ)
41	40	adantr 480	. . . . . . 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 12686	. . . . . 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 12688	. . . . 5 ⊢ ((𝑛 ∈ ℕ ∧ 3 ∥ ((4↑𝑛) + 5)) → (((4↑𝑛) + 5) · 4) ∈ ℤ)
46	35, 42	zmulcld 12688	. . . . 5 ⊢ ((𝑛 ∈ ℕ ∧ 3 ∥ ((4↑𝑛) + 5)) → (3 · 5) ∈ ℤ)
47		simpr 484	. . . . . 6 ⊢ ((𝑛 ∈ ℕ ∧ 3 ∥ ((4↑𝑛) + 5)) → 3 ∥ ((4↑𝑛) + 5))
48	35, 43, 44, 47	dvdsmultr1d 16259	. . . . 5 ⊢ ((𝑛 ∈ ℕ ∧ 3 ∥ ((4↑𝑛) + 5)) → 3 ∥ (((4↑𝑛) + 5) · 4))
49		dvdsmul1 16240	. . . . . . 7 ⊢ ((3 ∈ ℤ ∧ 5 ∈ ℤ) → 3 ∥ (3 · 5))
50	13, 19, 49	mp2an 691	. . . . . 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 16255	. . . 4 ⊢ ((𝑛 ∈ ℕ ∧ 3 ∥ ((4↑𝑛) + 5)) → 3 ∥ ((((4↑𝑛) + 5) · 4) − (3 · 5)))
53	39	nncnd 12244	. . . . . . . 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 11255	. . . . . . 7 ⊢ (𝑛 ∈ ℕ → (((4↑𝑛) + 5) · 4) = (((4↑𝑛) · 4) + (5 · 4)))
57	56	oveq1d 7429	. . . . . 6 ⊢ (𝑛 ∈ ℕ → ((((4↑𝑛) + 5) · 4) − ;15) = ((((4↑𝑛) · 4) + (5 · 4)) − ;15))
58		3cn 12309	. . . . . . . . 9 ⊢ 3 ∈ ℂ
59		5t3e15 12794	. . . . . . . . 9 ⊢ (5 · 3) = ;15
60	27, 58, 59	mulcomli 11239	. . . . . . . 8 ⊢ (3 · 5) = ;15
61	60	a1i 11	. . . . . . 7 ⊢ (𝑛 ∈ ℕ → (3 · 5) = ;15)
62	61	oveq2d 7430	. . . . . 6 ⊢ (𝑛 ∈ ℕ → ((((4↑𝑛) + 5) · 4) − (3 · 5)) = ((((4↑𝑛) + 5) · 4) − ;15))
63	55, 38	expp1d 14129	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → (4↑(𝑛 + 1)) = ((4↑𝑛) · 4))
64		ax-1cn 11182	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℂ
65		3p1e4 12373	. . . . . . . . . . . . . . . 16 ⊢ (3 + 1) = 4
66	58, 64, 65	addcomli 11422	. . . . . . . . . . . . . . 15 ⊢ (1 + 3) = 4
67	66	eqcomi 2736	. . . . . . . . . . . . . 14 ⊢ 4 = (1 + 3)
68	67	oveq1i 7424	. . . . . . . . . . . . 13 ⊢ (4 − 3) = ((1 + 3) − 3)
69	64, 58	pncan3oi 11492	. . . . . . . . . . . . 13 ⊢ ((1 + 3) − 3) = 1
70	68, 69	eqtri 2755	. . . . . . . . . . . 12 ⊢ (4 − 3) = 1
71	70	oveq2i 7425	. . . . . . . . . . 11 ⊢ (5 · (4 − 3)) = (5 · 1)
72	27, 28, 58	subdii 11679	. . . . . . . . . . 11 ⊢ (5 · (4 − 3)) = ((5 · 4) − (5 · 3))
73	27	mulridi 11234	. . . . . . . . . . 11 ⊢ (5 · 1) = 5
74	71, 72, 73	3eqtr3ri 2764	. . . . . . . . . 10 ⊢ 5 = ((5 · 4) − (5 · 3))
75	59	eqcomi 2736	. . . . . . . . . . 11 ⊢ ;15 = (5 · 3)
76	75	oveq2i 7425	. . . . . . . . . 10 ⊢ ((5 · 4) − ;15) = ((5 · 4) − (5 · 3))
77	74, 76	eqtr4i 2758	. . . . . . . . 9 ⊢ 5 = ((5 · 4) − ;15)
78	77	a1i 11	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → 5 = ((5 · 4) − ;15))
79	63, 78	oveq12d 7432	. . . . . . 7 ⊢ (𝑛 ∈ ℕ → ((4↑(𝑛 + 1)) + 5) = (((4↑𝑛) · 4) + ((5 · 4) − ;15)))
80	53, 55	mulcld 11250	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → ((4↑𝑛) · 4) ∈ ℂ)
81	54, 55	mulcld 11250	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → (5 · 4) ∈ ℂ)
82		5nn0 12508	. . . . . . . . . . 11 ⊢ 5 ∈ ℕ₀
83	15, 82	deccl 12708	. . . . . . . . . 10 ⊢ ;15 ∈ ℕ₀
84	83	nn0cni 12500	. . . . . . . . 9 ⊢ ;15 ∈ ℂ
85	84	a1i 11	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → ;15 ∈ ℂ)
86	80, 81, 85	addsubassd 11607	. . . . . . 7 ⊢ (𝑛 ∈ ℕ → ((((4↑𝑛) · 4) + (5 · 4)) − ;15) = (((4↑𝑛) · 4) + ((5 · 4) − ;15)))
87	79, 86	eqtr4d 2770	. . . . . 6 ⊢ (𝑛 ∈ ℕ → ((4↑(𝑛 + 1)) + 5) = ((((4↑𝑛) · 4) + (5 · 4)) − ;15))
88	57, 62, 87	3eqtr4rd 2778	. . . . 5 ⊢ (𝑛 ∈ ℕ → ((4↑(𝑛 + 1)) + 5) = ((((4↑𝑛) + 5) · 4) − (3 · 5)))
89	88	adantr 480	. . . 4 ⊢ ((𝑛 ∈ ℕ ∧ 3 ∥ ((4↑𝑛) + 5)) → ((4↑(𝑛 + 1)) + 5) = ((((4↑𝑛) + 5) · 4) − (3 · 5)))
90	52, 89	breqtrrd 5170	. . 3 ⊢ ((𝑛 ∈ ℕ ∧ 3 ∥ ((4↑𝑛) + 5)) → 3 ∥ ((4↑(𝑛 + 1)) + 5))
91	90	ex 412	. 2 ⊢ (𝑛 ∈ ℕ → (3 ∥ ((4↑𝑛) + 5) → 3 ∥ ((4↑(𝑛 + 1)) + 5)))
92	3, 6, 9, 12, 34, 91	nnind 12246	1 ⊢ (𝑁 ∈ ℕ → 3 ∥ ((4↑𝑁) + 5))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 class class class wbr 5142 (class class class)co 7414 ℂcc 11122 1c1 11125 + caddc 11127 · cmul 11129 − cmin 11460 ℕcn 12228 3c3 12284 4c4 12285 5c5 12286 9c9 12290 ℕ₀cn0 12488 ℤcz 12574 cdc 12693 ↑cexp 14044 ∥ cdvds 16216

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7863 df-2nd 7986 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-er 8716 df-en 8954 df-dom 8955 df-sdom 8956 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 df-sub 11462 df-neg 11463 df-nn 12229 df-2 12291 df-3 12292 df-4 12293 df-5 12294 df-6 12295 df-7 12296 df-8 12297 df-9 12298 df-n0 12489 df-z 12575 df-dec 12694 df-uz 12839 df-seq 13985 df-exp 14045 df-dvds 16217

This theorem is referenced by: (None)

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