inductionexd - Mathbox for Stanislas Polu

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Theorem inductionexd 44743

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 7408	. . . 4 ⊢ (𝑘 = 1 → (4↑𝑘) = (4↑1))
2	1	oveq1d 7415	. . 3 ⊢ (𝑘 = 1 → ((4↑𝑘) + 5) = ((4↑1) + 5))
3	2	breq2d 5117	. 2 ⊢ (𝑘 = 1 → (3 ∥ ((4↑𝑘) + 5) ↔ 3 ∥ ((4↑1) + 5)))
4		oveq2 7408	. . . 4 ⊢ (𝑘 = 𝑛 → (4↑𝑘) = (4↑𝑛))
5	4	oveq1d 7415	. . 3 ⊢ (𝑘 = 𝑛 → ((4↑𝑘) + 5) = ((4↑𝑛) + 5))
6	5	breq2d 5117	. 2 ⊢ (𝑘 = 𝑛 → (3 ∥ ((4↑𝑘) + 5) ↔ 3 ∥ ((4↑𝑛) + 5)))
7		oveq2 7408	. . . 4 ⊢ (𝑘 = (𝑛 + 1) → (4↑𝑘) = (4↑(𝑛 + 1)))
8	7	oveq1d 7415	. . 3 ⊢ (𝑘 = (𝑛 + 1) → ((4↑𝑘) + 5) = ((4↑(𝑛 + 1)) + 5))
9	8	breq2d 5117	. 2 ⊢ (𝑘 = (𝑛 + 1) → (3 ∥ ((4↑𝑘) + 5) ↔ 3 ∥ ((4↑(𝑛 + 1)) + 5)))
10		oveq2 7408	. . . 4 ⊢ (𝑘 = 𝑁 → (4↑𝑘) = (4↑𝑁))
11	10	oveq1d 7415	. . 3 ⊢ (𝑘 = 𝑁 → ((4↑𝑘) + 5) = ((4↑𝑁) + 5))
12	11	breq2d 5117	. 2 ⊢ (𝑘 = 𝑁 → (3 ∥ ((4↑𝑘) + 5) ↔ 3 ∥ ((4↑𝑁) + 5)))
13		3z 12618	. . . 4 ⊢ 3 ∈ ℤ
14		4z 12619	. . . . . 6 ⊢ 4 ∈ ℤ
15		1nn0 12511	. . . . . 6 ⊢ 1 ∈ ℕ₀
16		zexpcl 14103	. . . . . 6 ⊢ ((4 ∈ ℤ ∧ 1 ∈ ℕ₀) → (4↑1) ∈ ℤ)
17	14, 15, 16	mp2an 704	. . . . 5 ⊢ (4↑1) ∈ ℤ
18		5nn 12318	. . . . . 6 ⊢ 5 ∈ ℕ
19	18	nnzi 12609	. . . . 5 ⊢ 5 ∈ ℤ
20		zaddcl 12625	. . . . 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 1356	. . 3 ⊢ (3 ∈ ℤ ∧ 3 ∈ ℤ ∧ ((4↑1) + 5) ∈ ℤ)
23		3t3e9 12399	. . . 4 ⊢ (3 · 3) = 9
24		4nn0 12514	. . . . . . 7 ⊢ 4 ∈ ℕ₀
25	24	numexp1 17126	. . . . . 6 ⊢ (4↑1) = 4
26	25	oveq1i 7410	. . . . 5 ⊢ ((4↑1) + 5) = (4 + 5)
27		5cn 12320	. . . . . 6 ⊢ 5 ∈ ℂ
28		4cn 12317	. . . . . 6 ⊢ 4 ∈ ℂ
29		5p4e9 12389	. . . . . 6 ⊢ (5 + 4) = 9
30	27, 28, 29	addcomli 11390	. . . . 5 ⊢ (4 + 5) = 9
31	26, 30	eqtri 2788	. . . 4 ⊢ ((4↑1) + 5) = 9
32	23, 31	eqtr4i 2791	. . 3 ⊢ (3 · 3) = ((4↑1) + 5)
33		dvds0lem 16314	. . 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 12315	. . . . . . . . . . 11 ⊢ 4 ∈ ℕ
37	36	a1i 11	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → 4 ∈ ℕ)
38		nnnn0 12502	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ₀)
39	37, 38	nnexpcld 14272	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → (4↑𝑛) ∈ ℕ)
40	39	nnzd 12608	. . . . . . . 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 12695	. . . . . 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 12697	. . . . 5 ⊢ ((𝑛 ∈ ℕ ∧ 3 ∥ ((4↑𝑛) + 5)) → (((4↑𝑛) + 5) · 4) ∈ ℤ)
46	35, 42	zmulcld 12697	. . . . 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 16345	. . . . 5 ⊢ ((𝑛 ∈ ℕ ∧ 3 ∥ ((4↑𝑛) + 5)) → 3 ∥ (((4↑𝑛) + 5) · 4))
49		dvdsmul1 16325	. . . . . . 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 16341	. . . 4 ⊢ ((𝑛 ∈ ℕ ∧ 3 ∥ ((4↑𝑛) + 5)) → 3 ∥ ((((4↑𝑛) + 5) · 4) − (3 · 5)))
53	39	nncnd 12240	. . . . . . . 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 11222	. . . . . . 7 ⊢ (𝑛 ∈ ℕ → (((4↑𝑛) + 5) · 4) = (((4↑𝑛) · 4) + (5 · 4)))
57	56	oveq1d 7415	. . . . . 6 ⊢ (𝑛 ∈ ℕ → ((((4↑𝑛) + 5) · 4) − ;15) = ((((4↑𝑛) · 4) + (5 · 4)) − ;15))
58		3cn 12313	. . . . . . . . 9 ⊢ 3 ∈ ℂ
59		5t3e15 12808	. . . . . . . . 9 ⊢ (5 · 3) = ;15
60	27, 58, 59	mulcomli 11206	. . . . . . . 8 ⊢ (3 · 5) = ;15
61	60	a1i 11	. . . . . . 7 ⊢ (𝑛 ∈ ℕ → (3 · 5) = ;15)
62	61	oveq2d 7416	. . . . . 6 ⊢ (𝑛 ∈ ℕ → ((((4↑𝑛) + 5) · 4) − (3 · 5)) = ((((4↑𝑛) + 5) · 4) − ;15))
63	55, 38	expp1d 14174	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → (4↑(𝑛 + 1)) = ((4↑𝑛) · 4))
64		ax-1cn 11146	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℂ
65		3p1e4 12376	. . . . . . . . . . . . . . . 16 ⊢ (3 + 1) = 4
66	58, 64, 65	addcomli 11390	. . . . . . . . . . . . . . 15 ⊢ (1 + 3) = 4
67	66	eqcomi 2774	. . . . . . . . . . . . . 14 ⊢ 4 = (1 + 3)
68	67	oveq1i 7410	. . . . . . . . . . . . 13 ⊢ (4 − 3) = ((1 + 3) − 3)
69	64, 58	pncan3oi 11461	. . . . . . . . . . . . 13 ⊢ ((1 + 3) − 3) = 1
70	68, 69	eqtri 2788	. . . . . . . . . . . 12 ⊢ (4 − 3) = 1
71	70	oveq2i 7411	. . . . . . . . . . 11 ⊢ (5 · (4 − 3)) = (5 · 1)
72	27, 28, 58	subdii 11651	. . . . . . . . . . 11 ⊢ (5 · (4 − 3)) = ((5 · 4) − (5 · 3))
73	27	mulridi 11201	. . . . . . . . . . 11 ⊢ (5 · 1) = 5
74	71, 72, 73	3eqtr3ri 2797	. . . . . . . . . 10 ⊢ 5 = ((5 · 4) − (5 · 3))
75	59	eqcomi 2774	. . . . . . . . . . 11 ⊢ ;15 = (5 · 3)
76	75	oveq2i 7411	. . . . . . . . . 10 ⊢ ((5 · 4) − ;15) = ((5 · 4) − (5 · 3))
77	74, 76	eqtr4i 2791	. . . . . . . . 9 ⊢ 5 = ((5 · 4) − ;15)
78	77	a1i 11	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → 5 = ((5 · 4) − ;15))
79	63, 78	oveq12d 7418	. . . . . . 7 ⊢ (𝑛 ∈ ℕ → ((4↑(𝑛 + 1)) + 5) = (((4↑𝑛) · 4) + ((5 · 4) − ;15)))
80	53, 55	mulcld 11217	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → ((4↑𝑛) · 4) ∈ ℂ)
81	54, 55	mulcld 11217	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → (5 · 4) ∈ ℂ)
82		5nn0 12515	. . . . . . . . . . 11 ⊢ 5 ∈ ℕ₀
83	15, 82	deccl 12717	. . . . . . . . . 10 ⊢ ;15 ∈ ℕ₀
84	83	nn0cni 12507	. . . . . . . . 9 ⊢ ;15 ∈ ℂ
85	84	a1i 11	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → ;15 ∈ ℂ)
86	80, 81, 85	addsubassd 11577	. . . . . . 7 ⊢ (𝑛 ∈ ℕ → ((((4↑𝑛) · 4) + (5 · 4)) − ;15) = (((4↑𝑛) · 4) + ((5 · 4) − ;15)))
87	79, 86	eqtr4d 2803	. . . . . 6 ⊢ (𝑛 ∈ ℕ → ((4↑(𝑛 + 1)) + 5) = ((((4↑𝑛) · 4) + (5 · 4)) − ;15))
88	57, 62, 87	3eqtr4rd 2811	. . . . 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 5133	. . 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 12242	1 ⊢ (𝑁 ∈ ℕ → 3 ∥ ((4↑𝑁) + 5))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 400 ∧ w3a 1101 = wceq 1563 ∈ wcel 2145 class class class wbr 5105 (class class class)co 7400 ℂcc 11086 1c1 11089 + caddc 11091 · cmul 11093 − cmin 11429 ℕcn 12224 3c3 12287 4c4 12288 5c5 12289 9c9 12293 ℕ₀cn0 12495 ℤcz 12582 cdc 12702 ↑cexp 14088 ∥ cdvds 16300

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1818 ax-4 1832 ax-5 1933 ax-6 1990 ax-7 2031 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2737 ax-sep 5251 ax-nul 5261 ax-pow 5327 ax-pr 5395 ax-un 7722 ax-cnex 11144 ax-resscn 11145 ax-1cn 11146 ax-icn 11147 ax-addcl 11148 ax-addrcl 11149 ax-mulcl 11150 ax-mulrcl 11151 ax-mulcom 11152 ax-addass 11153 ax-mulass 11154 ax-distr 11155 ax-i2m1 11156 ax-1ne0 11157 ax-1rid 11158 ax-rnegex 11159 ax-rrecex 11160 ax-cnre 11161 ax-pre-lttri 11162 ax-pre-lttrn 11163 ax-pre-ltadd 11164 ax-pre-mulgt0 11165

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1566 df-fal 1576 df-ex 1803 df-nf 1807 df-sb 2094 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3065 df-ral 3080 df-rex 3090 df-reu 3371 df-rab 3418 df-v 3459 df-sbc 3748 df-csb 3856 df-dif 3910 df-un 3912 df-in 3914 df-ss 3924 df-pss 3927 df-nul 4289 df-if 4484 df-pw 4560 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4869 df-iun 4954 df-br 5106 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5547 df-eprel 5552 df-po 5560 df-so 5561 df-fr 5605 df-we 5607 df-xp 5658 df-rel 5659 df-cnv 5660 df-co 5661 df-dm 5662 df-rn 5663 df-res 5664 df-ima 5665 df-pred 6292 df-ord 6353 df-on 6354 df-lim 6355 df-suc 6356 df-iota 6481 df-fun 6527 df-fn 6528 df-f 6529 df-f1 6530 df-fo 6531 df-f1o 6532 df-fv 6533 df-riota 7357 df-ov 7403 df-oprab 7404 df-mpo 7405 df-om 7851 df-2nd 7975 df-frecs 8266 df-wrecs 8297 df-recs 8346 df-rdg 8385 df-er 8682 df-en 8932 df-dom 8933 df-sdom 8934 df-pnf 11233 df-mnf 11234 df-xr 11235 df-ltxr 11236 df-le 11237 df-sub 11431 df-neg 11432 df-nn 12225 df-2 12294 df-3 12295 df-4 12296 df-5 12297 df-6 12298 df-7 12299 df-8 12300 df-9 12301 df-n0 12496 df-z 12583 df-dec 12703 df-uz 12854 df-seq 14029 df-exp 14089 df-dvds 16301

This theorem is referenced by: (None)

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