inductionexd - Mathbox for Stanislas Polu

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Theorem inductionexd 44272

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 7360	. . . 4 ⊢ (𝑘 = 1 → (4↑𝑘) = (4↑1))
2	1	oveq1d 7367	. . 3 ⊢ (𝑘 = 1 → ((4↑𝑘) + 5) = ((4↑1) + 5))
3	2	breq2d 5105	. 2 ⊢ (𝑘 = 1 → (3 ∥ ((4↑𝑘) + 5) ↔ 3 ∥ ((4↑1) + 5)))
4		oveq2 7360	. . . 4 ⊢ (𝑘 = 𝑛 → (4↑𝑘) = (4↑𝑛))
5	4	oveq1d 7367	. . 3 ⊢ (𝑘 = 𝑛 → ((4↑𝑘) + 5) = ((4↑𝑛) + 5))
6	5	breq2d 5105	. 2 ⊢ (𝑘 = 𝑛 → (3 ∥ ((4↑𝑘) + 5) ↔ 3 ∥ ((4↑𝑛) + 5)))
7		oveq2 7360	. . . 4 ⊢ (𝑘 = (𝑛 + 1) → (4↑𝑘) = (4↑(𝑛 + 1)))
8	7	oveq1d 7367	. . 3 ⊢ (𝑘 = (𝑛 + 1) → ((4↑𝑘) + 5) = ((4↑(𝑛 + 1)) + 5))
9	8	breq2d 5105	. 2 ⊢ (𝑘 = (𝑛 + 1) → (3 ∥ ((4↑𝑘) + 5) ↔ 3 ∥ ((4↑(𝑛 + 1)) + 5)))
10		oveq2 7360	. . . 4 ⊢ (𝑘 = 𝑁 → (4↑𝑘) = (4↑𝑁))
11	10	oveq1d 7367	. . 3 ⊢ (𝑘 = 𝑁 → ((4↑𝑘) + 5) = ((4↑𝑁) + 5))
12	11	breq2d 5105	. 2 ⊢ (𝑘 = 𝑁 → (3 ∥ ((4↑𝑘) + 5) ↔ 3 ∥ ((4↑𝑁) + 5)))
13		3z 12511	. . . 4 ⊢ 3 ∈ ℤ
14		4z 12512	. . . . . 6 ⊢ 4 ∈ ℤ
15		1nn0 12404	. . . . . 6 ⊢ 1 ∈ ℕ₀
16		zexpcl 13985	. . . . . 6 ⊢ ((4 ∈ ℤ ∧ 1 ∈ ℕ₀) → (4↑1) ∈ ℤ)
17	14, 15, 16	mp2an 692	. . . . 5 ⊢ (4↑1) ∈ ℤ
18		5nn 12218	. . . . . 6 ⊢ 5 ∈ ℕ
19	18	nnzi 12502	. . . . 5 ⊢ 5 ∈ ℤ
20		zaddcl 12518	. . . . 5 ⊢ (((4↑1) ∈ ℤ ∧ 5 ∈ ℤ) → ((4↑1) + 5) ∈ ℤ)
21	17, 19, 20	mp2an 692	. . . 4 ⊢ ((4↑1) + 5) ∈ ℤ
22	13, 13, 21	3pm3.2i 1340	. . 3 ⊢ (3 ∈ ℤ ∧ 3 ∈ ℤ ∧ ((4↑1) + 5) ∈ ℤ)
23		3t3e9 12294	. . . 4 ⊢ (3 · 3) = 9
24		4nn0 12407	. . . . . . 7 ⊢ 4 ∈ ℕ₀
25	24	numexp1 16990	. . . . . 6 ⊢ (4↑1) = 4
26	25	oveq1i 7362	. . . . 5 ⊢ ((4↑1) + 5) = (4 + 5)
27		5cn 12220	. . . . . 6 ⊢ 5 ∈ ℂ
28		4cn 12217	. . . . . 6 ⊢ 4 ∈ ℂ
29		5p4e9 12285	. . . . . 6 ⊢ (5 + 4) = 9
30	27, 28, 29	addcomli 11312	. . . . 5 ⊢ (4 + 5) = 9
31	26, 30	eqtri 2756	. . . 4 ⊢ ((4↑1) + 5) = 9
32	23, 31	eqtr4i 2759	. . 3 ⊢ (3 · 3) = ((4↑1) + 5)
33		dvds0lem 16179	. . 3 ⊢ (((3 ∈ ℤ ∧ 3 ∈ ℤ ∧ ((4↑1) + 5) ∈ ℤ) ∧ (3 · 3) = ((4↑1) + 5)) → 3 ∥ ((4↑1) + 5))
34	22, 32, 33	mp2an 692	. 2 ⊢ 3 ∥ ((4↑1) + 5)
35	13	a1i 11	. . . . 5 ⊢ ((𝑛 ∈ ℕ ∧ 3 ∥ ((4↑𝑛) + 5)) → 3 ∈ ℤ)
36		4nn 12215	. . . . . . . . . . 11 ⊢ 4 ∈ ℕ
37	36	a1i 11	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → 4 ∈ ℕ)
38		nnnn0 12395	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ₀)
39	37, 38	nnexpcld 14154	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → (4↑𝑛) ∈ ℕ)
40	39	nnzd 12501	. . . . . . . 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 12587	. . . . . 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 12589	. . . . 5 ⊢ ((𝑛 ∈ ℕ ∧ 3 ∥ ((4↑𝑛) + 5)) → (((4↑𝑛) + 5) · 4) ∈ ℤ)
46	35, 42	zmulcld 12589	. . . . 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 16210	. . . . 5 ⊢ ((𝑛 ∈ ℕ ∧ 3 ∥ ((4↑𝑛) + 5)) → 3 ∥ (((4↑𝑛) + 5) · 4))
49		dvdsmul1 16190	. . . . . . 7 ⊢ ((3 ∈ ℤ ∧ 5 ∈ ℤ) → 3 ∥ (3 · 5))
50	13, 19, 49	mp2an 692	. . . . . 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 16206	. . . 4 ⊢ ((𝑛 ∈ ℕ ∧ 3 ∥ ((4↑𝑛) + 5)) → 3 ∥ ((((4↑𝑛) + 5) · 4) − (3 · 5)))
53	39	nncnd 12148	. . . . . . . 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 11144	. . . . . . 7 ⊢ (𝑛 ∈ ℕ → (((4↑𝑛) + 5) · 4) = (((4↑𝑛) · 4) + (5 · 4)))
57	56	oveq1d 7367	. . . . . 6 ⊢ (𝑛 ∈ ℕ → ((((4↑𝑛) + 5) · 4) − ;15) = ((((4↑𝑛) · 4) + (5 · 4)) − ;15))
58		3cn 12213	. . . . . . . . 9 ⊢ 3 ∈ ℂ
59		5t3e15 12695	. . . . . . . . 9 ⊢ (5 · 3) = ;15
60	27, 58, 59	mulcomli 11128	. . . . . . . 8 ⊢ (3 · 5) = ;15
61	60	a1i 11	. . . . . . 7 ⊢ (𝑛 ∈ ℕ → (3 · 5) = ;15)
62	61	oveq2d 7368	. . . . . 6 ⊢ (𝑛 ∈ ℕ → ((((4↑𝑛) + 5) · 4) − (3 · 5)) = ((((4↑𝑛) + 5) · 4) − ;15))
63	55, 38	expp1d 14056	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → (4↑(𝑛 + 1)) = ((4↑𝑛) · 4))
64		ax-1cn 11071	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℂ
65		3p1e4 12272	. . . . . . . . . . . . . . . 16 ⊢ (3 + 1) = 4
66	58, 64, 65	addcomli 11312	. . . . . . . . . . . . . . 15 ⊢ (1 + 3) = 4
67	66	eqcomi 2742	. . . . . . . . . . . . . 14 ⊢ 4 = (1 + 3)
68	67	oveq1i 7362	. . . . . . . . . . . . 13 ⊢ (4 − 3) = ((1 + 3) − 3)
69	64, 58	pncan3oi 11383	. . . . . . . . . . . . 13 ⊢ ((1 + 3) − 3) = 1
70	68, 69	eqtri 2756	. . . . . . . . . . . 12 ⊢ (4 − 3) = 1
71	70	oveq2i 7363	. . . . . . . . . . 11 ⊢ (5 · (4 − 3)) = (5 · 1)
72	27, 28, 58	subdii 11573	. . . . . . . . . . 11 ⊢ (5 · (4 − 3)) = ((5 · 4) − (5 · 3))
73	27	mulridi 11123	. . . . . . . . . . 11 ⊢ (5 · 1) = 5
74	71, 72, 73	3eqtr3ri 2765	. . . . . . . . . 10 ⊢ 5 = ((5 · 4) − (5 · 3))
75	59	eqcomi 2742	. . . . . . . . . . 11 ⊢ ;15 = (5 · 3)
76	75	oveq2i 7363	. . . . . . . . . 10 ⊢ ((5 · 4) − ;15) = ((5 · 4) − (5 · 3))
77	74, 76	eqtr4i 2759	. . . . . . . . 9 ⊢ 5 = ((5 · 4) − ;15)
78	77	a1i 11	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → 5 = ((5 · 4) − ;15))
79	63, 78	oveq12d 7370	. . . . . . 7 ⊢ (𝑛 ∈ ℕ → ((4↑(𝑛 + 1)) + 5) = (((4↑𝑛) · 4) + ((5 · 4) − ;15)))
80	53, 55	mulcld 11139	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → ((4↑𝑛) · 4) ∈ ℂ)
81	54, 55	mulcld 11139	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → (5 · 4) ∈ ℂ)
82		5nn0 12408	. . . . . . . . . . 11 ⊢ 5 ∈ ℕ₀
83	15, 82	deccl 12609	. . . . . . . . . 10 ⊢ ;15 ∈ ℕ₀
84	83	nn0cni 12400	. . . . . . . . 9 ⊢ ;15 ∈ ℂ
85	84	a1i 11	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → ;15 ∈ ℂ)
86	80, 81, 85	addsubassd 11499	. . . . . . 7 ⊢ (𝑛 ∈ ℕ → ((((4↑𝑛) · 4) + (5 · 4)) − ;15) = (((4↑𝑛) · 4) + ((5 · 4) − ;15)))
87	79, 86	eqtr4d 2771	. . . . . 6 ⊢ (𝑛 ∈ ℕ → ((4↑(𝑛 + 1)) + 5) = ((((4↑𝑛) · 4) + (5 · 4)) − ;15))
88	57, 62, 87	3eqtr4rd 2779	. . . . 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 5121	. . 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 12150	1 ⊢ (𝑁 ∈ ℕ → 3 ∥ ((4↑𝑁) + 5))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2113 class class class wbr 5093 (class class class)co 7352 ℂcc 11011 1c1 11014 + caddc 11016 · cmul 11018 − cmin 11351 ℕcn 12132 3c3 12188 4c4 12189 5c5 12190 9c9 12194 ℕ₀cn0 12388 ℤcz 12475 cdc 12594 ↑cexp 13970 ∥ cdvds 16165

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 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-sep 5236 ax-nul 5246 ax-pow 5305 ax-pr 5372 ax-un 7674 ax-cnex 11069 ax-resscn 11070 ax-1cn 11071 ax-icn 11072 ax-addcl 11073 ax-addrcl 11074 ax-mulcl 11075 ax-mulrcl 11076 ax-mulcom 11077 ax-addass 11078 ax-mulass 11079 ax-distr 11080 ax-i2m1 11081 ax-1ne0 11082 ax-1rid 11083 ax-rnegex 11084 ax-rrecex 11085 ax-cnre 11086 ax-pre-lttri 11087 ax-pre-lttrn 11088 ax-pre-ltadd 11089 ax-pre-mulgt0 11090

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-nel 3034 df-ral 3049 df-rex 3058 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4283 df-if 4475 df-pw 4551 df-sn 4576 df-pr 4578 df-op 4582 df-uni 4859 df-iun 4943 df-br 5094 df-opab 5156 df-mpt 5175 df-tr 5201 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-riota 7309 df-ov 7355 df-oprab 7356 df-mpo 7357 df-om 7803 df-2nd 7928 df-frecs 8217 df-wrecs 8248 df-recs 8297 df-rdg 8335 df-er 8628 df-en 8876 df-dom 8877 df-sdom 8878 df-pnf 11155 df-mnf 11156 df-xr 11157 df-ltxr 11158 df-le 11159 df-sub 11353 df-neg 11354 df-nn 12133 df-2 12195 df-3 12196 df-4 12197 df-5 12198 df-6 12199 df-7 12200 df-8 12201 df-9 12202 df-n0 12389 df-z 12476 df-dec 12595 df-uz 12739 df-seq 13911 df-exp 13971 df-dvds 16166

This theorem is referenced by: (None)

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