Theorem uzsinds 9428

Description: Strong (or "total") induction principle over an upper set of integers. (Contributed by Scott Fenton, 16-May-2014.)

Hypotheses

Ref	Expression
uzsinds.1	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
uzsinds.2	⊢ (𝑥 = 𝑁 → (𝜑 ↔ 𝜒))
uzsinds.3	⊢ (𝑥 ∈ (ℤ≥‘𝑀) → (∀𝑦 ∈ (𝑀...(𝑥 − 1))𝜓 → 𝜑))

Assertion

Ref	Expression
uzsinds	⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝜒)

Distinct variable groups:   𝜒,𝑥   𝑥,𝑀,𝑦   𝑥,𝑁   𝜑,𝑦   𝜓,𝑥

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)   𝜒(𝑦)   𝑁(𝑦)

Proof of Theorem uzsinds

Dummy variables 𝑘 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		uzsinds.2	. 2 ⊢ (𝑥 = 𝑁 → (𝜑 ↔ 𝜒))
2		oveq2 5540	. . . 4 ⊢ (𝑤 = 𝑀 → (𝑀...𝑤) = (𝑀...𝑀))
3	2	raleqdv 2555	. . 3 ⊢ (𝑤 = 𝑀 → (∀𝑥 ∈ (𝑀...𝑤)𝜑 ↔ ∀𝑥 ∈ (𝑀...𝑀)𝜑))
4		oveq2 5540	. . . 4 ⊢ (𝑤 = 𝑘 → (𝑀...𝑤) = (𝑀...𝑘))
5	4	raleqdv 2555	. . 3 ⊢ (𝑤 = 𝑘 → (∀𝑥 ∈ (𝑀...𝑤)𝜑 ↔ ∀𝑥 ∈ (𝑀...𝑘)𝜑))
6		oveq2 5540	. . . 4 ⊢ (𝑤 = (𝑘 + 1) → (𝑀...𝑤) = (𝑀...(𝑘 + 1)))
7	6	raleqdv 2555	. . 3 ⊢ (𝑤 = (𝑘 + 1) → (∀𝑥 ∈ (𝑀...𝑤)𝜑 ↔ ∀𝑥 ∈ (𝑀...(𝑘 + 1))𝜑))
8		oveq2 5540	. . . 4 ⊢ (𝑤 = 𝑁 → (𝑀...𝑤) = (𝑀...𝑁))
9	8	raleqdv 2555	. . 3 ⊢ (𝑤 = 𝑁 → (∀𝑥 ∈ (𝑀...𝑤)𝜑 ↔ ∀𝑥 ∈ (𝑀...𝑁)𝜑))
10		ral0 3342	. . . . . . 7 ⊢ ∀𝑦 ∈ ∅ 𝜓
11		zre 8355	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ)
12	11	ltm1d 8010	. . . . . . . . 9 ⊢ (𝑀 ∈ ℤ → (𝑀 − 1) < 𝑀)
13		peano2zm 8389	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℤ → (𝑀 − 1) ∈ ℤ)
14		fzn 9061	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤ ∧ (𝑀 − 1) ∈ ℤ) → ((𝑀 − 1) < 𝑀 ↔ (𝑀...(𝑀 − 1)) = ∅))
15	13, 14	mpdan 412	. . . . . . . . 9 ⊢ (𝑀 ∈ ℤ → ((𝑀 − 1) < 𝑀 ↔ (𝑀...(𝑀 − 1)) = ∅))
16	12, 15	mpbid 145	. . . . . . . 8 ⊢ (𝑀 ∈ ℤ → (𝑀...(𝑀 − 1)) = ∅)
17	16	raleqdv 2555	. . . . . . 7 ⊢ (𝑀 ∈ ℤ → (∀𝑦 ∈ (𝑀...(𝑀 − 1))𝜓 ↔ ∀𝑦 ∈ ∅ 𝜓))
18	10, 17	mpbiri 166	. . . . . 6 ⊢ (𝑀 ∈ ℤ → ∀𝑦 ∈ (𝑀...(𝑀 − 1))𝜓)
19		uzid 8633	. . . . . . 7 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ≥‘𝑀))
20		uzsinds.3	. . . . . . . 8 ⊢ (𝑥 ∈ (ℤ≥‘𝑀) → (∀𝑦 ∈ (𝑀...(𝑥 − 1))𝜓 → 𝜑))
21	20	rgen 2416	. . . . . . 7 ⊢ ∀𝑥 ∈ (ℤ≥‘𝑀)(∀𝑦 ∈ (𝑀...(𝑥 − 1))𝜓 → 𝜑)
22		nfv 1461	. . . . . . . . 9 ⊢ Ⅎ𝑥∀𝑦 ∈ (𝑀...(𝑀 − 1))𝜓
23		nfsbc1v 2833	. . . . . . . . 9 ⊢ Ⅎ𝑥[𝑀 / 𝑥]𝜑
24	22, 23	nfim 1504	. . . . . . . 8 ⊢ Ⅎ𝑥(∀𝑦 ∈ (𝑀...(𝑀 − 1))𝜓 → [𝑀 / 𝑥]𝜑)
25		oveq1 5539	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑀 → (𝑥 − 1) = (𝑀 − 1))
26	25	oveq2d 5548	. . . . . . . . . 10 ⊢ (𝑥 = 𝑀 → (𝑀...(𝑥 − 1)) = (𝑀...(𝑀 − 1)))
27	26	raleqdv 2555	. . . . . . . . 9 ⊢ (𝑥 = 𝑀 → (∀𝑦 ∈ (𝑀...(𝑥 − 1))𝜓 ↔ ∀𝑦 ∈ (𝑀...(𝑀 − 1))𝜓))
28		sbceq1a 2824	. . . . . . . . 9 ⊢ (𝑥 = 𝑀 → (𝜑 ↔ [𝑀 / 𝑥]𝜑))
29	27, 28	imbi12d 232	. . . . . . . 8 ⊢ (𝑥 = 𝑀 → ((∀𝑦 ∈ (𝑀...(𝑥 − 1))𝜓 → 𝜑) ↔ (∀𝑦 ∈ (𝑀...(𝑀 − 1))𝜓 → [𝑀 / 𝑥]𝜑)))
30	24, 29	rspc 2695	. . . . . . 7 ⊢ (𝑀 ∈ (ℤ≥‘𝑀) → (∀𝑥 ∈ (ℤ≥‘𝑀)(∀𝑦 ∈ (𝑀...(𝑥 − 1))𝜓 → 𝜑) → (∀𝑦 ∈ (𝑀...(𝑀 − 1))𝜓 → [𝑀 / 𝑥]𝜑)))
31	19, 21, 30	mpisyl 1375	. . . . . 6 ⊢ (𝑀 ∈ ℤ → (∀𝑦 ∈ (𝑀...(𝑀 − 1))𝜓 → [𝑀 / 𝑥]𝜑))
32	18, 31	mpd 13	. . . . 5 ⊢ (𝑀 ∈ ℤ → [𝑀 / 𝑥]𝜑)
33		ralsns 3431	. . . . 5 ⊢ (𝑀 ∈ ℤ → (∀𝑥 ∈ {𝑀}𝜑 ↔ [𝑀 / 𝑥]𝜑))
34	32, 33	mpbird 165	. . . 4 ⊢ (𝑀 ∈ ℤ → ∀𝑥 ∈ {𝑀}𝜑)
35		fzsn 9084	. . . . 5 ⊢ (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀})
36	35	raleqdv 2555	. . . 4 ⊢ (𝑀 ∈ ℤ → (∀𝑥 ∈ (𝑀...𝑀)𝜑 ↔ ∀𝑥 ∈ {𝑀}𝜑))
37	34, 36	mpbird 165	. . 3 ⊢ (𝑀 ∈ ℤ → ∀𝑥 ∈ (𝑀...𝑀)𝜑)
38		simpr 108	. . . . . 6 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → ∀𝑥 ∈ (𝑀...𝑘)𝜑)
39		uzsinds.1	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
40	39	cbvralv 2577	. . . . . . . . 9 ⊢ (∀𝑥 ∈ (𝑀...𝑘)𝜑 ↔ ∀𝑦 ∈ (𝑀...𝑘)𝜓)
41	38, 40	sylib 120	. . . . . . . 8 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → ∀𝑦 ∈ (𝑀...𝑘)𝜓)
42		eluzelz 8628	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → 𝑘 ∈ ℤ)
43	42	adantr 270	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → 𝑘 ∈ ℤ)
44	43	zcnd 8470	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → 𝑘 ∈ ℂ)
45		1cnd 7135	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → 1 ∈ ℂ)
46	44, 45	pncand 7420	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → ((𝑘 + 1) − 1) = 𝑘)
47	46	oveq2d 5548	. . . . . . . . . 10 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → (𝑀...((𝑘 + 1) − 1)) = (𝑀...𝑘))
48	47	raleqdv 2555	. . . . . . . . 9 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → (∀𝑦 ∈ (𝑀...((𝑘 + 1) − 1))𝜓 ↔ ∀𝑦 ∈ (𝑀...𝑘)𝜓))
49		peano2uz 8671	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → (𝑘 + 1) ∈ (ℤ≥‘𝑀))
50	49	adantr 270	. . . . . . . . . 10 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → (𝑘 + 1) ∈ (ℤ≥‘𝑀))
51		nfv 1461	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥∀𝑦 ∈ (𝑀...((𝑘 + 1) − 1))𝜓
52		nfsbc1v 2833	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥[(𝑘 + 1) / 𝑥]𝜑
53	51, 52	nfim 1504	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(∀𝑦 ∈ (𝑀...((𝑘 + 1) − 1))𝜓 → [(𝑘 + 1) / 𝑥]𝜑)
54		oveq1 5539	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑘 + 1) → (𝑥 − 1) = ((𝑘 + 1) − 1))
55	54	oveq2d 5548	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝑘 + 1) → (𝑀...(𝑥 − 1)) = (𝑀...((𝑘 + 1) − 1)))
56	55	raleqdv 2555	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑘 + 1) → (∀𝑦 ∈ (𝑀...(𝑥 − 1))𝜓 ↔ ∀𝑦 ∈ (𝑀...((𝑘 + 1) − 1))𝜓))
57		sbceq1a 2824	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑘 + 1) → (𝜑 ↔ [(𝑘 + 1) / 𝑥]𝜑))
58	56, 57	imbi12d 232	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑘 + 1) → ((∀𝑦 ∈ (𝑀...(𝑥 − 1))𝜓 → 𝜑) ↔ (∀𝑦 ∈ (𝑀...((𝑘 + 1) − 1))𝜓 → [(𝑘 + 1) / 𝑥]𝜑)))
59	53, 58	rspc 2695	. . . . . . . . . 10 ⊢ ((𝑘 + 1) ∈ (ℤ≥‘𝑀) → (∀𝑥 ∈ (ℤ≥‘𝑀)(∀𝑦 ∈ (𝑀...(𝑥 − 1))𝜓 → 𝜑) → (∀𝑦 ∈ (𝑀...((𝑘 + 1) − 1))𝜓 → [(𝑘 + 1) / 𝑥]𝜑)))
60	50, 21, 59	mpisyl 1375	. . . . . . . . 9 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → (∀𝑦 ∈ (𝑀...((𝑘 + 1) − 1))𝜓 → [(𝑘 + 1) / 𝑥]𝜑))
61	48, 60	sylbird 168	. . . . . . . 8 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → (∀𝑦 ∈ (𝑀...𝑘)𝜓 → [(𝑘 + 1) / 𝑥]𝜑))
62	41, 61	mpd 13	. . . . . . 7 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → [(𝑘 + 1) / 𝑥]𝜑)
63	42	peano2zd 8472	. . . . . . . . 9 ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → (𝑘 + 1) ∈ ℤ)
64	63	adantr 270	. . . . . . . 8 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → (𝑘 + 1) ∈ ℤ)
65		ralsns 3431	. . . . . . . 8 ⊢ ((𝑘 + 1) ∈ ℤ → (∀𝑥 ∈ {(𝑘 + 1)}𝜑 ↔ [(𝑘 + 1) / 𝑥]𝜑))
66	64, 65	syl 14	. . . . . . 7 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → (∀𝑥 ∈ {(𝑘 + 1)}𝜑 ↔ [(𝑘 + 1) / 𝑥]𝜑))
67	62, 66	mpbird 165	. . . . . 6 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → ∀𝑥 ∈ {(𝑘 + 1)}𝜑)
68		ralun 3154	. . . . . 6 ⊢ ((∀𝑥 ∈ (𝑀...𝑘)𝜑 ∧ ∀𝑥 ∈ {(𝑘 + 1)}𝜑) → ∀𝑥 ∈ ((𝑀...𝑘) ∪ {(𝑘 + 1)})𝜑)
69	38, 67, 68	syl2anc 403	. . . . 5 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → ∀𝑥 ∈ ((𝑀...𝑘) ∪ {(𝑘 + 1)})𝜑)
70		fzsuc 9086	. . . . . . 7 ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → (𝑀...(𝑘 + 1)) = ((𝑀...𝑘) ∪ {(𝑘 + 1)}))
71	70	raleqdv 2555	. . . . . 6 ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → (∀𝑥 ∈ (𝑀...(𝑘 + 1))𝜑 ↔ ∀𝑥 ∈ ((𝑀...𝑘) ∪ {(𝑘 + 1)})𝜑))
72	71	adantr 270	. . . . 5 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → (∀𝑥 ∈ (𝑀...(𝑘 + 1))𝜑 ↔ ∀𝑥 ∈ ((𝑀...𝑘) ∪ {(𝑘 + 1)})𝜑))
73	69, 72	mpbird 165	. . . 4 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → ∀𝑥 ∈ (𝑀...(𝑘 + 1))𝜑)
74	73	ex 113	. . 3 ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → (∀𝑥 ∈ (𝑀...𝑘)𝜑 → ∀𝑥 ∈ (𝑀...(𝑘 + 1))𝜑))
75	3, 5, 7, 9, 37, 74	uzind4 8676	. 2 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → ∀𝑥 ∈ (𝑀...𝑁)𝜑)
76		eluzfz2 9051	. 2 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁))
77	1, 75, 76	rspcdva 2707	1 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝜒)

Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103   = wceq 1284   ∈ wcel 1433  ∀wral 2348  [wsbc 2815   ∪ cun 2971  ∅c0 3251  {csn 3398   class class class wbr 3785  ‘cfv 4922  (class class class)co 5532  1c1 6982   + caddc 6984   < clt 7153   − cmin 7279  ℤcz 8351  ℤ_≥cuz 8619  ...cfz 9029

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-cnex 7067  ax-resscn 7068  ax-1cn 7069  ax-1re 7070  ax-icn 7071  ax-addcl 7072  ax-addrcl 7073  ax-mulcl 7074  ax-addcom 7076  ax-addass 7078  ax-distr 7080  ax-i2m1 7081  ax-0lt1 7082  ax-0id 7084  ax-rnegex 7085  ax-cnre 7087  ax-pre-ltirr 7088  ax-pre-ltwlin 7089  ax-pre-lttrn 7090  ax-pre-apti 7091  ax-pre-ltadd 7092

This theorem depends on definitions:  df-bi 115  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-nel 2340  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-br 3786  df-opab 3840  df-mpt 3841  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-fv 4930  df-riota 5488  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-pnf 7155  df-mnf 7156  df-xr 7157  df-ltxr 7158  df-le 7159  df-sub 7281  df-neg 7282  df-inn 8040  df-n0 8289  df-z 8352  df-uz 8620  df-fz 9030

This theorem is referenced by:  nnsinds  9429  nn0sinds  9430