Theorem uzsinds 10881

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 6093	. . . 4 ⊢ (𝑤 = 𝑀 → (𝑀...𝑤) = (𝑀...𝑀))
3	2	raleqdv 2755	. . 3 ⊢ (𝑤 = 𝑀 → (∀𝑥 ∈ (𝑀...𝑤)𝜑 ↔ ∀𝑥 ∈ (𝑀...𝑀)𝜑))
4		oveq2 6093	. . . 4 ⊢ (𝑤 = 𝑘 → (𝑀...𝑤) = (𝑀...𝑘))
5	4	raleqdv 2755	. . 3 ⊢ (𝑤 = 𝑘 → (∀𝑥 ∈ (𝑀...𝑤)𝜑 ↔ ∀𝑥 ∈ (𝑀...𝑘)𝜑))
6		oveq2 6093	. . . 4 ⊢ (𝑤 = (𝑘 + 1) → (𝑀...𝑤) = (𝑀...(𝑘 + 1)))
7	6	raleqdv 2755	. . 3 ⊢ (𝑤 = (𝑘 + 1) → (∀𝑥 ∈ (𝑀...𝑤)𝜑 ↔ ∀𝑥 ∈ (𝑀...(𝑘 + 1))𝜑))
8		oveq2 6093	. . . 4 ⊢ (𝑤 = 𝑁 → (𝑀...𝑤) = (𝑀...𝑁))
9	8	raleqdv 2755	. . 3 ⊢ (𝑤 = 𝑁 → (∀𝑥 ∈ (𝑀...𝑤)𝜑 ↔ ∀𝑥 ∈ (𝑀...𝑁)𝜑))
10		ral0 3629	. . . . . . 7 ⊢ ∀𝑦 ∈ ∅ 𝜓
11		zre 9648	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ)
12	11	ltm1d 9262	. . . . . . . . 9 ⊢ (𝑀 ∈ ℤ → (𝑀 − 1) < 𝑀)
13		peano2zm 9682	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℤ → (𝑀 − 1) ∈ ℤ)
14		fzn 10446	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤ ∧ (𝑀 − 1) ∈ ℤ) → ((𝑀 − 1) < 𝑀 ↔ (𝑀...(𝑀 − 1)) = ∅))
15	13, 14	mpdan 425	. . . . . . . . 9 ⊢ (𝑀 ∈ ℤ → ((𝑀 − 1) < 𝑀 ↔ (𝑀...(𝑀 − 1)) = ∅))
16	12, 15	mpbid 147	. . . . . . . 8 ⊢ (𝑀 ∈ ℤ → (𝑀...(𝑀 − 1)) = ∅)
17	16	raleqdv 2755	. . . . . . 7 ⊢ (𝑀 ∈ ℤ → (∀𝑦 ∈ (𝑀...(𝑀 − 1))𝜓 ↔ ∀𝑦 ∈ ∅ 𝜓))
18	10, 17	mpbiri 168	. . . . . 6 ⊢ (𝑀 ∈ ℤ → ∀𝑦 ∈ (𝑀...(𝑀 − 1))𝜓)
19		uzid 9936	. . . . . . 7 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ≥‘𝑀))
20		uzsinds.3	. . . . . . . 8 ⊢ (𝑥 ∈ (ℤ≥‘𝑀) → (∀𝑦 ∈ (𝑀...(𝑥 − 1))𝜓 → 𝜑))
21	20	rgen 2603	. . . . . . 7 ⊢ ∀𝑥 ∈ (ℤ≥‘𝑀)(∀𝑦 ∈ (𝑀...(𝑥 − 1))𝜓 → 𝜑)
22		nfv 1581	. . . . . . . . 9 ⊢ Ⅎ𝑥∀𝑦 ∈ (𝑀...(𝑀 − 1))𝜓
23		nfsbc1v 3070	. . . . . . . . 9 ⊢ Ⅎ𝑥[𝑀 / 𝑥]𝜑
24	22, 23	nfim 1625	. . . . . . . 8 ⊢ Ⅎ𝑥(∀𝑦 ∈ (𝑀...(𝑀 − 1))𝜓 → [𝑀 / 𝑥]𝜑)
25		oveq1 6092	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑀 → (𝑥 − 1) = (𝑀 − 1))
26	25	oveq2d 6101	. . . . . . . . . 10 ⊢ (𝑥 = 𝑀 → (𝑀...(𝑥 − 1)) = (𝑀...(𝑀 − 1)))
27	26	raleqdv 2755	. . . . . . . . 9 ⊢ (𝑥 = 𝑀 → (∀𝑦 ∈ (𝑀...(𝑥 − 1))𝜓 ↔ ∀𝑦 ∈ (𝑀...(𝑀 − 1))𝜓))
28		sbceq1a 3061	. . . . . . . . 9 ⊢ (𝑥 = 𝑀 → (𝜑 ↔ [𝑀 / 𝑥]𝜑))
29	27, 28	imbi12d 234	. . . . . . . 8 ⊢ (𝑥 = 𝑀 → ((∀𝑦 ∈ (𝑀...(𝑥 − 1))𝜓 → 𝜑) ↔ (∀𝑦 ∈ (𝑀...(𝑀 − 1))𝜓 → [𝑀 / 𝑥]𝜑)))
30	24, 29	rspc 2923	. . . . . . 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 1496	. . . . . 6 ⊢ (𝑀 ∈ ℤ → (∀𝑦 ∈ (𝑀...(𝑀 − 1))𝜓 → [𝑀 / 𝑥]𝜑))
32	18, 31	mpd 13	. . . . 5 ⊢ (𝑀 ∈ ℤ → [𝑀 / 𝑥]𝜑)
33		ralsns 3747	. . . . 5 ⊢ (𝑀 ∈ ℤ → (∀𝑥 ∈ {𝑀}𝜑 ↔ [𝑀 / 𝑥]𝜑))
34	32, 33	mpbird 167	. . . 4 ⊢ (𝑀 ∈ ℤ → ∀𝑥 ∈ {𝑀}𝜑)
35		fzsn 10472	. . . . 5 ⊢ (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀})
36	35	raleqdv 2755	. . . 4 ⊢ (𝑀 ∈ ℤ → (∀𝑥 ∈ (𝑀...𝑀)𝜑 ↔ ∀𝑥 ∈ {𝑀}𝜑))
37	34, 36	mpbird 167	. . 3 ⊢ (𝑀 ∈ ℤ → ∀𝑥 ∈ (𝑀...𝑀)𝜑)
38		simpr 110	. . . . . 6 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → ∀𝑥 ∈ (𝑀...𝑘)𝜑)
39		uzsinds.1	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
40	39	cbvralv 2786	. . . . . . . . 9 ⊢ (∀𝑥 ∈ (𝑀...𝑘)𝜑 ↔ ∀𝑦 ∈ (𝑀...𝑘)𝜓)
41	38, 40	sylib 122	. . . . . . . 8 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → ∀𝑦 ∈ (𝑀...𝑘)𝜓)
42		eluzelz 9931	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → 𝑘 ∈ ℤ)
43	42	adantr 276	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → 𝑘 ∈ ℤ)
44	43	zcnd 9769	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → 𝑘 ∈ ℂ)
45		1cnd 8342	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → 1 ∈ ℂ)
46	44, 45	pncand 8638	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → ((𝑘 + 1) − 1) = 𝑘)
47	46	oveq2d 6101	. . . . . . . . . 10 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → (𝑀...((𝑘 + 1) − 1)) = (𝑀...𝑘))
48	47	raleqdv 2755	. . . . . . . . 9 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → (∀𝑦 ∈ (𝑀...((𝑘 + 1) − 1))𝜓 ↔ ∀𝑦 ∈ (𝑀...𝑘)𝜓))
49		peano2uz 9983	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → (𝑘 + 1) ∈ (ℤ≥‘𝑀))
50	49	adantr 276	. . . . . . . . . 10 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → (𝑘 + 1) ∈ (ℤ≥‘𝑀))
51		nfv 1581	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥∀𝑦 ∈ (𝑀...((𝑘 + 1) − 1))𝜓
52		nfsbc1v 3070	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥[(𝑘 + 1) / 𝑥]𝜑
53	51, 52	nfim 1625	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(∀𝑦 ∈ (𝑀...((𝑘 + 1) − 1))𝜓 → [(𝑘 + 1) / 𝑥]𝜑)
54		oveq1 6092	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑘 + 1) → (𝑥 − 1) = ((𝑘 + 1) − 1))
55	54	oveq2d 6101	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝑘 + 1) → (𝑀...(𝑥 − 1)) = (𝑀...((𝑘 + 1) − 1)))
56	55	raleqdv 2755	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑘 + 1) → (∀𝑦 ∈ (𝑀...(𝑥 − 1))𝜓 ↔ ∀𝑦 ∈ (𝑀...((𝑘 + 1) − 1))𝜓))
57		sbceq1a 3061	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑘 + 1) → (𝜑 ↔ [(𝑘 + 1) / 𝑥]𝜑))
58	56, 57	imbi12d 234	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑘 + 1) → ((∀𝑦 ∈ (𝑀...(𝑥 − 1))𝜓 → 𝜑) ↔ (∀𝑦 ∈ (𝑀...((𝑘 + 1) − 1))𝜓 → [(𝑘 + 1) / 𝑥]𝜑)))
59	53, 58	rspc 2923	. . . . . . . . . 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 1496	. . . . . . . . 9 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → (∀𝑦 ∈ (𝑀...((𝑘 + 1) − 1))𝜓 → [(𝑘 + 1) / 𝑥]𝜑))
61	48, 60	sylbird 170	. . . . . . . 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 9771	. . . . . . . . 9 ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → (𝑘 + 1) ∈ ℤ)
64	63	adantr 276	. . . . . . . 8 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → (𝑘 + 1) ∈ ℤ)
65		ralsns 3747	. . . . . . . 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 167	. . . . . 6 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → ∀𝑥 ∈ {(𝑘 + 1)}𝜑)
68		ralun 3411	. . . . . 6 ⊢ ((∀𝑥 ∈ (𝑀...𝑘)𝜑 ∧ ∀𝑥 ∈ {(𝑘 + 1)}𝜑) → ∀𝑥 ∈ ((𝑀...𝑘) ∪ {(𝑘 + 1)})𝜑)
69	38, 67, 68	syl2anc 415	. . . . 5 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → ∀𝑥 ∈ ((𝑀...𝑘) ∪ {(𝑘 + 1)})𝜑)
70		fzsuc 10475	. . . . . . 7 ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → (𝑀...(𝑘 + 1)) = ((𝑀...𝑘) ∪ {(𝑘 + 1)}))
71	70	raleqdv 2755	. . . . . 6 ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → (∀𝑥 ∈ (𝑀...(𝑘 + 1))𝜑 ↔ ∀𝑥 ∈ ((𝑀...𝑘) ∪ {(𝑘 + 1)})𝜑))
72	71	adantr 276	. . . . 5 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → (∀𝑥 ∈ (𝑀...(𝑘 + 1))𝜑 ↔ ∀𝑥 ∈ ((𝑀...𝑘) ∪ {(𝑘 + 1)})𝜑))
73	69, 72	mpbird 167	. . . 4 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → ∀𝑥 ∈ (𝑀...(𝑘 + 1))𝜑)
74	73	ex 115	. . 3 ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → (∀𝑥 ∈ (𝑀...𝑘)𝜑 → ∀𝑥 ∈ (𝑀...(𝑘 + 1))𝜑))
75	3, 5, 7, 9, 37, 74	uzind4 9988	. 2 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → ∀𝑥 ∈ (𝑀...𝑁)𝜑)
76		eluzfz2 10436	. 2 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁))
77	1, 75, 76	rspcdva 2934	1 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝜒)

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1402   ∈ wcel 2209  ∀wral 2528  [wsbc 3051   ∪ cun 3218  ∅c0 3520  {csn 3709   class class class wbr 4130  ‘cfv 5377  (class class class)co 6085  1c1 8180   + caddc 8182   < clt 8360   − cmin 8497  ℤcz 9644  ℤ_≥cuz 9921  ...cfz 10411

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-0id 8287  ax-rnegex 8288  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295

This proof depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-inn 9305  df-n0 9564  df-z 9645  df-uz 9922  df-fz 10412

This theorem is used by:  nnsinds  10882  nn0sinds  10883