Theorem uzsinds 10589

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 5952	. . . 4 ⊢ (𝑤 = 𝑀 → (𝑀...𝑤) = (𝑀...𝑀))
3	2	raleqdv 2708	. . 3 ⊢ (𝑤 = 𝑀 → (∀𝑥 ∈ (𝑀...𝑤)𝜑 ↔ ∀𝑥 ∈ (𝑀...𝑀)𝜑))
4		oveq2 5952	. . . 4 ⊢ (𝑤 = 𝑘 → (𝑀...𝑤) = (𝑀...𝑘))
5	4	raleqdv 2708	. . 3 ⊢ (𝑤 = 𝑘 → (∀𝑥 ∈ (𝑀...𝑤)𝜑 ↔ ∀𝑥 ∈ (𝑀...𝑘)𝜑))
6		oveq2 5952	. . . 4 ⊢ (𝑤 = (𝑘 + 1) → (𝑀...𝑤) = (𝑀...(𝑘 + 1)))
7	6	raleqdv 2708	. . 3 ⊢ (𝑤 = (𝑘 + 1) → (∀𝑥 ∈ (𝑀...𝑤)𝜑 ↔ ∀𝑥 ∈ (𝑀...(𝑘 + 1))𝜑))
8		oveq2 5952	. . . 4 ⊢ (𝑤 = 𝑁 → (𝑀...𝑤) = (𝑀...𝑁))
9	8	raleqdv 2708	. . 3 ⊢ (𝑤 = 𝑁 → (∀𝑥 ∈ (𝑀...𝑤)𝜑 ↔ ∀𝑥 ∈ (𝑀...𝑁)𝜑))
10		ral0 3562	. . . . . . 7 ⊢ ∀𝑦 ∈ ∅ 𝜓
11		zre 9376	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ)
12	11	ltm1d 9005	. . . . . . . . 9 ⊢ (𝑀 ∈ ℤ → (𝑀 − 1) < 𝑀)
13		peano2zm 9410	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℤ → (𝑀 − 1) ∈ ℤ)
14		fzn 10164	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤ ∧ (𝑀 − 1) ∈ ℤ) → ((𝑀 − 1) < 𝑀 ↔ (𝑀...(𝑀 − 1)) = ∅))
15	13, 14	mpdan 421	. . . . . . . . 9 ⊢ (𝑀 ∈ ℤ → ((𝑀 − 1) < 𝑀 ↔ (𝑀...(𝑀 − 1)) = ∅))
16	12, 15	mpbid 147	. . . . . . . 8 ⊢ (𝑀 ∈ ℤ → (𝑀...(𝑀 − 1)) = ∅)
17	16	raleqdv 2708	. . . . . . 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 9662	. . . . . . 7 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ≥‘𝑀))
20		uzsinds.3	. . . . . . . 8 ⊢ (𝑥 ∈ (ℤ≥‘𝑀) → (∀𝑦 ∈ (𝑀...(𝑥 − 1))𝜓 → 𝜑))
21	20	rgen 2559	. . . . . . 7 ⊢ ∀𝑥 ∈ (ℤ≥‘𝑀)(∀𝑦 ∈ (𝑀...(𝑥 − 1))𝜓 → 𝜑)
22		nfv 1551	. . . . . . . . 9 ⊢ Ⅎ𝑥∀𝑦 ∈ (𝑀...(𝑀 − 1))𝜓
23		nfsbc1v 3017	. . . . . . . . 9 ⊢ Ⅎ𝑥[𝑀 / 𝑥]𝜑
24	22, 23	nfim 1595	. . . . . . . 8 ⊢ Ⅎ𝑥(∀𝑦 ∈ (𝑀...(𝑀 − 1))𝜓 → [𝑀 / 𝑥]𝜑)
25		oveq1 5951	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑀 → (𝑥 − 1) = (𝑀 − 1))
26	25	oveq2d 5960	. . . . . . . . . 10 ⊢ (𝑥 = 𝑀 → (𝑀...(𝑥 − 1)) = (𝑀...(𝑀 − 1)))
27	26	raleqdv 2708	. . . . . . . . 9 ⊢ (𝑥 = 𝑀 → (∀𝑦 ∈ (𝑀...(𝑥 − 1))𝜓 ↔ ∀𝑦 ∈ (𝑀...(𝑀 − 1))𝜓))
28		sbceq1a 3008	. . . . . . . . 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 2871	. . . . . . 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 1466	. . . . . 6 ⊢ (𝑀 ∈ ℤ → (∀𝑦 ∈ (𝑀...(𝑀 − 1))𝜓 → [𝑀 / 𝑥]𝜑))
32	18, 31	mpd 13	. . . . 5 ⊢ (𝑀 ∈ ℤ → [𝑀 / 𝑥]𝜑)
33		ralsns 3671	. . . . 5 ⊢ (𝑀 ∈ ℤ → (∀𝑥 ∈ {𝑀}𝜑 ↔ [𝑀 / 𝑥]𝜑))
34	32, 33	mpbird 167	. . . 4 ⊢ (𝑀 ∈ ℤ → ∀𝑥 ∈ {𝑀}𝜑)
35		fzsn 10188	. . . . 5 ⊢ (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀})
36	35	raleqdv 2708	. . . 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 2738	. . . . . . . . 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 9657	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → 𝑘 ∈ ℤ)
43	42	adantr 276	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → 𝑘 ∈ ℤ)
44	43	zcnd 9496	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → 𝑘 ∈ ℂ)
45		1cnd 8088	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → 1 ∈ ℂ)
46	44, 45	pncand 8384	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → ((𝑘 + 1) − 1) = 𝑘)
47	46	oveq2d 5960	. . . . . . . . . 10 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → (𝑀...((𝑘 + 1) − 1)) = (𝑀...𝑘))
48	47	raleqdv 2708	. . . . . . . . 9 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → (∀𝑦 ∈ (𝑀...((𝑘 + 1) − 1))𝜓 ↔ ∀𝑦 ∈ (𝑀...𝑘)𝜓))
49		peano2uz 9704	. . . . . . . . . . 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 1551	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥∀𝑦 ∈ (𝑀...((𝑘 + 1) − 1))𝜓
52		nfsbc1v 3017	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥[(𝑘 + 1) / 𝑥]𝜑
53	51, 52	nfim 1595	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(∀𝑦 ∈ (𝑀...((𝑘 + 1) − 1))𝜓 → [(𝑘 + 1) / 𝑥]𝜑)
54		oveq1 5951	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑘 + 1) → (𝑥 − 1) = ((𝑘 + 1) − 1))
55	54	oveq2d 5960	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝑘 + 1) → (𝑀...(𝑥 − 1)) = (𝑀...((𝑘 + 1) − 1)))
56	55	raleqdv 2708	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑘 + 1) → (∀𝑦 ∈ (𝑀...(𝑥 − 1))𝜓 ↔ ∀𝑦 ∈ (𝑀...((𝑘 + 1) − 1))𝜓))
57		sbceq1a 3008	. . . . . . . . . . . 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 2871	. . . . . . . . . 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 1466	. . . . . . . . 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 9498	. . . . . . . . 9 ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → (𝑘 + 1) ∈ ℤ)
64	63	adantr 276	. . . . . . . 8 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → (𝑘 + 1) ∈ ℤ)
65		ralsns 3671	. . . . . . . 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 3355	. . . . . 6 ⊢ ((∀𝑥 ∈ (𝑀...𝑘)𝜑 ∧ ∀𝑥 ∈ {(𝑘 + 1)}𝜑) → ∀𝑥 ∈ ((𝑀...𝑘) ∪ {(𝑘 + 1)})𝜑)
69	38, 67, 68	syl2anc 411	. . . . 5 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → ∀𝑥 ∈ ((𝑀...𝑘) ∪ {(𝑘 + 1)})𝜑)
70		fzsuc 10191	. . . . . . 7 ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → (𝑀...(𝑘 + 1)) = ((𝑀...𝑘) ∪ {(𝑘 + 1)}))
71	70	raleqdv 2708	. . . . . 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 9709	. 2 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → ∀𝑥 ∈ (𝑀...𝑁)𝜑)
76		eluzfz2 10154	. 2 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁))
77	1, 75, 76	rspcdva 2882	1 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝜒)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1373   ∈ wcel 2176  ∀wral 2484  [wsbc 2998   ∪ cun 3164  ∅c0 3460  {csn 3633   class class class wbr 4044  ‘cfv 5271  (class class class)co 5944  1c1 7926   + caddc 7928   < clt 8107   − cmin 8243  ℤcz 9372  ℤ_≥cuz 9648  ...cfz 10130

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-cnex 8016  ax-resscn 8017  ax-1cn 8018  ax-1re 8019  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-addcom 8025  ax-addass 8027  ax-distr 8029  ax-i2m1 8030  ax-0lt1 8031  ax-0id 8033  ax-rnegex 8034  ax-cnre 8036  ax-pre-ltirr 8037  ax-pre-ltwlin 8038  ax-pre-lttrn 8039  ax-pre-apti 8040  ax-pre-ltadd 8041

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-pnf 8109  df-mnf 8110  df-xr 8111  df-ltxr 8112  df-le 8113  df-sub 8245  df-neg 8246  df-inn 9037  df-n0 9296  df-z 9373  df-uz 9649  df-fz 10131

This theorem is referenced by:  nnsinds  10590  nn0sinds  10591