Theorem uzsinds 10246

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 5790	. . . 4 ⊢ (𝑤 = 𝑀 → (𝑀...𝑤) = (𝑀...𝑀))
3	2	raleqdv 2635	. . 3 ⊢ (𝑤 = 𝑀 → (∀𝑥 ∈ (𝑀...𝑤)𝜑 ↔ ∀𝑥 ∈ (𝑀...𝑀)𝜑))
4		oveq2 5790	. . . 4 ⊢ (𝑤 = 𝑘 → (𝑀...𝑤) = (𝑀...𝑘))
5	4	raleqdv 2635	. . 3 ⊢ (𝑤 = 𝑘 → (∀𝑥 ∈ (𝑀...𝑤)𝜑 ↔ ∀𝑥 ∈ (𝑀...𝑘)𝜑))
6		oveq2 5790	. . . 4 ⊢ (𝑤 = (𝑘 + 1) → (𝑀...𝑤) = (𝑀...(𝑘 + 1)))
7	6	raleqdv 2635	. . 3 ⊢ (𝑤 = (𝑘 + 1) → (∀𝑥 ∈ (𝑀...𝑤)𝜑 ↔ ∀𝑥 ∈ (𝑀...(𝑘 + 1))𝜑))
8		oveq2 5790	. . . 4 ⊢ (𝑤 = 𝑁 → (𝑀...𝑤) = (𝑀...𝑁))
9	8	raleqdv 2635	. . 3 ⊢ (𝑤 = 𝑁 → (∀𝑥 ∈ (𝑀...𝑤)𝜑 ↔ ∀𝑥 ∈ (𝑀...𝑁)𝜑))
10		ral0 3469	. . . . . . 7 ⊢ ∀𝑦 ∈ ∅ 𝜓
11		zre 9082	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ)
12	11	ltm1d 8714	. . . . . . . . 9 ⊢ (𝑀 ∈ ℤ → (𝑀 − 1) < 𝑀)
13		peano2zm 9116	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℤ → (𝑀 − 1) ∈ ℤ)
14		fzn 9853	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤ ∧ (𝑀 − 1) ∈ ℤ) → ((𝑀 − 1) < 𝑀 ↔ (𝑀...(𝑀 − 1)) = ∅))
15	13, 14	mpdan 418	. . . . . . . . 9 ⊢ (𝑀 ∈ ℤ → ((𝑀 − 1) < 𝑀 ↔ (𝑀...(𝑀 − 1)) = ∅))
16	12, 15	mpbid 146	. . . . . . . 8 ⊢ (𝑀 ∈ ℤ → (𝑀...(𝑀 − 1)) = ∅)
17	16	raleqdv 2635	. . . . . . 7 ⊢ (𝑀 ∈ ℤ → (∀𝑦 ∈ (𝑀...(𝑀 − 1))𝜓 ↔ ∀𝑦 ∈ ∅ 𝜓))
18	10, 17	mpbiri 167	. . . . . 6 ⊢ (𝑀 ∈ ℤ → ∀𝑦 ∈ (𝑀...(𝑀 − 1))𝜓)
19		uzid 9364	. . . . . . 7 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ≥‘𝑀))
20		uzsinds.3	. . . . . . . 8 ⊢ (𝑥 ∈ (ℤ≥‘𝑀) → (∀𝑦 ∈ (𝑀...(𝑥 − 1))𝜓 → 𝜑))
21	20	rgen 2488	. . . . . . 7 ⊢ ∀𝑥 ∈ (ℤ≥‘𝑀)(∀𝑦 ∈ (𝑀...(𝑥 − 1))𝜓 → 𝜑)
22		nfv 1509	. . . . . . . . 9 ⊢ Ⅎ𝑥∀𝑦 ∈ (𝑀...(𝑀 − 1))𝜓
23		nfsbc1v 2931	. . . . . . . . 9 ⊢ Ⅎ𝑥[𝑀 / 𝑥]𝜑
24	22, 23	nfim 1552	. . . . . . . 8 ⊢ Ⅎ𝑥(∀𝑦 ∈ (𝑀...(𝑀 − 1))𝜓 → [𝑀 / 𝑥]𝜑)
25		oveq1 5789	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑀 → (𝑥 − 1) = (𝑀 − 1))
26	25	oveq2d 5798	. . . . . . . . . 10 ⊢ (𝑥 = 𝑀 → (𝑀...(𝑥 − 1)) = (𝑀...(𝑀 − 1)))
27	26	raleqdv 2635	. . . . . . . . 9 ⊢ (𝑥 = 𝑀 → (∀𝑦 ∈ (𝑀...(𝑥 − 1))𝜓 ↔ ∀𝑦 ∈ (𝑀...(𝑀 − 1))𝜓))
28		sbceq1a 2922	. . . . . . . . 9 ⊢ (𝑥 = 𝑀 → (𝜑 ↔ [𝑀 / 𝑥]𝜑))
29	27, 28	imbi12d 233	. . . . . . . 8 ⊢ (𝑥 = 𝑀 → ((∀𝑦 ∈ (𝑀...(𝑥 − 1))𝜓 → 𝜑) ↔ (∀𝑦 ∈ (𝑀...(𝑀 − 1))𝜓 → [𝑀 / 𝑥]𝜑)))
30	24, 29	rspc 2787	. . . . . . 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 1423	. . . . . 6 ⊢ (𝑀 ∈ ℤ → (∀𝑦 ∈ (𝑀...(𝑀 − 1))𝜓 → [𝑀 / 𝑥]𝜑))
32	18, 31	mpd 13	. . . . 5 ⊢ (𝑀 ∈ ℤ → [𝑀 / 𝑥]𝜑)
33		ralsns 3569	. . . . 5 ⊢ (𝑀 ∈ ℤ → (∀𝑥 ∈ {𝑀}𝜑 ↔ [𝑀 / 𝑥]𝜑))
34	32, 33	mpbird 166	. . . 4 ⊢ (𝑀 ∈ ℤ → ∀𝑥 ∈ {𝑀}𝜑)
35		fzsn 9877	. . . . 5 ⊢ (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀})
36	35	raleqdv 2635	. . . 4 ⊢ (𝑀 ∈ ℤ → (∀𝑥 ∈ (𝑀...𝑀)𝜑 ↔ ∀𝑥 ∈ {𝑀}𝜑))
37	34, 36	mpbird 166	. . 3 ⊢ (𝑀 ∈ ℤ → ∀𝑥 ∈ (𝑀...𝑀)𝜑)
38		simpr 109	. . . . . 6 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → ∀𝑥 ∈ (𝑀...𝑘)𝜑)
39		uzsinds.1	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
40	39	cbvralv 2657	. . . . . . . . 9 ⊢ (∀𝑥 ∈ (𝑀...𝑘)𝜑 ↔ ∀𝑦 ∈ (𝑀...𝑘)𝜓)
41	38, 40	sylib 121	. . . . . . . 8 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → ∀𝑦 ∈ (𝑀...𝑘)𝜓)
42		eluzelz 9359	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → 𝑘 ∈ ℤ)
43	42	adantr 274	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → 𝑘 ∈ ℤ)
44	43	zcnd 9198	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → 𝑘 ∈ ℂ)
45		1cnd 7806	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → 1 ∈ ℂ)
46	44, 45	pncand 8098	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → ((𝑘 + 1) − 1) = 𝑘)
47	46	oveq2d 5798	. . . . . . . . . 10 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → (𝑀...((𝑘 + 1) − 1)) = (𝑀...𝑘))
48	47	raleqdv 2635	. . . . . . . . 9 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → (∀𝑦 ∈ (𝑀...((𝑘 + 1) − 1))𝜓 ↔ ∀𝑦 ∈ (𝑀...𝑘)𝜓))
49		peano2uz 9405	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → (𝑘 + 1) ∈ (ℤ≥‘𝑀))
50	49	adantr 274	. . . . . . . . . 10 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → (𝑘 + 1) ∈ (ℤ≥‘𝑀))
51		nfv 1509	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥∀𝑦 ∈ (𝑀...((𝑘 + 1) − 1))𝜓
52		nfsbc1v 2931	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥[(𝑘 + 1) / 𝑥]𝜑
53	51, 52	nfim 1552	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(∀𝑦 ∈ (𝑀...((𝑘 + 1) − 1))𝜓 → [(𝑘 + 1) / 𝑥]𝜑)
54		oveq1 5789	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑘 + 1) → (𝑥 − 1) = ((𝑘 + 1) − 1))
55	54	oveq2d 5798	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝑘 + 1) → (𝑀...(𝑥 − 1)) = (𝑀...((𝑘 + 1) − 1)))
56	55	raleqdv 2635	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑘 + 1) → (∀𝑦 ∈ (𝑀...(𝑥 − 1))𝜓 ↔ ∀𝑦 ∈ (𝑀...((𝑘 + 1) − 1))𝜓))
57		sbceq1a 2922	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑘 + 1) → (𝜑 ↔ [(𝑘 + 1) / 𝑥]𝜑))
58	56, 57	imbi12d 233	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑘 + 1) → ((∀𝑦 ∈ (𝑀...(𝑥 − 1))𝜓 → 𝜑) ↔ (∀𝑦 ∈ (𝑀...((𝑘 + 1) − 1))𝜓 → [(𝑘 + 1) / 𝑥]𝜑)))
59	53, 58	rspc 2787	. . . . . . . . . 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 1423	. . . . . . . . 9 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → (∀𝑦 ∈ (𝑀...((𝑘 + 1) − 1))𝜓 → [(𝑘 + 1) / 𝑥]𝜑))
61	48, 60	sylbird 169	. . . . . . . 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 9200	. . . . . . . . 9 ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → (𝑘 + 1) ∈ ℤ)
64	63	adantr 274	. . . . . . . 8 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → (𝑘 + 1) ∈ ℤ)
65		ralsns 3569	. . . . . . . 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 166	. . . . . 6 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → ∀𝑥 ∈ {(𝑘 + 1)}𝜑)
68		ralun 3263	. . . . . 6 ⊢ ((∀𝑥 ∈ (𝑀...𝑘)𝜑 ∧ ∀𝑥 ∈ {(𝑘 + 1)}𝜑) → ∀𝑥 ∈ ((𝑀...𝑘) ∪ {(𝑘 + 1)})𝜑)
69	38, 67, 68	syl2anc 409	. . . . 5 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → ∀𝑥 ∈ ((𝑀...𝑘) ∪ {(𝑘 + 1)})𝜑)
70		fzsuc 9880	. . . . . . 7 ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → (𝑀...(𝑘 + 1)) = ((𝑀...𝑘) ∪ {(𝑘 + 1)}))
71	70	raleqdv 2635	. . . . . 6 ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → (∀𝑥 ∈ (𝑀...(𝑘 + 1))𝜑 ↔ ∀𝑥 ∈ ((𝑀...𝑘) ∪ {(𝑘 + 1)})𝜑))
72	71	adantr 274	. . . . 5 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → (∀𝑥 ∈ (𝑀...(𝑘 + 1))𝜑 ↔ ∀𝑥 ∈ ((𝑀...𝑘) ∪ {(𝑘 + 1)})𝜑))
73	69, 72	mpbird 166	. . . 4 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → ∀𝑥 ∈ (𝑀...(𝑘 + 1))𝜑)
74	73	ex 114	. . 3 ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → (∀𝑥 ∈ (𝑀...𝑘)𝜑 → ∀𝑥 ∈ (𝑀...(𝑘 + 1))𝜑))
75	3, 5, 7, 9, 37, 74	uzind4 9410	. 2 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → ∀𝑥 ∈ (𝑀...𝑁)𝜑)
76		eluzfz2 9843	. 2 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁))
77	1, 75, 76	rspcdva 2798	1 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝜒)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1332   ∈ wcel 1481  ∀wral 2417  [wsbc 2913   ∪ cun 3074  ∅c0 3368  {csn 3532   class class class wbr 3937  ‘cfv 5131  (class class class)co 5782  1c1 7645   + caddc 7647   < clt 7824   − cmin 7957  ℤcz 9078  ℤ_≥cuz 9350  ...cfz 9821

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-cnex 7735  ax-resscn 7736  ax-1cn 7737  ax-1re 7738  ax-icn 7739  ax-addcl 7740  ax-addrcl 7741  ax-mulcl 7742  ax-addcom 7744  ax-addass 7746  ax-distr 7748  ax-i2m1 7749  ax-0lt1 7750  ax-0id 7752  ax-rnegex 7753  ax-cnre 7755  ax-pre-ltirr 7756  ax-pre-ltwlin 7757  ax-pre-lttrn 7758  ax-pre-apti 7759  ax-pre-ltadd 7760

This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2914  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-br 3938  df-opab 3998  df-mpt 3999  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-fv 5139  df-riota 5738  df-ov 5785  df-oprab 5786  df-mpo 5787  df-pnf 7826  df-mnf 7827  df-xr 7828  df-ltxr 7829  df-le 7830  df-sub 7959  df-neg 7960  df-inn 8745  df-n0 9002  df-z 9079  df-uz 9351  df-fz 9822

This theorem is referenced by:  nnsinds  10247  nn0sinds  10248