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Theorem uzsinds 10227
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 → (𝜑𝜓))
uzsinds.2 (𝑥 = 𝑁 → (𝜑𝜒))
uzsinds.3 (𝑥 ∈ (ℤ𝑀) → (∀𝑦 ∈ (𝑀...(𝑥 − 1))𝜓𝜑))
Assertion
Ref Expression
uzsinds (𝑁 ∈ (ℤ𝑀) → 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.
StepHypRef Expression
1 uzsinds.2 . 2 (𝑥 = 𝑁 → (𝜑𝜒))
2 oveq2 5782 . . . 4 (𝑤 = 𝑀 → (𝑀...𝑤) = (𝑀...𝑀))
32raleqdv 2632 . . 3 (𝑤 = 𝑀 → (∀𝑥 ∈ (𝑀...𝑤)𝜑 ↔ ∀𝑥 ∈ (𝑀...𝑀)𝜑))
4 oveq2 5782 . . . 4 (𝑤 = 𝑘 → (𝑀...𝑤) = (𝑀...𝑘))
54raleqdv 2632 . . 3 (𝑤 = 𝑘 → (∀𝑥 ∈ (𝑀...𝑤)𝜑 ↔ ∀𝑥 ∈ (𝑀...𝑘)𝜑))
6 oveq2 5782 . . . 4 (𝑤 = (𝑘 + 1) → (𝑀...𝑤) = (𝑀...(𝑘 + 1)))
76raleqdv 2632 . . 3 (𝑤 = (𝑘 + 1) → (∀𝑥 ∈ (𝑀...𝑤)𝜑 ↔ ∀𝑥 ∈ (𝑀...(𝑘 + 1))𝜑))
8 oveq2 5782 . . . 4 (𝑤 = 𝑁 → (𝑀...𝑤) = (𝑀...𝑁))
98raleqdv 2632 . . 3 (𝑤 = 𝑁 → (∀𝑥 ∈ (𝑀...𝑤)𝜑 ↔ ∀𝑥 ∈ (𝑀...𝑁)𝜑))
10 ral0 3464 . . . . . . 7 𝑦 ∈ ∅ 𝜓
11 zre 9070 . . . . . . . . . 10 (𝑀 ∈ ℤ → 𝑀 ∈ ℝ)
1211ltm1d 8702 . . . . . . . . 9 (𝑀 ∈ ℤ → (𝑀 − 1) < 𝑀)
13 peano2zm 9104 . . . . . . . . . 10 (𝑀 ∈ ℤ → (𝑀 − 1) ∈ ℤ)
14 fzn 9834 . . . . . . . . . 10 ((𝑀 ∈ ℤ ∧ (𝑀 − 1) ∈ ℤ) → ((𝑀 − 1) < 𝑀 ↔ (𝑀...(𝑀 − 1)) = ∅))
1513, 14mpdan 417 . . . . . . . . 9 (𝑀 ∈ ℤ → ((𝑀 − 1) < 𝑀 ↔ (𝑀...(𝑀 − 1)) = ∅))
1612, 15mpbid 146 . . . . . . . 8 (𝑀 ∈ ℤ → (𝑀...(𝑀 − 1)) = ∅)
1716raleqdv 2632 . . . . . . 7 (𝑀 ∈ ℤ → (∀𝑦 ∈ (𝑀...(𝑀 − 1))𝜓 ↔ ∀𝑦 ∈ ∅ 𝜓))
1810, 17mpbiri 167 . . . . . 6 (𝑀 ∈ ℤ → ∀𝑦 ∈ (𝑀...(𝑀 − 1))𝜓)
19 uzid 9352 . . . . . . 7 (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ𝑀))
20 uzsinds.3 . . . . . . . 8 (𝑥 ∈ (ℤ𝑀) → (∀𝑦 ∈ (𝑀...(𝑥 − 1))𝜓𝜑))
2120rgen 2485 . . . . . . 7 𝑥 ∈ (ℤ𝑀)(∀𝑦 ∈ (𝑀...(𝑥 − 1))𝜓𝜑)
22 nfv 1508 . . . . . . . . 9 𝑥𝑦 ∈ (𝑀...(𝑀 − 1))𝜓
23 nfsbc1v 2927 . . . . . . . . 9 𝑥[𝑀 / 𝑥]𝜑
2422, 23nfim 1551 . . . . . . . 8 𝑥(∀𝑦 ∈ (𝑀...(𝑀 − 1))𝜓[𝑀 / 𝑥]𝜑)
25 oveq1 5781 . . . . . . . . . . 11 (𝑥 = 𝑀 → (𝑥 − 1) = (𝑀 − 1))
2625oveq2d 5790 . . . . . . . . . 10 (𝑥 = 𝑀 → (𝑀...(𝑥 − 1)) = (𝑀...(𝑀 − 1)))
2726raleqdv 2632 . . . . . . . . 9 (𝑥 = 𝑀 → (∀𝑦 ∈ (𝑀...(𝑥 − 1))𝜓 ↔ ∀𝑦 ∈ (𝑀...(𝑀 − 1))𝜓))
28 sbceq1a 2918 . . . . . . . . 9 (𝑥 = 𝑀 → (𝜑[𝑀 / 𝑥]𝜑))
2927, 28imbi12d 233 . . . . . . . 8 (𝑥 = 𝑀 → ((∀𝑦 ∈ (𝑀...(𝑥 − 1))𝜓𝜑) ↔ (∀𝑦 ∈ (𝑀...(𝑀 − 1))𝜓[𝑀 / 𝑥]𝜑)))
3024, 29rspc 2783 . . . . . . 7 (𝑀 ∈ (ℤ𝑀) → (∀𝑥 ∈ (ℤ𝑀)(∀𝑦 ∈ (𝑀...(𝑥 − 1))𝜓𝜑) → (∀𝑦 ∈ (𝑀...(𝑀 − 1))𝜓[𝑀 / 𝑥]𝜑)))
3119, 21, 30mpisyl 1422 . . . . . 6 (𝑀 ∈ ℤ → (∀𝑦 ∈ (𝑀...(𝑀 − 1))𝜓[𝑀 / 𝑥]𝜑))
3218, 31mpd 13 . . . . 5 (𝑀 ∈ ℤ → [𝑀 / 𝑥]𝜑)
33 ralsns 3562 . . . . 5 (𝑀 ∈ ℤ → (∀𝑥 ∈ {𝑀}𝜑[𝑀 / 𝑥]𝜑))
3432, 33mpbird 166 . . . 4 (𝑀 ∈ ℤ → ∀𝑥 ∈ {𝑀}𝜑)
35 fzsn 9858 . . . . 5 (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀})
3635raleqdv 2632 . . . 4 (𝑀 ∈ ℤ → (∀𝑥 ∈ (𝑀...𝑀)𝜑 ↔ ∀𝑥 ∈ {𝑀}𝜑))
3734, 36mpbird 166 . . 3 (𝑀 ∈ ℤ → ∀𝑥 ∈ (𝑀...𝑀)𝜑)
38 simpr 109 . . . . . 6 ((𝑘 ∈ (ℤ𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → ∀𝑥 ∈ (𝑀...𝑘)𝜑)
39 uzsinds.1 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝜑𝜓))
4039cbvralv 2654 . . . . . . . . 9 (∀𝑥 ∈ (𝑀...𝑘)𝜑 ↔ ∀𝑦 ∈ (𝑀...𝑘)𝜓)
4138, 40sylib 121 . . . . . . . 8 ((𝑘 ∈ (ℤ𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → ∀𝑦 ∈ (𝑀...𝑘)𝜓)
42 eluzelz 9347 . . . . . . . . . . . . . 14 (𝑘 ∈ (ℤ𝑀) → 𝑘 ∈ ℤ)
4342adantr 274 . . . . . . . . . . . . 13 ((𝑘 ∈ (ℤ𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → 𝑘 ∈ ℤ)
4443zcnd 9186 . . . . . . . . . . . 12 ((𝑘 ∈ (ℤ𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → 𝑘 ∈ ℂ)
45 1cnd 7794 . . . . . . . . . . . 12 ((𝑘 ∈ (ℤ𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → 1 ∈ ℂ)
4644, 45pncand 8086 . . . . . . . . . . 11 ((𝑘 ∈ (ℤ𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → ((𝑘 + 1) − 1) = 𝑘)
4746oveq2d 5790 . . . . . . . . . 10 ((𝑘 ∈ (ℤ𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → (𝑀...((𝑘 + 1) − 1)) = (𝑀...𝑘))
4847raleqdv 2632 . . . . . . . . 9 ((𝑘 ∈ (ℤ𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → (∀𝑦 ∈ (𝑀...((𝑘 + 1) − 1))𝜓 ↔ ∀𝑦 ∈ (𝑀...𝑘)𝜓))
49 peano2uz 9390 . . . . . . . . . . 11 (𝑘 ∈ (ℤ𝑀) → (𝑘 + 1) ∈ (ℤ𝑀))
5049adantr 274 . . . . . . . . . 10 ((𝑘 ∈ (ℤ𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → (𝑘 + 1) ∈ (ℤ𝑀))
51 nfv 1508 . . . . . . . . . . . 12 𝑥𝑦 ∈ (𝑀...((𝑘 + 1) − 1))𝜓
52 nfsbc1v 2927 . . . . . . . . . . . 12 𝑥[(𝑘 + 1) / 𝑥]𝜑
5351, 52nfim 1551 . . . . . . . . . . 11 𝑥(∀𝑦 ∈ (𝑀...((𝑘 + 1) − 1))𝜓[(𝑘 + 1) / 𝑥]𝜑)
54 oveq1 5781 . . . . . . . . . . . . . 14 (𝑥 = (𝑘 + 1) → (𝑥 − 1) = ((𝑘 + 1) − 1))
5554oveq2d 5790 . . . . . . . . . . . . 13 (𝑥 = (𝑘 + 1) → (𝑀...(𝑥 − 1)) = (𝑀...((𝑘 + 1) − 1)))
5655raleqdv 2632 . . . . . . . . . . . 12 (𝑥 = (𝑘 + 1) → (∀𝑦 ∈ (𝑀...(𝑥 − 1))𝜓 ↔ ∀𝑦 ∈ (𝑀...((𝑘 + 1) − 1))𝜓))
57 sbceq1a 2918 . . . . . . . . . . . 12 (𝑥 = (𝑘 + 1) → (𝜑[(𝑘 + 1) / 𝑥]𝜑))
5856, 57imbi12d 233 . . . . . . . . . . 11 (𝑥 = (𝑘 + 1) → ((∀𝑦 ∈ (𝑀...(𝑥 − 1))𝜓𝜑) ↔ (∀𝑦 ∈ (𝑀...((𝑘 + 1) − 1))𝜓[(𝑘 + 1) / 𝑥]𝜑)))
5953, 58rspc 2783 . . . . . . . . . 10 ((𝑘 + 1) ∈ (ℤ𝑀) → (∀𝑥 ∈ (ℤ𝑀)(∀𝑦 ∈ (𝑀...(𝑥 − 1))𝜓𝜑) → (∀𝑦 ∈ (𝑀...((𝑘 + 1) − 1))𝜓[(𝑘 + 1) / 𝑥]𝜑)))
6050, 21, 59mpisyl 1422 . . . . . . . . 9 ((𝑘 ∈ (ℤ𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → (∀𝑦 ∈ (𝑀...((𝑘 + 1) − 1))𝜓[(𝑘 + 1) / 𝑥]𝜑))
6148, 60sylbird 169 . . . . . . . 8 ((𝑘 ∈ (ℤ𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → (∀𝑦 ∈ (𝑀...𝑘)𝜓[(𝑘 + 1) / 𝑥]𝜑))
6241, 61mpd 13 . . . . . . 7 ((𝑘 ∈ (ℤ𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → [(𝑘 + 1) / 𝑥]𝜑)
6342peano2zd 9188 . . . . . . . . 9 (𝑘 ∈ (ℤ𝑀) → (𝑘 + 1) ∈ ℤ)
6463adantr 274 . . . . . . . 8 ((𝑘 ∈ (ℤ𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → (𝑘 + 1) ∈ ℤ)
65 ralsns 3562 . . . . . . . 8 ((𝑘 + 1) ∈ ℤ → (∀𝑥 ∈ {(𝑘 + 1)}𝜑[(𝑘 + 1) / 𝑥]𝜑))
6664, 65syl 14 . . . . . . 7 ((𝑘 ∈ (ℤ𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → (∀𝑥 ∈ {(𝑘 + 1)}𝜑[(𝑘 + 1) / 𝑥]𝜑))
6762, 66mpbird 166 . . . . . 6 ((𝑘 ∈ (ℤ𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → ∀𝑥 ∈ {(𝑘 + 1)}𝜑)
68 ralun 3258 . . . . . 6 ((∀𝑥 ∈ (𝑀...𝑘)𝜑 ∧ ∀𝑥 ∈ {(𝑘 + 1)}𝜑) → ∀𝑥 ∈ ((𝑀...𝑘) ∪ {(𝑘 + 1)})𝜑)
6938, 67, 68syl2anc 408 . . . . 5 ((𝑘 ∈ (ℤ𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → ∀𝑥 ∈ ((𝑀...𝑘) ∪ {(𝑘 + 1)})𝜑)
70 fzsuc 9861 . . . . . . 7 (𝑘 ∈ (ℤ𝑀) → (𝑀...(𝑘 + 1)) = ((𝑀...𝑘) ∪ {(𝑘 + 1)}))
7170raleqdv 2632 . . . . . 6 (𝑘 ∈ (ℤ𝑀) → (∀𝑥 ∈ (𝑀...(𝑘 + 1))𝜑 ↔ ∀𝑥 ∈ ((𝑀...𝑘) ∪ {(𝑘 + 1)})𝜑))
7271adantr 274 . . . . 5 ((𝑘 ∈ (ℤ𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → (∀𝑥 ∈ (𝑀...(𝑘 + 1))𝜑 ↔ ∀𝑥 ∈ ((𝑀...𝑘) ∪ {(𝑘 + 1)})𝜑))
7369, 72mpbird 166 . . . 4 ((𝑘 ∈ (ℤ𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → ∀𝑥 ∈ (𝑀...(𝑘 + 1))𝜑)
7473ex 114 . . 3 (𝑘 ∈ (ℤ𝑀) → (∀𝑥 ∈ (𝑀...𝑘)𝜑 → ∀𝑥 ∈ (𝑀...(𝑘 + 1))𝜑))
753, 5, 7, 9, 37, 74uzind4 9395 . 2 (𝑁 ∈ (ℤ𝑀) → ∀𝑥 ∈ (𝑀...𝑁)𝜑)
76 eluzfz2 9824 . 2 (𝑁 ∈ (ℤ𝑀) → 𝑁 ∈ (𝑀...𝑁))
771, 75, 76rspcdva 2794 1 (𝑁 ∈ (ℤ𝑀) → 𝜒)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1331  wcel 1480  wral 2416  [wsbc 2909  cun 3069  c0 3363  {csn 3527   class class class wbr 3929  cfv 5123  (class class class)co 5774  1c1 7633   + caddc 7635   < clt 7812  cmin 7945  cz 9066  cuz 9338  ...cfz 9802
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7723  ax-resscn 7724  ax-1cn 7725  ax-1re 7726  ax-icn 7727  ax-addcl 7728  ax-addrcl 7729  ax-mulcl 7730  ax-addcom 7732  ax-addass 7734  ax-distr 7736  ax-i2m1 7737  ax-0lt1 7738  ax-0id 7740  ax-rnegex 7741  ax-cnre 7743  ax-pre-ltirr 7744  ax-pre-ltwlin 7745  ax-pre-lttrn 7746  ax-pre-apti 7747  ax-pre-ltadd 7748
This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-pnf 7814  df-mnf 7815  df-xr 7816  df-ltxr 7817  df-le 7818  df-sub 7947  df-neg 7948  df-inn 8733  df-n0 8990  df-z 9067  df-uz 9339  df-fz 9803
This theorem is referenced by:  nnsinds  10228  nn0sinds  10229
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