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Theorem uzsinds 9997
 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 5698 . . . 4 (𝑤 = 𝑀 → (𝑀...𝑤) = (𝑀...𝑀))
32raleqdv 2582 . . 3 (𝑤 = 𝑀 → (∀𝑥 ∈ (𝑀...𝑤)𝜑 ↔ ∀𝑥 ∈ (𝑀...𝑀)𝜑))
4 oveq2 5698 . . . 4 (𝑤 = 𝑘 → (𝑀...𝑤) = (𝑀...𝑘))
54raleqdv 2582 . . 3 (𝑤 = 𝑘 → (∀𝑥 ∈ (𝑀...𝑤)𝜑 ↔ ∀𝑥 ∈ (𝑀...𝑘)𝜑))
6 oveq2 5698 . . . 4 (𝑤 = (𝑘 + 1) → (𝑀...𝑤) = (𝑀...(𝑘 + 1)))
76raleqdv 2582 . . 3 (𝑤 = (𝑘 + 1) → (∀𝑥 ∈ (𝑀...𝑤)𝜑 ↔ ∀𝑥 ∈ (𝑀...(𝑘 + 1))𝜑))
8 oveq2 5698 . . . 4 (𝑤 = 𝑁 → (𝑀...𝑤) = (𝑀...𝑁))
98raleqdv 2582 . . 3 (𝑤 = 𝑁 → (∀𝑥 ∈ (𝑀...𝑤)𝜑 ↔ ∀𝑥 ∈ (𝑀...𝑁)𝜑))
10 ral0 3403 . . . . . . 7 𝑦 ∈ ∅ 𝜓
11 zre 8852 . . . . . . . . . 10 (𝑀 ∈ ℤ → 𝑀 ∈ ℝ)
1211ltm1d 8490 . . . . . . . . 9 (𝑀 ∈ ℤ → (𝑀 − 1) < 𝑀)
13 peano2zm 8886 . . . . . . . . . 10 (𝑀 ∈ ℤ → (𝑀 − 1) ∈ ℤ)
14 fzn 9605 . . . . . . . . . 10 ((𝑀 ∈ ℤ ∧ (𝑀 − 1) ∈ ℤ) → ((𝑀 − 1) < 𝑀 ↔ (𝑀...(𝑀 − 1)) = ∅))
1513, 14mpdan 413 . . . . . . . . 9 (𝑀 ∈ ℤ → ((𝑀 − 1) < 𝑀 ↔ (𝑀...(𝑀 − 1)) = ∅))
1612, 15mpbid 146 . . . . . . . 8 (𝑀 ∈ ℤ → (𝑀...(𝑀 − 1)) = ∅)
1716raleqdv 2582 . . . . . . 7 (𝑀 ∈ ℤ → (∀𝑦 ∈ (𝑀...(𝑀 − 1))𝜓 ↔ ∀𝑦 ∈ ∅ 𝜓))
1810, 17mpbiri 167 . . . . . 6 (𝑀 ∈ ℤ → ∀𝑦 ∈ (𝑀...(𝑀 − 1))𝜓)
19 uzid 9132 . . . . . . 7 (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ𝑀))
20 uzsinds.3 . . . . . . . 8 (𝑥 ∈ (ℤ𝑀) → (∀𝑦 ∈ (𝑀...(𝑥 − 1))𝜓𝜑))
2120rgen 2439 . . . . . . 7 𝑥 ∈ (ℤ𝑀)(∀𝑦 ∈ (𝑀...(𝑥 − 1))𝜓𝜑)
22 nfv 1473 . . . . . . . . 9 𝑥𝑦 ∈ (𝑀...(𝑀 − 1))𝜓
23 nfsbc1v 2872 . . . . . . . . 9 𝑥[𝑀 / 𝑥]𝜑
2422, 23nfim 1516 . . . . . . . 8 𝑥(∀𝑦 ∈ (𝑀...(𝑀 − 1))𝜓[𝑀 / 𝑥]𝜑)
25 oveq1 5697 . . . . . . . . . . 11 (𝑥 = 𝑀 → (𝑥 − 1) = (𝑀 − 1))
2625oveq2d 5706 . . . . . . . . . 10 (𝑥 = 𝑀 → (𝑀...(𝑥 − 1)) = (𝑀...(𝑀 − 1)))
2726raleqdv 2582 . . . . . . . . 9 (𝑥 = 𝑀 → (∀𝑦 ∈ (𝑀...(𝑥 − 1))𝜓 ↔ ∀𝑦 ∈ (𝑀...(𝑀 − 1))𝜓))
28 sbceq1a 2863 . . . . . . . . 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 2730 . . . . . . 7 (𝑀 ∈ (ℤ𝑀) → (∀𝑥 ∈ (ℤ𝑀)(∀𝑦 ∈ (𝑀...(𝑥 − 1))𝜓𝜑) → (∀𝑦 ∈ (𝑀...(𝑀 − 1))𝜓[𝑀 / 𝑥]𝜑)))
3119, 21, 30mpisyl 1387 . . . . . 6 (𝑀 ∈ ℤ → (∀𝑦 ∈ (𝑀...(𝑀 − 1))𝜓[𝑀 / 𝑥]𝜑))
3218, 31mpd 13 . . . . 5 (𝑀 ∈ ℤ → [𝑀 / 𝑥]𝜑)
33 ralsns 3501 . . . . 5 (𝑀 ∈ ℤ → (∀𝑥 ∈ {𝑀}𝜑[𝑀 / 𝑥]𝜑))
3432, 33mpbird 166 . . . 4 (𝑀 ∈ ℤ → ∀𝑥 ∈ {𝑀}𝜑)
35 fzsn 9629 . . . . 5 (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀})
3635raleqdv 2582 . . . 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 2604 . . . . . . . . 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 9127 . . . . . . . . . . . . . 14 (𝑘 ∈ (ℤ𝑀) → 𝑘 ∈ ℤ)
4342adantr 271 . . . . . . . . . . . . 13 ((𝑘 ∈ (ℤ𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → 𝑘 ∈ ℤ)
4443zcnd 8968 . . . . . . . . . . . 12 ((𝑘 ∈ (ℤ𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → 𝑘 ∈ ℂ)
45 1cnd 7601 . . . . . . . . . . . 12 ((𝑘 ∈ (ℤ𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → 1 ∈ ℂ)
4644, 45pncand 7891 . . . . . . . . . . 11 ((𝑘 ∈ (ℤ𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → ((𝑘 + 1) − 1) = 𝑘)
4746oveq2d 5706 . . . . . . . . . 10 ((𝑘 ∈ (ℤ𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → (𝑀...((𝑘 + 1) − 1)) = (𝑀...𝑘))
4847raleqdv 2582 . . . . . . . . 9 ((𝑘 ∈ (ℤ𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → (∀𝑦 ∈ (𝑀...((𝑘 + 1) − 1))𝜓 ↔ ∀𝑦 ∈ (𝑀...𝑘)𝜓))
49 peano2uz 9170 . . . . . . . . . . 11 (𝑘 ∈ (ℤ𝑀) → (𝑘 + 1) ∈ (ℤ𝑀))
5049adantr 271 . . . . . . . . . 10 ((𝑘 ∈ (ℤ𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → (𝑘 + 1) ∈ (ℤ𝑀))
51 nfv 1473 . . . . . . . . . . . 12 𝑥𝑦 ∈ (𝑀...((𝑘 + 1) − 1))𝜓
52 nfsbc1v 2872 . . . . . . . . . . . 12 𝑥[(𝑘 + 1) / 𝑥]𝜑
5351, 52nfim 1516 . . . . . . . . . . 11 𝑥(∀𝑦 ∈ (𝑀...((𝑘 + 1) − 1))𝜓[(𝑘 + 1) / 𝑥]𝜑)
54 oveq1 5697 . . . . . . . . . . . . . 14 (𝑥 = (𝑘 + 1) → (𝑥 − 1) = ((𝑘 + 1) − 1))
5554oveq2d 5706 . . . . . . . . . . . . 13 (𝑥 = (𝑘 + 1) → (𝑀...(𝑥 − 1)) = (𝑀...((𝑘 + 1) − 1)))
5655raleqdv 2582 . . . . . . . . . . . 12 (𝑥 = (𝑘 + 1) → (∀𝑦 ∈ (𝑀...(𝑥 − 1))𝜓 ↔ ∀𝑦 ∈ (𝑀...((𝑘 + 1) − 1))𝜓))
57 sbceq1a 2863 . . . . . . . . . . . 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 2730 . . . . . . . . . 10 ((𝑘 + 1) ∈ (ℤ𝑀) → (∀𝑥 ∈ (ℤ𝑀)(∀𝑦 ∈ (𝑀...(𝑥 − 1))𝜓𝜑) → (∀𝑦 ∈ (𝑀...((𝑘 + 1) − 1))𝜓[(𝑘 + 1) / 𝑥]𝜑)))
6050, 21, 59mpisyl 1387 . . . . . . . . 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 8970 . . . . . . . . 9 (𝑘 ∈ (ℤ𝑀) → (𝑘 + 1) ∈ ℤ)
6463adantr 271 . . . . . . . 8 ((𝑘 ∈ (ℤ𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → (𝑘 + 1) ∈ ℤ)
65 ralsns 3501 . . . . . . . 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 3197 . . . . . 6 ((∀𝑥 ∈ (𝑀...𝑘)𝜑 ∧ ∀𝑥 ∈ {(𝑘 + 1)}𝜑) → ∀𝑥 ∈ ((𝑀...𝑘) ∪ {(𝑘 + 1)})𝜑)
6938, 67, 68syl2anc 404 . . . . 5 ((𝑘 ∈ (ℤ𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → ∀𝑥 ∈ ((𝑀...𝑘) ∪ {(𝑘 + 1)})𝜑)
70 fzsuc 9632 . . . . . . 7 (𝑘 ∈ (ℤ𝑀) → (𝑀...(𝑘 + 1)) = ((𝑀...𝑘) ∪ {(𝑘 + 1)}))
7170raleqdv 2582 . . . . . 6 (𝑘 ∈ (ℤ𝑀) → (∀𝑥 ∈ (𝑀...(𝑘 + 1))𝜑 ↔ ∀𝑥 ∈ ((𝑀...𝑘) ∪ {(𝑘 + 1)})𝜑))
7271adantr 271 . . . . 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 9175 . 2 (𝑁 ∈ (ℤ𝑀) → ∀𝑥 ∈ (𝑀...𝑁)𝜑)
76 eluzfz2 9595 . 2 (𝑁 ∈ (ℤ𝑀) → 𝑁 ∈ (𝑀...𝑁))
771, 75, 76rspcdva 2741 1 (𝑁 ∈ (ℤ𝑀) → 𝜒)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1296   ∈ wcel 1445  ∀wral 2370  [wsbc 2854   ∪ cun 3011  ∅c0 3302  {csn 3466   class class class wbr 3867  ‘cfv 5049  (class class class)co 5690  1c1 7448   + caddc 7450   < clt 7619   − cmin 7750  ℤcz 8848  ℤ≥cuz 9118  ...cfz 9573 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060  ax-un 4284  ax-setind 4381  ax-cnex 7533  ax-resscn 7534  ax-1cn 7535  ax-1re 7536  ax-icn 7537  ax-addcl 7538  ax-addrcl 7539  ax-mulcl 7540  ax-addcom 7542  ax-addass 7544  ax-distr 7546  ax-i2m1 7547  ax-0lt1 7548  ax-0id 7550  ax-rnegex 7551  ax-cnre 7553  ax-pre-ltirr 7554  ax-pre-ltwlin 7555  ax-pre-lttrn 7556  ax-pre-apti 7557  ax-pre-ltadd 7558 This theorem depends on definitions:  df-bi 116  df-3or 928  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-nel 2358  df-ral 2375  df-rex 2376  df-reu 2377  df-rab 2379  df-v 2635  df-sbc 2855  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-nul 3303  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-int 3711  df-br 3868  df-opab 3922  df-mpt 3923  df-id 4144  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-fv 5057  df-riota 5646  df-ov 5693  df-oprab 5694  df-mpt2 5695  df-pnf 7621  df-mnf 7622  df-xr 7623  df-ltxr 7624  df-le 7625  df-sub 7752  df-neg 7753  df-inn 8521  df-n0 8772  df-z 8849  df-uz 9119  df-fz 9574 This theorem is referenced by:  nnsinds  9998  nn0sinds  9999
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