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Theorem uzsinds 10377
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 5850 . . . 4 (𝑤 = 𝑀 → (𝑀...𝑤) = (𝑀...𝑀))
32raleqdv 2667 . . 3 (𝑤 = 𝑀 → (∀𝑥 ∈ (𝑀...𝑤)𝜑 ↔ ∀𝑥 ∈ (𝑀...𝑀)𝜑))
4 oveq2 5850 . . . 4 (𝑤 = 𝑘 → (𝑀...𝑤) = (𝑀...𝑘))
54raleqdv 2667 . . 3 (𝑤 = 𝑘 → (∀𝑥 ∈ (𝑀...𝑤)𝜑 ↔ ∀𝑥 ∈ (𝑀...𝑘)𝜑))
6 oveq2 5850 . . . 4 (𝑤 = (𝑘 + 1) → (𝑀...𝑤) = (𝑀...(𝑘 + 1)))
76raleqdv 2667 . . 3 (𝑤 = (𝑘 + 1) → (∀𝑥 ∈ (𝑀...𝑤)𝜑 ↔ ∀𝑥 ∈ (𝑀...(𝑘 + 1))𝜑))
8 oveq2 5850 . . . 4 (𝑤 = 𝑁 → (𝑀...𝑤) = (𝑀...𝑁))
98raleqdv 2667 . . 3 (𝑤 = 𝑁 → (∀𝑥 ∈ (𝑀...𝑤)𝜑 ↔ ∀𝑥 ∈ (𝑀...𝑁)𝜑))
10 ral0 3510 . . . . . . 7 𝑦 ∈ ∅ 𝜓
11 zre 9195 . . . . . . . . . 10 (𝑀 ∈ ℤ → 𝑀 ∈ ℝ)
1211ltm1d 8827 . . . . . . . . 9 (𝑀 ∈ ℤ → (𝑀 − 1) < 𝑀)
13 peano2zm 9229 . . . . . . . . . 10 (𝑀 ∈ ℤ → (𝑀 − 1) ∈ ℤ)
14 fzn 9977 . . . . . . . . . 10 ((𝑀 ∈ ℤ ∧ (𝑀 − 1) ∈ ℤ) → ((𝑀 − 1) < 𝑀 ↔ (𝑀...(𝑀 − 1)) = ∅))
1513, 14mpdan 418 . . . . . . . . 9 (𝑀 ∈ ℤ → ((𝑀 − 1) < 𝑀 ↔ (𝑀...(𝑀 − 1)) = ∅))
1612, 15mpbid 146 . . . . . . . 8 (𝑀 ∈ ℤ → (𝑀...(𝑀 − 1)) = ∅)
1716raleqdv 2667 . . . . . . 7 (𝑀 ∈ ℤ → (∀𝑦 ∈ (𝑀...(𝑀 − 1))𝜓 ↔ ∀𝑦 ∈ ∅ 𝜓))
1810, 17mpbiri 167 . . . . . 6 (𝑀 ∈ ℤ → ∀𝑦 ∈ (𝑀...(𝑀 − 1))𝜓)
19 uzid 9480 . . . . . . 7 (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ𝑀))
20 uzsinds.3 . . . . . . . 8 (𝑥 ∈ (ℤ𝑀) → (∀𝑦 ∈ (𝑀...(𝑥 − 1))𝜓𝜑))
2120rgen 2519 . . . . . . 7 𝑥 ∈ (ℤ𝑀)(∀𝑦 ∈ (𝑀...(𝑥 − 1))𝜓𝜑)
22 nfv 1516 . . . . . . . . 9 𝑥𝑦 ∈ (𝑀...(𝑀 − 1))𝜓
23 nfsbc1v 2969 . . . . . . . . 9 𝑥[𝑀 / 𝑥]𝜑
2422, 23nfim 1560 . . . . . . . 8 𝑥(∀𝑦 ∈ (𝑀...(𝑀 − 1))𝜓[𝑀 / 𝑥]𝜑)
25 oveq1 5849 . . . . . . . . . . 11 (𝑥 = 𝑀 → (𝑥 − 1) = (𝑀 − 1))
2625oveq2d 5858 . . . . . . . . . 10 (𝑥 = 𝑀 → (𝑀...(𝑥 − 1)) = (𝑀...(𝑀 − 1)))
2726raleqdv 2667 . . . . . . . . 9 (𝑥 = 𝑀 → (∀𝑦 ∈ (𝑀...(𝑥 − 1))𝜓 ↔ ∀𝑦 ∈ (𝑀...(𝑀 − 1))𝜓))
28 sbceq1a 2960 . . . . . . . . 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 2824 . . . . . . 7 (𝑀 ∈ (ℤ𝑀) → (∀𝑥 ∈ (ℤ𝑀)(∀𝑦 ∈ (𝑀...(𝑥 − 1))𝜓𝜑) → (∀𝑦 ∈ (𝑀...(𝑀 − 1))𝜓[𝑀 / 𝑥]𝜑)))
3119, 21, 30mpisyl 1434 . . . . . 6 (𝑀 ∈ ℤ → (∀𝑦 ∈ (𝑀...(𝑀 − 1))𝜓[𝑀 / 𝑥]𝜑))
3218, 31mpd 13 . . . . 5 (𝑀 ∈ ℤ → [𝑀 / 𝑥]𝜑)
33 ralsns 3614 . . . . 5 (𝑀 ∈ ℤ → (∀𝑥 ∈ {𝑀}𝜑[𝑀 / 𝑥]𝜑))
3432, 33mpbird 166 . . . 4 (𝑀 ∈ ℤ → ∀𝑥 ∈ {𝑀}𝜑)
35 fzsn 10001 . . . . 5 (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀})
3635raleqdv 2667 . . . 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 2692 . . . . . . . . 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 9475 . . . . . . . . . . . . . 14 (𝑘 ∈ (ℤ𝑀) → 𝑘 ∈ ℤ)
4342adantr 274 . . . . . . . . . . . . 13 ((𝑘 ∈ (ℤ𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → 𝑘 ∈ ℤ)
4443zcnd 9314 . . . . . . . . . . . 12 ((𝑘 ∈ (ℤ𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → 𝑘 ∈ ℂ)
45 1cnd 7915 . . . . . . . . . . . 12 ((𝑘 ∈ (ℤ𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → 1 ∈ ℂ)
4644, 45pncand 8210 . . . . . . . . . . 11 ((𝑘 ∈ (ℤ𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → ((𝑘 + 1) − 1) = 𝑘)
4746oveq2d 5858 . . . . . . . . . 10 ((𝑘 ∈ (ℤ𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → (𝑀...((𝑘 + 1) − 1)) = (𝑀...𝑘))
4847raleqdv 2667 . . . . . . . . 9 ((𝑘 ∈ (ℤ𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → (∀𝑦 ∈ (𝑀...((𝑘 + 1) − 1))𝜓 ↔ ∀𝑦 ∈ (𝑀...𝑘)𝜓))
49 peano2uz 9521 . . . . . . . . . . 11 (𝑘 ∈ (ℤ𝑀) → (𝑘 + 1) ∈ (ℤ𝑀))
5049adantr 274 . . . . . . . . . 10 ((𝑘 ∈ (ℤ𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → (𝑘 + 1) ∈ (ℤ𝑀))
51 nfv 1516 . . . . . . . . . . . 12 𝑥𝑦 ∈ (𝑀...((𝑘 + 1) − 1))𝜓
52 nfsbc1v 2969 . . . . . . . . . . . 12 𝑥[(𝑘 + 1) / 𝑥]𝜑
5351, 52nfim 1560 . . . . . . . . . . 11 𝑥(∀𝑦 ∈ (𝑀...((𝑘 + 1) − 1))𝜓[(𝑘 + 1) / 𝑥]𝜑)
54 oveq1 5849 . . . . . . . . . . . . . 14 (𝑥 = (𝑘 + 1) → (𝑥 − 1) = ((𝑘 + 1) − 1))
5554oveq2d 5858 . . . . . . . . . . . . 13 (𝑥 = (𝑘 + 1) → (𝑀...(𝑥 − 1)) = (𝑀...((𝑘 + 1) − 1)))
5655raleqdv 2667 . . . . . . . . . . . 12 (𝑥 = (𝑘 + 1) → (∀𝑦 ∈ (𝑀...(𝑥 − 1))𝜓 ↔ ∀𝑦 ∈ (𝑀...((𝑘 + 1) − 1))𝜓))
57 sbceq1a 2960 . . . . . . . . . . . 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 2824 . . . . . . . . . 10 ((𝑘 + 1) ∈ (ℤ𝑀) → (∀𝑥 ∈ (ℤ𝑀)(∀𝑦 ∈ (𝑀...(𝑥 − 1))𝜓𝜑) → (∀𝑦 ∈ (𝑀...((𝑘 + 1) − 1))𝜓[(𝑘 + 1) / 𝑥]𝜑)))
6050, 21, 59mpisyl 1434 . . . . . . . . 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 9316 . . . . . . . . 9 (𝑘 ∈ (ℤ𝑀) → (𝑘 + 1) ∈ ℤ)
6463adantr 274 . . . . . . . 8 ((𝑘 ∈ (ℤ𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → (𝑘 + 1) ∈ ℤ)
65 ralsns 3614 . . . . . . . 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 3304 . . . . . 6 ((∀𝑥 ∈ (𝑀...𝑘)𝜑 ∧ ∀𝑥 ∈ {(𝑘 + 1)}𝜑) → ∀𝑥 ∈ ((𝑀...𝑘) ∪ {(𝑘 + 1)})𝜑)
6938, 67, 68syl2anc 409 . . . . 5 ((𝑘 ∈ (ℤ𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → ∀𝑥 ∈ ((𝑀...𝑘) ∪ {(𝑘 + 1)})𝜑)
70 fzsuc 10004 . . . . . . 7 (𝑘 ∈ (ℤ𝑀) → (𝑀...(𝑘 + 1)) = ((𝑀...𝑘) ∪ {(𝑘 + 1)}))
7170raleqdv 2667 . . . . . 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 9526 . 2 (𝑁 ∈ (ℤ𝑀) → ∀𝑥 ∈ (𝑀...𝑁)𝜑)
76 eluzfz2 9967 . 2 (𝑁 ∈ (ℤ𝑀) → 𝑁 ∈ (𝑀...𝑁))
771, 75, 76rspcdva 2835 1 (𝑁 ∈ (ℤ𝑀) → 𝜒)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1343  wcel 2136  wral 2444  [wsbc 2951  cun 3114  c0 3409  {csn 3576   class class class wbr 3982  cfv 5188  (class class class)co 5842  1c1 7754   + caddc 7756   < clt 7933  cmin 8069  cz 9191  cuz 9466  ...cfz 9944
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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-addcom 7853  ax-addass 7855  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-0id 7861  ax-rnegex 7862  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869
This theorem depends on definitions:  df-bi 116  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-inn 8858  df-n0 9115  df-z 9192  df-uz 9467  df-fz 9945
This theorem is referenced by:  nnsinds  10378  nn0sinds  10379
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