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Theorem uzsinds 9570
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 5571 . . . 4 (𝑤 = 𝑀 → (𝑀...𝑤) = (𝑀...𝑀))
32raleqdv 2560 . . 3 (𝑤 = 𝑀 → (∀𝑥 ∈ (𝑀...𝑤)𝜑 ↔ ∀𝑥 ∈ (𝑀...𝑀)𝜑))
4 oveq2 5571 . . . 4 (𝑤 = 𝑘 → (𝑀...𝑤) = (𝑀...𝑘))
54raleqdv 2560 . . 3 (𝑤 = 𝑘 → (∀𝑥 ∈ (𝑀...𝑤)𝜑 ↔ ∀𝑥 ∈ (𝑀...𝑘)𝜑))
6 oveq2 5571 . . . 4 (𝑤 = (𝑘 + 1) → (𝑀...𝑤) = (𝑀...(𝑘 + 1)))
76raleqdv 2560 . . 3 (𝑤 = (𝑘 + 1) → (∀𝑥 ∈ (𝑀...𝑤)𝜑 ↔ ∀𝑥 ∈ (𝑀...(𝑘 + 1))𝜑))
8 oveq2 5571 . . . 4 (𝑤 = 𝑁 → (𝑀...𝑤) = (𝑀...𝑁))
98raleqdv 2560 . . 3 (𝑤 = 𝑁 → (∀𝑥 ∈ (𝑀...𝑤)𝜑 ↔ ∀𝑥 ∈ (𝑀...𝑁)𝜑))
10 ral0 3359 . . . . . . 7 𝑦 ∈ ∅ 𝜓
11 zre 8488 . . . . . . . . . 10 (𝑀 ∈ ℤ → 𝑀 ∈ ℝ)
1211ltm1d 8129 . . . . . . . . 9 (𝑀 ∈ ℤ → (𝑀 − 1) < 𝑀)
13 peano2zm 8522 . . . . . . . . . 10 (𝑀 ∈ ℤ → (𝑀 − 1) ∈ ℤ)
14 fzn 9189 . . . . . . . . . 10 ((𝑀 ∈ ℤ ∧ (𝑀 − 1) ∈ ℤ) → ((𝑀 − 1) < 𝑀 ↔ (𝑀...(𝑀 − 1)) = ∅))
1513, 14mpdan 412 . . . . . . . . 9 (𝑀 ∈ ℤ → ((𝑀 − 1) < 𝑀 ↔ (𝑀...(𝑀 − 1)) = ∅))
1612, 15mpbid 145 . . . . . . . 8 (𝑀 ∈ ℤ → (𝑀...(𝑀 − 1)) = ∅)
1716raleqdv 2560 . . . . . . 7 (𝑀 ∈ ℤ → (∀𝑦 ∈ (𝑀...(𝑀 − 1))𝜓 ↔ ∀𝑦 ∈ ∅ 𝜓))
1810, 17mpbiri 166 . . . . . 6 (𝑀 ∈ ℤ → ∀𝑦 ∈ (𝑀...(𝑀 − 1))𝜓)
19 uzid 8766 . . . . . . 7 (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ𝑀))
20 uzsinds.3 . . . . . . . 8 (𝑥 ∈ (ℤ𝑀) → (∀𝑦 ∈ (𝑀...(𝑥 − 1))𝜓𝜑))
2120rgen 2421 . . . . . . 7 𝑥 ∈ (ℤ𝑀)(∀𝑦 ∈ (𝑀...(𝑥 − 1))𝜓𝜑)
22 nfv 1462 . . . . . . . . 9 𝑥𝑦 ∈ (𝑀...(𝑀 − 1))𝜓
23 nfsbc1v 2842 . . . . . . . . 9 𝑥[𝑀 / 𝑥]𝜑
2422, 23nfim 1505 . . . . . . . 8 𝑥(∀𝑦 ∈ (𝑀...(𝑀 − 1))𝜓[𝑀 / 𝑥]𝜑)
25 oveq1 5570 . . . . . . . . . . 11 (𝑥 = 𝑀 → (𝑥 − 1) = (𝑀 − 1))
2625oveq2d 5579 . . . . . . . . . 10 (𝑥 = 𝑀 → (𝑀...(𝑥 − 1)) = (𝑀...(𝑀 − 1)))
2726raleqdv 2560 . . . . . . . . 9 (𝑥 = 𝑀 → (∀𝑦 ∈ (𝑀...(𝑥 − 1))𝜓 ↔ ∀𝑦 ∈ (𝑀...(𝑀 − 1))𝜓))
28 sbceq1a 2833 . . . . . . . . 9 (𝑥 = 𝑀 → (𝜑[𝑀 / 𝑥]𝜑))
2927, 28imbi12d 232 . . . . . . . 8 (𝑥 = 𝑀 → ((∀𝑦 ∈ (𝑀...(𝑥 − 1))𝜓𝜑) ↔ (∀𝑦 ∈ (𝑀...(𝑀 − 1))𝜓[𝑀 / 𝑥]𝜑)))
3024, 29rspc 2704 . . . . . . 7 (𝑀 ∈ (ℤ𝑀) → (∀𝑥 ∈ (ℤ𝑀)(∀𝑦 ∈ (𝑀...(𝑥 − 1))𝜓𝜑) → (∀𝑦 ∈ (𝑀...(𝑀 − 1))𝜓[𝑀 / 𝑥]𝜑)))
3119, 21, 30mpisyl 1376 . . . . . 6 (𝑀 ∈ ℤ → (∀𝑦 ∈ (𝑀...(𝑀 − 1))𝜓[𝑀 / 𝑥]𝜑))
3218, 31mpd 13 . . . . 5 (𝑀 ∈ ℤ → [𝑀 / 𝑥]𝜑)
33 ralsns 3449 . . . . 5 (𝑀 ∈ ℤ → (∀𝑥 ∈ {𝑀}𝜑[𝑀 / 𝑥]𝜑))
3432, 33mpbird 165 . . . 4 (𝑀 ∈ ℤ → ∀𝑥 ∈ {𝑀}𝜑)
35 fzsn 9212 . . . . 5 (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀})
3635raleqdv 2560 . . . 4 (𝑀 ∈ ℤ → (∀𝑥 ∈ (𝑀...𝑀)𝜑 ↔ ∀𝑥 ∈ {𝑀}𝜑))
3734, 36mpbird 165 . . 3 (𝑀 ∈ ℤ → ∀𝑥 ∈ (𝑀...𝑀)𝜑)
38 simpr 108 . . . . . 6 ((𝑘 ∈ (ℤ𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → ∀𝑥 ∈ (𝑀...𝑘)𝜑)
39 uzsinds.1 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝜑𝜓))
4039cbvralv 2582 . . . . . . . . 9 (∀𝑥 ∈ (𝑀...𝑘)𝜑 ↔ ∀𝑦 ∈ (𝑀...𝑘)𝜓)
4138, 40sylib 120 . . . . . . . 8 ((𝑘 ∈ (ℤ𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → ∀𝑦 ∈ (𝑀...𝑘)𝜓)
42 eluzelz 8761 . . . . . . . . . . . . . 14 (𝑘 ∈ (ℤ𝑀) → 𝑘 ∈ ℤ)
4342adantr 270 . . . . . . . . . . . . 13 ((𝑘 ∈ (ℤ𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → 𝑘 ∈ ℤ)
4443zcnd 8603 . . . . . . . . . . . 12 ((𝑘 ∈ (ℤ𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → 𝑘 ∈ ℂ)
45 1cnd 7249 . . . . . . . . . . . 12 ((𝑘 ∈ (ℤ𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → 1 ∈ ℂ)
4644, 45pncand 7539 . . . . . . . . . . 11 ((𝑘 ∈ (ℤ𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → ((𝑘 + 1) − 1) = 𝑘)
4746oveq2d 5579 . . . . . . . . . 10 ((𝑘 ∈ (ℤ𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → (𝑀...((𝑘 + 1) − 1)) = (𝑀...𝑘))
4847raleqdv 2560 . . . . . . . . 9 ((𝑘 ∈ (ℤ𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → (∀𝑦 ∈ (𝑀...((𝑘 + 1) − 1))𝜓 ↔ ∀𝑦 ∈ (𝑀...𝑘)𝜓))
49 peano2uz 8804 . . . . . . . . . . 11 (𝑘 ∈ (ℤ𝑀) → (𝑘 + 1) ∈ (ℤ𝑀))
5049adantr 270 . . . . . . . . . 10 ((𝑘 ∈ (ℤ𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → (𝑘 + 1) ∈ (ℤ𝑀))
51 nfv 1462 . . . . . . . . . . . 12 𝑥𝑦 ∈ (𝑀...((𝑘 + 1) − 1))𝜓
52 nfsbc1v 2842 . . . . . . . . . . . 12 𝑥[(𝑘 + 1) / 𝑥]𝜑
5351, 52nfim 1505 . . . . . . . . . . 11 𝑥(∀𝑦 ∈ (𝑀...((𝑘 + 1) − 1))𝜓[(𝑘 + 1) / 𝑥]𝜑)
54 oveq1 5570 . . . . . . . . . . . . . 14 (𝑥 = (𝑘 + 1) → (𝑥 − 1) = ((𝑘 + 1) − 1))
5554oveq2d 5579 . . . . . . . . . . . . 13 (𝑥 = (𝑘 + 1) → (𝑀...(𝑥 − 1)) = (𝑀...((𝑘 + 1) − 1)))
5655raleqdv 2560 . . . . . . . . . . . 12 (𝑥 = (𝑘 + 1) → (∀𝑦 ∈ (𝑀...(𝑥 − 1))𝜓 ↔ ∀𝑦 ∈ (𝑀...((𝑘 + 1) − 1))𝜓))
57 sbceq1a 2833 . . . . . . . . . . . 12 (𝑥 = (𝑘 + 1) → (𝜑[(𝑘 + 1) / 𝑥]𝜑))
5856, 57imbi12d 232 . . . . . . . . . . 11 (𝑥 = (𝑘 + 1) → ((∀𝑦 ∈ (𝑀...(𝑥 − 1))𝜓𝜑) ↔ (∀𝑦 ∈ (𝑀...((𝑘 + 1) − 1))𝜓[(𝑘 + 1) / 𝑥]𝜑)))
5953, 58rspc 2704 . . . . . . . . . 10 ((𝑘 + 1) ∈ (ℤ𝑀) → (∀𝑥 ∈ (ℤ𝑀)(∀𝑦 ∈ (𝑀...(𝑥 − 1))𝜓𝜑) → (∀𝑦 ∈ (𝑀...((𝑘 + 1) − 1))𝜓[(𝑘 + 1) / 𝑥]𝜑)))
6050, 21, 59mpisyl 1376 . . . . . . . . 9 ((𝑘 ∈ (ℤ𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → (∀𝑦 ∈ (𝑀...((𝑘 + 1) − 1))𝜓[(𝑘 + 1) / 𝑥]𝜑))
6148, 60sylbird 168 . . . . . . . 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 8605 . . . . . . . . 9 (𝑘 ∈ (ℤ𝑀) → (𝑘 + 1) ∈ ℤ)
6463adantr 270 . . . . . . . 8 ((𝑘 ∈ (ℤ𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → (𝑘 + 1) ∈ ℤ)
65 ralsns 3449 . . . . . . . 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 165 . . . . . 6 ((𝑘 ∈ (ℤ𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → ∀𝑥 ∈ {(𝑘 + 1)}𝜑)
68 ralun 3164 . . . . . 6 ((∀𝑥 ∈ (𝑀...𝑘)𝜑 ∧ ∀𝑥 ∈ {(𝑘 + 1)}𝜑) → ∀𝑥 ∈ ((𝑀...𝑘) ∪ {(𝑘 + 1)})𝜑)
6938, 67, 68syl2anc 403 . . . . 5 ((𝑘 ∈ (ℤ𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → ∀𝑥 ∈ ((𝑀...𝑘) ∪ {(𝑘 + 1)})𝜑)
70 fzsuc 9214 . . . . . . 7 (𝑘 ∈ (ℤ𝑀) → (𝑀...(𝑘 + 1)) = ((𝑀...𝑘) ∪ {(𝑘 + 1)}))
7170raleqdv 2560 . . . . . 6 (𝑘 ∈ (ℤ𝑀) → (∀𝑥 ∈ (𝑀...(𝑘 + 1))𝜑 ↔ ∀𝑥 ∈ ((𝑀...𝑘) ∪ {(𝑘 + 1)})𝜑))
7271adantr 270 . . . . 5 ((𝑘 ∈ (ℤ𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → (∀𝑥 ∈ (𝑀...(𝑘 + 1))𝜑 ↔ ∀𝑥 ∈ ((𝑀...𝑘) ∪ {(𝑘 + 1)})𝜑))
7369, 72mpbird 165 . . . 4 ((𝑘 ∈ (ℤ𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → ∀𝑥 ∈ (𝑀...(𝑘 + 1))𝜑)
7473ex 113 . . 3 (𝑘 ∈ (ℤ𝑀) → (∀𝑥 ∈ (𝑀...𝑘)𝜑 → ∀𝑥 ∈ (𝑀...(𝑘 + 1))𝜑))
753, 5, 7, 9, 37, 74uzind4 8809 . 2 (𝑁 ∈ (ℤ𝑀) → ∀𝑥 ∈ (𝑀...𝑁)𝜑)
76 eluzfz2 9179 . 2 (𝑁 ∈ (ℤ𝑀) → 𝑁 ∈ (𝑀...𝑁))
771, 75, 76rspcdva 2715 1 (𝑁 ∈ (ℤ𝑀) → 𝜒)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   = wceq 1285  wcel 1434  wral 2353  [wsbc 2824  cun 2980  c0 3267  {csn 3416   class class class wbr 3805  cfv 4952  (class class class)co 5563  1c1 7096   + caddc 7098   < clt 7267  cmin 7398  cz 8484  cuz 8752  ...cfz 9157
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-cnex 7181  ax-resscn 7182  ax-1cn 7183  ax-1re 7184  ax-icn 7185  ax-addcl 7186  ax-addrcl 7187  ax-mulcl 7188  ax-addcom 7190  ax-addass 7192  ax-distr 7194  ax-i2m1 7195  ax-0lt1 7196  ax-0id 7198  ax-rnegex 7199  ax-cnre 7201  ax-pre-ltirr 7202  ax-pre-ltwlin 7203  ax-pre-lttrn 7204  ax-pre-apti 7205  ax-pre-ltadd 7206
This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2612  df-sbc 2825  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-int 3657  df-br 3806  df-opab 3860  df-mpt 3861  df-id 4076  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-fv 4960  df-riota 5519  df-ov 5566  df-oprab 5567  df-mpt2 5568  df-pnf 7269  df-mnf 7270  df-xr 7271  df-ltxr 7272  df-le 7273  df-sub 7400  df-neg 7401  df-inn 8159  df-n0 8408  df-z 8485  df-uz 8753  df-fz 9158
This theorem is referenced by:  nnsinds  9571  nn0sinds  9572
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