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Theorem uzsinds 10428
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 5877 . . . 4 (𝑤 = 𝑀 → (𝑀...𝑤) = (𝑀...𝑀))
32raleqdv 2678 . . 3 (𝑤 = 𝑀 → (∀𝑥 ∈ (𝑀...𝑤)𝜑 ↔ ∀𝑥 ∈ (𝑀...𝑀)𝜑))
4 oveq2 5877 . . . 4 (𝑤 = 𝑘 → (𝑀...𝑤) = (𝑀...𝑘))
54raleqdv 2678 . . 3 (𝑤 = 𝑘 → (∀𝑥 ∈ (𝑀...𝑤)𝜑 ↔ ∀𝑥 ∈ (𝑀...𝑘)𝜑))
6 oveq2 5877 . . . 4 (𝑤 = (𝑘 + 1) → (𝑀...𝑤) = (𝑀...(𝑘 + 1)))
76raleqdv 2678 . . 3 (𝑤 = (𝑘 + 1) → (∀𝑥 ∈ (𝑀...𝑤)𝜑 ↔ ∀𝑥 ∈ (𝑀...(𝑘 + 1))𝜑))
8 oveq2 5877 . . . 4 (𝑤 = 𝑁 → (𝑀...𝑤) = (𝑀...𝑁))
98raleqdv 2678 . . 3 (𝑤 = 𝑁 → (∀𝑥 ∈ (𝑀...𝑤)𝜑 ↔ ∀𝑥 ∈ (𝑀...𝑁)𝜑))
10 ral0 3524 . . . . . . 7 𝑦 ∈ ∅ 𝜓
11 zre 9246 . . . . . . . . . 10 (𝑀 ∈ ℤ → 𝑀 ∈ ℝ)
1211ltm1d 8878 . . . . . . . . 9 (𝑀 ∈ ℤ → (𝑀 − 1) < 𝑀)
13 peano2zm 9280 . . . . . . . . . 10 (𝑀 ∈ ℤ → (𝑀 − 1) ∈ ℤ)
14 fzn 10028 . . . . . . . . . 10 ((𝑀 ∈ ℤ ∧ (𝑀 − 1) ∈ ℤ) → ((𝑀 − 1) < 𝑀 ↔ (𝑀...(𝑀 − 1)) = ∅))
1513, 14mpdan 421 . . . . . . . . 9 (𝑀 ∈ ℤ → ((𝑀 − 1) < 𝑀 ↔ (𝑀...(𝑀 − 1)) = ∅))
1612, 15mpbid 147 . . . . . . . 8 (𝑀 ∈ ℤ → (𝑀...(𝑀 − 1)) = ∅)
1716raleqdv 2678 . . . . . . 7 (𝑀 ∈ ℤ → (∀𝑦 ∈ (𝑀...(𝑀 − 1))𝜓 ↔ ∀𝑦 ∈ ∅ 𝜓))
1810, 17mpbiri 168 . . . . . 6 (𝑀 ∈ ℤ → ∀𝑦 ∈ (𝑀...(𝑀 − 1))𝜓)
19 uzid 9531 . . . . . . 7 (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ𝑀))
20 uzsinds.3 . . . . . . . 8 (𝑥 ∈ (ℤ𝑀) → (∀𝑦 ∈ (𝑀...(𝑥 − 1))𝜓𝜑))
2120rgen 2530 . . . . . . 7 𝑥 ∈ (ℤ𝑀)(∀𝑦 ∈ (𝑀...(𝑥 − 1))𝜓𝜑)
22 nfv 1528 . . . . . . . . 9 𝑥𝑦 ∈ (𝑀...(𝑀 − 1))𝜓
23 nfsbc1v 2981 . . . . . . . . 9 𝑥[𝑀 / 𝑥]𝜑
2422, 23nfim 1572 . . . . . . . 8 𝑥(∀𝑦 ∈ (𝑀...(𝑀 − 1))𝜓[𝑀 / 𝑥]𝜑)
25 oveq1 5876 . . . . . . . . . . 11 (𝑥 = 𝑀 → (𝑥 − 1) = (𝑀 − 1))
2625oveq2d 5885 . . . . . . . . . 10 (𝑥 = 𝑀 → (𝑀...(𝑥 − 1)) = (𝑀...(𝑀 − 1)))
2726raleqdv 2678 . . . . . . . . 9 (𝑥 = 𝑀 → (∀𝑦 ∈ (𝑀...(𝑥 − 1))𝜓 ↔ ∀𝑦 ∈ (𝑀...(𝑀 − 1))𝜓))
28 sbceq1a 2972 . . . . . . . . 9 (𝑥 = 𝑀 → (𝜑[𝑀 / 𝑥]𝜑))
2927, 28imbi12d 234 . . . . . . . 8 (𝑥 = 𝑀 → ((∀𝑦 ∈ (𝑀...(𝑥 − 1))𝜓𝜑) ↔ (∀𝑦 ∈ (𝑀...(𝑀 − 1))𝜓[𝑀 / 𝑥]𝜑)))
3024, 29rspc 2835 . . . . . . 7 (𝑀 ∈ (ℤ𝑀) → (∀𝑥 ∈ (ℤ𝑀)(∀𝑦 ∈ (𝑀...(𝑥 − 1))𝜓𝜑) → (∀𝑦 ∈ (𝑀...(𝑀 − 1))𝜓[𝑀 / 𝑥]𝜑)))
3119, 21, 30mpisyl 1446 . . . . . 6 (𝑀 ∈ ℤ → (∀𝑦 ∈ (𝑀...(𝑀 − 1))𝜓[𝑀 / 𝑥]𝜑))
3218, 31mpd 13 . . . . 5 (𝑀 ∈ ℤ → [𝑀 / 𝑥]𝜑)
33 ralsns 3629 . . . . 5 (𝑀 ∈ ℤ → (∀𝑥 ∈ {𝑀}𝜑[𝑀 / 𝑥]𝜑))
3432, 33mpbird 167 . . . 4 (𝑀 ∈ ℤ → ∀𝑥 ∈ {𝑀}𝜑)
35 fzsn 10052 . . . . 5 (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀})
3635raleqdv 2678 . . . 4 (𝑀 ∈ ℤ → (∀𝑥 ∈ (𝑀...𝑀)𝜑 ↔ ∀𝑥 ∈ {𝑀}𝜑))
3734, 36mpbird 167 . . 3 (𝑀 ∈ ℤ → ∀𝑥 ∈ (𝑀...𝑀)𝜑)
38 simpr 110 . . . . . 6 ((𝑘 ∈ (ℤ𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → ∀𝑥 ∈ (𝑀...𝑘)𝜑)
39 uzsinds.1 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝜑𝜓))
4039cbvralv 2703 . . . . . . . . 9 (∀𝑥 ∈ (𝑀...𝑘)𝜑 ↔ ∀𝑦 ∈ (𝑀...𝑘)𝜓)
4138, 40sylib 122 . . . . . . . 8 ((𝑘 ∈ (ℤ𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → ∀𝑦 ∈ (𝑀...𝑘)𝜓)
42 eluzelz 9526 . . . . . . . . . . . . . 14 (𝑘 ∈ (ℤ𝑀) → 𝑘 ∈ ℤ)
4342adantr 276 . . . . . . . . . . . . 13 ((𝑘 ∈ (ℤ𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → 𝑘 ∈ ℤ)
4443zcnd 9365 . . . . . . . . . . . 12 ((𝑘 ∈ (ℤ𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → 𝑘 ∈ ℂ)
45 1cnd 7964 . . . . . . . . . . . 12 ((𝑘 ∈ (ℤ𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → 1 ∈ ℂ)
4644, 45pncand 8259 . . . . . . . . . . 11 ((𝑘 ∈ (ℤ𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → ((𝑘 + 1) − 1) = 𝑘)
4746oveq2d 5885 . . . . . . . . . 10 ((𝑘 ∈ (ℤ𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → (𝑀...((𝑘 + 1) − 1)) = (𝑀...𝑘))
4847raleqdv 2678 . . . . . . . . 9 ((𝑘 ∈ (ℤ𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → (∀𝑦 ∈ (𝑀...((𝑘 + 1) − 1))𝜓 ↔ ∀𝑦 ∈ (𝑀...𝑘)𝜓))
49 peano2uz 9572 . . . . . . . . . . 11 (𝑘 ∈ (ℤ𝑀) → (𝑘 + 1) ∈ (ℤ𝑀))
5049adantr 276 . . . . . . . . . 10 ((𝑘 ∈ (ℤ𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → (𝑘 + 1) ∈ (ℤ𝑀))
51 nfv 1528 . . . . . . . . . . . 12 𝑥𝑦 ∈ (𝑀...((𝑘 + 1) − 1))𝜓
52 nfsbc1v 2981 . . . . . . . . . . . 12 𝑥[(𝑘 + 1) / 𝑥]𝜑
5351, 52nfim 1572 . . . . . . . . . . 11 𝑥(∀𝑦 ∈ (𝑀...((𝑘 + 1) − 1))𝜓[(𝑘 + 1) / 𝑥]𝜑)
54 oveq1 5876 . . . . . . . . . . . . . 14 (𝑥 = (𝑘 + 1) → (𝑥 − 1) = ((𝑘 + 1) − 1))
5554oveq2d 5885 . . . . . . . . . . . . 13 (𝑥 = (𝑘 + 1) → (𝑀...(𝑥 − 1)) = (𝑀...((𝑘 + 1) − 1)))
5655raleqdv 2678 . . . . . . . . . . . 12 (𝑥 = (𝑘 + 1) → (∀𝑦 ∈ (𝑀...(𝑥 − 1))𝜓 ↔ ∀𝑦 ∈ (𝑀...((𝑘 + 1) − 1))𝜓))
57 sbceq1a 2972 . . . . . . . . . . . 12 (𝑥 = (𝑘 + 1) → (𝜑[(𝑘 + 1) / 𝑥]𝜑))
5856, 57imbi12d 234 . . . . . . . . . . 11 (𝑥 = (𝑘 + 1) → ((∀𝑦 ∈ (𝑀...(𝑥 − 1))𝜓𝜑) ↔ (∀𝑦 ∈ (𝑀...((𝑘 + 1) − 1))𝜓[(𝑘 + 1) / 𝑥]𝜑)))
5953, 58rspc 2835 . . . . . . . . . 10 ((𝑘 + 1) ∈ (ℤ𝑀) → (∀𝑥 ∈ (ℤ𝑀)(∀𝑦 ∈ (𝑀...(𝑥 − 1))𝜓𝜑) → (∀𝑦 ∈ (𝑀...((𝑘 + 1) − 1))𝜓[(𝑘 + 1) / 𝑥]𝜑)))
6050, 21, 59mpisyl 1446 . . . . . . . . 9 ((𝑘 ∈ (ℤ𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → (∀𝑦 ∈ (𝑀...((𝑘 + 1) − 1))𝜓[(𝑘 + 1) / 𝑥]𝜑))
6148, 60sylbird 170 . . . . . . . 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 9367 . . . . . . . . 9 (𝑘 ∈ (ℤ𝑀) → (𝑘 + 1) ∈ ℤ)
6463adantr 276 . . . . . . . 8 ((𝑘 ∈ (ℤ𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → (𝑘 + 1) ∈ ℤ)
65 ralsns 3629 . . . . . . . 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 167 . . . . . 6 ((𝑘 ∈ (ℤ𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → ∀𝑥 ∈ {(𝑘 + 1)}𝜑)
68 ralun 3317 . . . . . 6 ((∀𝑥 ∈ (𝑀...𝑘)𝜑 ∧ ∀𝑥 ∈ {(𝑘 + 1)}𝜑) → ∀𝑥 ∈ ((𝑀...𝑘) ∪ {(𝑘 + 1)})𝜑)
6938, 67, 68syl2anc 411 . . . . 5 ((𝑘 ∈ (ℤ𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → ∀𝑥 ∈ ((𝑀...𝑘) ∪ {(𝑘 + 1)})𝜑)
70 fzsuc 10055 . . . . . . 7 (𝑘 ∈ (ℤ𝑀) → (𝑀...(𝑘 + 1)) = ((𝑀...𝑘) ∪ {(𝑘 + 1)}))
7170raleqdv 2678 . . . . . 6 (𝑘 ∈ (ℤ𝑀) → (∀𝑥 ∈ (𝑀...(𝑘 + 1))𝜑 ↔ ∀𝑥 ∈ ((𝑀...𝑘) ∪ {(𝑘 + 1)})𝜑))
7271adantr 276 . . . . 5 ((𝑘 ∈ (ℤ𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → (∀𝑥 ∈ (𝑀...(𝑘 + 1))𝜑 ↔ ∀𝑥 ∈ ((𝑀...𝑘) ∪ {(𝑘 + 1)})𝜑))
7369, 72mpbird 167 . . . 4 ((𝑘 ∈ (ℤ𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → ∀𝑥 ∈ (𝑀...(𝑘 + 1))𝜑)
7473ex 115 . . 3 (𝑘 ∈ (ℤ𝑀) → (∀𝑥 ∈ (𝑀...𝑘)𝜑 → ∀𝑥 ∈ (𝑀...(𝑘 + 1))𝜑))
753, 5, 7, 9, 37, 74uzind4 9577 . 2 (𝑁 ∈ (ℤ𝑀) → ∀𝑥 ∈ (𝑀...𝑁)𝜑)
76 eluzfz2 10018 . 2 (𝑁 ∈ (ℤ𝑀) → 𝑁 ∈ (𝑀...𝑁))
771, 75, 76rspcdva 2846 1 (𝑁 ∈ (ℤ𝑀) → 𝜒)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1353  wcel 2148  wral 2455  [wsbc 2962  cun 3127  c0 3422  {csn 3591   class class class wbr 4000  cfv 5212  (class class class)co 5869  1c1 7803   + caddc 7805   < clt 7982  cmin 8118  cz 9242  cuz 9517  ...cfz 9995
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-addcom 7902  ax-addass 7904  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-0id 7910  ax-rnegex 7911  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918
This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-fv 5220  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-inn 8909  df-n0 9166  df-z 9243  df-uz 9518  df-fz 9996
This theorem is referenced by:  nnsinds  10429  nn0sinds  10430
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