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Theorem uzind4i 9399
 Description: Induction on the upper integers that start at 𝑀. The first four give us the substitution instances we need, and the last two are the basis and the induction step. This is a stronger version of uzind4 9395 assuming that 𝜓 holds unconditionally. Notice that 𝑁 ∈ (ℤ≥‘𝑀) implies that the lower bound 𝑀 is an integer (𝑀 ∈ ℤ, see eluzel2 9343). (Contributed by NM, 4-Sep-2005.) (Revised by AV, 13-Jul-2022.)
Hypotheses
Ref Expression
uzind4i.1 (𝑗 = 𝑀 → (𝜑𝜓))
uzind4i.2 (𝑗 = 𝑘 → (𝜑𝜒))
uzind4i.3 (𝑗 = (𝑘 + 1) → (𝜑𝜃))
uzind4i.4 (𝑗 = 𝑁 → (𝜑𝜏))
uzind4i.5 𝜓
uzind4i.6 (𝑘 ∈ (ℤ𝑀) → (𝜒𝜃))
Assertion
Ref Expression
uzind4i (𝑁 ∈ (ℤ𝑀) → 𝜏)
Distinct variable groups:   𝑗,𝑁   𝜓,𝑗   𝜒,𝑗   𝜃,𝑗   𝜏,𝑗   𝜑,𝑘   𝑗,𝑘,𝑀
Allowed substitution hints:   𝜑(𝑗)   𝜓(𝑘)   𝜒(𝑘)   𝜃(𝑘)   𝜏(𝑘)   𝑁(𝑘)

Proof of Theorem uzind4i
StepHypRef Expression
1 uzind4i.1 . 2 (𝑗 = 𝑀 → (𝜑𝜓))
2 uzind4i.2 . 2 (𝑗 = 𝑘 → (𝜑𝜒))
3 uzind4i.3 . 2 (𝑗 = (𝑘 + 1) → (𝜑𝜃))
4 uzind4i.4 . 2 (𝑗 = 𝑁 → (𝜑𝜏))
5 uzind4i.5 . . 3 𝜓
65a1i 9 . 2 (𝑀 ∈ ℤ → 𝜓)
7 uzind4i.6 . 2 (𝑘 ∈ (ℤ𝑀) → (𝜒𝜃))
81, 2, 3, 4, 6, 7uzind4 9395 1 (𝑁 ∈ (ℤ𝑀) → 𝜏)
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 104   = wceq 1331   ∈ wcel 1480  ‘cfv 5123  (class class class)co 5774  1c1 7633   + caddc 7635  ℤcz 9066  ℤ≥cuz 9338 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-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-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 This theorem is referenced by:  rebtwn2zlemshrink  10043
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