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Theorem uzind4s2 9379
 Description: Induction on the upper set of integers that starts at an integer 𝑀, using explicit substitution. The hypotheses are the basis and the induction step. Use this instead of uzind4s 9378 when 𝑗 and 𝑘 must be distinct in [(𝑘 + 1) / 𝑗]𝜑. (Contributed by NM, 16-Nov-2005.)
Hypotheses
Ref Expression
uzind4s2.1 (𝑀 ∈ ℤ → [𝑀 / 𝑗]𝜑)
uzind4s2.2 (𝑘 ∈ (ℤ𝑀) → ([𝑘 / 𝑗]𝜑[(𝑘 + 1) / 𝑗]𝜑))
Assertion
Ref Expression
uzind4s2 (𝑁 ∈ (ℤ𝑀) → [𝑁 / 𝑗]𝜑)
Distinct variable groups:   𝑘,𝑀   𝜑,𝑘   𝑗,𝑘
Allowed substitution hints:   𝜑(𝑗)   𝑀(𝑗)   𝑁(𝑗,𝑘)

Proof of Theorem uzind4s2
Dummy variables 𝑚 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfsbcq 2906 . 2 (𝑚 = 𝑀 → ([𝑚 / 𝑗]𝜑[𝑀 / 𝑗]𝜑))
2 dfsbcq 2906 . 2 (𝑚 = 𝑛 → ([𝑚 / 𝑗]𝜑[𝑛 / 𝑗]𝜑))
3 dfsbcq 2906 . 2 (𝑚 = (𝑛 + 1) → ([𝑚 / 𝑗]𝜑[(𝑛 + 1) / 𝑗]𝜑))
4 dfsbcq 2906 . 2 (𝑚 = 𝑁 → ([𝑚 / 𝑗]𝜑[𝑁 / 𝑗]𝜑))
5 uzind4s2.1 . 2 (𝑀 ∈ ℤ → [𝑀 / 𝑗]𝜑)
6 dfsbcq 2906 . . . 4 (𝑘 = 𝑛 → ([𝑘 / 𝑗]𝜑[𝑛 / 𝑗]𝜑))
7 oveq1 5774 . . . . 5 (𝑘 = 𝑛 → (𝑘 + 1) = (𝑛 + 1))
87sbceq1d 2909 . . . 4 (𝑘 = 𝑛 → ([(𝑘 + 1) / 𝑗]𝜑[(𝑛 + 1) / 𝑗]𝜑))
96, 8imbi12d 233 . . 3 (𝑘 = 𝑛 → (([𝑘 / 𝑗]𝜑[(𝑘 + 1) / 𝑗]𝜑) ↔ ([𝑛 / 𝑗]𝜑[(𝑛 + 1) / 𝑗]𝜑)))
10 uzind4s2.2 . . 3 (𝑘 ∈ (ℤ𝑀) → ([𝑘 / 𝑗]𝜑[(𝑘 + 1) / 𝑗]𝜑))
119, 10vtoclga 2747 . 2 (𝑛 ∈ (ℤ𝑀) → ([𝑛 / 𝑗]𝜑[(𝑛 + 1) / 𝑗]𝜑))
121, 2, 3, 4, 5, 11uzind4 9376 1 (𝑁 ∈ (ℤ𝑀) → [𝑁 / 𝑗]𝜑)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∈ wcel 1480  [wsbc 2904  ‘cfv 5118  (class class class)co 5767  1c1 7614   + caddc 7616  ℤcz 9047  ℤ≥cuz 9319 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 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-cnex 7704  ax-resscn 7705  ax-1cn 7706  ax-1re 7707  ax-icn 7708  ax-addcl 7709  ax-addrcl 7710  ax-mulcl 7711  ax-addcom 7713  ax-addass 7715  ax-distr 7717  ax-i2m1 7718  ax-0lt1 7719  ax-0id 7721  ax-rnegex 7722  ax-cnre 7724  ax-pre-ltirr 7725  ax-pre-ltwlin 7726  ax-pre-lttrn 7727  ax-pre-ltadd 7729 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 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-fv 5126  df-riota 5723  df-ov 5770  df-oprab 5771  df-mpo 5772  df-pnf 7795  df-mnf 7796  df-xr 7797  df-ltxr 7798  df-le 7799  df-sub 7928  df-neg 7929  df-inn 8714  df-n0 8971  df-z 9048  df-uz 9320 This theorem is referenced by: (None)
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