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Theorem uzind4ALT 9131
 Description: Induction on the upper set of integers that starts at an integer 𝑀. The last four hypotheses give us the substitution instances we need; the first two are the basis and the induction step. Either uzind4 9130 or uzind4ALT 9131 may be used; see comment for nnind 8492. (Contributed by NM, 7-Sep-2005.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
uzind4ALT.5 (𝑀 ∈ ℤ → 𝜓)
uzind4ALT.6 (𝑘 ∈ (ℤ𝑀) → (𝜒𝜃))
uzind4ALT.1 (𝑗 = 𝑀 → (𝜑𝜓))
uzind4ALT.2 (𝑗 = 𝑘 → (𝜑𝜒))
uzind4ALT.3 (𝑗 = (𝑘 + 1) → (𝜑𝜃))
uzind4ALT.4 (𝑗 = 𝑁 → (𝜑𝜏))
Assertion
Ref Expression
uzind4ALT (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 uzind4ALT
StepHypRef Expression
1 uzind4ALT.1 . 2 (𝑗 = 𝑀 → (𝜑𝜓))
2 uzind4ALT.2 . 2 (𝑗 = 𝑘 → (𝜑𝜒))
3 uzind4ALT.3 . 2 (𝑗 = (𝑘 + 1) → (𝜑𝜃))
4 uzind4ALT.4 . 2 (𝑗 = 𝑁 → (𝜑𝜏))
5 uzind4ALT.5 . 2 (𝑀 ∈ ℤ → 𝜓)
6 uzind4ALT.6 . 2 (𝑘 ∈ (ℤ𝑀) → (𝜒𝜃))
71, 2, 3, 4, 5, 6uzind4 9130 1 (𝑁 ∈ (ℤ𝑀) → 𝜏)
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 104   = wceq 1290   ∈ wcel 1439  ‘cfv 5028  (class class class)co 5666  1c1 7405   + caddc 7407  ℤcz 8804  ℤ≥cuz 9073 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366  ax-cnex 7490  ax-resscn 7491  ax-1cn 7492  ax-1re 7493  ax-icn 7494  ax-addcl 7495  ax-addrcl 7496  ax-mulcl 7497  ax-addcom 7499  ax-addass 7501  ax-distr 7503  ax-i2m1 7504  ax-0lt1 7505  ax-0id 7507  ax-rnegex 7508  ax-cnre 7510  ax-pre-ltirr 7511  ax-pre-ltwlin 7512  ax-pre-lttrn 7513  ax-pre-ltadd 7515 This theorem depends on definitions:  df-bi 116  df-3or 926  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-nel 2352  df-ral 2365  df-rex 2366  df-reu 2367  df-rab 2369  df-v 2622  df-sbc 2842  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-int 3695  df-br 3852  df-opab 3906  df-mpt 3907  df-id 4129  df-xp 4457  df-rel 4458  df-cnv 4459  df-co 4460  df-dm 4461  df-rn 4462  df-res 4463  df-ima 4464  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-fv 5036  df-riota 5622  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-pnf 7578  df-mnf 7579  df-xr 7580  df-ltxr 7581  df-le 7582  df-sub 7709  df-neg 7710  df-inn 8477  df-n0 8728  df-z 8805  df-uz 9074 This theorem is referenced by: (None)
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