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Theorem uzind4i 9563
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 9559 assuming that 𝜓 holds unconditionally. Notice that 𝑁 ∈ (ℤ𝑀) implies that the lower bound 𝑀 is an integer (𝑀 ∈ ℤ, see eluzel2 9504). (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 9559 1 (𝑁 ∈ (ℤ𝑀) → 𝜏)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105   = wceq 1353  wcel 2146  cfv 5208  (class class class)co 5865  1c1 7787   + caddc 7789  cz 9224  cuz 9499
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 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-cnex 7877  ax-resscn 7878  ax-1cn 7879  ax-1re 7880  ax-icn 7881  ax-addcl 7882  ax-addrcl 7883  ax-mulcl 7884  ax-addcom 7886  ax-addass 7888  ax-distr 7890  ax-i2m1 7891  ax-0lt1 7892  ax-0id 7894  ax-rnegex 7895  ax-cnre 7897  ax-pre-ltirr 7898  ax-pre-ltwlin 7899  ax-pre-lttrn 7900  ax-pre-ltadd 7902
This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-nel 2441  df-ral 2458  df-rex 2459  df-reu 2460  df-rab 2462  df-v 2737  df-sbc 2961  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-br 3999  df-opab 4060  df-mpt 4061  df-id 4287  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-fv 5216  df-riota 5821  df-ov 5868  df-oprab 5869  df-mpo 5870  df-pnf 7968  df-mnf 7969  df-xr 7970  df-ltxr 7971  df-le 7972  df-sub 8104  df-neg 8105  df-inn 8891  df-n0 9148  df-z 9225  df-uz 9500
This theorem is referenced by:  rebtwn2zlemshrink  10222
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