ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  uzind3 GIF version

Theorem uzind3 9364
Description: Induction on the upper integers that start at an integer 𝑀. The first four hypotheses give us the substitution instances we need, and the last two are the basis and the induction step. (Contributed by NM, 26-Jul-2005.)
Hypotheses
Ref Expression
uzind3.1 (𝑗 = 𝑀 → (𝜑𝜓))
uzind3.2 (𝑗 = 𝑚 → (𝜑𝜒))
uzind3.3 (𝑗 = (𝑚 + 1) → (𝜑𝜃))
uzind3.4 (𝑗 = 𝑁 → (𝜑𝜏))
uzind3.5 (𝑀 ∈ ℤ → 𝜓)
uzind3.6 ((𝑀 ∈ ℤ ∧ 𝑚 ∈ {𝑘 ∈ ℤ ∣ 𝑀𝑘}) → (𝜒𝜃))
Assertion
Ref Expression
uzind3 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ {𝑘 ∈ ℤ ∣ 𝑀𝑘}) → 𝜏)
Distinct variable groups:   𝑗,𝑘,𝑁   𝜓,𝑗   𝜒,𝑗   𝜃,𝑗   𝜏,𝑗   𝜑,𝑚   𝑗,𝑚,𝑀,𝑘
Allowed substitution hints:   𝜑(𝑗,𝑘)   𝜓(𝑘,𝑚)   𝜒(𝑘,𝑚)   𝜃(𝑘,𝑚)   𝜏(𝑘,𝑚)   𝑁(𝑚)

Proof of Theorem uzind3
StepHypRef Expression
1 breq2 4007 . . 3 (𝑘 = 𝑁 → (𝑀𝑘𝑀𝑁))
21elrab 2893 . 2 (𝑁 ∈ {𝑘 ∈ ℤ ∣ 𝑀𝑘} ↔ (𝑁 ∈ ℤ ∧ 𝑀𝑁))
3 uzind3.1 . . . 4 (𝑗 = 𝑀 → (𝜑𝜓))
4 uzind3.2 . . . 4 (𝑗 = 𝑚 → (𝜑𝜒))
5 uzind3.3 . . . 4 (𝑗 = (𝑚 + 1) → (𝜑𝜃))
6 uzind3.4 . . . 4 (𝑗 = 𝑁 → (𝜑𝜏))
7 uzind3.5 . . . 4 (𝑀 ∈ ℤ → 𝜓)
8 breq2 4007 . . . . . . 7 (𝑘 = 𝑚 → (𝑀𝑘𝑀𝑚))
98elrab 2893 . . . . . 6 (𝑚 ∈ {𝑘 ∈ ℤ ∣ 𝑀𝑘} ↔ (𝑚 ∈ ℤ ∧ 𝑀𝑚))
10 uzind3.6 . . . . . 6 ((𝑀 ∈ ℤ ∧ 𝑚 ∈ {𝑘 ∈ ℤ ∣ 𝑀𝑘}) → (𝜒𝜃))
119, 10sylan2br 288 . . . . 5 ((𝑀 ∈ ℤ ∧ (𝑚 ∈ ℤ ∧ 𝑀𝑚)) → (𝜒𝜃))
12113impb 1199 . . . 4 ((𝑀 ∈ ℤ ∧ 𝑚 ∈ ℤ ∧ 𝑀𝑚) → (𝜒𝜃))
133, 4, 5, 6, 7, 12uzind 9362 . . 3 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀𝑁) → 𝜏)
14133expb 1204 . 2 ((𝑀 ∈ ℤ ∧ (𝑁 ∈ ℤ ∧ 𝑀𝑁)) → 𝜏)
152, 14sylan2b 287 1 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ {𝑘 ∈ ℤ ∣ 𝑀𝑘}) → 𝜏)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1353  wcel 2148  {crab 2459   class class class wbr 4003  (class class class)co 5874  1c1 7811   + caddc 7813  cle 7991  cz 9251
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 4121  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-addcom 7910  ax-addass 7912  ax-distr 7914  ax-i2m1 7915  ax-0lt1 7916  ax-0id 7918  ax-rnegex 7919  ax-cnre 7921  ax-pre-ltirr 7922  ax-pre-ltwlin 7923  ax-pre-lttrn 7924  ax-pre-ltadd 7926
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-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-br 4004  df-opab 4065  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-iota 5178  df-fun 5218  df-fv 5224  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-pnf 7992  df-mnf 7993  df-xr 7994  df-ltxr 7995  df-le 7996  df-sub 8128  df-neg 8129  df-inn 8918  df-n0 9175  df-z 9252
This theorem is referenced by:  uzind4  9586  algfx  12046
  Copyright terms: Public domain W3C validator