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

Theorem odzval 12243
Description: Value of the order function. This is a function of functions; the inner argument selects the base (i.e., mod 𝑁 for some 𝑁, often prime) and the outer argument selects the integer or equivalence class (if you want to think about it that way) from the integers mod 𝑁. In order to ensure the supremum is well-defined, we only define the expression when 𝐴 and 𝑁 are coprime. (Contributed by Mario Carneiro, 23-Feb-2014.) (Revised by AV, 26-Sep-2020.)
Assertion
Ref Expression
odzval ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) → ((od𝑁)‘𝐴) = inf({𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝐴𝑛) − 1)}, ℝ, < ))
Distinct variable groups:   𝑛,𝑁   𝐴,𝑛

Proof of Theorem odzval
Dummy variables 𝑚 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 5885 . . . . . . . . 9 (𝑚 = 𝑁 → (𝑥 gcd 𝑚) = (𝑥 gcd 𝑁))
21eqeq1d 2186 . . . . . . . 8 (𝑚 = 𝑁 → ((𝑥 gcd 𝑚) = 1 ↔ (𝑥 gcd 𝑁) = 1))
32rabbidv 2728 . . . . . . 7 (𝑚 = 𝑁 → {𝑥 ∈ ℤ ∣ (𝑥 gcd 𝑚) = 1} = {𝑥 ∈ ℤ ∣ (𝑥 gcd 𝑁) = 1})
4 oveq1 5884 . . . . . . . . 9 (𝑛 = 𝑥 → (𝑛 gcd 𝑁) = (𝑥 gcd 𝑁))
54eqeq1d 2186 . . . . . . . 8 (𝑛 = 𝑥 → ((𝑛 gcd 𝑁) = 1 ↔ (𝑥 gcd 𝑁) = 1))
65cbvrabv 2738 . . . . . . 7 {𝑛 ∈ ℤ ∣ (𝑛 gcd 𝑁) = 1} = {𝑥 ∈ ℤ ∣ (𝑥 gcd 𝑁) = 1}
73, 6eqtr4di 2228 . . . . . 6 (𝑚 = 𝑁 → {𝑥 ∈ ℤ ∣ (𝑥 gcd 𝑚) = 1} = {𝑛 ∈ ℤ ∣ (𝑛 gcd 𝑁) = 1})
8 breq1 4008 . . . . . . . 8 (𝑚 = 𝑁 → (𝑚 ∥ ((𝑥𝑛) − 1) ↔ 𝑁 ∥ ((𝑥𝑛) − 1)))
98rabbidv 2728 . . . . . . 7 (𝑚 = 𝑁 → {𝑛 ∈ ℕ ∣ 𝑚 ∥ ((𝑥𝑛) − 1)} = {𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝑥𝑛) − 1)})
109infeq1d 7013 . . . . . 6 (𝑚 = 𝑁 → inf({𝑛 ∈ ℕ ∣ 𝑚 ∥ ((𝑥𝑛) − 1)}, ℝ, < ) = inf({𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝑥𝑛) − 1)}, ℝ, < ))
117, 10mpteq12dv 4087 . . . . 5 (𝑚 = 𝑁 → (𝑥 ∈ {𝑥 ∈ ℤ ∣ (𝑥 gcd 𝑚) = 1} ↦ inf({𝑛 ∈ ℕ ∣ 𝑚 ∥ ((𝑥𝑛) − 1)}, ℝ, < )) = (𝑥 ∈ {𝑛 ∈ ℤ ∣ (𝑛 gcd 𝑁) = 1} ↦ inf({𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝑥𝑛) − 1)}, ℝ, < )))
12 df-odz 12212 . . . . 5 od = (𝑚 ∈ ℕ ↦ (𝑥 ∈ {𝑥 ∈ ℤ ∣ (𝑥 gcd 𝑚) = 1} ↦ inf({𝑛 ∈ ℕ ∣ 𝑚 ∥ ((𝑥𝑛) − 1)}, ℝ, < )))
13 zex 9264 . . . . . 6 ℤ ∈ V
1413mptrabex 5746 . . . . 5 (𝑥 ∈ {𝑛 ∈ ℤ ∣ (𝑛 gcd 𝑁) = 1} ↦ inf({𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝑥𝑛) − 1)}, ℝ, < )) ∈ V
1511, 12, 14fvmpt 5595 . . . 4 (𝑁 ∈ ℕ → (od𝑁) = (𝑥 ∈ {𝑛 ∈ ℤ ∣ (𝑛 gcd 𝑁) = 1} ↦ inf({𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝑥𝑛) − 1)}, ℝ, < )))
1615fveq1d 5519 . . 3 (𝑁 ∈ ℕ → ((od𝑁)‘𝐴) = ((𝑥 ∈ {𝑛 ∈ ℤ ∣ (𝑛 gcd 𝑁) = 1} ↦ inf({𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝑥𝑛) − 1)}, ℝ, < ))‘𝐴))
17 oveq1 5884 . . . . . 6 (𝑛 = 𝐴 → (𝑛 gcd 𝑁) = (𝐴 gcd 𝑁))
1817eqeq1d 2186 . . . . 5 (𝑛 = 𝐴 → ((𝑛 gcd 𝑁) = 1 ↔ (𝐴 gcd 𝑁) = 1))
1918elrab 2895 . . . 4 (𝐴 ∈ {𝑛 ∈ ℤ ∣ (𝑛 gcd 𝑁) = 1} ↔ (𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1))
20 oveq1 5884 . . . . . . . . 9 (𝑥 = 𝐴 → (𝑥𝑛) = (𝐴𝑛))
2120oveq1d 5892 . . . . . . . 8 (𝑥 = 𝐴 → ((𝑥𝑛) − 1) = ((𝐴𝑛) − 1))
2221breq2d 4017 . . . . . . 7 (𝑥 = 𝐴 → (𝑁 ∥ ((𝑥𝑛) − 1) ↔ 𝑁 ∥ ((𝐴𝑛) − 1)))
2322rabbidv 2728 . . . . . 6 (𝑥 = 𝐴 → {𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝑥𝑛) − 1)} = {𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝐴𝑛) − 1)})
2423infeq1d 7013 . . . . 5 (𝑥 = 𝐴 → inf({𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝑥𝑛) − 1)}, ℝ, < ) = inf({𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝐴𝑛) − 1)}, ℝ, < ))
25 eqid 2177 . . . . 5 (𝑥 ∈ {𝑛 ∈ ℤ ∣ (𝑛 gcd 𝑁) = 1} ↦ inf({𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝑥𝑛) − 1)}, ℝ, < )) = (𝑥 ∈ {𝑛 ∈ ℤ ∣ (𝑛 gcd 𝑁) = 1} ↦ inf({𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝑥𝑛) − 1)}, ℝ, < ))
26 reex 7947 . . . . . 6 ℝ ∈ V
27 infex2g 7035 . . . . . 6 (ℝ ∈ V → inf({𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝐴𝑛) − 1)}, ℝ, < ) ∈ V)
2826, 27ax-mp 5 . . . . 5 inf({𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝐴𝑛) − 1)}, ℝ, < ) ∈ V
2924, 25, 28fvmpt 5595 . . . 4 (𝐴 ∈ {𝑛 ∈ ℤ ∣ (𝑛 gcd 𝑁) = 1} → ((𝑥 ∈ {𝑛 ∈ ℤ ∣ (𝑛 gcd 𝑁) = 1} ↦ inf({𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝑥𝑛) − 1)}, ℝ, < ))‘𝐴) = inf({𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝐴𝑛) − 1)}, ℝ, < ))
3019, 29sylbir 135 . . 3 ((𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) → ((𝑥 ∈ {𝑛 ∈ ℤ ∣ (𝑛 gcd 𝑁) = 1} ↦ inf({𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝑥𝑛) − 1)}, ℝ, < ))‘𝐴) = inf({𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝐴𝑛) − 1)}, ℝ, < ))
3116, 30sylan9eq 2230 . 2 ((𝑁 ∈ ℕ ∧ (𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1)) → ((od𝑁)‘𝐴) = inf({𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝐴𝑛) − 1)}, ℝ, < ))
32313impb 1199 1 ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) → ((od𝑁)‘𝐴) = inf({𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝐴𝑛) − 1)}, ℝ, < ))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  w3a 978   = wceq 1353  wcel 2148  {crab 2459  Vcvv 2739   class class class wbr 4005  cmpt 4066  cfv 5218  (class class class)co 5877  infcinf 6984  cr 7812  1c1 7814   < clt 7994  cmin 8130  cn 8921  cz 9255  cexp 10521  cdvds 11796   gcd cgcd 11945  odcodz 12210
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-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-coll 4120  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-cnex 7904  ax-resscn 7905
This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  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-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-ov 5880  df-sup 6985  df-inf 6986  df-neg 8133  df-z 9256  df-odz 12212
This theorem is referenced by:  odzcllem  12244  odzdvds  12247
  Copyright terms: Public domain W3C validator