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Theorem odzval 15703
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 6804 . . . . . . . . 9 (𝑚 = 𝑁 → (𝑥 gcd 𝑚) = (𝑥 gcd 𝑁))
21eqeq1d 2773 . . . . . . . 8 (𝑚 = 𝑁 → ((𝑥 gcd 𝑚) = 1 ↔ (𝑥 gcd 𝑁) = 1))
32rabbidv 3339 . . . . . . 7 (𝑚 = 𝑁 → {𝑥 ∈ ℤ ∣ (𝑥 gcd 𝑚) = 1} = {𝑥 ∈ ℤ ∣ (𝑥 gcd 𝑁) = 1})
4 oveq1 6803 . . . . . . . . 9 (𝑛 = 𝑥 → (𝑛 gcd 𝑁) = (𝑥 gcd 𝑁))
54eqeq1d 2773 . . . . . . . 8 (𝑛 = 𝑥 → ((𝑛 gcd 𝑁) = 1 ↔ (𝑥 gcd 𝑁) = 1))
65cbvrabv 3349 . . . . . . 7 {𝑛 ∈ ℤ ∣ (𝑛 gcd 𝑁) = 1} = {𝑥 ∈ ℤ ∣ (𝑥 gcd 𝑁) = 1}
73, 6syl6eqr 2823 . . . . . 6 (𝑚 = 𝑁 → {𝑥 ∈ ℤ ∣ (𝑥 gcd 𝑚) = 1} = {𝑛 ∈ ℤ ∣ (𝑛 gcd 𝑁) = 1})
8 breq1 4790 . . . . . . . 8 (𝑚 = 𝑁 → (𝑚 ∥ ((𝑥𝑛) − 1) ↔ 𝑁 ∥ ((𝑥𝑛) − 1)))
98rabbidv 3339 . . . . . . 7 (𝑚 = 𝑁 → {𝑛 ∈ ℕ ∣ 𝑚 ∥ ((𝑥𝑛) − 1)} = {𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝑥𝑛) − 1)})
109infeq1d 8543 . . . . . 6 (𝑚 = 𝑁 → inf({𝑛 ∈ ℕ ∣ 𝑚 ∥ ((𝑥𝑛) − 1)}, ℝ, < ) = inf({𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝑥𝑛) − 1)}, ℝ, < ))
117, 10mpteq12dv 4868 . . . . 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 15677 . . . . 5 od = (𝑚 ∈ ℕ ↦ (𝑥 ∈ {𝑥 ∈ ℤ ∣ (𝑥 gcd 𝑚) = 1} ↦ inf({𝑛 ∈ ℕ ∣ 𝑚 ∥ ((𝑥𝑛) − 1)}, ℝ, < )))
13 zex 11593 . . . . . 6 ℤ ∈ V
1413mptrabex 6635 . . . . 5 (𝑥 ∈ {𝑛 ∈ ℤ ∣ (𝑛 gcd 𝑁) = 1} ↦ inf({𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝑥𝑛) − 1)}, ℝ, < )) ∈ V
1511, 12, 14fvmpt 6426 . . . 4 (𝑁 ∈ ℕ → (od𝑁) = (𝑥 ∈ {𝑛 ∈ ℤ ∣ (𝑛 gcd 𝑁) = 1} ↦ inf({𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝑥𝑛) − 1)}, ℝ, < )))
1615fveq1d 6335 . . 3 (𝑁 ∈ ℕ → ((od𝑁)‘𝐴) = ((𝑥 ∈ {𝑛 ∈ ℤ ∣ (𝑛 gcd 𝑁) = 1} ↦ inf({𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝑥𝑛) − 1)}, ℝ, < ))‘𝐴))
17 oveq1 6803 . . . . . 6 (𝑛 = 𝐴 → (𝑛 gcd 𝑁) = (𝐴 gcd 𝑁))
1817eqeq1d 2773 . . . . 5 (𝑛 = 𝐴 → ((𝑛 gcd 𝑁) = 1 ↔ (𝐴 gcd 𝑁) = 1))
1918elrab 3515 . . . 4 (𝐴 ∈ {𝑛 ∈ ℤ ∣ (𝑛 gcd 𝑁) = 1} ↔ (𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1))
20 oveq1 6803 . . . . . . . . 9 (𝑥 = 𝐴 → (𝑥𝑛) = (𝐴𝑛))
2120oveq1d 6811 . . . . . . . 8 (𝑥 = 𝐴 → ((𝑥𝑛) − 1) = ((𝐴𝑛) − 1))
2221breq2d 4799 . . . . . . 7 (𝑥 = 𝐴 → (𝑁 ∥ ((𝑥𝑛) − 1) ↔ 𝑁 ∥ ((𝐴𝑛) − 1)))
2322rabbidv 3339 . . . . . 6 (𝑥 = 𝐴 → {𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝑥𝑛) − 1)} = {𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝐴𝑛) − 1)})
2423infeq1d 8543 . . . . 5 (𝑥 = 𝐴 → inf({𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝑥𝑛) − 1)}, ℝ, < ) = inf({𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝐴𝑛) − 1)}, ℝ, < ))
25 eqid 2771 . . . . 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 ltso 10324 . . . . . 6 < Or ℝ
2726infex 8559 . . . . 5 inf({𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝐴𝑛) − 1)}, ℝ, < ) ∈ V
2824, 25, 27fvmpt 6426 . . . 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, < ))
2919, 28sylbir 225 . . 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, < ))
3016, 29sylan9eq 2825 . 2 ((𝑁 ∈ ℕ ∧ (𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1)) → ((od𝑁)‘𝐴) = inf({𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝐴𝑛) − 1)}, ℝ, < ))
31303impb 1107 1 ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) → ((od𝑁)‘𝐴) = inf({𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝐴𝑛) − 1)}, ℝ, < ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1071   = wceq 1631  wcel 2145  {crab 3065   class class class wbr 4787  cmpt 4864  cfv 6030  (class class class)co 6796  infcinf 8507  cr 10141  1c1 10143   < clt 10280  cmin 10472  cn 11226  cz 11584  cexp 13067  cdvds 15189   gcd cgcd 15424  odcodz 15675
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100  ax-cnex 10198  ax-resscn 10199  ax-pre-lttri 10216  ax-pre-lttrn 10217
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-po 5171  df-so 5172  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-ov 6799  df-er 7900  df-en 8114  df-dom 8115  df-sdom 8116  df-sup 8508  df-inf 8509  df-pnf 10282  df-mnf 10283  df-ltxr 10285  df-neg 10475  df-z 11585  df-odz 15677
This theorem is referenced by:  odzcllem  15704  odzdvds  15707
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