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Theorem odzval 16768
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 7427 . . . . . . . . 9 (𝑚 = 𝑁 → (𝑥 gcd 𝑚) = (𝑥 gcd 𝑁))
21eqeq1d 2727 . . . . . . . 8 (𝑚 = 𝑁 → ((𝑥 gcd 𝑚) = 1 ↔ (𝑥 gcd 𝑁) = 1))
32rabbidv 3426 . . . . . . 7 (𝑚 = 𝑁 → {𝑥 ∈ ℤ ∣ (𝑥 gcd 𝑚) = 1} = {𝑥 ∈ ℤ ∣ (𝑥 gcd 𝑁) = 1})
4 oveq1 7426 . . . . . . . . 9 (𝑛 = 𝑥 → (𝑛 gcd 𝑁) = (𝑥 gcd 𝑁))
54eqeq1d 2727 . . . . . . . 8 (𝑛 = 𝑥 → ((𝑛 gcd 𝑁) = 1 ↔ (𝑥 gcd 𝑁) = 1))
65cbvrabv 3429 . . . . . . 7 {𝑛 ∈ ℤ ∣ (𝑛 gcd 𝑁) = 1} = {𝑥 ∈ ℤ ∣ (𝑥 gcd 𝑁) = 1}
73, 6eqtr4di 2783 . . . . . 6 (𝑚 = 𝑁 → {𝑥 ∈ ℤ ∣ (𝑥 gcd 𝑚) = 1} = {𝑛 ∈ ℤ ∣ (𝑛 gcd 𝑁) = 1})
8 breq1 5152 . . . . . . . 8 (𝑚 = 𝑁 → (𝑚 ∥ ((𝑥𝑛) − 1) ↔ 𝑁 ∥ ((𝑥𝑛) − 1)))
98rabbidv 3426 . . . . . . 7 (𝑚 = 𝑁 → {𝑛 ∈ ℕ ∣ 𝑚 ∥ ((𝑥𝑛) − 1)} = {𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝑥𝑛) − 1)})
109infeq1d 9507 . . . . . 6 (𝑚 = 𝑁 → inf({𝑛 ∈ ℕ ∣ 𝑚 ∥ ((𝑥𝑛) − 1)}, ℝ, < ) = inf({𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝑥𝑛) − 1)}, ℝ, < ))
117, 10mpteq12dv 5240 . . . . 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 16742 . . . . 5 od = (𝑚 ∈ ℕ ↦ (𝑥 ∈ {𝑥 ∈ ℤ ∣ (𝑥 gcd 𝑚) = 1} ↦ inf({𝑛 ∈ ℕ ∣ 𝑚 ∥ ((𝑥𝑛) − 1)}, ℝ, < )))
13 zex 12605 . . . . . 6 ℤ ∈ V
1413mptrabex 7237 . . . . 5 (𝑥 ∈ {𝑛 ∈ ℤ ∣ (𝑛 gcd 𝑁) = 1} ↦ inf({𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝑥𝑛) − 1)}, ℝ, < )) ∈ V
1511, 12, 14fvmpt 7004 . . . 4 (𝑁 ∈ ℕ → (od𝑁) = (𝑥 ∈ {𝑛 ∈ ℤ ∣ (𝑛 gcd 𝑁) = 1} ↦ inf({𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝑥𝑛) − 1)}, ℝ, < )))
1615fveq1d 6898 . . 3 (𝑁 ∈ ℕ → ((od𝑁)‘𝐴) = ((𝑥 ∈ {𝑛 ∈ ℤ ∣ (𝑛 gcd 𝑁) = 1} ↦ inf({𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝑥𝑛) − 1)}, ℝ, < ))‘𝐴))
17 oveq1 7426 . . . . . 6 (𝑛 = 𝐴 → (𝑛 gcd 𝑁) = (𝐴 gcd 𝑁))
1817eqeq1d 2727 . . . . 5 (𝑛 = 𝐴 → ((𝑛 gcd 𝑁) = 1 ↔ (𝐴 gcd 𝑁) = 1))
1918elrab 3679 . . . 4 (𝐴 ∈ {𝑛 ∈ ℤ ∣ (𝑛 gcd 𝑁) = 1} ↔ (𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1))
20 oveq1 7426 . . . . . . . . 9 (𝑥 = 𝐴 → (𝑥𝑛) = (𝐴𝑛))
2120oveq1d 7434 . . . . . . . 8 (𝑥 = 𝐴 → ((𝑥𝑛) − 1) = ((𝐴𝑛) − 1))
2221breq2d 5161 . . . . . . 7 (𝑥 = 𝐴 → (𝑁 ∥ ((𝑥𝑛) − 1) ↔ 𝑁 ∥ ((𝐴𝑛) − 1)))
2322rabbidv 3426 . . . . . 6 (𝑥 = 𝐴 → {𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝑥𝑛) − 1)} = {𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝐴𝑛) − 1)})
2423infeq1d 9507 . . . . 5 (𝑥 = 𝐴 → inf({𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝑥𝑛) − 1)}, ℝ, < ) = inf({𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝐴𝑛) − 1)}, ℝ, < ))
25 eqid 2725 . . . . 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 11331 . . . . . 6 < Or ℝ
2726infex 9523 . . . . 5 inf({𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝐴𝑛) − 1)}, ℝ, < ) ∈ V
2824, 25, 27fvmpt 7004 . . . 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 234 . . 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 2785 . 2 ((𝑁 ∈ ℕ ∧ (𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1)) → ((od𝑁)‘𝐴) = inf({𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝐴𝑛) − 1)}, ℝ, < ))
31303impb 1112 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 394  w3a 1084   = wceq 1533  wcel 2098  {crab 3418   class class class wbr 5149  cmpt 5232  cfv 6549  (class class class)co 7419  infcinf 9471  cr 11144  1c1 11146   < clt 11285  cmin 11481  cn 12250  cz 12596  cexp 14067  cdvds 16239   gcd cgcd 16477  odcodz 16740
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-cnex 11201  ax-resscn 11202  ax-pre-lttri 11219  ax-pre-lttrn 11220
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-po 5590  df-so 5591  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-ov 7422  df-er 8725  df-en 8965  df-dom 8966  df-sdom 8967  df-sup 9472  df-inf 9473  df-pnf 11287  df-mnf 11288  df-ltxr 11290  df-neg 11484  df-z 12597  df-odz 16742
This theorem is referenced by:  odzcllem  16769  odzdvds  16772  hashscontpow1  41726
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