Theorem odzval 12255

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) → ((odℤ‘𝑁)‘𝐴) = inf({𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝐴↑𝑛) − 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.

Step	Hyp	Ref	Expression
1		oveq2 5896	. . . . . . . . 9 ⊢ (𝑚 = 𝑁 → (𝑥 gcd 𝑚) = (𝑥 gcd 𝑁))
2	1	eqeq1d 2196	. . . . . . . 8 ⊢ (𝑚 = 𝑁 → ((𝑥 gcd 𝑚) = 1 ↔ (𝑥 gcd 𝑁) = 1))
3	2	rabbidv 2738	. . . . . . 7 ⊢ (𝑚 = 𝑁 → {𝑥 ∈ ℤ ∣ (𝑥 gcd 𝑚) = 1} = {𝑥 ∈ ℤ ∣ (𝑥 gcd 𝑁) = 1})
4		oveq1 5895	. . . . . . . . 9 ⊢ (𝑛 = 𝑥 → (𝑛 gcd 𝑁) = (𝑥 gcd 𝑁))
5	4	eqeq1d 2196	. . . . . . . 8 ⊢ (𝑛 = 𝑥 → ((𝑛 gcd 𝑁) = 1 ↔ (𝑥 gcd 𝑁) = 1))
6	5	cbvrabv 2748	. . . . . . 7 ⊢ {𝑛 ∈ ℤ ∣ (𝑛 gcd 𝑁) = 1} = {𝑥 ∈ ℤ ∣ (𝑥 gcd 𝑁) = 1}
7	3, 6	eqtr4di 2238	. . . . . 6 ⊢ (𝑚 = 𝑁 → {𝑥 ∈ ℤ ∣ (𝑥 gcd 𝑚) = 1} = {𝑛 ∈ ℤ ∣ (𝑛 gcd 𝑁) = 1})
8		breq1 4018	. . . . . . . 8 ⊢ (𝑚 = 𝑁 → (𝑚 ∥ ((𝑥↑𝑛) − 1) ↔ 𝑁 ∥ ((𝑥↑𝑛) − 1)))
9	8	rabbidv 2738	. . . . . . 7 ⊢ (𝑚 = 𝑁 → {𝑛 ∈ ℕ ∣ 𝑚 ∥ ((𝑥↑𝑛) − 1)} = {𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝑥↑𝑛) − 1)})
10	9	infeq1d 7025	. . . . . 6 ⊢ (𝑚 = 𝑁 → inf({𝑛 ∈ ℕ ∣ 𝑚 ∥ ((𝑥↑𝑛) − 1)}, ℝ, < ) = inf({𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝑥↑𝑛) − 1)}, ℝ, < ))
11	7, 10	mpteq12dv 4097	. . . . 5 ⊢ (𝑚 = 𝑁 → (𝑥 ∈ {𝑥 ∈ ℤ ∣ (𝑥 gcd 𝑚) = 1} ↦ inf({𝑛 ∈ ℕ ∣ 𝑚 ∥ ((𝑥↑𝑛) − 1)}, ℝ, < )) = (𝑥 ∈ {𝑛 ∈ ℤ ∣ (𝑛 gcd 𝑁) = 1} ↦ inf({𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝑥↑𝑛) − 1)}, ℝ, < )))
12		df-odz 12224	. . . . 5 ⊢ odℤ = (𝑚 ∈ ℕ ↦ (𝑥 ∈ {𝑥 ∈ ℤ ∣ (𝑥 gcd 𝑚) = 1} ↦ inf({𝑛 ∈ ℕ ∣ 𝑚 ∥ ((𝑥↑𝑛) − 1)}, ℝ, < )))
13		zex 9276	. . . . . 6 ⊢ ℤ ∈ V
14	13	mptrabex 5757	. . . . 5 ⊢ (𝑥 ∈ {𝑛 ∈ ℤ ∣ (𝑛 gcd 𝑁) = 1} ↦ inf({𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝑥↑𝑛) − 1)}, ℝ, < )) ∈ V
15	11, 12, 14	fvmpt 5606	. . . 4 ⊢ (𝑁 ∈ ℕ → (odℤ‘𝑁) = (𝑥 ∈ {𝑛 ∈ ℤ ∣ (𝑛 gcd 𝑁) = 1} ↦ inf({𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝑥↑𝑛) − 1)}, ℝ, < )))
16	15	fveq1d 5529	. . 3 ⊢ (𝑁 ∈ ℕ → ((odℤ‘𝑁)‘𝐴) = ((𝑥 ∈ {𝑛 ∈ ℤ ∣ (𝑛 gcd 𝑁) = 1} ↦ inf({𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝑥↑𝑛) − 1)}, ℝ, < ))‘𝐴))
17		oveq1 5895	. . . . . 6 ⊢ (𝑛 = 𝐴 → (𝑛 gcd 𝑁) = (𝐴 gcd 𝑁))
18	17	eqeq1d 2196	. . . . 5 ⊢ (𝑛 = 𝐴 → ((𝑛 gcd 𝑁) = 1 ↔ (𝐴 gcd 𝑁) = 1))
19	18	elrab 2905	. . . 4 ⊢ (𝐴 ∈ {𝑛 ∈ ℤ ∣ (𝑛 gcd 𝑁) = 1} ↔ (𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1))
20		oveq1 5895	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → (𝑥↑𝑛) = (𝐴↑𝑛))
21	20	oveq1d 5903	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → ((𝑥↑𝑛) − 1) = ((𝐴↑𝑛) − 1))
22	21	breq2d 4027	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑁 ∥ ((𝑥↑𝑛) − 1) ↔ 𝑁 ∥ ((𝐴↑𝑛) − 1)))
23	22	rabbidv 2738	. . . . . 6 ⊢ (𝑥 = 𝐴 → {𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝑥↑𝑛) − 1)} = {𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝐴↑𝑛) − 1)})
24	23	infeq1d 7025	. . . . 5 ⊢ (𝑥 = 𝐴 → inf({𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝑥↑𝑛) − 1)}, ℝ, < ) = inf({𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝐴↑𝑛) − 1)}, ℝ, < ))
25		eqid 2187	. . . . 5 ⊢ (𝑥 ∈ {𝑛 ∈ ℤ ∣ (𝑛 gcd 𝑁) = 1} ↦ inf({𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝑥↑𝑛) − 1)}, ℝ, < )) = (𝑥 ∈ {𝑛 ∈ ℤ ∣ (𝑛 gcd 𝑁) = 1} ↦ inf({𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝑥↑𝑛) − 1)}, ℝ, < ))
26		reex 7959	. . . . . 6 ⊢ ℝ ∈ V
27		infex2g 7047	. . . . . 6 ⊢ (ℝ ∈ V → inf({𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝐴↑𝑛) − 1)}, ℝ, < ) ∈ V)
28	26, 27	ax-mp 5	. . . . 5 ⊢ inf({𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝐴↑𝑛) − 1)}, ℝ, < ) ∈ V
29	24, 25, 28	fvmpt 5606	. . . 4 ⊢ (𝐴 ∈ {𝑛 ∈ ℤ ∣ (𝑛 gcd 𝑁) = 1} → ((𝑥 ∈ {𝑛 ∈ ℤ ∣ (𝑛 gcd 𝑁) = 1} ↦ inf({𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝑥↑𝑛) − 1)}, ℝ, < ))‘𝐴) = inf({𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝐴↑𝑛) − 1)}, ℝ, < ))
30	19, 29	sylbir 135	. . 3 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) → ((𝑥 ∈ {𝑛 ∈ ℤ ∣ (𝑛 gcd 𝑁) = 1} ↦ inf({𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝑥↑𝑛) − 1)}, ℝ, < ))‘𝐴) = inf({𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝐴↑𝑛) − 1)}, ℝ, < ))
31	16, 30	sylan9eq 2240	. 2 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1)) → ((odℤ‘𝑁)‘𝐴) = inf({𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝐴↑𝑛) − 1)}, ℝ, < ))
32	31	3impb 1200	1 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) → ((odℤ‘𝑁)‘𝐴) = inf({𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝐴↑𝑛) − 1)}, ℝ, < ))

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 979   = wceq 1363   ∈ wcel 2158  {crab 2469  Vcvv 2749   class class class wbr 4015   ↦ cmpt 4076  ‘cfv 5228  (class class class)co 5888  infcinf 6996  ℝcr 7824  1c1 7826   < clt 8006   − cmin 8142  ℕcn 8933  ℤcz 9267  ↑cexp 10533   ∥ cdvds 11808   gcd cgcd 11957  od_ℤcodz 12222

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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-cnex 7916  ax-resscn 7917

This theorem depends on definitions:  df-bi 117  df-3or 980  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-reu 2472  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-ov 5891  df-sup 6997  df-inf 6998  df-neg 8145  df-z 9268  df-odz 12224

This theorem is referenced by:  odzcllem  12256  odzdvds  12259