Theorem odzval 16855

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 7418	. . . . . . . . 9 ⊢ (𝑚 = 𝑁 → (𝑥 gcd 𝑚) = (𝑥 gcd 𝑁))
2	1	eqeq1d 2765	. . . . . . . 8 ⊢ (𝑚 = 𝑁 → ((𝑥 gcd 𝑚) = 1 ↔ (𝑥 gcd 𝑁) = 1))
3	2	rabbidv 3423	. . . . . . 7 ⊢ (𝑚 = 𝑁 → {𝑥 ∈ ℤ ∣ (𝑥 gcd 𝑚) = 1} = {𝑥 ∈ ℤ ∣ (𝑥 gcd 𝑁) = 1})
4		oveq1 7417	. . . . . . . . 9 ⊢ (𝑛 = 𝑥 → (𝑛 gcd 𝑁) = (𝑥 gcd 𝑁))
5	4	eqeq1d 2765	. . . . . . . 8 ⊢ (𝑛 = 𝑥 → ((𝑛 gcd 𝑁) = 1 ↔ (𝑥 gcd 𝑁) = 1))
6	5	cbvrabv 3426	. . . . . . 7 ⊢ {𝑛 ∈ ℤ ∣ (𝑛 gcd 𝑁) = 1} = {𝑥 ∈ ℤ ∣ (𝑥 gcd 𝑁) = 1}
7	3, 6	eqtr4di 2816	. . . . . 6 ⊢ (𝑚 = 𝑁 → {𝑥 ∈ ℤ ∣ (𝑥 gcd 𝑚) = 1} = {𝑛 ∈ ℤ ∣ (𝑛 gcd 𝑁) = 1})
8		breq1 5112	. . . . . . . 8 ⊢ (𝑚 = 𝑁 → (𝑚 ∥ ((𝑥↑𝑛) − 1) ↔ 𝑁 ∥ ((𝑥↑𝑛) − 1)))
9	8	rabbidv 3423	. . . . . . 7 ⊢ (𝑚 = 𝑁 → {𝑛 ∈ ℕ ∣ 𝑚 ∥ ((𝑥↑𝑛) − 1)} = {𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝑥↑𝑛) − 1)})
10	9	infeq1d 9434	. . . . . 6 ⊢ (𝑚 = 𝑁 → inf({𝑛 ∈ ℕ ∣ 𝑚 ∥ ((𝑥↑𝑛) − 1)}, ℝ, < ) = inf({𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝑥↑𝑛) − 1)}, ℝ, < ))
11	7, 10	mpteq12dv 5198	. . . . 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 16828	. . . . 5 ⊢ odℤ = (𝑚 ∈ ℕ ↦ (𝑥 ∈ {𝑥 ∈ ℤ ∣ (𝑥 gcd 𝑚) = 1} ↦ inf({𝑛 ∈ ℕ ∣ 𝑚 ∥ ((𝑥↑𝑛) − 1)}, ℝ, < )))
13		zex 12604	. . . . . 6 ⊢ ℤ ∈ V
14	13	mptrabex 7223	. . . . 5 ⊢ (𝑥 ∈ {𝑛 ∈ ℤ ∣ (𝑛 gcd 𝑁) = 1} ↦ inf({𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝑥↑𝑛) − 1)}, ℝ, < )) ∈ V
15	11, 12, 14	fvmpt 6989	. . . 4 ⊢ (𝑁 ∈ ℕ → (odℤ‘𝑁) = (𝑥 ∈ {𝑛 ∈ ℤ ∣ (𝑛 gcd 𝑁) = 1} ↦ inf({𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝑥↑𝑛) − 1)}, ℝ, < )))
16	15	fveq1d 6883	. . 3 ⊢ (𝑁 ∈ ℕ → ((odℤ‘𝑁)‘𝐴) = ((𝑥 ∈ {𝑛 ∈ ℤ ∣ (𝑛 gcd 𝑁) = 1} ↦ inf({𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝑥↑𝑛) − 1)}, ℝ, < ))‘𝐴))
17		oveq1 7417	. . . . . 6 ⊢ (𝑛 = 𝐴 → (𝑛 gcd 𝑁) = (𝐴 gcd 𝑁))
18	17	eqeq1d 2765	. . . . 5 ⊢ (𝑛 = 𝐴 → ((𝑛 gcd 𝑁) = 1 ↔ (𝐴 gcd 𝑁) = 1))
19	18	elrab 3650	. . . 4 ⊢ (𝐴 ∈ {𝑛 ∈ ℤ ∣ (𝑛 gcd 𝑁) = 1} ↔ (𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1))
20		oveq1 7417	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → (𝑥↑𝑛) = (𝐴↑𝑛))
21	20	oveq1d 7425	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → ((𝑥↑𝑛) − 1) = ((𝐴↑𝑛) − 1))
22	21	breq2d 5121	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑁 ∥ ((𝑥↑𝑛) − 1) ↔ 𝑁 ∥ ((𝐴↑𝑛) − 1)))
23	22	rabbidv 3423	. . . . . 6 ⊢ (𝑥 = 𝐴 → {𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝑥↑𝑛) − 1)} = {𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝐴↑𝑛) − 1)})
24	23	infeq1d 9434	. . . . 5 ⊢ (𝑥 = 𝐴 → inf({𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝑥↑𝑛) − 1)}, ℝ, < ) = inf({𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝐴↑𝑛) − 1)}, ℝ, < ))
25		eqid 2763	. . . . 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		ltso 11294	. . . . . 6 ⊢ < Or ℝ
27	26	infex 9451	. . . . 5 ⊢ inf({𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝐴↑𝑛) − 1)}, ℝ, < ) ∈ V
28	24, 25, 27	fvmpt 6989	. . . 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, < ))
29	19, 28	sylbir 238	. . 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, < ))
30	16, 29	sylan9eq 2818	. 2 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1)) → ((odℤ‘𝑁)‘𝐴) = inf({𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝐴↑𝑛) − 1)}, ℝ, < ))
31	30	3impb 1132	1 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) → ((odℤ‘𝑁)‘𝐴) = inf({𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝐴↑𝑛) − 1)}, ℝ, < ))

This proof depends on syntax axioms:   → wi 4   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  {crab 3416   class class class wbr 5109   ↦ cmpt 5192  ‘cfv 6536  (class class class)co 7410  infcinf 9397  ℝcr 11103  1c1 11105   < clt 11247   − cmin 11445  ℕcn 12237  ℤcz 12595  ↑cexp 14102   ∥ cdvds 16314   gcd cgcd 16556  od_ℤcodz 16826

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-cnex 11160  ax-resscn 11161  ax-pre-lttri 11178  ax-pre-lttrn 11179

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-po 5569  df-so 5570  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-er 8690  df-en 8940  df-dom 8941  df-sdom 8942  df-sup 9398  df-inf 9399  df-pnf 11249  df-mnf 11250  df-ltxr 11252  df-neg 11448  df-z 12596  df-odz 16828

This theorem is used by:  odzcllem  16856  odzdvds  16859  hashscontpow1  42916