Theorem odzval 12195

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 5861	. . . . . . . . 9 ⊢ (𝑚 = 𝑁 → (𝑥 gcd 𝑚) = (𝑥 gcd 𝑁))
2	1	eqeq1d 2179	. . . . . . . 8 ⊢ (𝑚 = 𝑁 → ((𝑥 gcd 𝑚) = 1 ↔ (𝑥 gcd 𝑁) = 1))
3	2	rabbidv 2719	. . . . . . 7 ⊢ (𝑚 = 𝑁 → {𝑥 ∈ ℤ ∣ (𝑥 gcd 𝑚) = 1} = {𝑥 ∈ ℤ ∣ (𝑥 gcd 𝑁) = 1})
4		oveq1 5860	. . . . . . . . 9 ⊢ (𝑛 = 𝑥 → (𝑛 gcd 𝑁) = (𝑥 gcd 𝑁))
5	4	eqeq1d 2179	. . . . . . . 8 ⊢ (𝑛 = 𝑥 → ((𝑛 gcd 𝑁) = 1 ↔ (𝑥 gcd 𝑁) = 1))
6	5	cbvrabv 2729	. . . . . . 7 ⊢ {𝑛 ∈ ℤ ∣ (𝑛 gcd 𝑁) = 1} = {𝑥 ∈ ℤ ∣ (𝑥 gcd 𝑁) = 1}
7	3, 6	eqtr4di 2221	. . . . . 6 ⊢ (𝑚 = 𝑁 → {𝑥 ∈ ℤ ∣ (𝑥 gcd 𝑚) = 1} = {𝑛 ∈ ℤ ∣ (𝑛 gcd 𝑁) = 1})
8		breq1 3992	. . . . . . . 8 ⊢ (𝑚 = 𝑁 → (𝑚 ∥ ((𝑥↑𝑛) − 1) ↔ 𝑁 ∥ ((𝑥↑𝑛) − 1)))
9	8	rabbidv 2719	. . . . . . 7 ⊢ (𝑚 = 𝑁 → {𝑛 ∈ ℕ ∣ 𝑚 ∥ ((𝑥↑𝑛) − 1)} = {𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝑥↑𝑛) − 1)})
10	9	infeq1d 6989	. . . . . 6 ⊢ (𝑚 = 𝑁 → inf({𝑛 ∈ ℕ ∣ 𝑚 ∥ ((𝑥↑𝑛) − 1)}, ℝ, < ) = inf({𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝑥↑𝑛) − 1)}, ℝ, < ))
11	7, 10	mpteq12dv 4071	. . . . 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 12164	. . . . 5 ⊢ odℤ = (𝑚 ∈ ℕ ↦ (𝑥 ∈ {𝑥 ∈ ℤ ∣ (𝑥 gcd 𝑚) = 1} ↦ inf({𝑛 ∈ ℕ ∣ 𝑚 ∥ ((𝑥↑𝑛) − 1)}, ℝ, < )))
13		zex 9221	. . . . . 6 ⊢ ℤ ∈ V
14	13	mptrabex 5724	. . . . 5 ⊢ (𝑥 ∈ {𝑛 ∈ ℤ ∣ (𝑛 gcd 𝑁) = 1} ↦ inf({𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝑥↑𝑛) − 1)}, ℝ, < )) ∈ V
15	11, 12, 14	fvmpt 5573	. . . 4 ⊢ (𝑁 ∈ ℕ → (odℤ‘𝑁) = (𝑥 ∈ {𝑛 ∈ ℤ ∣ (𝑛 gcd 𝑁) = 1} ↦ inf({𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝑥↑𝑛) − 1)}, ℝ, < )))
16	15	fveq1d 5498	. . 3 ⊢ (𝑁 ∈ ℕ → ((odℤ‘𝑁)‘𝐴) = ((𝑥 ∈ {𝑛 ∈ ℤ ∣ (𝑛 gcd 𝑁) = 1} ↦ inf({𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝑥↑𝑛) − 1)}, ℝ, < ))‘𝐴))
17		oveq1 5860	. . . . . 6 ⊢ (𝑛 = 𝐴 → (𝑛 gcd 𝑁) = (𝐴 gcd 𝑁))
18	17	eqeq1d 2179	. . . . 5 ⊢ (𝑛 = 𝐴 → ((𝑛 gcd 𝑁) = 1 ↔ (𝐴 gcd 𝑁) = 1))
19	18	elrab 2886	. . . 4 ⊢ (𝐴 ∈ {𝑛 ∈ ℤ ∣ (𝑛 gcd 𝑁) = 1} ↔ (𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1))
20		oveq1 5860	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → (𝑥↑𝑛) = (𝐴↑𝑛))
21	20	oveq1d 5868	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → ((𝑥↑𝑛) − 1) = ((𝐴↑𝑛) − 1))
22	21	breq2d 4001	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑁 ∥ ((𝑥↑𝑛) − 1) ↔ 𝑁 ∥ ((𝐴↑𝑛) − 1)))
23	22	rabbidv 2719	. . . . . 6 ⊢ (𝑥 = 𝐴 → {𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝑥↑𝑛) − 1)} = {𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝐴↑𝑛) − 1)})
24	23	infeq1d 6989	. . . . 5 ⊢ (𝑥 = 𝐴 → inf({𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝑥↑𝑛) − 1)}, ℝ, < ) = inf({𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝐴↑𝑛) − 1)}, ℝ, < ))
25		eqid 2170	. . . . 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 7908	. . . . . 6 ⊢ ℝ ∈ V
27		infex2g 7011	. . . . . 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 5573	. . . 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 134	. . 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 2223	. 2 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1)) → ((odℤ‘𝑁)‘𝐴) = inf({𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝐴↑𝑛) − 1)}, ℝ, < ))
32	31	3impb 1194	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 103   ∧ w3a 973   = wceq 1348   ∈ wcel 2141  {crab 2452  Vcvv 2730   class class class wbr 3989   ↦ cmpt 4050  ‘cfv 5198  (class class class)co 5853  infcinf 6960  ℝcr 7773  1c1 7775   < clt 7954   − cmin 8090  ℕcn 8878  ℤcz 9212  ↑cexp 10475   ∥ cdvds 11749   gcd cgcd 11897  od_ℤcodz 12162

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-cnex 7865  ax-resscn 7866

This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-sup 6961  df-inf 6962  df-neg 8093  df-z 9213  df-odz 12164

This theorem is referenced by:  odzcllem  12196  odzdvds  12199