Theorem odzval 16721

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 7366	. . . . . . . . 9 ⊢ (𝑚 = 𝑁 → (𝑥 gcd 𝑚) = (𝑥 gcd 𝑁))
2	1	eqeq1d 2738	. . . . . . . 8 ⊢ (𝑚 = 𝑁 → ((𝑥 gcd 𝑚) = 1 ↔ (𝑥 gcd 𝑁) = 1))
3	2	rabbidv 3406	. . . . . . 7 ⊢ (𝑚 = 𝑁 → {𝑥 ∈ ℤ ∣ (𝑥 gcd 𝑚) = 1} = {𝑥 ∈ ℤ ∣ (𝑥 gcd 𝑁) = 1})
4		oveq1 7365	. . . . . . . . 9 ⊢ (𝑛 = 𝑥 → (𝑛 gcd 𝑁) = (𝑥 gcd 𝑁))
5	4	eqeq1d 2738	. . . . . . . 8 ⊢ (𝑛 = 𝑥 → ((𝑛 gcd 𝑁) = 1 ↔ (𝑥 gcd 𝑁) = 1))
6	5	cbvrabv 3409	. . . . . . 7 ⊢ {𝑛 ∈ ℤ ∣ (𝑛 gcd 𝑁) = 1} = {𝑥 ∈ ℤ ∣ (𝑥 gcd 𝑁) = 1}
7	3, 6	eqtr4di 2789	. . . . . 6 ⊢ (𝑚 = 𝑁 → {𝑥 ∈ ℤ ∣ (𝑥 gcd 𝑚) = 1} = {𝑛 ∈ ℤ ∣ (𝑛 gcd 𝑁) = 1})
8		breq1 5101	. . . . . . . 8 ⊢ (𝑚 = 𝑁 → (𝑚 ∥ ((𝑥↑𝑛) − 1) ↔ 𝑁 ∥ ((𝑥↑𝑛) − 1)))
9	8	rabbidv 3406	. . . . . . 7 ⊢ (𝑚 = 𝑁 → {𝑛 ∈ ℕ ∣ 𝑚 ∥ ((𝑥↑𝑛) − 1)} = {𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝑥↑𝑛) − 1)})
10	9	infeq1d 9383	. . . . . 6 ⊢ (𝑚 = 𝑁 → inf({𝑛 ∈ ℕ ∣ 𝑚 ∥ ((𝑥↑𝑛) − 1)}, ℝ, < ) = inf({𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝑥↑𝑛) − 1)}, ℝ, < ))
11	7, 10	mpteq12dv 5185	. . . . 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 16694	. . . . 5 ⊢ odℤ = (𝑚 ∈ ℕ ↦ (𝑥 ∈ {𝑥 ∈ ℤ ∣ (𝑥 gcd 𝑚) = 1} ↦ inf({𝑛 ∈ ℕ ∣ 𝑚 ∥ ((𝑥↑𝑛) − 1)}, ℝ, < )))
13		zex 12499	. . . . . 6 ⊢ ℤ ∈ V
14	13	mptrabex 7171	. . . . 5 ⊢ (𝑥 ∈ {𝑛 ∈ ℤ ∣ (𝑛 gcd 𝑁) = 1} ↦ inf({𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝑥↑𝑛) − 1)}, ℝ, < )) ∈ V
15	11, 12, 14	fvmpt 6941	. . . 4 ⊢ (𝑁 ∈ ℕ → (odℤ‘𝑁) = (𝑥 ∈ {𝑛 ∈ ℤ ∣ (𝑛 gcd 𝑁) = 1} ↦ inf({𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝑥↑𝑛) − 1)}, ℝ, < )))
16	15	fveq1d 6836	. . 3 ⊢ (𝑁 ∈ ℕ → ((odℤ‘𝑁)‘𝐴) = ((𝑥 ∈ {𝑛 ∈ ℤ ∣ (𝑛 gcd 𝑁) = 1} ↦ inf({𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝑥↑𝑛) − 1)}, ℝ, < ))‘𝐴))
17		oveq1 7365	. . . . . 6 ⊢ (𝑛 = 𝐴 → (𝑛 gcd 𝑁) = (𝐴 gcd 𝑁))
18	17	eqeq1d 2738	. . . . 5 ⊢ (𝑛 = 𝐴 → ((𝑛 gcd 𝑁) = 1 ↔ (𝐴 gcd 𝑁) = 1))
19	18	elrab 3646	. . . 4 ⊢ (𝐴 ∈ {𝑛 ∈ ℤ ∣ (𝑛 gcd 𝑁) = 1} ↔ (𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1))
20		oveq1 7365	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → (𝑥↑𝑛) = (𝐴↑𝑛))
21	20	oveq1d 7373	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → ((𝑥↑𝑛) − 1) = ((𝐴↑𝑛) − 1))
22	21	breq2d 5110	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑁 ∥ ((𝑥↑𝑛) − 1) ↔ 𝑁 ∥ ((𝐴↑𝑛) − 1)))
23	22	rabbidv 3406	. . . . . 6 ⊢ (𝑥 = 𝐴 → {𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝑥↑𝑛) − 1)} = {𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝐴↑𝑛) − 1)})
24	23	infeq1d 9383	. . . . 5 ⊢ (𝑥 = 𝐴 → inf({𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝑥↑𝑛) − 1)}, ℝ, < ) = inf({𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝐴↑𝑛) − 1)}, ℝ, < ))
25		eqid 2736	. . . . 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 11215	. . . . . 6 ⊢ < Or ℝ
27	26	infex 9400	. . . . 5 ⊢ inf({𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝐴↑𝑛) − 1)}, ℝ, < ) ∈ V
28	24, 25, 27	fvmpt 6941	. . . 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 235	. . 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 2791	. 2 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1)) → ((odℤ‘𝑁)‘𝐴) = inf({𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝐴↑𝑛) − 1)}, ℝ, < ))
31	30	3impb 1114	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 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  {crab 3399   class class class wbr 5098   ↦ cmpt 5179  ‘cfv 6492  (class class class)co 7358  infcinf 9346  ℝcr 11027  1c1 11029   < clt 11168   − cmin 11366  ℕcn 12147  ℤcz 12490  ↑cexp 13986   ∥ cdvds 16181   gcd cgcd 16423  od_ℤcodz 16692

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-cnex 11084  ax-resscn 11085  ax-pre-lttri 11102  ax-pre-lttrn 11103

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-po 5532  df-so 5533  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-er 8635  df-en 8886  df-dom 8887  df-sdom 8888  df-sup 9347  df-inf 9348  df-pnf 11170  df-mnf 11171  df-ltxr 11173  df-neg 11369  df-z 12491  df-odz 16694

This theorem is referenced by:  odzcllem  16722  odzdvds  16725  hashscontpow1  42397