Theorem odzval 15496

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 e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) -> ( ( odZ ` N ) ` A ) = inf ( { n e. NN | N || ( ( A ^ n ) - 1 ) } , RR , < ) )

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 6658	. . . . . . . . 9 |- ( m = N -> ( x gcd m ) = ( x gcd N ) )
2	1	eqeq1d 2624	. . . . . . . 8 |- ( m = N -> ( ( x gcd m ) = 1 <-> ( x gcd N ) = 1 ) )
3	2	rabbidv 3189	. . . . . . 7 |- ( m = N -> { x e. ZZ | ( x gcd m ) = 1 } = { x e. ZZ | ( x gcd N ) = 1 } )
4		oveq1 6657	. . . . . . . . 9 |- ( n = x -> ( n gcd N ) = ( x gcd N ) )
5	4	eqeq1d 2624	. . . . . . . 8 |- ( n = x -> ( ( n gcd N ) = 1 <-> ( x gcd N ) = 1 ) )
6	5	cbvrabv 3199	. . . . . . 7 |- { n e. ZZ | ( n gcd N ) = 1 } = { x e. ZZ | ( x gcd N ) = 1 }
7	3, 6	syl6eqr 2674	. . . . . 6 |- ( m = N -> { x e. ZZ | ( x gcd m ) = 1 } = { n e. ZZ | ( n gcd N ) = 1 } )
8		breq1 4656	. . . . . . . 8 |- ( m = N -> ( m || ( ( x ^ n ) - 1 ) <-> N || ( ( x ^ n ) - 1 ) ) )
9	8	rabbidv 3189	. . . . . . 7 |- ( m = N -> { n e. NN | m || ( ( x ^ n ) - 1 ) } = { n e. NN | N || ( ( x ^ n ) - 1 ) } )
10	9	infeq1d 8383	. . . . . 6 |- ( m = N -> inf ( { n e. NN | m || ( ( x ^ n ) - 1 ) } , RR , < ) = inf ( { n e. NN | N || ( ( x ^ n ) - 1 ) } , RR , < ) )
11	7, 10	mpteq12dv 4733	. . . . 5 |- ( m = N -> ( x e. { x e. ZZ | ( x gcd m ) = 1 } |-> inf ( { n e. NN | m || ( ( x ^ n ) - 1 ) } , RR , < ) ) = ( x e. { n e. ZZ | ( n gcd N ) = 1 } |-> inf ( { n e. NN | N || ( ( x ^ n ) - 1 ) } , RR , < ) ) )
12		df-odz 15470	. . . . 5 |- odZ = ( m e. NN |-> ( x e. { x e. ZZ | ( x gcd m ) = 1 } |-> inf ( { n e. NN | m || ( ( x ^ n ) - 1 ) } , RR , < ) ) )
13		zex 11386	. . . . . 6 |- ZZ e. _V
14	13	mptrabex 6488	. . . . 5 |- ( x e. { n e. ZZ | ( n gcd N ) = 1 } |-> inf ( { n e. NN | N || ( ( x ^ n ) - 1 ) } , RR , < ) ) e. _V
15	11, 12, 14	fvmpt 6282	. . . 4 |- ( N e. NN -> ( odZ ` N ) = ( x e. { n e. ZZ | ( n gcd N ) = 1 } |-> inf ( { n e. NN | N || ( ( x ^ n ) - 1 ) } , RR , < ) ) )
16	15	fveq1d 6193	. . 3 |- ( N e. NN -> ( ( odZ ` N ) ` A ) = ( ( x e. { n e. ZZ | ( n gcd N ) = 1 } |-> inf ( { n e. NN | N || ( ( x ^ n ) - 1 ) } , RR , < ) ) ` A ) )
17		oveq1 6657	. . . . . 6 |- ( n = A -> ( n gcd N ) = ( A gcd N ) )
18	17	eqeq1d 2624	. . . . 5 |- ( n = A -> ( ( n gcd N ) = 1 <-> ( A gcd N ) = 1 ) )
19	18	elrab 3363	. . . 4 |- ( A e. { n e. ZZ | ( n gcd N ) = 1 } <-> ( A e. ZZ /\ ( A gcd N ) = 1 ) )
20		oveq1 6657	. . . . . . . . 9 |- ( x = A -> ( x ^ n ) = ( A ^ n ) )
21	20	oveq1d 6665	. . . . . . . 8 |- ( x = A -> ( ( x ^ n ) - 1 ) = ( ( A ^ n ) - 1 ) )
22	21	breq2d 4665	. . . . . . 7 |- ( x = A -> ( N || ( ( x ^ n ) - 1 ) <-> N || ( ( A ^ n ) - 1 ) ) )
23	22	rabbidv 3189	. . . . . 6 |- ( x = A -> { n e. NN | N || ( ( x ^ n ) - 1 ) } = { n e. NN | N || ( ( A ^ n ) - 1 ) } )
24	23	infeq1d 8383	. . . . 5 |- ( x = A -> inf ( { n e. NN | N || ( ( x ^ n ) - 1 ) } , RR , < ) = inf ( { n e. NN | N || ( ( A ^ n ) - 1 ) } , RR , < ) )
25		eqid 2622	. . . . 5 |- ( x e. { n e. ZZ | ( n gcd N ) = 1 } |-> inf ( { n e. NN | N || ( ( x ^ n ) - 1 ) } , RR , < ) ) = ( x e. { n e. ZZ | ( n gcd N ) = 1 } |-> inf ( { n e. NN | N || ( ( x ^ n ) - 1 ) } , RR , < ) )
26		ltso 10118	. . . . . 6 |- < Or RR
27	26	infex 8399	. . . . 5 |- inf ( { n e. NN | N || ( ( A ^ n ) - 1 ) } , RR , < ) e. _V
28	24, 25, 27	fvmpt 6282	. . . 4 |- ( A e. { n e. ZZ | ( n gcd N ) = 1 } -> ( ( x e. { n e. ZZ | ( n gcd N ) = 1 } |-> inf ( { n e. NN | N || ( ( x ^ n ) - 1 ) } , RR , < ) ) ` A ) = inf ( { n e. NN | N || ( ( A ^ n ) - 1 ) } , RR , < ) )
29	19, 28	sylbir 225	. . 3 |- ( ( A e. ZZ /\ ( A gcd N ) = 1 ) -> ( ( x e. { n e. ZZ | ( n gcd N ) = 1 } |-> inf ( { n e. NN | N || ( ( x ^ n ) - 1 ) } , RR , < ) ) ` A ) = inf ( { n e. NN | N || ( ( A ^ n ) - 1 ) } , RR , < ) )
30	16, 29	sylan9eq 2676	. 2 |- ( ( N e. NN /\ ( A e. ZZ /\ ( A gcd N ) = 1 ) ) -> ( ( odZ ` N ) ` A ) = inf ( { n e. NN | N || ( ( A ^ n ) - 1 ) } , RR , < ) )
31	30	3impb 1260	1 |- ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) -> ( ( odZ ` N ) ` A ) = inf ( { n e. NN | N || ( ( A ^ n ) - 1 ) } , RR , < ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   {crab 2916   class class class wbr 4653   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650  infcinf 8347   RRcr 9935   1c1 9937   < clt 10074   - cmin 10266   NNcn 11020   ZZcz 11377   ^cexp 12860   || cdvds 14983   gcd cgcd 15216   odZcodz 15468

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-pre-lttri 10010  ax-pre-lttrn 10011

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-ltxr 10079  df-neg 10269  df-z 11378  df-odz 15470

This theorem is referenced by:  odzcllem  15497  odzdvds  15500