Theorem odzval 13020

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 6093	. . . . . . . . 9 |- ( m = N -> ( x gcd m ) = ( x gcd N ) )
2	1	eqeq1d 2247	. . . . . . . 8 |- ( m = N -> ( ( x gcd m ) = 1 <-> ( x gcd N ) = 1 ) )
3	2	rabbidv 2810	. . . . . . 7 |- ( m = N -> { x e. ZZ | ( x gcd m ) = 1 } = { x e. ZZ | ( x gcd N ) = 1 } )
4		oveq1 6092	. . . . . . . . 9 |- ( n = x -> ( n gcd N ) = ( x gcd N ) )
5	4	eqeq1d 2247	. . . . . . . 8 |- ( n = x -> ( ( n gcd N ) = 1 <-> ( x gcd N ) = 1 ) )
6	5	cbvrabv 2820	. . . . . . 7 |- { n e. ZZ | ( n gcd N ) = 1 } = { x e. ZZ | ( x gcd N ) = 1 }
7	3, 6	eqtr4di 2289	. . . . . 6 |- ( m = N -> { x e. ZZ | ( x gcd m ) = 1 } = { n e. ZZ | ( n gcd N ) = 1 } )
8		breq1 4133	. . . . . . . 8 |- ( m = N -> ( m || ( ( x ^ n ) - 1 ) <-> N || ( ( x ^ n ) - 1 ) ) )
9	8	rabbidv 2810	. . . . . . 7 |- ( m = N -> { n e. NN | m || ( ( x ^ n ) - 1 ) } = { n e. NN | N || ( ( x ^ n ) - 1 ) } )
10	9	infeq1d 7352	. . . . . 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 4213	. . . . 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 12988	. . . . 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 9653	. . . . . 6 |- ZZ e. _V
14	13	mptrabex 5946	. . . . 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 5782	. . . 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 5697	. . 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 6092	. . . . . 6 |- ( n = A -> ( n gcd N ) = ( A gcd N ) )
18	17	eqeq1d 2247	. . . . 5 |- ( n = A -> ( ( n gcd N ) = 1 <-> ( A gcd N ) = 1 ) )
19	18	elrab 2982	. . . 4 |- ( A e. { n e. ZZ | ( n gcd N ) = 1 } <-> ( A e. ZZ /\ ( A gcd N ) = 1 ) )
20		oveq1 6092	. . . . . . . . 9 |- ( x = A -> ( x ^ n ) = ( A ^ n ) )
21	20	oveq1d 6100	. . . . . . . 8 |- ( x = A -> ( ( x ^ n ) - 1 ) = ( ( A ^ n ) - 1 ) )
22	21	breq2d 4142	. . . . . . 7 |- ( x = A -> ( N || ( ( x ^ n ) - 1 ) <-> N || ( ( A ^ n ) - 1 ) ) )
23	22	rabbidv 2810	. . . . . 6 |- ( x = A -> { n e. NN | N || ( ( x ^ n ) - 1 ) } = { n e. NN | N || ( ( A ^ n ) - 1 ) } )
24	23	infeq1d 7352	. . . . 5 |- ( x = A -> inf ( { n e. NN | N || ( ( x ^ n ) - 1 ) } , RR , < ) = inf ( { n e. NN | N || ( ( A ^ n ) - 1 ) } , RR , < ) )
25		eqid 2238	. . . . 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		reex 8313	. . . . . 6 |- RR e. _V
27		infex2g 7374	. . . . . 6 |- ( RR e. _V -> inf ( { n e. NN | N || ( ( A ^ n ) - 1 ) } , RR , < ) e. _V )
28	26, 27	ax-mp 5	. . . . 5 |- inf ( { n e. NN | N || ( ( A ^ n ) - 1 ) } , RR , < ) e. _V
29	24, 25, 28	fvmpt 5782	. . . 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 , < ) )
30	19, 29	sylbir 135	. . 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 , < ) )
31	16, 30	sylan9eq 2291	. 2 |- ( ( N e. NN /\ ( A e. ZZ /\ ( A gcd N ) = 1 ) ) -> ( ( odZ ` N ) ` A ) = inf ( { n e. NN | N || ( ( A ^ n ) - 1 ) } , RR , < ) )
32	31	3impb 1230	1 |- ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) -> ( ( odZ ` N ) ` A ) = inf ( { n e. NN | N || ( ( A ^ n ) - 1 ) } , RR , < ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   /\ w3a 1009   = wceq 1402   e. wcel 2209   {crab 2532   _Vcvv 2821   class class class wbr 4130   |-> cmpt 4192   ` cfv 5377  (class class class)co 6085  infcinf 7323   RRcr 8178   1c1 8180   < clt 8360   - cmin 8497   NNcn 9304   ZZcz 9644   ^cexp 10975   || cdvds 12554   gcd cgcd 12730   odZcodz 12986

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-cnex 8270  ax-resscn 8271

This proof depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-ov 6088  df-sup 7324  df-inf 7325  df-neg 8500  df-z 9645  df-odz 12988

This theorem is used by:  odzcllem  13021  odzdvds  13024