Theorem odzval 12939

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 6058	. . . . . . . . 9 |- ( m = N -> ( x gcd m ) = ( x gcd N ) )
2	1	eqeq1d 2241	. . . . . . . 8 |- ( m = N -> ( ( x gcd m ) = 1 <-> ( x gcd N ) = 1 ) )
3	2	rabbidv 2802	. . . . . . 7 |- ( m = N -> { x e. ZZ | ( x gcd m ) = 1 } = { x e. ZZ | ( x gcd N ) = 1 } )
4		oveq1 6057	. . . . . . . . 9 |- ( n = x -> ( n gcd N ) = ( x gcd N ) )
5	4	eqeq1d 2241	. . . . . . . 8 |- ( n = x -> ( ( n gcd N ) = 1 <-> ( x gcd N ) = 1 ) )
6	5	cbvrabv 2812	. . . . . . 7 |- { n e. ZZ | ( n gcd N ) = 1 } = { x e. ZZ | ( x gcd N ) = 1 }
7	3, 6	eqtr4di 2283	. . . . . 6 |- ( m = N -> { x e. ZZ | ( x gcd m ) = 1 } = { n e. ZZ | ( n gcd N ) = 1 } )
8		breq1 4112	. . . . . . . 8 |- ( m = N -> ( m || ( ( x ^ n ) - 1 ) <-> N || ( ( x ^ n ) - 1 ) ) )
9	8	rabbidv 2802	. . . . . . 7 |- ( m = N -> { n e. NN | m || ( ( x ^ n ) - 1 ) } = { n e. NN | N || ( ( x ^ n ) - 1 ) } )
10	9	infeq1d 7303	. . . . . 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 4192	. . . . 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 12907	. . . . 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 9586	. . . . . 6 |- ZZ e. _V
14	13	mptrabex 5914	. . . . 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 5754	. . . 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 5672	. . 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 6057	. . . . . 6 |- ( n = A -> ( n gcd N ) = ( A gcd N ) )
18	17	eqeq1d 2241	. . . . 5 |- ( n = A -> ( ( n gcd N ) = 1 <-> ( A gcd N ) = 1 ) )
19	18	elrab 2973	. . . 4 |- ( A e. { n e. ZZ | ( n gcd N ) = 1 } <-> ( A e. ZZ /\ ( A gcd N ) = 1 ) )
20		oveq1 6057	. . . . . . . . 9 |- ( x = A -> ( x ^ n ) = ( A ^ n ) )
21	20	oveq1d 6065	. . . . . . . 8 |- ( x = A -> ( ( x ^ n ) - 1 ) = ( ( A ^ n ) - 1 ) )
22	21	breq2d 4121	. . . . . . 7 |- ( x = A -> ( N || ( ( x ^ n ) - 1 ) <-> N || ( ( A ^ n ) - 1 ) ) )
23	22	rabbidv 2802	. . . . . 6 |- ( x = A -> { n e. NN | N || ( ( x ^ n ) - 1 ) } = { n e. NN | N || ( ( A ^ n ) - 1 ) } )
24	23	infeq1d 7303	. . . . 5 |- ( x = A -> inf ( { n e. NN | N || ( ( x ^ n ) - 1 ) } , RR , < ) = inf ( { n e. NN | N || ( ( A ^ n ) - 1 ) } , RR , < ) )
25		eqid 2232	. . . . 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 8261	. . . . . 6 |- RR e. _V
27		infex2g 7325	. . . . . 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 5754	. . . 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 2285	. 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 1226	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 104   /\ w3a 1005   = wceq 1398   e. wcel 2203   {crab 2524   _Vcvv 2813   class class class wbr 4109   |-> cmpt 4171   ` cfv 5352  (class class class)co 6050  infcinf 7274   RRcr 8126   1c1 8128   < clt 8308   - cmin 8444   NNcn 9237   ZZcz 9577   ^cexp 10900   || cdvds 12473   gcd cgcd 12649   odZcodz 12905

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-cnex 8218  ax-resscn 8219

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-ov 6053  df-sup 7275  df-inf 7276  df-neg 8447  df-z 9578  df-odz 12907

This theorem is referenced by:  odzcllem  12940  odzdvds  12943