Theorem odzval 13049

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 6096	. . . . . . . . 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 6095	. . . . . . . . 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 4134	. . . . . . . 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 7358	. . . . . 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 4214	. . . . 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 13017	. . . . 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 9664	. . . . . 6 |- ZZ e. _V
14	13	mptrabex 5949	. . . . 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 5785	. . . 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 5700	. . 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 6095	. . . . . 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 6095	. . . . . . . . 9 |- ( x = A -> ( x ^ n ) = ( A ^ n ) )
21	20	oveq1d 6103	. . . . . . . 8 |- ( x = A -> ( ( x ^ n ) - 1 ) = ( ( A ^ n ) - 1 ) )
22	21	breq2d 4143	. . . . . . 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 7358	. . . . 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 8319	. . . . . 6 |- RR e. _V
27		infex2g 7380	. . . . . 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 5785	. . . 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 4131   |-> cmpt 4193   ` cfv 5380  (class class class)co 6088  infcinf 7329   RRcr 8184   1c1 8186   < clt 8366   - cmin 8504   NNcn 9313   ZZcz 9655   ^cexp 10996   || cdvds 12579   gcd cgcd 12755   odZcodz 13015

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 4248  ax-sep 4251  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-cnex 8276  ax-resscn 8277

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 4132  df-opab 4194  df-mpt 4195  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-ov 6091  df-sup 7330  df-inf 7331  df-neg 8507  df-z 9656  df-odz 13017

This theorem is used by:  odzcllem  13050  odzdvds  13053