Theorem odzval 12998

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 6083	. . . . . . . . 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 6082	. . . . . . . . 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 4128	. . . . . . . 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 7342	. . . . . 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 4208	. . . . 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 12966	. . . . 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 9632	. . . . . 6 |- ZZ e. _V
14	13	mptrabex 5936	. . . . 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 5776	. . . 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 5692	. . 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 6082	. . . . . 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 6082	. . . . . . . . 9 |- ( x = A -> ( x ^ n ) = ( A ^ n ) )
21	20	oveq1d 6090	. . . . . . . 8 |- ( x = A -> ( ( x ^ n ) - 1 ) = ( ( A ^ n ) - 1 ) )
22	21	breq2d 4137	. . . . . . 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 7342	. . . . 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 8303	. . . . . 6 |- RR e. _V
27		infex2g 7364	. . . . . 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 5776	. . . 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 , < ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1009   = wceq 1402   e. wcel 2209   {crab 2532   _Vcvv 2821   class class class wbr 4125   |-> cmpt 4187   ` cfv 5372  (class class class)co 6075  infcinf 7313   RRcr 8168   1c1 8170   < clt 8350   - cmin 8487   NNcn 9283   ZZcz 9623   ^cexp 10953   || cdvds 12532   gcd cgcd 12708   odZcodz 12964

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 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 4241  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-cnex 8260  ax-resscn 8261

This theorem 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 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-ov 6078  df-sup 7314  df-inf 7315  df-neg 8490  df-z 9624  df-odz 12966

This theorem is referenced by:  odzcllem  12999  odzdvds  13002