Theorem hashgcdeq 12811

Description: Number of initial positive integers with specified divisors. (Contributed by Stefan O'Rear, 12-Sep-2015.)

Assertion

Ref	Expression
hashgcdeq	|- ( ( M e. NN /\ N e. NN ) -> (♯ ` { x e. ( 0..^ M ) | ( x gcd M ) = N } ) = if ( N || M , ( phi ` ( M / N ) ) , 0 ) )

Distinct variable groups:   x, M   x, N

Proof of Theorem hashgcdeq

Dummy variables z y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq2 2241	. 2 |- ( ( phi ` ( M / N ) ) = if ( N || M , ( phi ` ( M / N ) ) , 0 ) -> ( (♯ ` { x e. ( 0..^ M ) | ( x gcd M ) = N } ) = ( phi ` ( M / N ) ) <-> (♯ ` { x e. ( 0..^ M ) | ( x gcd M ) = N } ) = if ( N || M , ( phi ` ( M / N ) ) , 0 ) ) )
2		eqeq2 2241	. 2 |- ( 0 = if ( N || M , ( phi ` ( M / N ) ) , 0 ) -> ( (♯ ` { x e. ( 0..^ M ) | ( x gcd M ) = N } ) = 0 <-> (♯ ` { x e. ( 0..^ M ) | ( x gcd M ) = N } ) = if ( N || M , ( phi ` ( M / N ) ) , 0 ) ) )
3		nndivdvds 12356	. . . . 5 |- ( ( M e. NN /\ N e. NN ) -> ( N || M <-> ( M / N ) e. NN ) )
4	3	biimpa 296	. . . 4 |- ( ( ( M e. NN /\ N e. NN ) /\ N || M ) -> ( M / N ) e. NN )
5		dfphi2 12791	. . . 4 |- ( ( M / N ) e. NN -> ( phi ` ( M / N ) ) = (♯ ` { y e. ( 0..^ ( M / N ) ) | ( y gcd ( M / N ) ) = 1 } ) )
6	4, 5	syl 14	. . 3 |- ( ( ( M e. NN /\ N e. NN ) /\ N || M ) -> ( phi ` ( M / N ) ) = (♯ ` { y e. ( 0..^ ( M / N ) ) | ( y gcd ( M / N ) ) = 1 } ) )
7		0z 9489	. . . . . 6 |- 0 e. ZZ
8	4	nnzd 9600	. . . . . 6 |- ( ( ( M e. NN /\ N e. NN ) /\ N || M ) -> ( M / N ) e. ZZ )
9		fzofig 10693	. . . . . 6 |- ( ( 0 e. ZZ /\ ( M / N ) e. ZZ ) -> ( 0..^ ( M / N ) ) e. Fin )
10	7, 8, 9	sylancr 414	. . . . 5 |- ( ( ( M e. NN /\ N e. NN ) /\ N || M ) -> ( 0..^ ( M / N ) ) e. Fin )
11		elfzoelz 10381	. . . . . . . . . 10 |- ( y e. ( 0..^ ( M / N ) ) -> y e. ZZ )
12	11	adantl 277	. . . . . . . . 9 |- ( ( ( ( M e. NN /\ N e. NN ) /\ N || M ) /\ y e. ( 0..^ ( M / N ) ) ) -> y e. ZZ )
13	8	adantr 276	. . . . . . . . 9 |- ( ( ( ( M e. NN /\ N e. NN ) /\ N || M ) /\ y e. ( 0..^ ( M / N ) ) ) -> ( M / N ) e. ZZ )
14	12, 13	gcdcld 12538	. . . . . . . 8 |- ( ( ( ( M e. NN /\ N e. NN ) /\ N || M ) /\ y e. ( 0..^ ( M / N ) ) ) -> ( y gcd ( M / N ) ) e. NN0 )
15	14	nn0zd 9599	. . . . . . 7 |- ( ( ( ( M e. NN /\ N e. NN ) /\ N || M ) /\ y e. ( 0..^ ( M / N ) ) ) -> ( y gcd ( M / N ) ) e. ZZ )
16		1zzd 9505	. . . . . . 7 |- ( ( ( ( M e. NN /\ N e. NN ) /\ N || M ) /\ y e. ( 0..^ ( M / N ) ) ) -> 1 e. ZZ )
17		zdceq 9554	. . . . . . 7 |- ( ( ( y gcd ( M / N ) ) e. ZZ /\ 1 e. ZZ ) -> DECID ( y gcd ( M / N ) ) = 1 )
18	15, 16, 17	syl2anc 411	. . . . . 6 |- ( ( ( ( M e. NN /\ N e. NN ) /\ N || M ) /\ y e. ( 0..^ ( M / N ) ) ) -> DECID ( y gcd ( M / N ) ) = 1 )
19	18	ralrimiva 2605	. . . . 5 |- ( ( ( M e. NN /\ N e. NN ) /\ N || M ) -> A. y e. ( 0..^ ( M / N ) )DECID ( y gcd ( M / N ) ) = 1 )
20	10, 19	ssfirab 7128	. . . 4 |- ( ( ( M e. NN /\ N e. NN ) /\ N || M ) -> { y e. ( 0..^ ( M / N ) ) | ( y gcd ( M / N ) ) = 1 } e. Fin )
21		eqid 2231	. . . . . 6 |- { y e. ( 0..^ ( M / N ) ) | ( y gcd ( M / N ) ) = 1 } = { y e. ( 0..^ ( M / N ) ) | ( y gcd ( M / N ) ) = 1 }
22		eqid 2231	. . . . . 6 |- { x e. ( 0..^ M ) | ( x gcd M ) = N } = { x e. ( 0..^ M ) | ( x gcd M ) = N }
23		eqid 2231	. . . . . 6 |- ( z e. { y e. ( 0..^ ( M / N ) ) | ( y gcd ( M / N ) ) = 1 } |-> ( z x. N ) ) = ( z e. { y e. ( 0..^ ( M / N ) ) | ( y gcd ( M / N ) ) = 1 } |-> ( z x. N ) )
24	21, 22, 23	hashgcdlem 12809	. . . . 5 |- ( ( M e. NN /\ N e. NN /\ N || M ) -> ( z e. { y e. ( 0..^ ( M / N ) ) | ( y gcd ( M / N ) ) = 1 } |-> ( z x. N ) ) : { y e. ( 0..^ ( M / N ) ) | ( y gcd ( M / N ) ) = 1 } -1-1-onto-> { x e. ( 0..^ M ) | ( x gcd M ) = N } )
25	24	3expa 1229	. . . 4 |- ( ( ( M e. NN /\ N e. NN ) /\ N || M ) -> ( z e. { y e. ( 0..^ ( M / N ) ) | ( y gcd ( M / N ) ) = 1 } |-> ( z x. N ) ) : { y e. ( 0..^ ( M / N ) ) | ( y gcd ( M / N ) ) = 1 } -1-1-onto-> { x e. ( 0..^ M ) | ( x gcd M ) = N } )
26	20, 25	fihasheqf1od 11050	. . 3 |- ( ( ( M e. NN /\ N e. NN ) /\ N || M ) -> (♯ ` { y e. ( 0..^ ( M / N ) ) | ( y gcd ( M / N ) ) = 1 } ) = (♯ ` { x e. ( 0..^ M ) | ( x gcd M ) = N } ) )
27	6, 26	eqtr2d 2265	. 2 |- ( ( ( M e. NN /\ N e. NN ) /\ N || M ) -> (♯ ` { x e. ( 0..^ M ) | ( x gcd M ) = N } ) = ( phi ` ( M / N ) ) )
28		simprr 533	. . . . . . . . . 10 |- ( ( ( M e. NN /\ N e. NN ) /\ ( x e. ( 0..^ M ) /\ ( x gcd M ) = N ) ) -> ( x gcd M ) = N )
29		elfzoelz 10381	. . . . . . . . . . . . 13 |- ( x e. ( 0..^ M ) -> x e. ZZ )
30	29	ad2antrl 490	. . . . . . . . . . . 12 |- ( ( ( M e. NN /\ N e. NN ) /\ ( x e. ( 0..^ M ) /\ ( x gcd M ) = N ) ) -> x e. ZZ )
31		nnz 9497	. . . . . . . . . . . . 13 |- ( M e. NN -> M e. ZZ )
32	31	ad2antrr 488	. . . . . . . . . . . 12 |- ( ( ( M e. NN /\ N e. NN ) /\ ( x e. ( 0..^ M ) /\ ( x gcd M ) = N ) ) -> M e. ZZ )
33		gcddvds 12533	. . . . . . . . . . . 12 |- ( ( x e. ZZ /\ M e. ZZ ) -> ( ( x gcd M ) || x /\ ( x gcd M ) || M ) )
34	30, 32, 33	syl2anc 411	. . . . . . . . . . 11 |- ( ( ( M e. NN /\ N e. NN ) /\ ( x e. ( 0..^ M ) /\ ( x gcd M ) = N ) ) -> ( ( x gcd M ) || x /\ ( x gcd M ) || M ) )
35	34	simprd 114	. . . . . . . . . 10 |- ( ( ( M e. NN /\ N e. NN ) /\ ( x e. ( 0..^ M ) /\ ( x gcd M ) = N ) ) -> ( x gcd M ) || M )
36	28, 35	eqbrtrrd 4112	. . . . . . . . 9 |- ( ( ( M e. NN /\ N e. NN ) /\ ( x e. ( 0..^ M ) /\ ( x gcd M ) = N ) ) -> N || M )
37	36	expr 375	. . . . . . . 8 |- ( ( ( M e. NN /\ N e. NN ) /\ x e. ( 0..^ M ) ) -> ( ( x gcd M ) = N -> N || M ) )
38	37	con3d 636	. . . . . . 7 |- ( ( ( M e. NN /\ N e. NN ) /\ x e. ( 0..^ M ) ) -> ( -. N || M -> -. ( x gcd M ) = N ) )
39	38	impancom 260	. . . . . 6 |- ( ( ( M e. NN /\ N e. NN ) /\ -. N || M ) -> ( x e. ( 0..^ M ) -> -. ( x gcd M ) = N ) )
40	39	ralrimiv 2604	. . . . 5 |- ( ( ( M e. NN /\ N e. NN ) /\ -. N || M ) -> A. x e. ( 0..^ M ) -. ( x gcd M ) = N )
41		rabeq0 3524	. . . . 5 |- ( { x e. ( 0..^ M ) | ( x gcd M ) = N } = (/) <-> A. x e. ( 0..^ M ) -. ( x gcd M ) = N )
42	40, 41	sylibr 134	. . . 4 |- ( ( ( M e. NN /\ N e. NN ) /\ -. N || M ) -> { x e. ( 0..^ M ) | ( x gcd M ) = N } = (/) )
43	42	fveq2d 5643	. . 3 |- ( ( ( M e. NN /\ N e. NN ) /\ -. N || M ) -> (♯ ` { x e. ( 0..^ M ) | ( x gcd M ) = N } ) = (♯ ` (/) ) )
44		hash0 11057	. . 3 |- (♯ ` (/) ) = 0
45	43, 44	eqtrdi 2280	. 2 |- ( ( ( M e. NN /\ N e. NN ) /\ -. N || M ) -> (♯ ` { x e. ( 0..^ M ) | ( x gcd M ) = N } ) = 0 )
46		dvdsdc 12358	. . . 4 |- ( ( N e. NN /\ M e. ZZ ) -> DECID N || M )
47	31, 46	sylan2 286	. . 3 |- ( ( N e. NN /\ M e. NN ) -> DECID N || M )
48	47	ancoms 268	. 2 |- ( ( M e. NN /\ N e. NN ) -> DECID N || M )
49	1, 2, 27, 45, 48	ifbothdadc 3639	1 |- ( ( M e. NN /\ N e. NN ) -> (♯ ` { x e. ( 0..^ M ) | ( x gcd M ) = N } ) = if ( N || M , ( phi ` ( M / N ) ) , 0 ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104  DECID wdc 841   = wceq 1397   e. wcel 2202   A.wral 2510   {crab 2514   (/)c0 3494   ifcif 3605   class class class wbr 4088   |-> cmpt 4150   -1-1-onto->wf1o 5325   ` cfv 5326  (class class class)co 6017   Fincfn 6908   0cc0 8031   1c1 8032   x. cmul 8036   / cdiv 8851   NNcn 9142   ZZcz 9478  ..^cfzo 10376  ♯chash 11036   || cdvds 12347   gcd cgcd 12523   phicphi 12780

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-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-mulrcl 8130  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-precex 8141  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147  ax-pre-mulgt0 8148  ax-pre-mulext 8149  ax-arch 8150  ax-caucvg 8151

This theorem depends on definitions:  df-bi 117  df-stab 838  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-recs 6470  df-frec 6556  df-1o 6581  df-er 6701  df-en 6909  df-dom 6910  df-fin 6911  df-sup 7182  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-reap 8754  df-ap 8761  df-div 8852  df-inn 9143  df-2 9201  df-3 9202  df-4 9203  df-n0 9402  df-z 9479  df-uz 9755  df-q 9853  df-rp 9888  df-fz 10243  df-fzo 10377  df-fl 10529  df-mod 10584  df-seqfrec 10709  df-exp 10800  df-ihash 11037  df-cj 11402  df-re 11403  df-im 11404  df-rsqrt 11558  df-abs 11559  df-dvds 12348  df-gcd 12524  df-phi 12782

This theorem is referenced by:  phisum  12812