Theorem hashgcdeq 12962

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 2244	. 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 2244	. 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 12507	. . . . 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 12942	. . . 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 9605	. . . . . 6 |- 0 e. ZZ
8	4	nnzd 9717	. . . . . 6 |- ( ( ( M e. NN /\ N e. NN ) /\ N || M ) -> ( M / N ) e. ZZ )
9		fzofig 10818	. . . . . 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 10503	. . . . . . . . . 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 12689	. . . . . . . 8 |- ( ( ( ( M e. NN /\ N e. NN ) /\ N || M ) /\ y e. ( 0..^ ( M / N ) ) ) -> ( y gcd ( M / N ) ) e. NN0 )
15	14	nn0zd 9716	. . . . . . 7 |- ( ( ( ( M e. NN /\ N e. NN ) /\ N || M ) /\ y e. ( 0..^ ( M / N ) ) ) -> ( y gcd ( M / N ) ) e. ZZ )
16		1zzd 9621	. . . . . . 7 |- ( ( ( ( M e. NN /\ N e. NN ) /\ N || M ) /\ y e. ( 0..^ ( M / N ) ) ) -> 1 e. ZZ )
17		zdceq 9670	. . . . . . 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 2617	. . . . 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 7210	. . . 4 |- ( ( ( M e. NN /\ N e. NN ) /\ N || M ) -> { y e. ( 0..^ ( M / N ) ) | ( y gcd ( M / N ) ) = 1 } e. Fin )
21		eqid 2234	. . . . . 6 |- { y e. ( 0..^ ( M / N ) ) | ( y gcd ( M / N ) ) = 1 } = { y e. ( 0..^ ( M / N ) ) | ( y gcd ( M / N ) ) = 1 }
22		eqid 2234	. . . . . 6 |- { x e. ( 0..^ M ) | ( x gcd M ) = N } = { x e. ( 0..^ M ) | ( x gcd M ) = N }
23		eqid 2234	. . . . . 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 12960	. . . . 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 1230	. . . 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 11177	. . 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 2268	. 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 10503	. . . . . . . . . . . . 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 9613	. . . . . . . . . . . . 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 12684	. . . . . . . . . . . 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 4138	. . . . . . . . 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 2616	. . . . 5 |- ( ( ( M e. NN /\ N e. NN ) /\ -. N || M ) -> A. x e. ( 0..^ M ) -. ( x gcd M ) = N )
41		rabeq0 3542	. . . . 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 5679	. . 3 |- ( ( ( M e. NN /\ N e. NN ) /\ -. N || M ) -> (♯ ` { x e. ( 0..^ M ) | ( x gcd M ) = N } ) = (♯ ` (/) ) )
44		hash0 11184	. . 3 |- (♯ ` (/) ) = 0
45	43, 44	eqtrdi 2283	. 2 |- ( ( ( M e. NN /\ N e. NN ) /\ -. N || M ) -> (♯ ` { x e. ( 0..^ M ) | ( x gcd M ) = N } ) = 0 )
46		dvdsdc 12509	. . . 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 3660	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 842   = wceq 1398   e. wcel 2205   A.wral 2522   {crab 2526   (/)c0 3512   ifcif 3624   class class class wbr 4114   |-> cmpt 4176   -1-1-onto->wf1o 5356   ` cfv 5357  (class class class)co 6058   Fincfn 6988   0cc0 8143   1c1 8144   x. cmul 8148   / cdiv 8963   NNcn 9254   ZZcz 9594  ..^cfzo 10498  ♯chash 11163   || cdvds 12498   gcd cgcd 12674   phicphi 12931

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 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 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261  ax-arch 8262  ax-caucvg 8263

This theorem depends on definitions:  df-bi 117  df-stab 839  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-1o 6660  df-er 6780  df-en 6989  df-dom 6990  df-fin 6991  df-sup 7288  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-div 8964  df-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-n0 9514  df-z 9595  df-uz 9872  df-q 9970  df-rp 10005  df-fz 10362  df-fzo 10499  df-fl 10654  df-mod 10709  df-seqfrec 10834  df-exp 10925  df-ihash 11164  df-cj 11552  df-re 11553  df-im 11554  df-rsqrt 11708  df-abs 11709  df-dvds 12499  df-gcd 12675  df-phi 12933

This theorem is referenced by:  phisum  12963