Theorem hashgcdeq 12802

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 2239	. 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 2239	. 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 12347	. . . . 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 12782	. . . 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 9480	. . . . . 6 |- 0 e. ZZ
8	4	nnzd 9591	. . . . . 6 |- ( ( ( M e. NN /\ N e. NN ) /\ N || M ) -> ( M / N ) e. ZZ )
9		fzofig 10684	. . . . . 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 10372	. . . . . . . . . 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 12529	. . . . . . . 8 |- ( ( ( ( M e. NN /\ N e. NN ) /\ N || M ) /\ y e. ( 0..^ ( M / N ) ) ) -> ( y gcd ( M / N ) ) e. NN0 )
15	14	nn0zd 9590	. . . . . . 7 |- ( ( ( ( M e. NN /\ N e. NN ) /\ N || M ) /\ y e. ( 0..^ ( M / N ) ) ) -> ( y gcd ( M / N ) ) e. ZZ )
16		1zzd 9496	. . . . . . 7 |- ( ( ( ( M e. NN /\ N e. NN ) /\ N || M ) /\ y e. ( 0..^ ( M / N ) ) ) -> 1 e. ZZ )
17		zdceq 9545	. . . . . . 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 2603	. . . . 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 7121	. . . 4 |- ( ( ( M e. NN /\ N e. NN ) /\ N || M ) -> { y e. ( 0..^ ( M / N ) ) | ( y gcd ( M / N ) ) = 1 } e. Fin )
21		eqid 2229	. . . . . 6 |- { y e. ( 0..^ ( M / N ) ) | ( y gcd ( M / N ) ) = 1 } = { y e. ( 0..^ ( M / N ) ) | ( y gcd ( M / N ) ) = 1 }
22		eqid 2229	. . . . . 6 |- { x e. ( 0..^ M ) | ( x gcd M ) = N } = { x e. ( 0..^ M ) | ( x gcd M ) = N }
23		eqid 2229	. . . . . 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 12800	. . . . 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 1227	. . . 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 11041	. . 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 2263	. 2 |- ( ( ( M e. NN /\ N e. NN ) /\ N || M ) -> (♯ ` { x e. ( 0..^ M ) | ( x gcd M ) = N } ) = ( phi ` ( M / N ) ) )
28		simprr 531	. . . . . . . . . 10 |- ( ( ( M e. NN /\ N e. NN ) /\ ( x e. ( 0..^ M ) /\ ( x gcd M ) = N ) ) -> ( x gcd M ) = N )
29		elfzoelz 10372	. . . . . . . . . . . . 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 9488	. . . . . . . . . . . . 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 12524	. . . . . . . . . . . 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 4110	. . . . . . . . 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 634	. . . . . . 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 2602	. . . . 5 |- ( ( ( M e. NN /\ N e. NN ) /\ -. N || M ) -> A. x e. ( 0..^ M ) -. ( x gcd M ) = N )
41		rabeq0 3522	. . . . 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 5639	. . 3 |- ( ( ( M e. NN /\ N e. NN ) /\ -. N || M ) -> (♯ ` { x e. ( 0..^ M ) | ( x gcd M ) = N } ) = (♯ ` (/) ) )
44		hash0 11048	. . 3 |- (♯ ` (/) ) = 0
45	43, 44	eqtrdi 2278	. 2 |- ( ( ( M e. NN /\ N e. NN ) /\ -. N || M ) -> (♯ ` { x e. ( 0..^ M ) | ( x gcd M ) = N } ) = 0 )
46		dvdsdc 12349	. . . 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 3637	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 839   = wceq 1395   e. wcel 2200   A.wral 2508   {crab 2512   (/)c0 3492   ifcif 3603   class class class wbr 4086   |-> cmpt 4148   -1-1-onto->wf1o 5323   ` cfv 5324  (class class class)co 6013   Fincfn 6904   0cc0 8022   1c1 8023   x. cmul 8027   / cdiv 8842   NNcn 9133   ZZcz 9469  ..^cfzo 10367  ♯chash 11027   || cdvds 12338   gcd cgcd 12514   phicphi 12771

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-mulrcl 8121  ax-addcom 8122  ax-mulcom 8123  ax-addass 8124  ax-mulass 8125  ax-distr 8126  ax-i2m1 8127  ax-0lt1 8128  ax-1rid 8129  ax-0id 8130  ax-rnegex 8131  ax-precex 8132  ax-cnre 8133  ax-pre-ltirr 8134  ax-pre-ltwlin 8135  ax-pre-lttrn 8136  ax-pre-apti 8137  ax-pre-ltadd 8138  ax-pre-mulgt0 8139  ax-pre-mulext 8140  ax-arch 8141  ax-caucvg 8142

This theorem depends on definitions:  df-bi 117  df-stab 836  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-po 4391  df-iso 4392  df-iord 4461  df-on 4463  df-ilim 4464  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-frec 6552  df-1o 6577  df-er 6697  df-en 6905  df-dom 6906  df-fin 6907  df-sup 7174  df-pnf 8206  df-mnf 8207  df-xr 8208  df-ltxr 8209  df-le 8210  df-sub 8342  df-neg 8343  df-reap 8745  df-ap 8752  df-div 8843  df-inn 9134  df-2 9192  df-3 9193  df-4 9194  df-n0 9393  df-z 9470  df-uz 9746  df-q 9844  df-rp 9879  df-fz 10234  df-fzo 10368  df-fl 10520  df-mod 10575  df-seqfrec 10700  df-exp 10791  df-ihash 11028  df-cj 11393  df-re 11394  df-im 11395  df-rsqrt 11549  df-abs 11550  df-dvds 12339  df-gcd 12515  df-phi 12773

This theorem is referenced by:  phisum  12803