Theorem phicl2 15473

Description: Bounds and closure for the value of the Euler phi function. (Contributed by Mario Carneiro, 23-Feb-2014.)

Assertion

Ref	Expression
phicl2	|- ( N e. NN -> ( phi ` N ) e. ( 1 ... N ) )

Proof of Theorem phicl2

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		phival 15472	. 2 |- ( N e. NN -> ( phi ` N ) = ( # ` { x e. ( 1 ... N ) | ( x gcd N ) = 1 } ) )
2		fzfi 12771	. . . . . . 7 |- ( 1 ... N ) e. Fin
3		ssrab2 3687	. . . . . . 7 |- { x e. ( 1 ... N ) | ( x gcd N ) = 1 } C_ ( 1 ... N )
4		ssfi 8180	. . . . . . 7 |- ( ( ( 1 ... N ) e. Fin /\ { x e. ( 1 ... N ) | ( x gcd N ) = 1 } C_ ( 1 ... N ) ) -> { x e. ( 1 ... N ) | ( x gcd N ) = 1 } e. Fin )
5	2, 3, 4	mp2an 708	. . . . . 6 |- { x e. ( 1 ... N ) | ( x gcd N ) = 1 } e. Fin
6		hashcl 13147	. . . . . 6 |- ( { x e. ( 1 ... N ) | ( x gcd N ) = 1 } e. Fin -> ( # ` { x e. ( 1 ... N ) | ( x gcd N ) = 1 } ) e. NN0 )
7	5, 6	ax-mp 5	. . . . 5 |- ( # ` { x e. ( 1 ... N ) | ( x gcd N ) = 1 } ) e. NN0
8	7	nn0zi 11402	. . . 4 |- ( # ` { x e. ( 1 ... N ) | ( x gcd N ) = 1 } ) e. ZZ
9	8	a1i 11	. . 3 |- ( N e. NN -> ( # ` { x e. ( 1 ... N ) | ( x gcd N ) = 1 } ) e. ZZ )
10		1z 11407	. . . . 5 |- 1 e. ZZ
11		hashsng 13159	. . . . 5 |- ( 1 e. ZZ -> ( # ` { 1 } ) = 1 )
12	10, 11	ax-mp 5	. . . 4 |- ( # ` { 1 } ) = 1
13		ovex 6678	. . . . . . 7 |- ( 1 ... N ) e. _V
14	13	rabex 4813	. . . . . 6 |- { x e. ( 1 ... N ) | ( x gcd N ) = 1 } e. _V
15		eluzfz1 12348	. . . . . . . . 9 |- ( N e. ( ZZ>= ` 1 ) -> 1 e. ( 1 ... N ) )
16		nnuz 11723	. . . . . . . . 9 |- NN = ( ZZ>= ` 1 )
17	15, 16	eleq2s 2719	. . . . . . . 8 |- ( N e. NN -> 1 e. ( 1 ... N ) )
18		nnz 11399	. . . . . . . . 9 |- ( N e. NN -> N e. ZZ )
19		1gcd 15254	. . . . . . . . 9 |- ( N e. ZZ -> ( 1 gcd N ) = 1 )
20	18, 19	syl 17	. . . . . . . 8 |- ( N e. NN -> ( 1 gcd N ) = 1 )
21		oveq1 6657	. . . . . . . . . 10 |- ( x = 1 -> ( x gcd N ) = ( 1 gcd N ) )
22	21	eqeq1d 2624	. . . . . . . . 9 |- ( x = 1 -> ( ( x gcd N ) = 1 <-> ( 1 gcd N ) = 1 ) )
23	22	elrab 3363	. . . . . . . 8 |- ( 1 e. { x e. ( 1 ... N ) | ( x gcd N ) = 1 } <-> ( 1 e. ( 1 ... N ) /\ ( 1 gcd N ) = 1 ) )
24	17, 20, 23	sylanbrc 698	. . . . . . 7 |- ( N e. NN -> 1 e. { x e. ( 1 ... N ) | ( x gcd N ) = 1 } )
25	24	snssd 4340	. . . . . 6 |- ( N e. NN -> { 1 } C_ { x e. ( 1 ... N ) | ( x gcd N ) = 1 } )
26		ssdomg 8001	. . . . . 6 |- ( { x e. ( 1 ... N ) | ( x gcd N ) = 1 } e. _V -> ( { 1 } C_ { x e. ( 1 ... N ) | ( x gcd N ) = 1 } -> { 1 } ~<_ { x e. ( 1 ... N ) | ( x gcd N ) = 1 } ) )
27	14, 25, 26	mpsyl 68	. . . . 5 |- ( N e. NN -> { 1 } ~<_ { x e. ( 1 ... N ) | ( x gcd N ) = 1 } )
28		snfi 8038	. . . . . 6 |- { 1 } e. Fin
29		hashdom 13168	. . . . . 6 |- ( ( { 1 } e. Fin /\ { x e. ( 1 ... N ) | ( x gcd N ) = 1 } e. Fin ) -> ( ( # ` { 1 } ) <_ ( # ` { x e. ( 1 ... N ) | ( x gcd N ) = 1 } ) <-> { 1 } ~<_ { x e. ( 1 ... N ) | ( x gcd N ) = 1 } ) )
30	28, 5, 29	mp2an 708	. . . . 5 |- ( ( # ` { 1 } ) <_ ( # ` { x e. ( 1 ... N ) | ( x gcd N ) = 1 } ) <-> { 1 } ~<_ { x e. ( 1 ... N ) | ( x gcd N ) = 1 } )
31	27, 30	sylibr 224	. . . 4 |- ( N e. NN -> ( # ` { 1 } ) <_ ( # ` { x e. ( 1 ... N ) | ( x gcd N ) = 1 } ) )
32	12, 31	syl5eqbrr 4689	. . 3 |- ( N e. NN -> 1 <_ ( # ` { x e. ( 1 ... N ) | ( x gcd N ) = 1 } ) )
33		ssdomg 8001	. . . . . 6 |- ( ( 1 ... N ) e. _V -> ( { x e. ( 1 ... N ) | ( x gcd N ) = 1 } C_ ( 1 ... N ) -> { x e. ( 1 ... N ) | ( x gcd N ) = 1 } ~<_ ( 1 ... N ) ) )
34	13, 3, 33	mp2 9	. . . . 5 |- { x e. ( 1 ... N ) | ( x gcd N ) = 1 } ~<_ ( 1 ... N )
35		hashdom 13168	. . . . . 6 |- ( ( { x e. ( 1 ... N ) | ( x gcd N ) = 1 } e. Fin /\ ( 1 ... N ) e. Fin ) -> ( ( # ` { x e. ( 1 ... N ) | ( x gcd N ) = 1 } ) <_ ( # ` ( 1 ... N ) ) <-> { x e. ( 1 ... N ) | ( x gcd N ) = 1 } ~<_ ( 1 ... N ) ) )
36	5, 2, 35	mp2an 708	. . . . 5 |- ( ( # ` { x e. ( 1 ... N ) | ( x gcd N ) = 1 } ) <_ ( # ` ( 1 ... N ) ) <-> { x e. ( 1 ... N ) | ( x gcd N ) = 1 } ~<_ ( 1 ... N ) )
37	34, 36	mpbir 221	. . . 4 |- ( # ` { x e. ( 1 ... N ) | ( x gcd N ) = 1 } ) <_ ( # ` ( 1 ... N ) )
38		nnnn0 11299	. . . . 5 |- ( N e. NN -> N e. NN0 )
39		hashfz1 13134	. . . . 5 |- ( N e. NN0 -> ( # ` ( 1 ... N ) ) = N )
40	38, 39	syl 17	. . . 4 |- ( N e. NN -> ( # ` ( 1 ... N ) ) = N )
41	37, 40	syl5breq 4690	. . 3 |- ( N e. NN -> ( # ` { x e. ( 1 ... N ) | ( x gcd N ) = 1 } ) <_ N )
42		elfz1 12331	. . . 4 |- ( ( 1 e. ZZ /\ N e. ZZ ) -> ( ( # ` { x e. ( 1 ... N ) | ( x gcd N ) = 1 } ) e. ( 1 ... N ) <-> ( ( # ` { x e. ( 1 ... N ) | ( x gcd N ) = 1 } ) e. ZZ /\ 1 <_ ( # ` { x e. ( 1 ... N ) | ( x gcd N ) = 1 } ) /\ ( # ` { x e. ( 1 ... N ) | ( x gcd N ) = 1 } ) <_ N ) ) )
43	10, 18, 42	sylancr 695	. . 3 |- ( N e. NN -> ( ( # ` { x e. ( 1 ... N ) | ( x gcd N ) = 1 } ) e. ( 1 ... N ) <-> ( ( # ` { x e. ( 1 ... N ) | ( x gcd N ) = 1 } ) e. ZZ /\ 1 <_ ( # ` { x e. ( 1 ... N ) | ( x gcd N ) = 1 } ) /\ ( # ` { x e. ( 1 ... N ) | ( x gcd N ) = 1 } ) <_ N ) ) )
44	9, 32, 41, 43	mpbir3and 1245	. 2 |- ( N e. NN -> ( # ` { x e. ( 1 ... N ) | ( x gcd N ) = 1 } ) e. ( 1 ... N ) )
45	1, 44	eqeltrd 2701	1 |- ( N e. NN -> ( phi ` N ) e. ( 1 ... N ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ w3a 1037   = wceq 1483   e. wcel 1990   {crab 2916   _Vcvv 3200   C_ wss 3574   {csn 4177   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   ~<_ cdom 7953   Fincfn 7955   1c1 9937   <_ cle 10075   NNcn 11020   NN0cn0 11292   ZZcz 11377   ZZ>=cuz 11687   ...cfz 12326   #chash 13117   gcd cgcd 15216   phicphi 15469

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-dvds 14984  df-gcd 15217  df-phi 15471

This theorem is referenced by:  phicl  15474  phi1  15478