Theorem prmdvdsfmtnof1 41499

Description: The mapping of a Fermat number to its smallest prime factor is a one-to-one function. (Contributed by AV, 4-Aug-2021.)

Hypothesis

Ref	Expression
prmdvdsfmtnof.1	|- F = ( f e. ran FermatNo |-> inf ( { p e. Prime | p || f } , RR , < ) )

Assertion

Ref	Expression
prmdvdsfmtnof1	|- F : ran FermatNo -1-1-> Prime

Distinct variable group:   f, p

Allowed substitution hints:   F( f, p)

Proof of Theorem prmdvdsfmtnof1

Dummy variables g h n are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		prmdvdsfmtnof.1	. . 3 |- F = ( f e. ran FermatNo |-> inf ( { p e. Prime | p || f } , RR , < ) )
2	1	prmdvdsfmtnof 41498	. 2 |- F : ran FermatNo --> Prime
3	1	a1i 11	. . . . . 6 |- ( g e. ran FermatNo -> F = ( f e. ran FermatNo |-> inf ( { p e. Prime | p || f } , RR , < ) ) )
4		breq2 4657	. . . . . . . . 9 |- ( f = g -> ( p || f <-> p || g ) )
5	4	rabbidv 3189	. . . . . . . 8 |- ( f = g -> { p e. Prime | p || f } = { p e. Prime | p || g } )
6	5	infeq1d 8383	. . . . . . 7 |- ( f = g -> inf ( { p e. Prime | p || f } , RR , < ) = inf ( { p e. Prime | p || g } , RR , < ) )
7	6	adantl 482	. . . . . 6 |- ( ( g e. ran FermatNo /\ f = g ) -> inf ( { p e. Prime | p || f } , RR , < ) = inf ( { p e. Prime | p || g } , RR , < ) )
8		id 22	. . . . . 6 |- ( g e. ran FermatNo -> g e. ran FermatNo )
9		ltso 10118	. . . . . . . 8 |- < Or RR
10	9	a1i 11	. . . . . . 7 |- ( g e. ran FermatNo -> < Or RR )
11	10	infexd 8389	. . . . . 6 |- ( g e. ran FermatNo -> inf ( { p e. Prime | p || g } , RR , < ) e. _V )
12	3, 7, 8, 11	fvmptd 6288	. . . . 5 |- ( g e. ran FermatNo -> ( F ` g ) = inf ( { p e. Prime | p || g } , RR , < ) )
13	1	a1i 11	. . . . . 6 |- ( h e. ran FermatNo -> F = ( f e. ran FermatNo |-> inf ( { p e. Prime | p || f } , RR , < ) ) )
14		breq2 4657	. . . . . . . . 9 |- ( f = h -> ( p || f <-> p || h ) )
15	14	rabbidv 3189	. . . . . . . 8 |- ( f = h -> { p e. Prime | p || f } = { p e. Prime | p || h } )
16	15	infeq1d 8383	. . . . . . 7 |- ( f = h -> inf ( { p e. Prime | p || f } , RR , < ) = inf ( { p e. Prime | p || h } , RR , < ) )
17	16	adantl 482	. . . . . 6 |- ( ( h e. ran FermatNo /\ f = h ) -> inf ( { p e. Prime | p || f } , RR , < ) = inf ( { p e. Prime | p || h } , RR , < ) )
18		id 22	. . . . . 6 |- ( h e. ran FermatNo -> h e. ran FermatNo )
19	9	a1i 11	. . . . . . 7 |- ( h e. ran FermatNo -> < Or RR )
20	19	infexd 8389	. . . . . 6 |- ( h e. ran FermatNo -> inf ( { p e. Prime | p || h } , RR , < ) e. _V )
21	13, 17, 18, 20	fvmptd 6288	. . . . 5 |- ( h e. ran FermatNo -> ( F ` h ) = inf ( { p e. Prime | p || h } , RR , < ) )
22	12, 21	eqeqan12d 2638	. . . 4 |- ( ( g e. ran FermatNo /\ h e. ran FermatNo ) -> ( ( F ` g ) = ( F ` h ) <-> inf ( { p e. Prime | p || g } , RR , < ) = inf ( { p e. Prime | p || h } , RR , < ) ) )
23		fmtnorn 41446	. . . . . . 7 |- ( g e. ran FermatNo <-> E. n e. NN0 (FermatNo ` n ) = g )
24		fmtnoge3 41442	. . . . . . . . . . 11 |- ( n e. NN0 -> (FermatNo ` n ) e. ( ZZ>= ` 3 ) )
25		uzuzle23 11729	. . . . . . . . . . 11 |- ( (FermatNo ` n ) e. ( ZZ>= ` 3 ) -> (FermatNo ` n ) e. ( ZZ>= ` 2 ) )
26	24, 25	syl 17	. . . . . . . . . 10 |- ( n e. NN0 -> (FermatNo ` n ) e. ( ZZ>= ` 2 ) )
27	26	adantr 481	. . . . . . . . 9 |- ( ( n e. NN0 /\ (FermatNo ` n ) = g ) -> (FermatNo ` n ) e. ( ZZ>= ` 2 ) )
28		eleq1 2689	. . . . . . . . . 10 |- ( (FermatNo ` n ) = g -> ( (FermatNo ` n ) e. ( ZZ>= ` 2 ) <-> g e. ( ZZ>= ` 2 ) ) )
29	28	adantl 482	. . . . . . . . 9 |- ( ( n e. NN0 /\ (FermatNo ` n ) = g ) -> ( (FermatNo ` n ) e. ( ZZ>= ` 2 ) <-> g e. ( ZZ>= ` 2 ) ) )
30	27, 29	mpbid 222	. . . . . . . 8 |- ( ( n e. NN0 /\ (FermatNo ` n ) = g ) -> g e. ( ZZ>= ` 2 ) )
31	30	rexlimiva 3028	. . . . . . 7 |- ( E. n e. NN0 (FermatNo ` n ) = g -> g e. ( ZZ>= ` 2 ) )
32	23, 31	sylbi 207	. . . . . 6 |- ( g e. ran FermatNo -> g e. ( ZZ>= ` 2 ) )
33		fmtnorn 41446	. . . . . . 7 |- ( h e. ran FermatNo <-> E. n e. NN0 (FermatNo ` n ) = h )
34	26	adantr 481	. . . . . . . . 9 |- ( ( n e. NN0 /\ (FermatNo ` n ) = h ) -> (FermatNo ` n ) e. ( ZZ>= ` 2 ) )
35		eleq1 2689	. . . . . . . . . 10 |- ( (FermatNo ` n ) = h -> ( (FermatNo ` n ) e. ( ZZ>= ` 2 ) <-> h e. ( ZZ>= ` 2 ) ) )
36	35	adantl 482	. . . . . . . . 9 |- ( ( n e. NN0 /\ (FermatNo ` n ) = h ) -> ( (FermatNo ` n ) e. ( ZZ>= ` 2 ) <-> h e. ( ZZ>= ` 2 ) ) )
37	34, 36	mpbid 222	. . . . . . . 8 |- ( ( n e. NN0 /\ (FermatNo ` n ) = h ) -> h e. ( ZZ>= ` 2 ) )
38	37	rexlimiva 3028	. . . . . . 7 |- ( E. n e. NN0 (FermatNo ` n ) = h -> h e. ( ZZ>= ` 2 ) )
39	33, 38	sylbi 207	. . . . . 6 |- ( h e. ran FermatNo -> h e. ( ZZ>= ` 2 ) )
40		eqid 2622	. . . . . . 7 |- inf ( { p e. Prime | p || g } , RR , < ) = inf ( { p e. Prime | p || g } , RR , < )
41		eqid 2622	. . . . . . 7 |- inf ( { p e. Prime | p || h } , RR , < ) = inf ( { p e. Prime | p || h } , RR , < )
42	40, 41	prmdvdsfmtnof1lem1 41496	. . . . . 6 |- ( ( g e. ( ZZ>= ` 2 ) /\ h e. ( ZZ>= ` 2 ) ) -> (inf ( { p e. Prime | p || g } , RR , < ) = inf ( { p e. Prime | p || h } , RR , < ) -> (inf ( { p e. Prime | p || g } , RR , < ) e. Prime /\ inf ( { p e. Prime | p || g } , RR , < ) || g /\ inf ( { p e. Prime | p || g } , RR , < ) || h ) ) )
43	32, 39, 42	syl2an 494	. . . . 5 |- ( ( g e. ran FermatNo /\ h e. ran FermatNo ) -> (inf ( { p e. Prime | p || g } , RR , < ) = inf ( { p e. Prime | p || h } , RR , < ) -> (inf ( { p e. Prime | p || g } , RR , < ) e. Prime /\ inf ( { p e. Prime | p || g } , RR , < ) || g /\ inf ( { p e. Prime | p || g } , RR , < ) || h ) ) )
44		prmdvdsfmtnof1lem2 41497	. . . . 5 |- ( ( g e. ran FermatNo /\ h e. ran FermatNo ) -> ( (inf ( { p e. Prime | p || g } , RR , < ) e. Prime /\ inf ( { p e. Prime | p || g } , RR , < ) || g /\ inf ( { p e. Prime | p || g } , RR , < ) || h ) -> g = h ) )
45	43, 44	syld 47	. . . 4 |- ( ( g e. ran FermatNo /\ h e. ran FermatNo ) -> (inf ( { p e. Prime | p || g } , RR , < ) = inf ( { p e. Prime | p || h } , RR , < ) -> g = h ) )
46	22, 45	sylbid 230	. . 3 |- ( ( g e. ran FermatNo /\ h e. ran FermatNo ) -> ( ( F ` g ) = ( F ` h ) -> g = h ) )
47	46	rgen2a 2977	. 2 |- A. g e. ran FermatNo A. h e. ran FermatNo ( ( F ` g ) = ( F ` h ) -> g = h )
48		dff13 6512	. 2 |- ( F : ran FermatNo -1-1-> Prime <-> ( F : ran FermatNo --> Prime /\ A. g e. ran FermatNo A. h e. ran FermatNo ( ( F ` g ) = ( F ` h ) -> g = h ) ) )
49	2, 47, 48	mpbir2an 955	1 |- F : ran FermatNo -1-1-> Prime

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   {crab 2916   _Vcvv 3200   class class class wbr 4653   |-> cmpt 4729   Or wor 5034   ran crn 5115   -->wf 5884   -1-1->wf1 5885   ` cfv 5888  infcinf 8347   RRcr 9935   < clt 10074   2c2 11070   3c3 11071   NN0cn0 11292   ZZ>=cuz 11687   || cdvds 14983   Primecprime 15385  FermatNocfmtno 41439

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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  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-fal 1489  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-se 5074  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-isom 5897  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-2o 7561  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-oi 8415  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-4 11081  df-5 11082  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  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-clim 14219  df-prod 14636  df-dvds 14984  df-gcd 15217  df-prm 15386  df-fmtno 41440

This theorem is referenced by:  prminf2  41500