Theorem fvprmselgcd1 15749

Description: The greatest common divisor of two values of the prime selection function for different arguments is 1. (Contributed by AV, 19-Aug-2020.)

Hypothesis

Ref	Expression
fvprmselelfz.f	|- F = ( m e. NN |-> if ( m e. Prime , m , 1 ) )

Assertion

Ref	Expression
fvprmselgcd1	|- ( ( X e. ( 1 ... N ) /\ Y e. ( 1 ... N ) /\ X =/= Y ) -> ( ( F ` X ) gcd ( F ` Y ) ) = 1 )

Distinct variable groups:   m, N   m, X   m, Y

Allowed substitution hint:   F( m)

Proof of Theorem fvprmselgcd1

Step	Hyp	Ref	Expression
1		fvprmselelfz.f	. . . . . . 7 |- F = ( m e. NN |-> if ( m e. Prime , m , 1 ) )
2	1	a1i 11	. . . . . 6 |- ( ( ( X e. Prime /\ Y e. Prime ) /\ ( X e. ( 1 ... N ) /\ Y e. ( 1 ... N ) /\ X =/= Y ) ) -> F = ( m e. NN |-> if ( m e. Prime , m , 1 ) ) )
3		eleq1 2689	. . . . . . . 8 |- ( m = X -> ( m e. Prime <-> X e. Prime ) )
4		id 22	. . . . . . . 8 |- ( m = X -> m = X )
5	3, 4	ifbieq1d 4109	. . . . . . 7 |- ( m = X -> if ( m e. Prime , m , 1 ) = if ( X e. Prime , X , 1 ) )
6		iftrue 4092	. . . . . . . 8 |- ( X e. Prime -> if ( X e. Prime , X , 1 ) = X )
7	6	ad2antrr 762	. . . . . . 7 |- ( ( ( X e. Prime /\ Y e. Prime ) /\ ( X e. ( 1 ... N ) /\ Y e. ( 1 ... N ) /\ X =/= Y ) ) -> if ( X e. Prime , X , 1 ) = X )
8	5, 7	sylan9eqr 2678	. . . . . 6 |- ( ( ( ( X e. Prime /\ Y e. Prime ) /\ ( X e. ( 1 ... N ) /\ Y e. ( 1 ... N ) /\ X =/= Y ) ) /\ m = X ) -> if ( m e. Prime , m , 1 ) = X )
9		elfznn 12370	. . . . . . . 8 |- ( X e. ( 1 ... N ) -> X e. NN )
10	9	3ad2ant1 1082	. . . . . . 7 |- ( ( X e. ( 1 ... N ) /\ Y e. ( 1 ... N ) /\ X =/= Y ) -> X e. NN )
11	10	adantl 482	. . . . . 6 |- ( ( ( X e. Prime /\ Y e. Prime ) /\ ( X e. ( 1 ... N ) /\ Y e. ( 1 ... N ) /\ X =/= Y ) ) -> X e. NN )
12		simpll 790	. . . . . 6 |- ( ( ( X e. Prime /\ Y e. Prime ) /\ ( X e. ( 1 ... N ) /\ Y e. ( 1 ... N ) /\ X =/= Y ) ) -> X e. Prime )
13	2, 8, 11, 12	fvmptd 6288	. . . . 5 |- ( ( ( X e. Prime /\ Y e. Prime ) /\ ( X e. ( 1 ... N ) /\ Y e. ( 1 ... N ) /\ X =/= Y ) ) -> ( F ` X ) = X )
14		eleq1 2689	. . . . . . . 8 |- ( m = Y -> ( m e. Prime <-> Y e. Prime ) )
15		id 22	. . . . . . . 8 |- ( m = Y -> m = Y )
16	14, 15	ifbieq1d 4109	. . . . . . 7 |- ( m = Y -> if ( m e. Prime , m , 1 ) = if ( Y e. Prime , Y , 1 ) )
17		iftrue 4092	. . . . . . . 8 |- ( Y e. Prime -> if ( Y e. Prime , Y , 1 ) = Y )
18	17	ad2antlr 763	. . . . . . 7 |- ( ( ( X e. Prime /\ Y e. Prime ) /\ ( X e. ( 1 ... N ) /\ Y e. ( 1 ... N ) /\ X =/= Y ) ) -> if ( Y e. Prime , Y , 1 ) = Y )
19	16, 18	sylan9eqr 2678	. . . . . 6 |- ( ( ( ( X e. Prime /\ Y e. Prime ) /\ ( X e. ( 1 ... N ) /\ Y e. ( 1 ... N ) /\ X =/= Y ) ) /\ m = Y ) -> if ( m e. Prime , m , 1 ) = Y )
20		elfznn 12370	. . . . . . . 8 |- ( Y e. ( 1 ... N ) -> Y e. NN )
21	20	3ad2ant2 1083	. . . . . . 7 |- ( ( X e. ( 1 ... N ) /\ Y e. ( 1 ... N ) /\ X =/= Y ) -> Y e. NN )
22	21	adantl 482	. . . . . 6 |- ( ( ( X e. Prime /\ Y e. Prime ) /\ ( X e. ( 1 ... N ) /\ Y e. ( 1 ... N ) /\ X =/= Y ) ) -> Y e. NN )
23		simplr 792	. . . . . 6 |- ( ( ( X e. Prime /\ Y e. Prime ) /\ ( X e. ( 1 ... N ) /\ Y e. ( 1 ... N ) /\ X =/= Y ) ) -> Y e. Prime )
24	2, 19, 22, 23	fvmptd 6288	. . . . 5 |- ( ( ( X e. Prime /\ Y e. Prime ) /\ ( X e. ( 1 ... N ) /\ Y e. ( 1 ... N ) /\ X =/= Y ) ) -> ( F ` Y ) = Y )
25	13, 24	oveq12d 6668	. . . 4 |- ( ( ( X e. Prime /\ Y e. Prime ) /\ ( X e. ( 1 ... N ) /\ Y e. ( 1 ... N ) /\ X =/= Y ) ) -> ( ( F ` X ) gcd ( F ` Y ) ) = ( X gcd Y ) )
26		prmrp 15424	. . . . . . 7 |- ( ( X e. Prime /\ Y e. Prime ) -> ( ( X gcd Y ) = 1 <-> X =/= Y ) )
27	26	biimprcd 240	. . . . . 6 |- ( X =/= Y -> ( ( X e. Prime /\ Y e. Prime ) -> ( X gcd Y ) = 1 ) )
28	27	3ad2ant3 1084	. . . . 5 |- ( ( X e. ( 1 ... N ) /\ Y e. ( 1 ... N ) /\ X =/= Y ) -> ( ( X e. Prime /\ Y e. Prime ) -> ( X gcd Y ) = 1 ) )
29	28	impcom 446	. . . 4 |- ( ( ( X e. Prime /\ Y e. Prime ) /\ ( X e. ( 1 ... N ) /\ Y e. ( 1 ... N ) /\ X =/= Y ) ) -> ( X gcd Y ) = 1 )
30	25, 29	eqtrd 2656	. . 3 |- ( ( ( X e. Prime /\ Y e. Prime ) /\ ( X e. ( 1 ... N ) /\ Y e. ( 1 ... N ) /\ X =/= Y ) ) -> ( ( F ` X ) gcd ( F ` Y ) ) = 1 )
31	30	ex 450	. 2 |- ( ( X e. Prime /\ Y e. Prime ) -> ( ( X e. ( 1 ... N ) /\ Y e. ( 1 ... N ) /\ X =/= Y ) -> ( ( F ` X ) gcd ( F ` Y ) ) = 1 ) )
32	1	a1i 11	. . . . . 6 |- ( ( ( X e. Prime /\ -. Y e. Prime ) /\ ( X e. ( 1 ... N ) /\ Y e. ( 1 ... N ) /\ X =/= Y ) ) -> F = ( m e. NN |-> if ( m e. Prime , m , 1 ) ) )
33	6	ad2antrr 762	. . . . . . 7 |- ( ( ( X e. Prime /\ -. Y e. Prime ) /\ ( X e. ( 1 ... N ) /\ Y e. ( 1 ... N ) /\ X =/= Y ) ) -> if ( X e. Prime , X , 1 ) = X )
34	5, 33	sylan9eqr 2678	. . . . . 6 |- ( ( ( ( X e. Prime /\ -. Y e. Prime ) /\ ( X e. ( 1 ... N ) /\ Y e. ( 1 ... N ) /\ X =/= Y ) ) /\ m = X ) -> if ( m e. Prime , m , 1 ) = X )
35	10	adantl 482	. . . . . 6 |- ( ( ( X e. Prime /\ -. Y e. Prime ) /\ ( X e. ( 1 ... N ) /\ Y e. ( 1 ... N ) /\ X =/= Y ) ) -> X e. NN )
36		simpll 790	. . . . . 6 |- ( ( ( X e. Prime /\ -. Y e. Prime ) /\ ( X e. ( 1 ... N ) /\ Y e. ( 1 ... N ) /\ X =/= Y ) ) -> X e. Prime )
37	32, 34, 35, 36	fvmptd 6288	. . . . 5 |- ( ( ( X e. Prime /\ -. Y e. Prime ) /\ ( X e. ( 1 ... N ) /\ Y e. ( 1 ... N ) /\ X =/= Y ) ) -> ( F ` X ) = X )
38		iffalse 4095	. . . . . . . 8 |- ( -. Y e. Prime -> if ( Y e. Prime , Y , 1 ) = 1 )
39	38	ad2antlr 763	. . . . . . 7 |- ( ( ( X e. Prime /\ -. Y e. Prime ) /\ ( X e. ( 1 ... N ) /\ Y e. ( 1 ... N ) /\ X =/= Y ) ) -> if ( Y e. Prime , Y , 1 ) = 1 )
40	16, 39	sylan9eqr 2678	. . . . . 6 |- ( ( ( ( X e. Prime /\ -. Y e. Prime ) /\ ( X e. ( 1 ... N ) /\ Y e. ( 1 ... N ) /\ X =/= Y ) ) /\ m = Y ) -> if ( m e. Prime , m , 1 ) = 1 )
41	21	adantl 482	. . . . . 6 |- ( ( ( X e. Prime /\ -. Y e. Prime ) /\ ( X e. ( 1 ... N ) /\ Y e. ( 1 ... N ) /\ X =/= Y ) ) -> Y e. NN )
42		1nn 11031	. . . . . . 7 |- 1 e. NN
43	42	a1i 11	. . . . . 6 |- ( ( ( X e. Prime /\ -. Y e. Prime ) /\ ( X e. ( 1 ... N ) /\ Y e. ( 1 ... N ) /\ X =/= Y ) ) -> 1 e. NN )
44	32, 40, 41, 43	fvmptd 6288	. . . . 5 |- ( ( ( X e. Prime /\ -. Y e. Prime ) /\ ( X e. ( 1 ... N ) /\ Y e. ( 1 ... N ) /\ X =/= Y ) ) -> ( F ` Y ) = 1 )
45	37, 44	oveq12d 6668	. . . 4 |- ( ( ( X e. Prime /\ -. Y e. Prime ) /\ ( X e. ( 1 ... N ) /\ Y e. ( 1 ... N ) /\ X =/= Y ) ) -> ( ( F ` X ) gcd ( F ` Y ) ) = ( X gcd 1 ) )
46		prmz 15389	. . . . . 6 |- ( X e. Prime -> X e. ZZ )
47		gcd1 15249	. . . . . 6 |- ( X e. ZZ -> ( X gcd 1 ) = 1 )
48	46, 47	syl 17	. . . . 5 |- ( X e. Prime -> ( X gcd 1 ) = 1 )
49	48	ad2antrr 762	. . . 4 |- ( ( ( X e. Prime /\ -. Y e. Prime ) /\ ( X e. ( 1 ... N ) /\ Y e. ( 1 ... N ) /\ X =/= Y ) ) -> ( X gcd 1 ) = 1 )
50	45, 49	eqtrd 2656	. . 3 |- ( ( ( X e. Prime /\ -. Y e. Prime ) /\ ( X e. ( 1 ... N ) /\ Y e. ( 1 ... N ) /\ X =/= Y ) ) -> ( ( F ` X ) gcd ( F ` Y ) ) = 1 )
51	50	ex 450	. 2 |- ( ( X e. Prime /\ -. Y e. Prime ) -> ( ( X e. ( 1 ... N ) /\ Y e. ( 1 ... N ) /\ X =/= Y ) -> ( ( F ` X ) gcd ( F ` Y ) ) = 1 ) )
52	1	a1i 11	. . . . . 6 |- ( ( ( -. X e. Prime /\ Y e. Prime ) /\ ( X e. ( 1 ... N ) /\ Y e. ( 1 ... N ) /\ X =/= Y ) ) -> F = ( m e. NN |-> if ( m e. Prime , m , 1 ) ) )
53		iffalse 4095	. . . . . . . 8 |- ( -. X e. Prime -> if ( X e. Prime , X , 1 ) = 1 )
54	53	ad2antrr 762	. . . . . . 7 |- ( ( ( -. X e. Prime /\ Y e. Prime ) /\ ( X e. ( 1 ... N ) /\ Y e. ( 1 ... N ) /\ X =/= Y ) ) -> if ( X e. Prime , X , 1 ) = 1 )
55	5, 54	sylan9eqr 2678	. . . . . 6 |- ( ( ( ( -. X e. Prime /\ Y e. Prime ) /\ ( X e. ( 1 ... N ) /\ Y e. ( 1 ... N ) /\ X =/= Y ) ) /\ m = X ) -> if ( m e. Prime , m , 1 ) = 1 )
56	10	adantl 482	. . . . . 6 |- ( ( ( -. X e. Prime /\ Y e. Prime ) /\ ( X e. ( 1 ... N ) /\ Y e. ( 1 ... N ) /\ X =/= Y ) ) -> X e. NN )
57	42	a1i 11	. . . . . 6 |- ( ( ( -. X e. Prime /\ Y e. Prime ) /\ ( X e. ( 1 ... N ) /\ Y e. ( 1 ... N ) /\ X =/= Y ) ) -> 1 e. NN )
58	52, 55, 56, 57	fvmptd 6288	. . . . 5 |- ( ( ( -. X e. Prime /\ Y e. Prime ) /\ ( X e. ( 1 ... N ) /\ Y e. ( 1 ... N ) /\ X =/= Y ) ) -> ( F ` X ) = 1 )
59	17	ad2antlr 763	. . . . . . 7 |- ( ( ( -. X e. Prime /\ Y e. Prime ) /\ ( X e. ( 1 ... N ) /\ Y e. ( 1 ... N ) /\ X =/= Y ) ) -> if ( Y e. Prime , Y , 1 ) = Y )
60	16, 59	sylan9eqr 2678	. . . . . 6 |- ( ( ( ( -. X e. Prime /\ Y e. Prime ) /\ ( X e. ( 1 ... N ) /\ Y e. ( 1 ... N ) /\ X =/= Y ) ) /\ m = Y ) -> if ( m e. Prime , m , 1 ) = Y )
61	21	adantl 482	. . . . . 6 |- ( ( ( -. X e. Prime /\ Y e. Prime ) /\ ( X e. ( 1 ... N ) /\ Y e. ( 1 ... N ) /\ X =/= Y ) ) -> Y e. NN )
62		simplr 792	. . . . . 6 |- ( ( ( -. X e. Prime /\ Y e. Prime ) /\ ( X e. ( 1 ... N ) /\ Y e. ( 1 ... N ) /\ X =/= Y ) ) -> Y e. Prime )
63	52, 60, 61, 62	fvmptd 6288	. . . . 5 |- ( ( ( -. X e. Prime /\ Y e. Prime ) /\ ( X e. ( 1 ... N ) /\ Y e. ( 1 ... N ) /\ X =/= Y ) ) -> ( F ` Y ) = Y )
64	58, 63	oveq12d 6668	. . . 4 |- ( ( ( -. X e. Prime /\ Y e. Prime ) /\ ( X e. ( 1 ... N ) /\ Y e. ( 1 ... N ) /\ X =/= Y ) ) -> ( ( F ` X ) gcd ( F ` Y ) ) = ( 1 gcd Y ) )
65		prmz 15389	. . . . . 6 |- ( Y e. Prime -> Y e. ZZ )
66		1gcd 15254	. . . . . 6 |- ( Y e. ZZ -> ( 1 gcd Y ) = 1 )
67	65, 66	syl 17	. . . . 5 |- ( Y e. Prime -> ( 1 gcd Y ) = 1 )
68	67	ad2antlr 763	. . . 4 |- ( ( ( -. X e. Prime /\ Y e. Prime ) /\ ( X e. ( 1 ... N ) /\ Y e. ( 1 ... N ) /\ X =/= Y ) ) -> ( 1 gcd Y ) = 1 )
69	64, 68	eqtrd 2656	. . 3 |- ( ( ( -. X e. Prime /\ Y e. Prime ) /\ ( X e. ( 1 ... N ) /\ Y e. ( 1 ... N ) /\ X =/= Y ) ) -> ( ( F ` X ) gcd ( F ` Y ) ) = 1 )
70	69	ex 450	. 2 |- ( ( -. X e. Prime /\ Y e. Prime ) -> ( ( X e. ( 1 ... N ) /\ Y e. ( 1 ... N ) /\ X =/= Y ) -> ( ( F ` X ) gcd ( F ` Y ) ) = 1 ) )
71	1	a1i 11	. . . . . 6 |- ( ( ( -. X e. Prime /\ -. Y e. Prime ) /\ ( X e. ( 1 ... N ) /\ Y e. ( 1 ... N ) /\ X =/= Y ) ) -> F = ( m e. NN |-> if ( m e. Prime , m , 1 ) ) )
72	53	ad2antrr 762	. . . . . . 7 |- ( ( ( -. X e. Prime /\ -. Y e. Prime ) /\ ( X e. ( 1 ... N ) /\ Y e. ( 1 ... N ) /\ X =/= Y ) ) -> if ( X e. Prime , X , 1 ) = 1 )
73	5, 72	sylan9eqr 2678	. . . . . 6 |- ( ( ( ( -. X e. Prime /\ -. Y e. Prime ) /\ ( X e. ( 1 ... N ) /\ Y e. ( 1 ... N ) /\ X =/= Y ) ) /\ m = X ) -> if ( m e. Prime , m , 1 ) = 1 )
74	10	adantl 482	. . . . . 6 |- ( ( ( -. X e. Prime /\ -. Y e. Prime ) /\ ( X e. ( 1 ... N ) /\ Y e. ( 1 ... N ) /\ X =/= Y ) ) -> X e. NN )
75	42	a1i 11	. . . . . 6 |- ( ( ( -. X e. Prime /\ -. Y e. Prime ) /\ ( X e. ( 1 ... N ) /\ Y e. ( 1 ... N ) /\ X =/= Y ) ) -> 1 e. NN )
76	71, 73, 74, 75	fvmptd 6288	. . . . 5 |- ( ( ( -. X e. Prime /\ -. Y e. Prime ) /\ ( X e. ( 1 ... N ) /\ Y e. ( 1 ... N ) /\ X =/= Y ) ) -> ( F ` X ) = 1 )
77	38	ad2antlr 763	. . . . . . 7 |- ( ( ( -. X e. Prime /\ -. Y e. Prime ) /\ ( X e. ( 1 ... N ) /\ Y e. ( 1 ... N ) /\ X =/= Y ) ) -> if ( Y e. Prime , Y , 1 ) = 1 )
78	16, 77	sylan9eqr 2678	. . . . . 6 |- ( ( ( ( -. X e. Prime /\ -. Y e. Prime ) /\ ( X e. ( 1 ... N ) /\ Y e. ( 1 ... N ) /\ X =/= Y ) ) /\ m = Y ) -> if ( m e. Prime , m , 1 ) = 1 )
79	21	adantl 482	. . . . . 6 |- ( ( ( -. X e. Prime /\ -. Y e. Prime ) /\ ( X e. ( 1 ... N ) /\ Y e. ( 1 ... N ) /\ X =/= Y ) ) -> Y e. NN )
80	71, 78, 79, 75	fvmptd 6288	. . . . 5 |- ( ( ( -. X e. Prime /\ -. Y e. Prime ) /\ ( X e. ( 1 ... N ) /\ Y e. ( 1 ... N ) /\ X =/= Y ) ) -> ( F ` Y ) = 1 )
81	76, 80	oveq12d 6668	. . . 4 |- ( ( ( -. X e. Prime /\ -. Y e. Prime ) /\ ( X e. ( 1 ... N ) /\ Y e. ( 1 ... N ) /\ X =/= Y ) ) -> ( ( F ` X ) gcd ( F ` Y ) ) = ( 1 gcd 1 ) )
82		1z 11407	. . . . 5 |- 1 e. ZZ
83		1gcd 15254	. . . . 5 |- ( 1 e. ZZ -> ( 1 gcd 1 ) = 1 )
84	82, 83	mp1i 13	. . . 4 |- ( ( ( -. X e. Prime /\ -. Y e. Prime ) /\ ( X e. ( 1 ... N ) /\ Y e. ( 1 ... N ) /\ X =/= Y ) ) -> ( 1 gcd 1 ) = 1 )
85	81, 84	eqtrd 2656	. . 3 |- ( ( ( -. X e. Prime /\ -. Y e. Prime ) /\ ( X e. ( 1 ... N ) /\ Y e. ( 1 ... N ) /\ X =/= Y ) ) -> ( ( F ` X ) gcd ( F ` Y ) ) = 1 )
86	85	ex 450	. 2 |- ( ( -. X e. Prime /\ -. Y e. Prime ) -> ( ( X e. ( 1 ... N ) /\ Y e. ( 1 ... N ) /\ X =/= Y ) -> ( ( F ` X ) gcd ( F ` Y ) ) = 1 ) )
87	31, 51, 70, 86	4cases 990	1 |- ( ( X e. ( 1 ... N ) /\ Y e. ( 1 ... N ) /\ X =/= Y ) -> ( ( F ` X ) gcd ( F ` Y ) ) = 1 )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   ifcif 4086   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   1c1 9937   NNcn 11020   ZZcz 11377   ...cfz 12326   gcd cgcd 15216   Primecprime 15385

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-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-2o 7561  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  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-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-dvds 14984  df-gcd 15217  df-prm 15386

This theorem is referenced by:  prmodvdslcmf  15751