lgslem1 - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > lgslem1		Unicode version

Theorem lgslem1 13695

Description: When

is coprime to the prime

is equivalent

, and so adding

makes it equivalent to

. (Contributed by Mario Carneiro, 4-Feb-2015.)

**Assertion**
Ref	Expression
lgslem1	$/\$ $\$ $/\$

**Proof of Theorem lgslem1**
Step	Hyp	Ref	Expression
1		eldifi 3249	. . . . . . . . 9 $\$
2	1	3ad2ant2 1014	. . . . . . . 8 $/\$ $\$ $/\$
3		prmnn 12064	. . . . . . . 8
4	2, 3	syl 14	. . . . . . 7 $/\$ $\$ $/\$
5		simp1 992	. . . . . . 7 $/\$ $\$ $/\$
6		prmz 12065	. . . . . . . . . 10
7	2, 6	syl 14	. . . . . . . . 9 $/\$ $\$ $/\$
8	5, 7	gcdcomd 11929	. . . . . . . 8 $/\$ $\$ $/\$
9		simp3 994	. . . . . . . . 9 $/\$ $\$ $/\$
10		coprm 12098	. . . . . . . . . 10 $/\$
11	2, 5, 10	syl2anc 409	. . . . . . . . 9 $/\$ $\$ $/\$
12	9, 11	mpbid 146	. . . . . . . 8 $/\$ $\$ $/\$
13	8, 12	eqtrd 2203	. . . . . . 7 $/\$ $\$ $/\$
14		eulerth 12187	. . . . . . 7 $/\$ $/\$
15	4, 5, 13, 14	syl3anc 1233	. . . . . 6 $/\$ $\$ $/\$
16		phiprm 12177	. . . . . . . . . 10
17	2, 16	syl 14	. . . . . . . . 9 $/\$ $\$ $/\$
18		nnm1nn0 9176	. . . . . . . . . 10
19	4, 18	syl 14	. . . . . . . . 9 $/\$ $\$ $/\$
20	17, 19	eqeltrd 2247	. . . . . . . 8 $/\$ $\$ $/\$
21		zexpcl 10491	. . . . . . . 8 $/\$
22	5, 20, 21	syl2anc 409	. . . . . . 7 $/\$ $\$ $/\$
23		1zzd 9239	. . . . . . 7 $/\$ $\$ $/\$
24		moddvds 11761	. . . . . . 7 $/\$ $/\$
25	4, 22, 23, 24	syl3anc 1233	. . . . . 6 $/\$ $\$ $/\$
26	15, 25	mpbid 146	. . . . 5 $/\$ $\$ $/\$
27	19	nn0cnd 9190	. . . . . . . . . . . 12 $/\$ $\$ $/\$
28		2cnd 8951	. . . . . . . . . . . 12 $/\$ $\$ $/\$
29		2ap0 8971	. . . . . . . . . . . . 13 #
30	29	a1i 9	. . . . . . . . . . . 12 $/\$ $\$ $/\$ #
31	27, 28, 30	divcanap1d 8708	. . . . . . . . . . 11 $/\$ $\$ $/\$
32	17, 31	eqtr4d 2206	. . . . . . . . . 10 $/\$ $\$ $/\$
33	32	oveq2d 5869	. . . . . . . . 9 $/\$ $\$ $/\$
34	5	zcnd 9335	. . . . . . . . . 10 $/\$ $\$ $/\$
35		2nn0 9152	. . . . . . . . . . 11
36	35	a1i 9	. . . . . . . . . 10 $/\$ $\$ $/\$
37		oddprm 12213	. . . . . . . . . . . 12 $\$
38	37	3ad2ant2 1014	. . . . . . . . . . 11 $/\$ $\$ $/\$
39	38	nnnn0d 9188	. . . . . . . . . 10 $/\$ $\$ $/\$
40	34, 36, 39	expmuld 10612	. . . . . . . . 9 $/\$ $\$ $/\$
41	33, 40	eqtrd 2203	. . . . . . . 8 $/\$ $\$ $/\$
42	41	oveq1d 5868	. . . . . . 7 $/\$ $\$ $/\$
43		sq1 10569	. . . . . . . 8
44	43	oveq2i 5864	. . . . . . 7
45	42, 44	eqtr4di 2221	. . . . . 6 $/\$ $\$ $/\$
46		zexpcl 10491	. . . . . . . . 9 $/\$
47	5, 39, 46	syl2anc 409	. . . . . . . 8 $/\$ $\$ $/\$
48	47	zcnd 9335	. . . . . . 7 $/\$ $\$ $/\$
49		ax-1cn 7867	. . . . . . 7
50		subsq 10582	. . . . . . 7 $/\$
51	48, 49, 50	sylancl 411	. . . . . 6 $/\$ $\$ $/\$
52	45, 51	eqtrd 2203	. . . . 5 $/\$ $\$ $/\$
53	26, 52	breqtrd 4015	. . . 4 $/\$ $\$ $/\$
54	47	peano2zd 9337	. . . . 5 $/\$ $\$ $/\$
55		peano2zm 9250	. . . . . 6
56	47, 55	syl 14	. . . . 5 $/\$ $\$ $/\$
57		euclemma 12100	. . . . 5 $/\$ $/\$ $\/$
58	2, 54, 56, 57	syl3anc 1233	. . . 4 $/\$ $\$ $/\$ $\/$
59	53, 58	mpbid 146	. . 3 $/\$ $\$ $/\$ $\/$
60		dvdsval3 11753	. . . . 5 $/\$
61	4, 54, 60	syl2anc 409	. . . 4 $/\$ $\$ $/\$
62		2z 9240	. . . . . . 7
63	62	a1i 9	. . . . . 6 $/\$ $\$ $/\$
64		moddvds 11761	. . . . . 6 $/\$ $/\$
65	4, 54, 63, 64	syl3anc 1233	. . . . 5 $/\$ $\$ $/\$
66		zq 9585	. . . . . . . 8
67	62, 66	mp1i 10	. . . . . . 7 $/\$ $\$ $/\$
68		zq 9585	. . . . . . . 8
69	7, 68	syl 14	. . . . . . 7 $/\$ $\$ $/\$
70		0le2 8968	. . . . . . . 8
71	70	a1i 9	. . . . . . 7 $/\$ $\$ $/\$
72		eldifsni 3712	. . . . . . . . . 10 $\$
73	72	3ad2ant2 1014	. . . . . . . . 9 $/\$ $\$ $/\$
74		zapne 9286	. . . . . . . . . 10 $/\$ #
75	7, 62, 74	sylancl 411	. . . . . . . . 9 $/\$ $\$ $/\$ #
76	73, 75	mpbird 166	. . . . . . . 8 $/\$ $\$ $/\$ #
77		2re 8948	. . . . . . . . . 10
78	77	a1i 9	. . . . . . . . 9 $/\$ $\$ $/\$
79	4	nnred 8891	. . . . . . . . 9 $/\$ $\$ $/\$
80		prmuz2 12085	. . . . . . . . . . 11
81	2, 80	syl 14	. . . . . . . . . 10 $/\$ $\$ $/\$
82		eluzle 9499	. . . . . . . . . 10
83	81, 82	syl 14	. . . . . . . . 9 $/\$ $\$ $/\$
84	78, 79, 83	leltapd 8558	. . . . . . . 8 $/\$ $\$ $/\$ #
85	76, 84	mpbird 166	. . . . . . 7 $/\$ $\$ $/\$
86		modqid 10305	. . . . . . 7 $/\$ $/\$ $/\$
87	67, 69, 71, 85, 86	syl22anc 1234	. . . . . 6 $/\$ $\$ $/\$
88	87	eqeq2d 2182	. . . . 5 $/\$ $\$ $/\$
89		df-2 8937	. . . . . . . 8
90	89	oveq2i 5864	. . . . . . 7
91	49	a1i 9	. . . . . . . 8 $/\$ $\$ $/\$
92	48, 91, 91	pnpcan2d 8268	. . . . . . 7 $/\$ $\$ $/\$
93	90, 92	eqtrid 2215	. . . . . 6 $/\$ $\$ $/\$
94	93	breq2d 4001	. . . . 5 $/\$ $\$ $/\$
95	65, 88, 94	3bitr3rd 218	. . . 4 $/\$ $\$ $/\$
96	61, 95	orbi12d 788	. . 3 $/\$ $\$ $/\$ $\/$ $\/$
97	59, 96	mpbid 146	. 2 $/\$ $\$ $/\$ $\/$
98	54, 4	zmodcld 10301	. . 3 $/\$ $\$ $/\$
99		elprg 3603	. . 3 $\/$
100	98, 99	syl 14	. 2 $/\$ $\$ $/\$ $\/$
101	97, 100	mpbird 166	1 $/\$ $\$ $/\$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4

wb 104 $\/$ wo 703 $/\$ w3a 973

wceq 1348

wcel 2141

wne 2340 $\$ cdif 3118

csn 3583

cpr 3584 class class class wbr 3989

cfv 5198 (class class class)co 5853

cc 7772

cr 7773

cc0 7774

c1 7775

caddc 7777

cmul 7779

clt 7954

cle 7955

cmin 8090 # cap 8500

cdiv 8589

cn 8878

c2 8929

cn0 9135

cz 9212

cuz 9487

cq 9578

cmo 10278

cexp 10475

cdvds 11749

cgcd 11897

cprime 12061

cphi 12163

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-mulrcl 7873 ax-addcom 7874 ax-mulcom 7875 ax-addass 7876 ax-mulass 7877 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-1rid 7881 ax-0id 7882 ax-rnegex 7883 ax-precex 7884 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-apti 7889 ax-pre-ltadd 7890 ax-pre-mulgt0 7891 ax-pre-mulext 7892 ax-arch 7893 ax-caucvg 7894

This theorem depends on definitions: df-bi 116 df-stab 826 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-xor 1371 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-if 3527 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-id 4278 df-po 4281 df-iso 4282 df-iord 4351 df-on 4353 df-ilim 4354 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-isom 5207 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-recs 6284 df-irdg 6349 df-frec 6370 df-1o 6395 df-2o 6396 df-oadd 6399 df-er 6513 df-en 6719 df-dom 6720 df-fin 6721 df-sup 6961 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-reap 8494 df-ap 8501 df-div 8590 df-inn 8879 df-2 8937 df-3 8938 df-4 8939 df-n0 9136 df-z 9213 df-uz 9488 df-q 9579 df-rp 9611 df-fz 9966 df-fzo 10099 df-fl 10226 df-mod 10279 df-seqfrec 10402 df-exp 10476 df-ihash 10710 df-cj 10806 df-re 10807 df-im 10808 df-rsqrt 10962 df-abs 10963 df-clim 11242 df-proddc 11514 df-dvds 11750 df-gcd 11898 df-prm 12062 df-phi 12165

This theorem is referenced by: lgslem4 13698

W3C validator