lgslem1 - Intuitionistic Logic Explorer

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Theorem lgslem1 15728

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 3329	. . . . . . . . 9 $\$
2	1	3ad2ant2 1045	. . . . . . . 8 $/\$ $\$ $/\$
3		prmnn 12681	. . . . . . . 8
4	2, 3	syl 14	. . . . . . 7 $/\$ $\$ $/\$
5		simp1 1023	. . . . . . 7 $/\$ $\$ $/\$
6		prmz 12682	. . . . . . . . . 10
7	2, 6	syl 14	. . . . . . . . 9 $/\$ $\$ $/\$
8	5, 7	gcdcomd 12544	. . . . . . . 8 $/\$ $\$ $/\$
9		simp3 1025	. . . . . . . . 9 $/\$ $\$ $/\$
10		coprm 12715	. . . . . . . . . 10 $/\$
11	2, 5, 10	syl2anc 411	. . . . . . . . 9 $/\$ $\$ $/\$
12	9, 11	mpbid 147	. . . . . . . 8 $/\$ $\$ $/\$
13	8, 12	eqtrd 2264	. . . . . . 7 $/\$ $\$ $/\$
14		eulerth 12804	. . . . . . 7 $/\$ $/\$
15	4, 5, 13, 14	syl3anc 1273	. . . . . 6 $/\$ $\$ $/\$
16		phiprm 12794	. . . . . . . . . 10
17	2, 16	syl 14	. . . . . . . . 9 $/\$ $\$ $/\$
18		nnm1nn0 9442	. . . . . . . . . 10
19	4, 18	syl 14	. . . . . . . . 9 $/\$ $\$ $/\$
20	17, 19	eqeltrd 2308	. . . . . . . 8 $/\$ $\$ $/\$
21		zexpcl 10815	. . . . . . . 8 $/\$
22	5, 20, 21	syl2anc 411	. . . . . . 7 $/\$ $\$ $/\$
23		1zzd 9505	. . . . . . 7 $/\$ $\$ $/\$
24		moddvds 12359	. . . . . . 7 $/\$ $/\$
25	4, 22, 23, 24	syl3anc 1273	. . . . . 6 $/\$ $\$ $/\$
26	15, 25	mpbid 147	. . . . 5 $/\$ $\$ $/\$
27	19	nn0cnd 9456	. . . . . . . . . . . 12 $/\$ $\$ $/\$
28		2cnd 9215	. . . . . . . . . . . 12 $/\$ $\$ $/\$
29		2ap0 9235	. . . . . . . . . . . . 13 #
30	29	a1i 9	. . . . . . . . . . . 12 $/\$ $\$ $/\$ #
31	27, 28, 30	divcanap1d 8970	. . . . . . . . . . 11 $/\$ $\$ $/\$
32	17, 31	eqtr4d 2267	. . . . . . . . . 10 $/\$ $\$ $/\$
33	32	oveq2d 6033	. . . . . . . . 9 $/\$ $\$ $/\$
34	5	zcnd 9602	. . . . . . . . . 10 $/\$ $\$ $/\$
35		2nn0 9418	. . . . . . . . . . 11
36	35	a1i 9	. . . . . . . . . 10 $/\$ $\$ $/\$
37		oddprm 12831	. . . . . . . . . . . 12 $\$
38	37	3ad2ant2 1045	. . . . . . . . . . 11 $/\$ $\$ $/\$
39	38	nnnn0d 9454	. . . . . . . . . 10 $/\$ $\$ $/\$
40	34, 36, 39	expmuld 10937	. . . . . . . . 9 $/\$ $\$ $/\$
41	33, 40	eqtrd 2264	. . . . . . . 8 $/\$ $\$ $/\$
42	41	oveq1d 6032	. . . . . . 7 $/\$ $\$ $/\$
43		sq1 10894	. . . . . . . 8
44	43	oveq2i 6028	. . . . . . 7
45	42, 44	eqtr4di 2282	. . . . . 6 $/\$ $\$ $/\$
46		zexpcl 10815	. . . . . . . . 9 $/\$
47	5, 39, 46	syl2anc 411	. . . . . . . 8 $/\$ $\$ $/\$
48	47	zcnd 9602	. . . . . . 7 $/\$ $\$ $/\$
49		ax-1cn 8124	. . . . . . 7
50		subsq 10907	. . . . . . 7 $/\$
51	48, 49, 50	sylancl 413	. . . . . 6 $/\$ $\$ $/\$
52	45, 51	eqtrd 2264	. . . . 5 $/\$ $\$ $/\$
53	26, 52	breqtrd 4114	. . . 4 $/\$ $\$ $/\$
54	47	peano2zd 9604	. . . . 5 $/\$ $\$ $/\$
55		peano2zm 9516	. . . . . 6
56	47, 55	syl 14	. . . . 5 $/\$ $\$ $/\$
57		euclemma 12717	. . . . 5 $/\$ $/\$ $\/$
58	2, 54, 56, 57	syl3anc 1273	. . . 4 $/\$ $\$ $/\$ $\/$
59	53, 58	mpbid 147	. . 3 $/\$ $\$ $/\$ $\/$
60		dvdsval3 12351	. . . . 5 $/\$
61	4, 54, 60	syl2anc 411	. . . 4 $/\$ $\$ $/\$
62		2z 9506	. . . . . . 7
63	62	a1i 9	. . . . . 6 $/\$ $\$ $/\$
64		moddvds 12359	. . . . . 6 $/\$ $/\$
65	4, 54, 63, 64	syl3anc 1273	. . . . 5 $/\$ $\$ $/\$
66		zq 9859	. . . . . . . 8
67	62, 66	mp1i 10	. . . . . . 7 $/\$ $\$ $/\$
68		zq 9859	. . . . . . . 8
69	7, 68	syl 14	. . . . . . 7 $/\$ $\$ $/\$
70		0le2 9232	. . . . . . . 8
71	70	a1i 9	. . . . . . 7 $/\$ $\$ $/\$
72		eldifsni 3802	. . . . . . . . . 10 $\$
73	72	3ad2ant2 1045	. . . . . . . . 9 $/\$ $\$ $/\$
74		zapne 9553	. . . . . . . . . 10 $/\$ #
75	7, 62, 74	sylancl 413	. . . . . . . . 9 $/\$ $\$ $/\$ #
76	73, 75	mpbird 167	. . . . . . . 8 $/\$ $\$ $/\$ #
77		2re 9212	. . . . . . . . . 10
78	77	a1i 9	. . . . . . . . 9 $/\$ $\$ $/\$
79	4	nnred 9155	. . . . . . . . 9 $/\$ $\$ $/\$
80		prmuz2 12702	. . . . . . . . . . 11
81	2, 80	syl 14	. . . . . . . . . 10 $/\$ $\$ $/\$
82		eluzle 9767	. . . . . . . . . 10
83	81, 82	syl 14	. . . . . . . . 9 $/\$ $\$ $/\$
84	78, 79, 83	leltapd 8818	. . . . . . . 8 $/\$ $\$ $/\$ #
85	76, 84	mpbird 167	. . . . . . 7 $/\$ $\$ $/\$
86		modqid 10610	. . . . . . 7 $/\$ $/\$ $/\$
87	67, 69, 71, 85, 86	syl22anc 1274	. . . . . 6 $/\$ $\$ $/\$
88	87	eqeq2d 2243	. . . . 5 $/\$ $\$ $/\$
89		df-2 9201	. . . . . . . 8
90	89	oveq2i 6028	. . . . . . 7
91	49	a1i 9	. . . . . . . 8 $/\$ $\$ $/\$
92	48, 91, 91	pnpcan2d 8527	. . . . . . 7 $/\$ $\$ $/\$
93	90, 92	eqtrid 2276	. . . . . 6 $/\$ $\$ $/\$
94	93	breq2d 4100	. . . . 5 $/\$ $\$ $/\$
95	65, 88, 94	3bitr3rd 219	. . . 4 $/\$ $\$ $/\$
96	61, 95	orbi12d 800	. . 3 $/\$ $\$ $/\$ $\/$ $\/$
97	59, 96	mpbid 147	. 2 $/\$ $\$ $/\$ $\/$
98	54, 4	zmodcld 10606	. . 3 $/\$ $\$ $/\$
99		elprg 3689	. . 3 $\/$
100	98, 99	syl 14	. 2 $/\$ $\$ $/\$ $\/$
101	97, 100	mpbird 167	1 $/\$ $\$ $/\$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4

wb 105 $\/$ wo 715 $/\$ w3a 1004

wceq 1397

wcel 2202

wne 2402 $\$ cdif 3197

csn 3669

cpr 3670 class class class wbr 4088

cfv 5326 (class class class)co 6017

cc 8029

cr 8030

cc0 8031

c1 8032

caddc 8034

cmul 8036

clt 8213

cle 8214

cmin 8349 # cap 8760

cdiv 8851

cn 9142

c2 9193

cn0 9401

cz 9478

cuz 9754

cq 9852

cmo 10583

cexp 10799

cdvds 12347

cgcd 12523

cprime 12678

cphi 12780

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 619 ax-in2 620 ax-io 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-13 2204 ax-14 2205 ax-ext 2213 ax-coll 4204 ax-sep 4207 ax-nul 4215 ax-pow 4264 ax-pr 4299 ax-un 4530 ax-setind 4635 ax-iinf 4686 ax-cnex 8122 ax-resscn 8123 ax-1cn 8124 ax-1re 8125 ax-icn 8126 ax-addcl 8127 ax-addrcl 8128 ax-mulcl 8129 ax-mulrcl 8130 ax-addcom 8131 ax-mulcom 8132 ax-addass 8133 ax-mulass 8134 ax-distr 8135 ax-i2m1 8136 ax-0lt1 8137 ax-1rid 8138 ax-0id 8139 ax-rnegex 8140 ax-precex 8141 ax-cnre 8142 ax-pre-ltirr 8143 ax-pre-ltwlin 8144 ax-pre-lttrn 8145 ax-pre-apti 8146 ax-pre-ltadd 8147 ax-pre-mulgt0 8148 ax-pre-mulext 8149 ax-arch 8150 ax-caucvg 8151

This theorem depends on definitions: df-bi 117 df-stab 838 df-dc 842 df-3or 1005 df-3an 1006 df-tru 1400 df-fal 1403 df-xor 1420 df-nf 1509 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2363 df-ne 2403 df-nel 2498 df-ral 2515 df-rex 2516 df-reu 2517 df-rmo 2518 df-rab 2519 df-v 2804 df-sbc 3032 df-csb 3128 df-dif 3202 df-un 3204 df-in 3206 df-ss 3213 df-nul 3495 df-if 3606 df-pw 3654 df-sn 3675 df-pr 3676 df-op 3678 df-uni 3894 df-int 3929 df-iun 3972 df-br 4089 df-opab 4151 df-mpt 4152 df-tr 4188 df-id 4390 df-po 4393 df-iso 4394 df-iord 4463 df-on 4465 df-ilim 4466 df-suc 4468 df-iom 4689 df-xp 4731 df-rel 4732 df-cnv 4733 df-co 4734 df-dm 4735 df-rn 4736 df-res 4737 df-ima 4738 df-iota 5286 df-fun 5328 df-fn 5329 df-f 5330 df-f1 5331 df-fo 5332 df-f1o 5333 df-fv 5334 df-isom 5335 df-riota 5970 df-ov 6020 df-oprab 6021 df-mpo 6022 df-1st 6302 df-2nd 6303 df-recs 6470 df-irdg 6535 df-frec 6556 df-1o 6581 df-2o 6582 df-oadd 6585 df-er 6701 df-en 6909 df-dom 6910 df-fin 6911 df-sup 7182 df-pnf 8215 df-mnf 8216 df-xr 8217 df-ltxr 8218 df-le 8219 df-sub 8351 df-neg 8352 df-reap 8754 df-ap 8761 df-div 8852 df-inn 9143 df-2 9201 df-3 9202 df-4 9203 df-n0 9402 df-z 9479 df-uz 9755 df-q 9853 df-rp 9888 df-fz 10243 df-fzo 10377 df-fl 10529 df-mod 10584 df-seqfrec 10709 df-exp 10800 df-ihash 11037 df-cj 11402 df-re 11403 df-im 11404 df-rsqrt 11558 df-abs 11559 df-clim 11839 df-proddc 12111 df-dvds 12348 df-gcd 12524 df-prm 12679 df-phi 12782

This theorem is referenced by: lgslem4 15731

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