lgslem1 - Intuitionistic Logic Explorer

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Theorem lgslem1 15873

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 3341	. . . . . . . . 9 $\$
2	1	3ad2ant2 1046	. . . . . . . 8 $/\$ $\$ $/\$
3		prmnn 12807	. . . . . . . 8
4	2, 3	syl 14	. . . . . . 7 $/\$ $\$ $/\$
5		simp1 1024	. . . . . . 7 $/\$ $\$ $/\$
6		prmz 12808	. . . . . . . . . 10
7	2, 6	syl 14	. . . . . . . . 9 $/\$ $\$ $/\$
8	5, 7	gcdcomd 12670	. . . . . . . 8 $/\$ $\$ $/\$
9		simp3 1026	. . . . . . . . 9 $/\$ $\$ $/\$
10		coprm 12841	. . . . . . . . . 10 $/\$
11	2, 5, 10	syl2anc 411	. . . . . . . . 9 $/\$ $\$ $/\$
12	9, 11	mpbid 147	. . . . . . . 8 $/\$ $\$ $/\$
13	8, 12	eqtrd 2265	. . . . . . 7 $/\$ $\$ $/\$
14		eulerth 12930	. . . . . . 7 $/\$ $/\$
15	4, 5, 13, 14	syl3anc 1274	. . . . . 6 $/\$ $\$ $/\$
16		phiprm 12920	. . . . . . . . . 10
17	2, 16	syl 14	. . . . . . . . 9 $/\$ $\$ $/\$
18		nnm1nn0 9537	. . . . . . . . . 10
19	4, 18	syl 14	. . . . . . . . 9 $/\$ $\$ $/\$
20	17, 19	eqeltrd 2309	. . . . . . . 8 $/\$ $\$ $/\$
21		zexpcl 10916	. . . . . . . 8 $/\$
22	5, 20, 21	syl2anc 411	. . . . . . 7 $/\$ $\$ $/\$
23		1zzd 9604	. . . . . . 7 $/\$ $\$ $/\$
24		moddvds 12485	. . . . . . 7 $/\$ $/\$
25	4, 22, 23, 24	syl3anc 1274	. . . . . 6 $/\$ $\$ $/\$
26	15, 25	mpbid 147	. . . . 5 $/\$ $\$ $/\$
27	19	nn0cnd 9555	. . . . . . . . . . . 12 $/\$ $\$ $/\$
28		2cnd 9310	. . . . . . . . . . . 12 $/\$ $\$ $/\$
29		2ap0 9330	. . . . . . . . . . . . 13 #
30	29	a1i 9	. . . . . . . . . . . 12 $/\$ $\$ $/\$ #
31	27, 28, 30	divcanap1d 9065	. . . . . . . . . . 11 $/\$ $\$ $/\$
32	17, 31	eqtr4d 2268	. . . . . . . . . 10 $/\$ $\$ $/\$
33	32	oveq2d 6066	. . . . . . . . 9 $/\$ $\$ $/\$
34	5	zcnd 9701	. . . . . . . . . 10 $/\$ $\$ $/\$
35		2nn0 9513	. . . . . . . . . . 11
36	35	a1i 9	. . . . . . . . . 10 $/\$ $\$ $/\$
37		oddprm 12957	. . . . . . . . . . . 12 $\$
38	37	3ad2ant2 1046	. . . . . . . . . . 11 $/\$ $\$ $/\$
39	38	nnnn0d 9553	. . . . . . . . . 10 $/\$ $\$ $/\$
40	34, 36, 39	expmuld 11038	. . . . . . . . 9 $/\$ $\$ $/\$
41	33, 40	eqtrd 2265	. . . . . . . 8 $/\$ $\$ $/\$
42	41	oveq1d 6065	. . . . . . 7 $/\$ $\$ $/\$
43		sq1 10995	. . . . . . . 8
44	43	oveq2i 6061	. . . . . . 7
45	42, 44	eqtr4di 2283	. . . . . 6 $/\$ $\$ $/\$
46		zexpcl 10916	. . . . . . . . 9 $/\$
47	5, 39, 46	syl2anc 411	. . . . . . . 8 $/\$ $\$ $/\$
48	47	zcnd 9701	. . . . . . 7 $/\$ $\$ $/\$
49		ax-1cn 8220	. . . . . . 7
50		subsq 11008	. . . . . . 7 $/\$
51	48, 49, 50	sylancl 413	. . . . . 6 $/\$ $\$ $/\$
52	45, 51	eqtrd 2265	. . . . 5 $/\$ $\$ $/\$
53	26, 52	breqtrd 4135	. . . 4 $/\$ $\$ $/\$
54	47	peano2zd 9703	. . . . 5 $/\$ $\$ $/\$
55		peano2zm 9615	. . . . . 6
56	47, 55	syl 14	. . . . 5 $/\$ $\$ $/\$
57		euclemma 12843	. . . . 5 $/\$ $/\$ $\/$
58	2, 54, 56, 57	syl3anc 1274	. . . 4 $/\$ $\$ $/\$ $\/$
59	53, 58	mpbid 147	. . 3 $/\$ $\$ $/\$ $\/$
60		dvdsval3 12477	. . . . 5 $/\$
61	4, 54, 60	syl2anc 411	. . . 4 $/\$ $\$ $/\$
62		2z 9605	. . . . . . 7
63	62	a1i 9	. . . . . 6 $/\$ $\$ $/\$
64		moddvds 12485	. . . . . 6 $/\$ $/\$
65	4, 54, 63, 64	syl3anc 1274	. . . . 5 $/\$ $\$ $/\$
66		zq 9958	. . . . . . . 8
67	62, 66	mp1i 10	. . . . . . 7 $/\$ $\$ $/\$
68		zq 9958	. . . . . . . 8
69	7, 68	syl 14	. . . . . . 7 $/\$ $\$ $/\$
70		0le2 9327	. . . . . . . 8
71	70	a1i 9	. . . . . . 7 $/\$ $\$ $/\$
72		eldifsni 3822	. . . . . . . . . 10 $\$
73	72	3ad2ant2 1046	. . . . . . . . 9 $/\$ $\$ $/\$
74		zapne 9652	. . . . . . . . . 10 $/\$ #
75	7, 62, 74	sylancl 413	. . . . . . . . 9 $/\$ $\$ $/\$ #
76	73, 75	mpbird 167	. . . . . . . 8 $/\$ $\$ $/\$ #
77		2re 9307	. . . . . . . . . 10
78	77	a1i 9	. . . . . . . . 9 $/\$ $\$ $/\$
79	4	nnred 9250	. . . . . . . . 9 $/\$ $\$ $/\$
80		prmuz2 12828	. . . . . . . . . . 11
81	2, 80	syl 14	. . . . . . . . . 10 $/\$ $\$ $/\$
82		eluzle 9866	. . . . . . . . . 10
83	81, 82	syl 14	. . . . . . . . 9 $/\$ $\$ $/\$
84	78, 79, 83	leltapd 8913	. . . . . . . 8 $/\$ $\$ $/\$ #
85	76, 84	mpbird 167	. . . . . . 7 $/\$ $\$ $/\$
86		modqid 10711	. . . . . . 7 $/\$ $/\$ $/\$
87	67, 69, 71, 85, 86	syl22anc 1275	. . . . . 6 $/\$ $\$ $/\$
88	87	eqeq2d 2244	. . . . 5 $/\$ $\$ $/\$
89		df-2 9296	. . . . . . . 8
90	89	oveq2i 6061	. . . . . . 7
91	49	a1i 9	. . . . . . . 8 $/\$ $\$ $/\$
92	48, 91, 91	pnpcan2d 8622	. . . . . . 7 $/\$ $\$ $/\$
93	90, 92	eqtrid 2277	. . . . . 6 $/\$ $\$ $/\$
94	93	breq2d 4121	. . . . 5 $/\$ $\$ $/\$
95	65, 88, 94	3bitr3rd 219	. . . 4 $/\$ $\$ $/\$
96	61, 95	orbi12d 801	. . 3 $/\$ $\$ $/\$ $\/$ $\/$
97	59, 96	mpbid 147	. 2 $/\$ $\$ $/\$ $\/$
98	54, 4	zmodcld 10707	. . 3 $/\$ $\$ $/\$
99		elprg 3709	. . 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 716 $/\$ w3a 1005

wceq 1398

wcel 2203

wne 2412 $\$ cdif 3208

csn 3689

cpr 3690 class class class wbr 4109

cfv 5352 (class class class)co 6050

cc 8125

cr 8126

cc0 8127

c1 8128

caddc 8130

cmul 8132

clt 8308

cle 8309

cmin 8444 # cap 8855

cdiv 8946

cn 9237

c2 9288

cn0 9496

cz 9577

cuz 9853

cq 9951

cmo 10684

cexp 10900

cdvds 12473

cgcd 12649

cprime 12804

cphi 12906

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 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2205 ax-14 2206 ax-ext 2214 ax-coll 4225 ax-sep 4228 ax-nul 4236 ax-pow 4287 ax-pr 4322 ax-un 4554 ax-setind 4659 ax-iinf 4710 ax-cnex 8218 ax-resscn 8219 ax-1cn 8220 ax-1re 8221 ax-icn 8222 ax-addcl 8223 ax-addrcl 8224 ax-mulcl 8225 ax-mulrcl 8226 ax-addcom 8227 ax-mulcom 8228 ax-addass 8229 ax-mulass 8230 ax-distr 8231 ax-i2m1 8232 ax-0lt1 8233 ax-1rid 8234 ax-0id 8235 ax-rnegex 8236 ax-precex 8237 ax-cnre 8238 ax-pre-ltirr 8239 ax-pre-ltwlin 8240 ax-pre-lttrn 8241 ax-pre-apti 8242 ax-pre-ltadd 8243 ax-pre-mulgt0 8244 ax-pre-mulext 8245 ax-arch 8246 ax-caucvg 8247

This theorem depends on definitions: df-bi 117 df-stab 839 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-xor 1421 df-nf 1510 df-sb 1812 df-eu 2083 df-mo 2084 df-clab 2219 df-cleq 2225 df-clel 2228 df-nfc 2373 df-ne 2413 df-nel 2508 df-ral 2525 df-rex 2526 df-reu 2527 df-rmo 2528 df-rab 2529 df-v 2815 df-sbc 3043 df-csb 3139 df-dif 3213 df-un 3215 df-in 3217 df-ss 3224 df-nul 3509 df-if 3621 df-pw 3671 df-sn 3695 df-pr 3696 df-op 3698 df-uni 3915 df-int 3950 df-iun 3993 df-br 4110 df-opab 4172 df-mpt 4173 df-tr 4209 df-id 4414 df-po 4417 df-iso 4418 df-iord 4487 df-on 4489 df-ilim 4490 df-suc 4492 df-iom 4713 df-xp 4755 df-rel 4756 df-cnv 4757 df-co 4758 df-dm 4759 df-rn 4760 df-res 4761 df-ima 4762 df-iota 5312 df-fun 5354 df-fn 5355 df-f 5356 df-f1 5357 df-fo 5358 df-f1o 5359 df-fv 5360 df-isom 5361 df-riota 6003 df-ov 6053 df-oprab 6054 df-mpo 6055 df-1st 6334 df-2nd 6335 df-recs 6536 df-irdg 6601 df-frec 6622 df-1o 6647 df-2o 6648 df-oadd 6651 df-er 6767 df-en 6976 df-dom 6977 df-fin 6978 df-sup 7275 df-pnf 8310 df-mnf 8311 df-xr 8312 df-ltxr 8313 df-le 8314 df-sub 8446 df-neg 8447 df-reap 8849 df-ap 8856 df-div 8947 df-inn 9238 df-2 9296 df-3 9297 df-4 9298 df-n0 9497 df-z 9578 df-uz 9854 df-q 9952 df-rp 9987 df-fz 10343 df-fzo 10477 df-fl 10630 df-mod 10685 df-seqfrec 10810 df-exp 10901 df-ihash 11139 df-cj 11527 df-re 11528 df-im 11529 df-rsqrt 11683 df-abs 11684 df-clim 11964 df-proddc 12237 df-dvds 12474 df-gcd 12650 df-prm 12805 df-phi 12908

This theorem is referenced by: lgslem4 15876

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