lgslem1 - Intuitionistic Logic Explorer

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Theorem lgslem1 15999

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 3345	. . . . . . . . 9 $\$
2	1	3ad2ant2 1046	. . . . . . . 8 $/\$ $\$ $/\$
3		prmnn 12832	. . . . . . . 8
4	2, 3	syl 14	. . . . . . 7 $/\$ $\$ $/\$
5		simp1 1024	. . . . . . 7 $/\$ $\$ $/\$
6		prmz 12833	. . . . . . . . . 10
7	2, 6	syl 14	. . . . . . . . 9 $/\$ $\$ $/\$
8	5, 7	gcdcomd 12695	. . . . . . . 8 $/\$ $\$ $/\$
9		simp3 1026	. . . . . . . . 9 $/\$ $\$ $/\$
10		coprm 12866	. . . . . . . . . 10 $/\$
11	2, 5, 10	syl2anc 411	. . . . . . . . 9 $/\$ $\$ $/\$
12	9, 11	mpbid 147	. . . . . . . 8 $/\$ $\$ $/\$
13	8, 12	eqtrd 2267	. . . . . . 7 $/\$ $\$ $/\$
14		eulerth 12955	. . . . . . 7 $/\$ $/\$
15	4, 5, 13, 14	syl3anc 1274	. . . . . 6 $/\$ $\$ $/\$
16		phiprm 12945	. . . . . . . . . 10
17	2, 16	syl 14	. . . . . . . . 9 $/\$ $\$ $/\$
18		nnm1nn0 9554	. . . . . . . . . 10
19	4, 18	syl 14	. . . . . . . . 9 $/\$ $\$ $/\$
20	17, 19	eqeltrd 2311	. . . . . . . 8 $/\$ $\$ $/\$
21		zexpcl 10940	. . . . . . . 8 $/\$
22	5, 20, 21	syl2anc 411	. . . . . . 7 $/\$ $\$ $/\$
23		1zzd 9621	. . . . . . 7 $/\$ $\$ $/\$
24		moddvds 12510	. . . . . . 7 $/\$ $/\$
25	4, 22, 23, 24	syl3anc 1274	. . . . . 6 $/\$ $\$ $/\$
26	15, 25	mpbid 147	. . . . 5 $/\$ $\$ $/\$
27	19	nn0cnd 9572	. . . . . . . . . . . 12 $/\$ $\$ $/\$
28		2cnd 9327	. . . . . . . . . . . 12 $/\$ $\$ $/\$
29		2ap0 9347	. . . . . . . . . . . . 13 #
30	29	a1i 9	. . . . . . . . . . . 12 $/\$ $\$ $/\$ #
31	27, 28, 30	divcanap1d 9082	. . . . . . . . . . 11 $/\$ $\$ $/\$
32	17, 31	eqtr4d 2270	. . . . . . . . . 10 $/\$ $\$ $/\$
33	32	oveq2d 6074	. . . . . . . . 9 $/\$ $\$ $/\$
34	5	zcnd 9719	. . . . . . . . . 10 $/\$ $\$ $/\$
35		2nn0 9530	. . . . . . . . . . 11
36	35	a1i 9	. . . . . . . . . 10 $/\$ $\$ $/\$
37		oddprm 12982	. . . . . . . . . . . 12 $\$
38	37	3ad2ant2 1046	. . . . . . . . . . 11 $/\$ $\$ $/\$
39	38	nnnn0d 9570	. . . . . . . . . 10 $/\$ $\$ $/\$
40	34, 36, 39	expmuld 11063	. . . . . . . . 9 $/\$ $\$ $/\$
41	33, 40	eqtrd 2267	. . . . . . . 8 $/\$ $\$ $/\$
42	41	oveq1d 6073	. . . . . . 7 $/\$ $\$ $/\$
43		sq1 11019	. . . . . . . 8
44	43	oveq2i 6069	. . . . . . 7
45	42, 44	eqtr4di 2285	. . . . . 6 $/\$ $\$ $/\$
46		zexpcl 10940	. . . . . . . . 9 $/\$
47	5, 39, 46	syl2anc 411	. . . . . . . 8 $/\$ $\$ $/\$
48	47	zcnd 9719	. . . . . . 7 $/\$ $\$ $/\$
49		ax-1cn 8236	. . . . . . 7
50		subsq 11032	. . . . . . 7 $/\$
51	48, 49, 50	sylancl 413	. . . . . 6 $/\$ $\$ $/\$
52	45, 51	eqtrd 2267	. . . . 5 $/\$ $\$ $/\$
53	26, 52	breqtrd 4140	. . . 4 $/\$ $\$ $/\$
54	47	peano2zd 9721	. . . . 5 $/\$ $\$ $/\$
55		peano2zm 9632	. . . . . 6
56	47, 55	syl 14	. . . . 5 $/\$ $\$ $/\$
57		euclemma 12868	. . . . 5 $/\$ $/\$ $\/$
58	2, 54, 56, 57	syl3anc 1274	. . . 4 $/\$ $\$ $/\$ $\/$
59	53, 58	mpbid 147	. . 3 $/\$ $\$ $/\$ $\/$
60		dvdsval3 12502	. . . . 5 $/\$
61	4, 54, 60	syl2anc 411	. . . 4 $/\$ $\$ $/\$
62		2z 9622	. . . . . . 7
63	62	a1i 9	. . . . . 6 $/\$ $\$ $/\$
64		moddvds 12510	. . . . . 6 $/\$ $/\$
65	4, 54, 63, 64	syl3anc 1274	. . . . 5 $/\$ $\$ $/\$
66		zq 9976	. . . . . . . 8
67	62, 66	mp1i 10	. . . . . . 7 $/\$ $\$ $/\$
68		zq 9976	. . . . . . . 8
69	7, 68	syl 14	. . . . . . 7 $/\$ $\$ $/\$
70		0le2 9344	. . . . . . . 8
71	70	a1i 9	. . . . . . 7 $/\$ $\$ $/\$
72		eldifsni 3827	. . . . . . . . . 10 $\$
73	72	3ad2ant2 1046	. . . . . . . . 9 $/\$ $\$ $/\$
74		zapne 9669	. . . . . . . . . 10 $/\$ #
75	7, 62, 74	sylancl 413	. . . . . . . . 9 $/\$ $\$ $/\$ #
76	73, 75	mpbird 167	. . . . . . . 8 $/\$ $\$ $/\$ #
77		2re 9324	. . . . . . . . . 10
78	77	a1i 9	. . . . . . . . 9 $/\$ $\$ $/\$
79	4	nnred 9267	. . . . . . . . 9 $/\$ $\$ $/\$
80		prmuz2 12853	. . . . . . . . . . 11
81	2, 80	syl 14	. . . . . . . . . 10 $/\$ $\$ $/\$
82		eluzle 9884	. . . . . . . . . 10
83	81, 82	syl 14	. . . . . . . . 9 $/\$ $\$ $/\$
84	78, 79, 83	leltapd 8930	. . . . . . . 8 $/\$ $\$ $/\$ #
85	76, 84	mpbird 167	. . . . . . 7 $/\$ $\$ $/\$
86		modqid 10735	. . . . . . 7 $/\$ $/\$ $/\$
87	67, 69, 71, 85, 86	syl22anc 1275	. . . . . 6 $/\$ $\$ $/\$
88	87	eqeq2d 2246	. . . . 5 $/\$ $\$ $/\$
89		df-2 9313	. . . . . . . 8
90	89	oveq2i 6069	. . . . . . 7
91	49	a1i 9	. . . . . . . 8 $/\$ $\$ $/\$
92	48, 91, 91	pnpcan2d 8638	. . . . . . 7 $/\$ $\$ $/\$
93	90, 92	eqtrid 2279	. . . . . 6 $/\$ $\$ $/\$
94	93	breq2d 4126	. . . . 5 $/\$ $\$ $/\$
95	65, 88, 94	3bitr3rd 219	. . . 4 $/\$ $\$ $/\$
96	61, 95	orbi12d 801	. . 3 $/\$ $\$ $/\$ $\/$ $\/$
97	59, 96	mpbid 147	. 2 $/\$ $\$ $/\$ $\/$
98	54, 4	zmodcld 10731	. . 3 $/\$ $\$ $/\$
99		elprg 3714	. . 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 2205

wne 2414 $\$ cdif 3211

csn 3694

cpr 3695 class class class wbr 4114

cfv 5357 (class class class)co 6058

cc 8141

cr 8142

cc0 8143

c1 8144

caddc 8146

cmul 8148

clt 8324

cle 8325

cmin 8460 # cap 8872

cdiv 8963

cn 9254

c2 9305

cn0 9513

cz 9594

cuz 9871

cq 9969

cmo 10708

cexp 10924

cdvds 12498

cgcd 12674

cprime 12829

cphi 12931

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 2207 ax-14 2208 ax-ext 2216 ax-coll 4230 ax-sep 4233 ax-nul 4241 ax-pow 4292 ax-pr 4327 ax-un 4559 ax-setind 4664 ax-iinf 4715 ax-cnex 8234 ax-resscn 8235 ax-1cn 8236 ax-1re 8237 ax-icn 8238 ax-addcl 8239 ax-addrcl 8240 ax-mulcl 8241 ax-mulrcl 8242 ax-addcom 8243 ax-mulcom 8244 ax-addass 8245 ax-mulass 8246 ax-distr 8247 ax-i2m1 8248 ax-0lt1 8249 ax-1rid 8250 ax-0id 8251 ax-rnegex 8252 ax-precex 8253 ax-cnre 8254 ax-pre-ltirr 8255 ax-pre-ltwlin 8256 ax-pre-lttrn 8257 ax-pre-apti 8258 ax-pre-ltadd 8259 ax-pre-mulgt0 8260 ax-pre-mulext 8261 ax-arch 8262 ax-caucvg 8263

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 2085 df-mo 2086 df-clab 2221 df-cleq 2227 df-clel 2230 df-nfc 2375 df-ne 2415 df-nel 2510 df-ral 2527 df-rex 2528 df-reu 2529 df-rmo 2530 df-rab 2531 df-v 2817 df-sbc 3046 df-csb 3142 df-dif 3216 df-un 3218 df-in 3220 df-ss 3227 df-nul 3513 df-if 3625 df-pw 3676 df-sn 3700 df-pr 3701 df-op 3703 df-uni 3920 df-int 3955 df-iun 3998 df-br 4115 df-opab 4177 df-mpt 4178 df-tr 4214 df-id 4419 df-po 4422 df-iso 4423 df-iord 4492 df-on 4494 df-ilim 4495 df-suc 4497 df-iom 4718 df-xp 4760 df-rel 4761 df-cnv 4762 df-co 4763 df-dm 4764 df-rn 4765 df-res 4766 df-ima 4767 df-iota 5317 df-fun 5359 df-fn 5360 df-f 5361 df-f1 5362 df-fo 5363 df-f1o 5364 df-fv 5365 df-isom 5366 df-riota 6011 df-ov 6061 df-oprab 6062 df-mpo 6063 df-1st 6347 df-2nd 6348 df-recs 6549 df-irdg 6614 df-frec 6635 df-1o 6660 df-2o 6661 df-oadd 6664 df-er 6780 df-en 6989 df-dom 6990 df-fin 6991 df-sup 7288 df-pnf 8326 df-mnf 8327 df-xr 8328 df-ltxr 8329 df-le 8330 df-sub 8462 df-neg 8463 df-reap 8866 df-ap 8873 df-div 8964 df-inn 9255 df-2 9313 df-3 9314 df-4 9315 df-n0 9514 df-z 9595 df-uz 9872 df-q 9970 df-rp 10005 df-fz 10362 df-fzo 10499 df-fl 10654 df-mod 10709 df-seqfrec 10834 df-exp 10925 df-ihash 11164 df-cj 11552 df-re 11553 df-im 11554 df-rsqrt 11708 df-abs 11709 df-clim 11989 df-proddc 12262 df-dvds 12499 df-gcd 12675 df-prm 12830 df-phi 12933

This theorem is referenced by: lgslem4 16002

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