lgslem1 - Intuitionistic Logic Explorer

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Theorem lgslem1 16119

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 3351	. . . . . . . . 9 $\$
2	1	3ad2ant2 1050	. . . . . . . 8 $/\$ $\$ $/\$
3		prmnn 12888	. . . . . . . 8
4	2, 3	syl 14	. . . . . . 7 $/\$ $\$ $/\$
5		simp1 1028	. . . . . . 7 $/\$ $\$ $/\$
6		prmz 12889	. . . . . . . . . 10
7	2, 6	syl 14	. . . . . . . . 9 $/\$ $\$ $/\$
8	5, 7	gcdcomd 12751	. . . . . . . 8 $/\$ $\$ $/\$
9		simp3 1030	. . . . . . . . 9 $/\$ $\$ $/\$
10		coprm 12922	. . . . . . . . . 10 $/\$
11	2, 5, 10	syl2anc 415	. . . . . . . . 9 $/\$ $\$ $/\$
12	9, 11	mpbid 147	. . . . . . . 8 $/\$ $\$ $/\$
13	8, 12	eqtrd 2271	. . . . . . 7 $/\$ $\$ $/\$
14		eulerth 13011	. . . . . . 7 $/\$ $/\$
15	4, 5, 13, 14	syl3anc 1278	. . . . . 6 $/\$ $\$ $/\$
16		phiprm 13001	. . . . . . . . . 10
17	2, 16	syl 14	. . . . . . . . 9 $/\$ $\$ $/\$
18		nnm1nn0 9604	. . . . . . . . . 10
19	4, 18	syl 14	. . . . . . . . 9 $/\$ $\$ $/\$
20	17, 19	eqeltrd 2315	. . . . . . . 8 $/\$ $\$ $/\$
21		zexpcl 10991	. . . . . . . 8 $/\$
22	5, 20, 21	syl2anc 415	. . . . . . 7 $/\$ $\$ $/\$
23		1zzd 9671	. . . . . . 7 $/\$ $\$ $/\$
24		moddvds 12566	. . . . . . 7 $/\$ $/\$
25	4, 22, 23, 24	syl3anc 1278	. . . . . 6 $/\$ $\$ $/\$
26	15, 25	mpbid 147	. . . . 5 $/\$ $\$ $/\$
27	19	nn0cnd 9622	. . . . . . . . . . . 12 $/\$ $\$ $/\$
28		2cnd 9377	. . . . . . . . . . . 12 $/\$ $\$ $/\$
29		2ap0 9397	. . . . . . . . . . . . 13 #
30	29	a1i 9	. . . . . . . . . . . 12 $/\$ $\$ $/\$ #
31	27, 28, 30	divcanap1d 9121	. . . . . . . . . . 11 $/\$ $\$ $/\$
32	17, 31	eqtr4d 2274	. . . . . . . . . 10 $/\$ $\$ $/\$
33	32	oveq2d 6101	. . . . . . . . 9 $/\$ $\$ $/\$
34	5	zcnd 9769	. . . . . . . . . 10 $/\$ $\$ $/\$
35		2nn0 9580	. . . . . . . . . . 11
36	35	a1i 9	. . . . . . . . . 10 $/\$ $\$ $/\$
37		oddprm 13038	. . . . . . . . . . . 12 $\$
38	37	3ad2ant2 1050	. . . . . . . . . . 11 $/\$ $\$ $/\$
39	38	nnnn0d 9620	. . . . . . . . . 10 $/\$ $\$ $/\$
40	34, 36, 39	expmuld 11114	. . . . . . . . 9 $/\$ $\$ $/\$
41	33, 40	eqtrd 2271	. . . . . . . 8 $/\$ $\$ $/\$
42	41	oveq1d 6100	. . . . . . 7 $/\$ $\$ $/\$
43		sq1 11070	. . . . . . . 8
44	43	oveq2i 6096	. . . . . . 7
45	42, 44	eqtr4di 2289	. . . . . 6 $/\$ $\$ $/\$
46		zexpcl 10991	. . . . . . . . 9 $/\$
47	5, 39, 46	syl2anc 415	. . . . . . . 8 $/\$ $\$ $/\$
48	47	zcnd 9769	. . . . . . 7 $/\$ $\$ $/\$
49		ax-1cn 8272	. . . . . . 7
50		subsq 11083	. . . . . . 7 $/\$
51	48, 49, 50	sylancl 417	. . . . . 6 $/\$ $\$ $/\$
52	45, 51	eqtrd 2271	. . . . 5 $/\$ $\$ $/\$
53	26, 52	breqtrd 4156	. . . 4 $/\$ $\$ $/\$
54	47	peano2zd 9771	. . . . 5 $/\$ $\$ $/\$
55		peano2zm 9682	. . . . . 6
56	47, 55	syl 14	. . . . 5 $/\$ $\$ $/\$
57		euclemma 12924	. . . . 5 $/\$ $/\$ $\/$
58	2, 54, 56, 57	syl3anc 1278	. . . 4 $/\$ $\$ $/\$ $\/$
59	53, 58	mpbid 147	. . 3 $/\$ $\$ $/\$ $\/$
60		dvdsval3 12558	. . . . 5 $/\$
61	4, 54, 60	syl2anc 415	. . . 4 $/\$ $\$ $/\$
62		2z 9672	. . . . . . 7
63	62	a1i 9	. . . . . 6 $/\$ $\$ $/\$
64		moddvds 12566	. . . . . 6 $/\$ $/\$
65	4, 54, 63, 64	syl3anc 1278	. . . . 5 $/\$ $\$ $/\$
66		zq 10026	. . . . . . . 8
67	62, 66	mp1i 10	. . . . . . 7 $/\$ $\$ $/\$
68		zq 10026	. . . . . . . 8
69	7, 68	syl 14	. . . . . . 7 $/\$ $\$ $/\$
70		0le2 9394	. . . . . . . 8
71	70	a1i 9	. . . . . . 7 $/\$ $\$ $/\$
72		eldifsni 3843	. . . . . . . . . 10 $\$
73	72	3ad2ant2 1050	. . . . . . . . 9 $/\$ $\$ $/\$
74		zapne 9719	. . . . . . . . . 10 $/\$ #
75	7, 62, 74	sylancl 417	. . . . . . . . 9 $/\$ $\$ $/\$ #
76	73, 75	mpbird 167	. . . . . . . 8 $/\$ $\$ $/\$ #
77		2re 9374	. . . . . . . . . 10
78	77	a1i 9	. . . . . . . . 9 $/\$ $\$ $/\$
79	4	nnred 9317	. . . . . . . . 9 $/\$ $\$ $/\$
80		prmuz2 12909	. . . . . . . . . . 11
81	2, 80	syl 14	. . . . . . . . . 10 $/\$ $\$ $/\$
82		eluzle 9934	. . . . . . . . . 10
83	81, 82	syl 14	. . . . . . . . 9 $/\$ $\$ $/\$
84	78, 79, 83	leltapd 8967	. . . . . . . 8 $/\$ $\$ $/\$ #
85	76, 84	mpbird 167	. . . . . . 7 $/\$ $\$ $/\$
86		modqid 10786	. . . . . . 7 $/\$ $/\$ $/\$
87	67, 69, 71, 85, 86	syl22anc 1279	. . . . . 6 $/\$ $\$ $/\$
88	87	eqeq2d 2250	. . . . 5 $/\$ $\$ $/\$
89		df-2 9363	. . . . . . . 8
90	89	oveq2i 6096	. . . . . . 7
91	49	a1i 9	. . . . . . . 8 $/\$ $\$ $/\$
92	48, 91, 91	pnpcan2d 8675	. . . . . . 7 $/\$ $\$ $/\$
93	90, 92	eqtrid 2283	. . . . . 6 $/\$ $\$ $/\$
94	93	breq2d 4142	. . . . 5 $/\$ $\$ $/\$
95	65, 88, 94	3bitr3rd 219	. . . 4 $/\$ $\$ $/\$
96	61, 95	orbi12d 805	. . 3 $/\$ $\$ $/\$ $\/$ $\/$
97	59, 96	mpbid 147	. 2 $/\$ $\$ $/\$ $\/$
98	54, 4	zmodcld 10782	. . 3 $/\$ $\$ $/\$
99		elprg 3729	. . 3 $\/$
100	98, 99	syl 14	. 2 $/\$ $\$ $/\$ $\/$
101	97, 100	mpbird 167	1 $/\$ $\$ $/\$

Colors of variables: wff set class

This proof depends on syntax axioms:

wn 3

wi 4

wb 105 $\/$ wo 720 $/\$ w3a 1009

wceq 1402

wcel 2209

wne 2420 $\$ cdif 3217

csn 3709

cpr 3710 class class class wbr 4130

cfv 5377 (class class class)co 6085

cc 8177

cr 8178

cc0 8179

c1 8180

caddc 8182

cmul 8184

clt 8360

cle 8361

cmin 8497 # cap 8909

cdiv 9002

cn 9304

c2 9355

cn0 9563

cz 9644

cuz 9921

cq 10019

cmo 10759

cexp 10975

cdvds 12554

cgcd 12730

cprime 12885

cphi 12987

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4246 ax-sep 4249 ax-nul 4259 ax-pow 4311 ax-pr 4346 ax-un 4578 ax-setind 4684 ax-iinf 4735 ax-cnex 8270 ax-resscn 8271 ax-1cn 8272 ax-1re 8273 ax-icn 8274 ax-addcl 8275 ax-addrcl 8276 ax-mulcl 8277 ax-mulrcl 8278 ax-addcom 8279 ax-mulcom 8280 ax-addass 8281 ax-mulass 8282 ax-distr 8283 ax-i2m1 8284 ax-0lt1 8285 ax-1rid 8286 ax-0id 8287 ax-rnegex 8288 ax-precex 8289 ax-cnre 8290 ax-pre-ltirr 8291 ax-pre-ltwlin 8292 ax-pre-lttrn 8293 ax-pre-apti 8294 ax-pre-ltadd 8295 ax-pre-mulgt0 8296 ax-pre-mulext 8297 ax-arch 8298 ax-caucvg 8299

This proof depends on definitions: df-bi 117 df-stab 843 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-xor 1425 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rmo 2536 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-if 3639 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-int 3971 df-iun 4014 df-br 4131 df-opab 4193 df-mpt 4194 df-tr 4230 df-id 4438 df-po 4441 df-iso 4442 df-iord 4511 df-on 4513 df-ilim 4514 df-suc 4516 df-iom 4738 df-xp 4780 df-rel 4781 df-cnv 4782 df-co 4783 df-dm 4784 df-rn 4785 df-res 4786 df-ima 4787 df-iota 5337 df-fun 5379 df-fn 5380 df-f 5381 df-f1 5382 df-fo 5383 df-f1o 5384 df-fv 5385 df-isom 5386 df-riota 6038 df-ov 6088 df-oprab 6089 df-mpo 6090 df-1st 6374 df-2nd 6375 df-recs 6576 df-irdg 6641 df-frec 6662 df-1o 6687 df-2o 6688 df-oadd 6691 df-er 6807 df-en 7023 df-dom 7024 df-fin 7025 df-sup 7324 df-pnf 8362 df-mnf 8363 df-xr 8364 df-ltxr 8365 df-le 8366 df-sub 8499 df-neg 8500 df-reap 8903 df-ap 8910 df-div 9003 df-inn 9305 df-2 9363 df-3 9364 df-4 9365 df-n0 9564 df-z 9645 df-uz 9922 df-q 10020 df-rp 10055 df-fz 10412 df-fzo 10550 df-fl 10705 df-mod 10760 df-seqfrec 10885 df-exp 10976 df-ihash 11215 df-cj 11607 df-re 11608 df-im 11609 df-rsqrt 11764 df-abs 11765 df-clim 12045 df-proddc 12318 df-dvds 12555 df-gcd 12731 df-prm 12886 df-phi 12989

This theorem is used by: lgslem4 16122

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