lgslem1 - Intuitionistic Logic Explorer

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Theorem lgslem1 15679

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 3326	. . . . . . . . 9 $\$
2	1	3ad2ant2 1043	. . . . . . . 8 $/\$ $\$ $/\$
3		prmnn 12632	. . . . . . . 8
4	2, 3	syl 14	. . . . . . 7 $/\$ $\$ $/\$
5		simp1 1021	. . . . . . 7 $/\$ $\$ $/\$
6		prmz 12633	. . . . . . . . . 10
7	2, 6	syl 14	. . . . . . . . 9 $/\$ $\$ $/\$
8	5, 7	gcdcomd 12495	. . . . . . . 8 $/\$ $\$ $/\$
9		simp3 1023	. . . . . . . . 9 $/\$ $\$ $/\$
10		coprm 12666	. . . . . . . . . 10 $/\$
11	2, 5, 10	syl2anc 411	. . . . . . . . 9 $/\$ $\$ $/\$
12	9, 11	mpbid 147	. . . . . . . 8 $/\$ $\$ $/\$
13	8, 12	eqtrd 2262	. . . . . . 7 $/\$ $\$ $/\$
14		eulerth 12755	. . . . . . 7 $/\$ $/\$
15	4, 5, 13, 14	syl3anc 1271	. . . . . 6 $/\$ $\$ $/\$
16		phiprm 12745	. . . . . . . . . 10
17	2, 16	syl 14	. . . . . . . . 9 $/\$ $\$ $/\$
18		nnm1nn0 9410	. . . . . . . . . 10
19	4, 18	syl 14	. . . . . . . . 9 $/\$ $\$ $/\$
20	17, 19	eqeltrd 2306	. . . . . . . 8 $/\$ $\$ $/\$
21		zexpcl 10776	. . . . . . . 8 $/\$
22	5, 20, 21	syl2anc 411	. . . . . . 7 $/\$ $\$ $/\$
23		1zzd 9473	. . . . . . 7 $/\$ $\$ $/\$
24		moddvds 12310	. . . . . . 7 $/\$ $/\$
25	4, 22, 23, 24	syl3anc 1271	. . . . . 6 $/\$ $\$ $/\$
26	15, 25	mpbid 147	. . . . 5 $/\$ $\$ $/\$
27	19	nn0cnd 9424	. . . . . . . . . . . 12 $/\$ $\$ $/\$
28		2cnd 9183	. . . . . . . . . . . 12 $/\$ $\$ $/\$
29		2ap0 9203	. . . . . . . . . . . . 13 #
30	29	a1i 9	. . . . . . . . . . . 12 $/\$ $\$ $/\$ #
31	27, 28, 30	divcanap1d 8938	. . . . . . . . . . 11 $/\$ $\$ $/\$
32	17, 31	eqtr4d 2265	. . . . . . . . . 10 $/\$ $\$ $/\$
33	32	oveq2d 6017	. . . . . . . . 9 $/\$ $\$ $/\$
34	5	zcnd 9570	. . . . . . . . . 10 $/\$ $\$ $/\$
35		2nn0 9386	. . . . . . . . . . 11
36	35	a1i 9	. . . . . . . . . 10 $/\$ $\$ $/\$
37		oddprm 12782	. . . . . . . . . . . 12 $\$
38	37	3ad2ant2 1043	. . . . . . . . . . 11 $/\$ $\$ $/\$
39	38	nnnn0d 9422	. . . . . . . . . 10 $/\$ $\$ $/\$
40	34, 36, 39	expmuld 10898	. . . . . . . . 9 $/\$ $\$ $/\$
41	33, 40	eqtrd 2262	. . . . . . . 8 $/\$ $\$ $/\$
42	41	oveq1d 6016	. . . . . . 7 $/\$ $\$ $/\$
43		sq1 10855	. . . . . . . 8
44	43	oveq2i 6012	. . . . . . 7
45	42, 44	eqtr4di 2280	. . . . . 6 $/\$ $\$ $/\$
46		zexpcl 10776	. . . . . . . . 9 $/\$
47	5, 39, 46	syl2anc 411	. . . . . . . 8 $/\$ $\$ $/\$
48	47	zcnd 9570	. . . . . . 7 $/\$ $\$ $/\$
49		ax-1cn 8092	. . . . . . 7
50		subsq 10868	. . . . . . 7 $/\$
51	48, 49, 50	sylancl 413	. . . . . 6 $/\$ $\$ $/\$
52	45, 51	eqtrd 2262	. . . . 5 $/\$ $\$ $/\$
53	26, 52	breqtrd 4109	. . . 4 $/\$ $\$ $/\$
54	47	peano2zd 9572	. . . . 5 $/\$ $\$ $/\$
55		peano2zm 9484	. . . . . 6
56	47, 55	syl 14	. . . . 5 $/\$ $\$ $/\$
57		euclemma 12668	. . . . 5 $/\$ $/\$ $\/$
58	2, 54, 56, 57	syl3anc 1271	. . . 4 $/\$ $\$ $/\$ $\/$
59	53, 58	mpbid 147	. . 3 $/\$ $\$ $/\$ $\/$
60		dvdsval3 12302	. . . . 5 $/\$
61	4, 54, 60	syl2anc 411	. . . 4 $/\$ $\$ $/\$
62		2z 9474	. . . . . . 7
63	62	a1i 9	. . . . . 6 $/\$ $\$ $/\$
64		moddvds 12310	. . . . . 6 $/\$ $/\$
65	4, 54, 63, 64	syl3anc 1271	. . . . 5 $/\$ $\$ $/\$
66		zq 9821	. . . . . . . 8
67	62, 66	mp1i 10	. . . . . . 7 $/\$ $\$ $/\$
68		zq 9821	. . . . . . . 8
69	7, 68	syl 14	. . . . . . 7 $/\$ $\$ $/\$
70		0le2 9200	. . . . . . . 8
71	70	a1i 9	. . . . . . 7 $/\$ $\$ $/\$
72		eldifsni 3797	. . . . . . . . . 10 $\$
73	72	3ad2ant2 1043	. . . . . . . . 9 $/\$ $\$ $/\$
74		zapne 9521	. . . . . . . . . 10 $/\$ #
75	7, 62, 74	sylancl 413	. . . . . . . . 9 $/\$ $\$ $/\$ #
76	73, 75	mpbird 167	. . . . . . . 8 $/\$ $\$ $/\$ #
77		2re 9180	. . . . . . . . . 10
78	77	a1i 9	. . . . . . . . 9 $/\$ $\$ $/\$
79	4	nnred 9123	. . . . . . . . 9 $/\$ $\$ $/\$
80		prmuz2 12653	. . . . . . . . . . 11
81	2, 80	syl 14	. . . . . . . . . 10 $/\$ $\$ $/\$
82		eluzle 9734	. . . . . . . . . 10
83	81, 82	syl 14	. . . . . . . . 9 $/\$ $\$ $/\$
84	78, 79, 83	leltapd 8786	. . . . . . . 8 $/\$ $\$ $/\$ #
85	76, 84	mpbird 167	. . . . . . 7 $/\$ $\$ $/\$
86		modqid 10571	. . . . . . 7 $/\$ $/\$ $/\$
87	67, 69, 71, 85, 86	syl22anc 1272	. . . . . 6 $/\$ $\$ $/\$
88	87	eqeq2d 2241	. . . . 5 $/\$ $\$ $/\$
89		df-2 9169	. . . . . . . 8
90	89	oveq2i 6012	. . . . . . 7
91	49	a1i 9	. . . . . . . 8 $/\$ $\$ $/\$
92	48, 91, 91	pnpcan2d 8495	. . . . . . 7 $/\$ $\$ $/\$
93	90, 92	eqtrid 2274	. . . . . 6 $/\$ $\$ $/\$
94	93	breq2d 4095	. . . . 5 $/\$ $\$ $/\$
95	65, 88, 94	3bitr3rd 219	. . . 4 $/\$ $\$ $/\$
96	61, 95	orbi12d 798	. . 3 $/\$ $\$ $/\$ $\/$ $\/$
97	59, 96	mpbid 147	. 2 $/\$ $\$ $/\$ $\/$
98	54, 4	zmodcld 10567	. . 3 $/\$ $\$ $/\$
99		elprg 3686	. . 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 713 $/\$ w3a 1002

wceq 1395

wcel 2200

wne 2400 $\$ cdif 3194

csn 3666

cpr 3667 class class class wbr 4083

cfv 5318 (class class class)co 6001

cc 7997

cr 7998

cc0 7999

c1 8000

caddc 8002

cmul 8004

clt 8181

cle 8182

cmin 8317 # cap 8728

cdiv 8819

cn 9110

c2 9161

cn0 9369

cz 9446

cuz 9722

cq 9814

cmo 10544

cexp 10760

cdvds 12298

cgcd 12474

cprime 12629

cphi 12731

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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4199 ax-sep 4202 ax-nul 4210 ax-pow 4258 ax-pr 4293 ax-un 4524 ax-setind 4629 ax-iinf 4680 ax-cnex 8090 ax-resscn 8091 ax-1cn 8092 ax-1re 8093 ax-icn 8094 ax-addcl 8095 ax-addrcl 8096 ax-mulcl 8097 ax-mulrcl 8098 ax-addcom 8099 ax-mulcom 8100 ax-addass 8101 ax-mulass 8102 ax-distr 8103 ax-i2m1 8104 ax-0lt1 8105 ax-1rid 8106 ax-0id 8107 ax-rnegex 8108 ax-precex 8109 ax-cnre 8110 ax-pre-ltirr 8111 ax-pre-ltwlin 8112 ax-pre-lttrn 8113 ax-pre-apti 8114 ax-pre-ltadd 8115 ax-pre-mulgt0 8116 ax-pre-mulext 8117 ax-arch 8118 ax-caucvg 8119

This theorem depends on definitions: df-bi 117 df-stab 836 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-xor 1418 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rmo 2516 df-rab 2517 df-v 2801 df-sbc 3029 df-csb 3125 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-nul 3492 df-if 3603 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3889 df-int 3924 df-iun 3967 df-br 4084 df-opab 4146 df-mpt 4147 df-tr 4183 df-id 4384 df-po 4387 df-iso 4388 df-iord 4457 df-on 4459 df-ilim 4460 df-suc 4462 df-iom 4683 df-xp 4725 df-rel 4726 df-cnv 4727 df-co 4728 df-dm 4729 df-rn 4730 df-res 4731 df-ima 4732 df-iota 5278 df-fun 5320 df-fn 5321 df-f 5322 df-f1 5323 df-fo 5324 df-f1o 5325 df-fv 5326 df-isom 5327 df-riota 5954 df-ov 6004 df-oprab 6005 df-mpo 6006 df-1st 6286 df-2nd 6287 df-recs 6451 df-irdg 6516 df-frec 6537 df-1o 6562 df-2o 6563 df-oadd 6566 df-er 6680 df-en 6888 df-dom 6889 df-fin 6890 df-sup 7151 df-pnf 8183 df-mnf 8184 df-xr 8185 df-ltxr 8186 df-le 8187 df-sub 8319 df-neg 8320 df-reap 8722 df-ap 8729 df-div 8820 df-inn 9111 df-2 9169 df-3 9170 df-4 9171 df-n0 9370 df-z 9447 df-uz 9723 df-q 9815 df-rp 9850 df-fz 10205 df-fzo 10339 df-fl 10490 df-mod 10545 df-seqfrec 10670 df-exp 10761 df-ihash 10998 df-cj 11353 df-re 11354 df-im 11355 df-rsqrt 11509 df-abs 11510 df-clim 11790 df-proddc 12062 df-dvds 12299 df-gcd 12475 df-prm 12630 df-phi 12733

This theorem is referenced by: lgslem4 15682

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