lgslem1 - Intuitionistic Logic Explorer

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Theorem lgslem1 15116

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 3281	. . . . . . . . 9 $\$
2	1	3ad2ant2 1021	. . . . . . . 8 $/\$ $\$ $/\$
3		prmnn 12248	. . . . . . . 8
4	2, 3	syl 14	. . . . . . 7 $/\$ $\$ $/\$
5		simp1 999	. . . . . . 7 $/\$ $\$ $/\$
6		prmz 12249	. . . . . . . . . 10
7	2, 6	syl 14	. . . . . . . . 9 $/\$ $\$ $/\$
8	5, 7	gcdcomd 12111	. . . . . . . 8 $/\$ $\$ $/\$
9		simp3 1001	. . . . . . . . 9 $/\$ $\$ $/\$
10		coprm 12282	. . . . . . . . . 10 $/\$
11	2, 5, 10	syl2anc 411	. . . . . . . . 9 $/\$ $\$ $/\$
12	9, 11	mpbid 147	. . . . . . . 8 $/\$ $\$ $/\$
13	8, 12	eqtrd 2226	. . . . . . 7 $/\$ $\$ $/\$
14		eulerth 12371	. . . . . . 7 $/\$ $/\$
15	4, 5, 13, 14	syl3anc 1249	. . . . . 6 $/\$ $\$ $/\$
16		phiprm 12361	. . . . . . . . . 10
17	2, 16	syl 14	. . . . . . . . 9 $/\$ $\$ $/\$
18		nnm1nn0 9281	. . . . . . . . . 10
19	4, 18	syl 14	. . . . . . . . 9 $/\$ $\$ $/\$
20	17, 19	eqeltrd 2270	. . . . . . . 8 $/\$ $\$ $/\$
21		zexpcl 10625	. . . . . . . 8 $/\$
22	5, 20, 21	syl2anc 411	. . . . . . 7 $/\$ $\$ $/\$
23		1zzd 9344	. . . . . . 7 $/\$ $\$ $/\$
24		moddvds 11942	. . . . . . 7 $/\$ $/\$
25	4, 22, 23, 24	syl3anc 1249	. . . . . 6 $/\$ $\$ $/\$
26	15, 25	mpbid 147	. . . . 5 $/\$ $\$ $/\$
27	19	nn0cnd 9295	. . . . . . . . . . . 12 $/\$ $\$ $/\$
28		2cnd 9055	. . . . . . . . . . . 12 $/\$ $\$ $/\$
29		2ap0 9075	. . . . . . . . . . . . 13 #
30	29	a1i 9	. . . . . . . . . . . 12 $/\$ $\$ $/\$ #
31	27, 28, 30	divcanap1d 8810	. . . . . . . . . . 11 $/\$ $\$ $/\$
32	17, 31	eqtr4d 2229	. . . . . . . . . 10 $/\$ $\$ $/\$
33	32	oveq2d 5934	. . . . . . . . 9 $/\$ $\$ $/\$
34	5	zcnd 9440	. . . . . . . . . 10 $/\$ $\$ $/\$
35		2nn0 9257	. . . . . . . . . . 11
36	35	a1i 9	. . . . . . . . . 10 $/\$ $\$ $/\$
37		oddprm 12397	. . . . . . . . . . . 12 $\$
38	37	3ad2ant2 1021	. . . . . . . . . . 11 $/\$ $\$ $/\$
39	38	nnnn0d 9293	. . . . . . . . . 10 $/\$ $\$ $/\$
40	34, 36, 39	expmuld 10747	. . . . . . . . 9 $/\$ $\$ $/\$
41	33, 40	eqtrd 2226	. . . . . . . 8 $/\$ $\$ $/\$
42	41	oveq1d 5933	. . . . . . 7 $/\$ $\$ $/\$
43		sq1 10704	. . . . . . . 8
44	43	oveq2i 5929	. . . . . . 7
45	42, 44	eqtr4di 2244	. . . . . 6 $/\$ $\$ $/\$
46		zexpcl 10625	. . . . . . . . 9 $/\$
47	5, 39, 46	syl2anc 411	. . . . . . . 8 $/\$ $\$ $/\$
48	47	zcnd 9440	. . . . . . 7 $/\$ $\$ $/\$
49		ax-1cn 7965	. . . . . . 7
50		subsq 10717	. . . . . . 7 $/\$
51	48, 49, 50	sylancl 413	. . . . . 6 $/\$ $\$ $/\$
52	45, 51	eqtrd 2226	. . . . 5 $/\$ $\$ $/\$
53	26, 52	breqtrd 4055	. . . 4 $/\$ $\$ $/\$
54	47	peano2zd 9442	. . . . 5 $/\$ $\$ $/\$
55		peano2zm 9355	. . . . . 6
56	47, 55	syl 14	. . . . 5 $/\$ $\$ $/\$
57		euclemma 12284	. . . . 5 $/\$ $/\$ $\/$
58	2, 54, 56, 57	syl3anc 1249	. . . 4 $/\$ $\$ $/\$ $\/$
59	53, 58	mpbid 147	. . 3 $/\$ $\$ $/\$ $\/$
60		dvdsval3 11934	. . . . 5 $/\$
61	4, 54, 60	syl2anc 411	. . . 4 $/\$ $\$ $/\$
62		2z 9345	. . . . . . 7
63	62	a1i 9	. . . . . 6 $/\$ $\$ $/\$
64		moddvds 11942	. . . . . 6 $/\$ $/\$
65	4, 54, 63, 64	syl3anc 1249	. . . . 5 $/\$ $\$ $/\$
66		zq 9691	. . . . . . . 8
67	62, 66	mp1i 10	. . . . . . 7 $/\$ $\$ $/\$
68		zq 9691	. . . . . . . 8
69	7, 68	syl 14	. . . . . . 7 $/\$ $\$ $/\$
70		0le2 9072	. . . . . . . 8
71	70	a1i 9	. . . . . . 7 $/\$ $\$ $/\$
72		eldifsni 3747	. . . . . . . . . 10 $\$
73	72	3ad2ant2 1021	. . . . . . . . 9 $/\$ $\$ $/\$
74		zapne 9391	. . . . . . . . . 10 $/\$ #
75	7, 62, 74	sylancl 413	. . . . . . . . 9 $/\$ $\$ $/\$ #
76	73, 75	mpbird 167	. . . . . . . 8 $/\$ $\$ $/\$ #
77		2re 9052	. . . . . . . . . 10
78	77	a1i 9	. . . . . . . . 9 $/\$ $\$ $/\$
79	4	nnred 8995	. . . . . . . . 9 $/\$ $\$ $/\$
80		prmuz2 12269	. . . . . . . . . . 11
81	2, 80	syl 14	. . . . . . . . . 10 $/\$ $\$ $/\$
82		eluzle 9604	. . . . . . . . . 10
83	81, 82	syl 14	. . . . . . . . 9 $/\$ $\$ $/\$
84	78, 79, 83	leltapd 8658	. . . . . . . 8 $/\$ $\$ $/\$ #
85	76, 84	mpbird 167	. . . . . . 7 $/\$ $\$ $/\$
86		modqid 10420	. . . . . . 7 $/\$ $/\$ $/\$
87	67, 69, 71, 85, 86	syl22anc 1250	. . . . . 6 $/\$ $\$ $/\$
88	87	eqeq2d 2205	. . . . 5 $/\$ $\$ $/\$
89		df-2 9041	. . . . . . . 8
90	89	oveq2i 5929	. . . . . . 7
91	49	a1i 9	. . . . . . . 8 $/\$ $\$ $/\$
92	48, 91, 91	pnpcan2d 8368	. . . . . . 7 $/\$ $\$ $/\$
93	90, 92	eqtrid 2238	. . . . . 6 $/\$ $\$ $/\$
94	93	breq2d 4041	. . . . 5 $/\$ $\$ $/\$
95	65, 88, 94	3bitr3rd 219	. . . 4 $/\$ $\$ $/\$
96	61, 95	orbi12d 794	. . 3 $/\$ $\$ $/\$ $\/$ $\/$
97	59, 96	mpbid 147	. 2 $/\$ $\$ $/\$ $\/$
98	54, 4	zmodcld 10416	. . 3 $/\$ $\$ $/\$
99		elprg 3638	. . 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 709 $/\$ w3a 980

wceq 1364

wcel 2164

wne 2364 $\$ cdif 3150

csn 3618

cpr 3619 class class class wbr 4029

cfv 5254 (class class class)co 5918

cc 7870

cr 7871

cc0 7872

c1 7873

caddc 7875

cmul 7877

clt 8054

cle 8055

cmin 8190 # cap 8600

cdiv 8691

cn 8982

c2 9033

cn0 9240

cz 9317

cuz 9592

cq 9684

cmo 10393

cexp 10609

cdvds 11930

cgcd 12079

cprime 12245

cphi 12347

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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-coll 4144 ax-sep 4147 ax-nul 4155 ax-pow 4203 ax-pr 4238 ax-un 4464 ax-setind 4569 ax-iinf 4620 ax-cnex 7963 ax-resscn 7964 ax-1cn 7965 ax-1re 7966 ax-icn 7967 ax-addcl 7968 ax-addrcl 7969 ax-mulcl 7970 ax-mulrcl 7971 ax-addcom 7972 ax-mulcom 7973 ax-addass 7974 ax-mulass 7975 ax-distr 7976 ax-i2m1 7977 ax-0lt1 7978 ax-1rid 7979 ax-0id 7980 ax-rnegex 7981 ax-precex 7982 ax-cnre 7983 ax-pre-ltirr 7984 ax-pre-ltwlin 7985 ax-pre-lttrn 7986 ax-pre-apti 7987 ax-pre-ltadd 7988 ax-pre-mulgt0 7989 ax-pre-mulext 7990 ax-arch 7991 ax-caucvg 7992

This theorem depends on definitions: df-bi 117 df-stab 832 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-xor 1387 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rmo 2480 df-rab 2481 df-v 2762 df-sbc 2986 df-csb 3081 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166 df-nul 3447 df-if 3558 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-int 3871 df-iun 3914 df-br 4030 df-opab 4091 df-mpt 4092 df-tr 4128 df-id 4324 df-po 4327 df-iso 4328 df-iord 4397 df-on 4399 df-ilim 4400 df-suc 4402 df-iom 4623 df-xp 4665 df-rel 4666 df-cnv 4667 df-co 4668 df-dm 4669 df-rn 4670 df-res 4671 df-ima 4672 df-iota 5215 df-fun 5256 df-fn 5257 df-f 5258 df-f1 5259 df-fo 5260 df-f1o 5261 df-fv 5262 df-isom 5263 df-riota 5873 df-ov 5921 df-oprab 5922 df-mpo 5923 df-1st 6193 df-2nd 6194 df-recs 6358 df-irdg 6423 df-frec 6444 df-1o 6469 df-2o 6470 df-oadd 6473 df-er 6587 df-en 6795 df-dom 6796 df-fin 6797 df-sup 7043 df-pnf 8056 df-mnf 8057 df-xr 8058 df-ltxr 8059 df-le 8060 df-sub 8192 df-neg 8193 df-reap 8594 df-ap 8601 df-div 8692 df-inn 8983 df-2 9041 df-3 9042 df-4 9043 df-n0 9241 df-z 9318 df-uz 9593 df-q 9685 df-rp 9720 df-fz 10075 df-fzo 10209 df-fl 10339 df-mod 10394 df-seqfrec 10519 df-exp 10610 df-ihash 10847 df-cj 10986 df-re 10987 df-im 10988 df-rsqrt 11142 df-abs 11143 df-clim 11422 df-proddc 11694 df-dvds 11931 df-gcd 12080 df-prm 12246 df-phi 12349

This theorem is referenced by: lgslem4 15119

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