lgslem1 - Intuitionistic Logic Explorer

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Theorem lgslem1 16033

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 12866	. . . . . . . 8
4	2, 3	syl 14	. . . . . . 7 $/\$ $\$ $/\$
5		simp1 1028	. . . . . . 7 $/\$ $\$ $/\$
6		prmz 12867	. . . . . . . . . 10
7	2, 6	syl 14	. . . . . . . . 9 $/\$ $\$ $/\$
8	5, 7	gcdcomd 12729	. . . . . . . 8 $/\$ $\$ $/\$
9		simp3 1030	. . . . . . . . 9 $/\$ $\$ $/\$
10		coprm 12900	. . . . . . . . . 10 $/\$
11	2, 5, 10	syl2anc 415	. . . . . . . . 9 $/\$ $\$ $/\$
12	9, 11	mpbid 147	. . . . . . . 8 $/\$ $\$ $/\$
13	8, 12	eqtrd 2271	. . . . . . 7 $/\$ $\$ $/\$
14		eulerth 12989	. . . . . . 7 $/\$ $/\$
15	4, 5, 13, 14	syl3anc 1278	. . . . . 6 $/\$ $\$ $/\$
16		phiprm 12979	. . . . . . . . . 10
17	2, 16	syl 14	. . . . . . . . 9 $/\$ $\$ $/\$
18		nnm1nn0 9583	. . . . . . . . . 10
19	4, 18	syl 14	. . . . . . . . 9 $/\$ $\$ $/\$
20	17, 19	eqeltrd 2315	. . . . . . . 8 $/\$ $\$ $/\$
21		zexpcl 10969	. . . . . . . 8 $/\$
22	5, 20, 21	syl2anc 415	. . . . . . 7 $/\$ $\$ $/\$
23		1zzd 9650	. . . . . . 7 $/\$ $\$ $/\$
24		moddvds 12544	. . . . . . 7 $/\$ $/\$
25	4, 22, 23, 24	syl3anc 1278	. . . . . 6 $/\$ $\$ $/\$
26	15, 25	mpbid 147	. . . . 5 $/\$ $\$ $/\$
27	19	nn0cnd 9601	. . . . . . . . . . . 12 $/\$ $\$ $/\$
28		2cnd 9356	. . . . . . . . . . . 12 $/\$ $\$ $/\$
29		2ap0 9376	. . . . . . . . . . . . 13 #
30	29	a1i 9	. . . . . . . . . . . 12 $/\$ $\$ $/\$ #
31	27, 28, 30	divcanap1d 9111	. . . . . . . . . . 11 $/\$ $\$ $/\$
32	17, 31	eqtr4d 2274	. . . . . . . . . 10 $/\$ $\$ $/\$
33	32	oveq2d 6091	. . . . . . . . 9 $/\$ $\$ $/\$
34	5	zcnd 9748	. . . . . . . . . 10 $/\$ $\$ $/\$
35		2nn0 9559	. . . . . . . . . . 11
36	35	a1i 9	. . . . . . . . . 10 $/\$ $\$ $/\$
37		oddprm 13016	. . . . . . . . . . . 12 $\$
38	37	3ad2ant2 1050	. . . . . . . . . . 11 $/\$ $\$ $/\$
39	38	nnnn0d 9599	. . . . . . . . . 10 $/\$ $\$ $/\$
40	34, 36, 39	expmuld 11092	. . . . . . . . 9 $/\$ $\$ $/\$
41	33, 40	eqtrd 2271	. . . . . . . 8 $/\$ $\$ $/\$
42	41	oveq1d 6090	. . . . . . 7 $/\$ $\$ $/\$
43		sq1 11048	. . . . . . . 8
44	43	oveq2i 6086	. . . . . . 7
45	42, 44	eqtr4di 2289	. . . . . 6 $/\$ $\$ $/\$
46		zexpcl 10969	. . . . . . . . 9 $/\$
47	5, 39, 46	syl2anc 415	. . . . . . . 8 $/\$ $\$ $/\$
48	47	zcnd 9748	. . . . . . 7 $/\$ $\$ $/\$
49		ax-1cn 8262	. . . . . . 7
50		subsq 11061	. . . . . . 7 $/\$
51	48, 49, 50	sylancl 417	. . . . . 6 $/\$ $\$ $/\$
52	45, 51	eqtrd 2271	. . . . 5 $/\$ $\$ $/\$
53	26, 52	breqtrd 4151	. . . 4 $/\$ $\$ $/\$
54	47	peano2zd 9750	. . . . 5 $/\$ $\$ $/\$
55		peano2zm 9661	. . . . . 6
56	47, 55	syl 14	. . . . 5 $/\$ $\$ $/\$
57		euclemma 12902	. . . . 5 $/\$ $/\$ $\/$
58	2, 54, 56, 57	syl3anc 1278	. . . 4 $/\$ $\$ $/\$ $\/$
59	53, 58	mpbid 147	. . 3 $/\$ $\$ $/\$ $\/$
60		dvdsval3 12536	. . . . 5 $/\$
61	4, 54, 60	syl2anc 415	. . . 4 $/\$ $\$ $/\$
62		2z 9651	. . . . . . 7
63	62	a1i 9	. . . . . 6 $/\$ $\$ $/\$
64		moddvds 12544	. . . . . 6 $/\$ $/\$
65	4, 54, 63, 64	syl3anc 1278	. . . . 5 $/\$ $\$ $/\$
66		zq 10005	. . . . . . . 8
67	62, 66	mp1i 10	. . . . . . 7 $/\$ $\$ $/\$
68		zq 10005	. . . . . . . 8
69	7, 68	syl 14	. . . . . . 7 $/\$ $\$ $/\$
70		0le2 9373	. . . . . . . 8
71	70	a1i 9	. . . . . . 7 $/\$ $\$ $/\$
72		eldifsni 3838	. . . . . . . . . 10 $\$
73	72	3ad2ant2 1050	. . . . . . . . 9 $/\$ $\$ $/\$
74		zapne 9698	. . . . . . . . . 10 $/\$ #
75	7, 62, 74	sylancl 417	. . . . . . . . 9 $/\$ $\$ $/\$ #
76	73, 75	mpbird 167	. . . . . . . 8 $/\$ $\$ $/\$ #
77		2re 9353	. . . . . . . . . 10
78	77	a1i 9	. . . . . . . . 9 $/\$ $\$ $/\$
79	4	nnred 9296	. . . . . . . . 9 $/\$ $\$ $/\$
80		prmuz2 12887	. . . . . . . . . . 11
81	2, 80	syl 14	. . . . . . . . . 10 $/\$ $\$ $/\$
82		eluzle 9913	. . . . . . . . . 10
83	81, 82	syl 14	. . . . . . . . 9 $/\$ $\$ $/\$
84	78, 79, 83	leltapd 8957	. . . . . . . 8 $/\$ $\$ $/\$ #
85	76, 84	mpbird 167	. . . . . . 7 $/\$ $\$ $/\$
86		modqid 10764	. . . . . . 7 $/\$ $/\$ $/\$
87	67, 69, 71, 85, 86	syl22anc 1279	. . . . . 6 $/\$ $\$ $/\$
88	87	eqeq2d 2250	. . . . 5 $/\$ $\$ $/\$
89		df-2 9342	. . . . . . . 8
90	89	oveq2i 6086	. . . . . . 7
91	49	a1i 9	. . . . . . . 8 $/\$ $\$ $/\$
92	48, 91, 91	pnpcan2d 8665	. . . . . . 7 $/\$ $\$ $/\$
93	90, 92	eqtrid 2283	. . . . . 6 $/\$ $\$ $/\$
94	93	breq2d 4137	. . . . 5 $/\$ $\$ $/\$
95	65, 88, 94	3bitr3rd 219	. . . 4 $/\$ $\$ $/\$
96	61, 95	orbi12d 805	. . 3 $/\$ $\$ $/\$ $\/$ $\/$
97	59, 96	mpbid 147	. 2 $/\$ $\$ $/\$ $\/$
98	54, 4	zmodcld 10760	. . 3 $/\$ $\$ $/\$
99		elprg 3725	. . 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 720 $/\$ w3a 1009

wceq 1402

wcel 2209

wne 2420 $\$ cdif 3217

csn 3705

cpr 3706 class class class wbr 4125

cfv 5372 (class class class)co 6075

cc 8167

cr 8168

cc0 8169

c1 8170

caddc 8172

cmul 8174

clt 8350

cle 8351

cmin 8487 # cap 8899

cdiv 8992

cn 9283

c2 9334

cn0 9542

cz 9623

cuz 9900

cq 9998

cmo 10737

cexp 10953

cdvds 12532

cgcd 12708

cprime 12863

cphi 12965

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 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 4241 ax-sep 4244 ax-nul 4254 ax-pow 4306 ax-pr 4341 ax-un 4573 ax-setind 4679 ax-iinf 4730 ax-cnex 8260 ax-resscn 8261 ax-1cn 8262 ax-1re 8263 ax-icn 8264 ax-addcl 8265 ax-addrcl 8266 ax-mulcl 8267 ax-mulrcl 8268 ax-addcom 8269 ax-mulcom 8270 ax-addass 8271 ax-mulass 8272 ax-distr 8273 ax-i2m1 8274 ax-0lt1 8275 ax-1rid 8276 ax-0id 8277 ax-rnegex 8278 ax-precex 8279 ax-cnre 8280 ax-pre-ltirr 8281 ax-pre-ltwlin 8282 ax-pre-lttrn 8283 ax-pre-apti 8284 ax-pre-ltadd 8285 ax-pre-mulgt0 8286 ax-pre-mulext 8287 ax-arch 8288 ax-caucvg 8289

This theorem 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 3636 df-pw 3687 df-sn 3711 df-pr 3712 df-op 3714 df-uni 3931 df-int 3966 df-iun 4009 df-br 4126 df-opab 4188 df-mpt 4189 df-tr 4225 df-id 4433 df-po 4436 df-iso 4437 df-iord 4506 df-on 4508 df-ilim 4509 df-suc 4511 df-iom 4733 df-xp 4775 df-rel 4776 df-cnv 4777 df-co 4778 df-dm 4779 df-rn 4780 df-res 4781 df-ima 4782 df-iota 5332 df-fun 5374 df-fn 5375 df-f 5376 df-f1 5377 df-fo 5378 df-f1o 5379 df-fv 5380 df-isom 5381 df-riota 6028 df-ov 6078 df-oprab 6079 df-mpo 6080 df-1st 6364 df-2nd 6365 df-recs 6566 df-irdg 6631 df-frec 6652 df-1o 6677 df-2o 6678 df-oadd 6681 df-er 6797 df-en 7013 df-dom 7014 df-fin 7015 df-sup 7314 df-pnf 8352 df-mnf 8353 df-xr 8354 df-ltxr 8355 df-le 8356 df-sub 8489 df-neg 8490 df-reap 8893 df-ap 8900 df-div 8993 df-inn 9284 df-2 9342 df-3 9343 df-4 9344 df-n0 9543 df-z 9624 df-uz 9901 df-q 9999 df-rp 10034 df-fz 10391 df-fzo 10528 df-fl 10683 df-mod 10738 df-seqfrec 10863 df-exp 10954 df-ihash 11193 df-cj 11585 df-re 11586 df-im 11587 df-rsqrt 11742 df-abs 11743 df-clim 12023 df-proddc 12296 df-dvds 12533 df-gcd 12709 df-prm 12864 df-phi 12967

This theorem is referenced by: lgslem4 16036

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