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Theorem lgsdir 25057

Description: The Legendre symbol is completely multiplicative in its left argument. Generalization of theorem 9.9(a) in [ApostolNT] p. 188 (which assumes that

and

are odd positive integers). Together with lgsqr 25076 this implies that the product of two quadratic residues or nonresidues is a residue, and the product of a residue and a nonresidue is a nonresidue. (Contributed by Mario Carneiro, 4-Feb-2015.)

**Assertion**
Ref	Expression
lgsdir	$/\$ $/\$ $/\$ $/\$

Proof of Theorem lgsdir Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-1cn 9994	. . . . . . 7
2		0cn 10032	. . . . . . 7
3	1, 2	keepel 4155	. . . . . 6
4	3	mulid2i 10043	. . . . 5
5		iftrue 4092	. . . . . . 7
6	5	adantl 482	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
7	6	oveq1d 6665	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
8		simpl1 1064	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
9	8	zcnd 11483	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
10	9	ad2antrr 762	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
11		simpl2 1065	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
12	11	zcnd 11483	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
13	12	ad2antrr 762	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
14	10, 13	sqmuld 13020	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
15		simpr 477	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
16	15	oveq1d 6665	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
17	12	sqcld 13006	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
18	17	ad2antrr 762	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
19	18	mulid2d 10058	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
20	14, 16, 19	3eqtrd 2660	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
21	20	eqeq1d 2624	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
22	21	ifbid 4108	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
23	4, 7, 22	3eqtr4a 2682	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
24	3	mul02i 10225	. . . . 5
25		iffalse 4095	. . . . . . 7
26	25	adantl 482	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
27	26	oveq1d 6665	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
28		dvdsmul1 15003	. . . . . . . . . . . 12 $/\$
29	8, 11, 28	syl2anc 693	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
30	8, 11	zmulcld 11488	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
31		dvdssq 15280	. . . . . . . . . . . 12 $/\$
32	8, 30, 31	syl2anc 693	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
33	29, 32	mpbid 222	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
34	33	adantr 481	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
35		breq2 4657	. . . . . . . . 9
36	34, 35	syl5ibcom 235	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
37		simprl 794	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$
38	37	neneqd 2799	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$
39		sqeq0 12927	. . . . . . . . . . . . . . . 16
40	9, 39	syl 17	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$
41	38, 40	mtbird 315	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
42		zsqcl2 12941	. . . . . . . . . . . . . . . . 17
43	8, 42	syl 17	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$
44		elnn0 11294	. . . . . . . . . . . . . . . 16 $\/$
45	43, 44	sylib 208	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $\/$
46	45	ord 392	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
47	41, 46	mt3d 140	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
48	47	adantr 481	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
49	48	nnzd 11481	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
50		1nn 11031	. . . . . . . . . . 11
51		dvdsle 15032	. . . . . . . . . . 11 $/\$
52	49, 50, 51	sylancl 694	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
53	48	nnge1d 11063	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
54	52, 53	jctird 567	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
55	48	nnred 11035	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
56		1re 10039	. . . . . . . . . 10
57		letri3 10123	. . . . . . . . . 10 $/\$ $/\$
58	55, 56, 57	sylancl 694	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
59	54, 58	sylibrd 249	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
60	36, 59	syld 47	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
61	60	con3dimp 457	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
62	61	iffalsed 4097	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
63	24, 27, 62	3eqtr4a 2682	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
64	23, 63	pm2.61dan 832	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
65		oveq2 6658	. . . . 5
66		lgs0 25035	. . . . . 6
67	8, 66	syl 17	. . . . 5 $/\$ $/\$ $/\$ $/\$
68	65, 67	sylan9eqr 2678	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
69		oveq2 6658	. . . . 5
70		lgs0 25035	. . . . . 6
71	11, 70	syl 17	. . . . 5 $/\$ $/\$ $/\$ $/\$
72	69, 71	sylan9eqr 2678	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
73	68, 72	oveq12d 6668	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
74		oveq2 6658	. . . 4
75		lgs0 25035	. . . . 5
76	30, 75	syl 17	. . . 4 $/\$ $/\$ $/\$ $/\$
77	74, 76	sylan9eqr 2678	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
78	64, 73, 77	3eqtr4rd 2667	. 2 $/\$ $/\$ $/\$ $/\$ $/\$
79		lgsdilem 25049	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
80	79	adantr 481	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
81		mulcl 10020	. . . . . 6 $/\$
82	81	adantl 482	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
83		mulcom 10022	. . . . . 6 $/\$
84	83	adantl 482	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
85		mulass 10024	. . . . . 6 $/\$ $/\$
86	85	adantl 482	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
87		simpl3 1066	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
88		nnabscl 14065	. . . . . . 7 $/\$
89	87, 88	sylan 488	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
90		nnuz 11723	. . . . . 6
91	89, 90	syl6eleq 2711	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
92		simpll1 1100	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
93		simpll3 1102	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
94		simpr 477	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
95		eqid 2622	. . . . . . . . 9
96	95	lgsfcl3 25043	. . . . . . . 8 $/\$ $/\$
97	92, 93, 94, 96	syl3anc 1326	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
98		elfznn 12370	. . . . . . 7
99		ffvelrn 6357	. . . . . . 7 $/\$
100	97, 98, 99	syl2an 494	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
101	100	zcnd 11483	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
102		simpll2 1101	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
103		eqid 2622	. . . . . . . . 9
104	103	lgsfcl3 25043	. . . . . . . 8 $/\$ $/\$
105	102, 93, 94, 104	syl3anc 1326	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
106		ffvelrn 6357	. . . . . . 7 $/\$
107	105, 98, 106	syl2an 494	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
108	107	zcnd 11483	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
109	92	adantr 481	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
110	102	adantr 481	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
111		simpr 477	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
112		lgsdirprm 25056	. . . . . . . . . . . 12 $/\$ $/\$
113	109, 110, 111, 112	syl3anc 1326	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
114	113	oveq1d 6665	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
115		prmz 15389	. . . . . . . . . . . . 13
116		lgscl 25036	. . . . . . . . . . . . 13 $/\$
117	92, 115, 116	syl2an 494	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
118	117	zcnd 11483	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
119		lgscl 25036	. . . . . . . . . . . . 13 $/\$
120	102, 115, 119	syl2an 494	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
121	120	zcnd 11483	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
122	93	adantr 481	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
123	94	adantr 481	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
124		pczcl 15553	. . . . . . . . . . . 12 $/\$ $/\$
125	111, 122, 123, 124	syl12anc 1324	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
126	118, 121, 125	mulexpd 13023	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
127	114, 126	eqtrd 2656	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
128		iftrue 4092	. . . . . . . . . 10
129	128	adantl 482	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
130		iftrue 4092	. . . . . . . . . . 11
131		iftrue 4092	. . . . . . . . . . 11
132	130, 131	oveq12d 6668	. . . . . . . . . 10
133	132	adantl 482	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
134	127, 129, 133	3eqtr4d 2666	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
135		1t1e1 11175	. . . . . . . . . . 11
136	135	eqcomi 2631	. . . . . . . . . 10
137		iffalse 4095	. . . . . . . . . 10
138		iffalse 4095	. . . . . . . . . . 11
139		iffalse 4095	. . . . . . . . . . 11
140	138, 139	oveq12d 6668	. . . . . . . . . 10
141	136, 137, 140	3eqtr4a 2682	. . . . . . . . 9
142	141	adantl 482	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
143	134, 142	pm2.61dan 832	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
144	143	adantr 481	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
145	98	adantl 482	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
146		eleq1 2689	. . . . . . . . 9
147		oveq2 6658	. . . . . . . . . 10
148		oveq1 6657	. . . . . . . . . 10
149	147, 148	oveq12d 6668	. . . . . . . . 9
150	146, 149	ifbieq1d 4109	. . . . . . . 8
151		eqid 2622	. . . . . . . 8
152		ovex 6678	. . . . . . . . 9
153		1ex 10035	. . . . . . . . 9
154	152, 153	ifex 4156	. . . . . . . 8
155	150, 151, 154	fvmpt 6282	. . . . . . 7
156	145, 155	syl 17	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
157		oveq2 6658	. . . . . . . . . . 11
158	157, 148	oveq12d 6668	. . . . . . . . . 10
159	146, 158	ifbieq1d 4109	. . . . . . . . 9
160		ovex 6678	. . . . . . . . . 10
161	160, 153	ifex 4156	. . . . . . . . 9
162	159, 95, 161	fvmpt 6282	. . . . . . . 8
163	145, 162	syl 17	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
164		oveq2 6658	. . . . . . . . . . 11
165	164, 148	oveq12d 6668	. . . . . . . . . 10
166	146, 165	ifbieq1d 4109	. . . . . . . . 9
167		ovex 6678	. . . . . . . . . 10
168	167, 153	ifex 4156	. . . . . . . . 9
169	166, 103, 168	fvmpt 6282	. . . . . . . 8
170	145, 169	syl 17	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
171	163, 170	oveq12d 6668	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
172	144, 156, 171	3eqtr4d 2666	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
173	82, 84, 86, 91, 101, 108, 172	seqcaopr 12838	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
174	80, 173	oveq12d 6668	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
175	30	adantr 481	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
176	151	lgsval4 25042	. . . 4 $/\$ $/\$ $/\$
177	175, 93, 94, 176	syl3anc 1326	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
178	95	lgsval4 25042	. . . . . 6 $/\$ $/\$ $/\$
179	92, 93, 94, 178	syl3anc 1326	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
180	103	lgsval4 25042	. . . . . 6 $/\$ $/\$ $/\$
181	102, 93, 94, 180	syl3anc 1326	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
182	179, 181	oveq12d 6668	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
183		neg1cn 11124	. . . . . . 7
184	183, 1	keepel 4155	. . . . . 6 $/\$
185	184	a1i 11	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
186		mulcl 10020	. . . . . . 7 $/\$
187	186	adantl 482	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
188	91, 101, 187	seqcl 12821	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
189	183, 1	keepel 4155	. . . . . 6 $/\$
190	189	a1i 11	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
191	91, 108, 187	seqcl 12821	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
192	185, 188, 190, 191	mul4d 10248	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
193	182, 192	eqtrd 2656	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
194	174, 177, 193	3eqtr4d 2666	. 2 $/\$ $/\$ $/\$ $/\$ $/\$
195	78, 194	pm2.61dane 2881	1 $/\$ $/\$ $/\$ $/\$

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4

wb 196 $\/$ wo 383 $/\$ wa 384 $/\$ w3a 1037

wceq 1483

wcel 1990

wne 2794

cif 4086 class class class wbr 4653

cmpt 4729

wf 5884

cfv 5888 (class class class)co 6650

cc 9934

cr 9935

cc0 9936

c1 9937

cmul 9941

clt 10074

cle 10075

cneg 10267

cn 11020

c2 11070

cn0 11292

cz 11377

cuz 11687

cfz 12326

cseq 12801

cexp 12860

cabs 13974

cdvds 14983

cprime 15385

cpc 15541

clgs 25019

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-inf 8349 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-xnn0 11364 df-z 11378 df-uz 11688 df-q 11789 df-rp 11833 df-fz 12327 df-fzo 12466 df-fl 12593 df-mod 12669 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-dvds 14984 df-gcd 15217 df-prm 15386 df-phi 15471 df-pc 15542 df-lgs 25020

This theorem is referenced by: lgssq 25062 lgsmulsqcoprm 25068 lgsdirnn0 25069 lgsquad2lem1 25109

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