lgsdi - Metamath Proof Explorer

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Theorem lgsdi 25059

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

and

are odd positive integers). (Contributed by Mario Carneiro, 5-Feb-2015.)

**Assertion**
Ref	Expression
lgsdi	$/\$ $/\$ $/\$ $/\$

Proof of Theorem lgsdi Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		3anrot 1043	. . . . 5 $/\$ $/\$ $/\$ $/\$
2		lgsdilem 25049	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
3	1, 2	sylanb 489	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
4		ancom 466	. . . . 5 $/\$ $/\$
5		ifbi 4107	. . . . 5 $/\$ $/\$ $/\$ $/\$
6	4, 5	ax-mp 5	. . . 4 $/\$ $/\$
7		ancom 466	. . . . . 6 $/\$ $/\$
8		ifbi 4107	. . . . . 6 $/\$ $/\$ $/\$ $/\$
9	7, 8	ax-mp 5	. . . . 5 $/\$ $/\$
10		ancom 466	. . . . . 6 $/\$ $/\$
11		ifbi 4107	. . . . . 6 $/\$ $/\$ $/\$ $/\$
12	10, 11	ax-mp 5	. . . . 5 $/\$ $/\$
13	9, 12	oveq12i 6662	. . . 4 $/\$ $/\$ $/\$ $/\$
14	3, 6, 13	3eqtr4g 2681	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
15		mulcl 10020	. . . . . 6 $/\$
16	15	adantl 482	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
17		mulcom 10022	. . . . . 6 $/\$
18	17	adantl 482	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
19		mulass 10024	. . . . . 6 $/\$ $/\$
20	19	adantl 482	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
21		simpl2 1065	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
22		simpl3 1066	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
23	21, 22	zmulcld 11488	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
24	21	zcnd 11483	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
25	22	zcnd 11483	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
26		simprl 794	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
27		simprr 796	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
28	24, 25, 26, 27	mulne0d 10679	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
29		nnabscl 14065	. . . . . . 7 $/\$
30	23, 28, 29	syl2anc 693	. . . . . 6 $/\$ $/\$ $/\$ $/\$
31		nnuz 11723	. . . . . 6
32	30, 31	syl6eleq 2711	. . . . 5 $/\$ $/\$ $/\$ $/\$
33		simpl1 1064	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
34		eqid 2622	. . . . . . . . 9
35	34	lgsfcl3 25043	. . . . . . . 8 $/\$ $/\$
36	33, 21, 26, 35	syl3anc 1326	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
37		elfznn 12370	. . . . . . 7
38		ffvelrn 6357	. . . . . . 7 $/\$
39	36, 37, 38	syl2an 494	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
40	39	zcnd 11483	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
41		eqid 2622	. . . . . . . . 9
42	41	lgsfcl3 25043	. . . . . . . 8 $/\$ $/\$
43	33, 22, 27, 42	syl3anc 1326	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
44		ffvelrn 6357	. . . . . . 7 $/\$
45	43, 37, 44	syl2an 494	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
46	45	zcnd 11483	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
47		simpr 477	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
48	21	ad2antrr 762	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
49	26	ad2antrr 762	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
50	22	ad2antrr 762	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
51	27	ad2antrr 762	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
52		pcmul 15556	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
53	47, 48, 49, 50, 51, 52	syl122anc 1335	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
54	53	oveq2d 6666	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
55	33	ad2antrr 762	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
56		prmz 15389	. . . . . . . . . . . . 13
57	56	adantl 482	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
58		lgscl 25036	. . . . . . . . . . . 12 $/\$
59	55, 57, 58	syl2anc 693	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
60	59	zcnd 11483	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
61		pczcl 15553	. . . . . . . . . . 11 $/\$ $/\$
62	47, 50, 51, 61	syl12anc 1324	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
63		pczcl 15553	. . . . . . . . . . 11 $/\$ $/\$
64	47, 48, 49, 63	syl12anc 1324	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
65	60, 62, 64	expaddd 13010	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
66	54, 65	eqtrd 2656	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
67		iftrue 4092	. . . . . . . . 9
68	67	adantl 482	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
69		iftrue 4092	. . . . . . . . . 10
70		iftrue 4092	. . . . . . . . . 10
71	69, 70	oveq12d 6668	. . . . . . . . 9
72	71	adantl 482	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
73	66, 68, 72	3eqtr4rd 2667	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
74		1t1e1 11175	. . . . . . . . 9
75		iffalse 4095	. . . . . . . . . 10
76		iffalse 4095	. . . . . . . . . 10
77	75, 76	oveq12d 6668	. . . . . . . . 9
78		iffalse 4095	. . . . . . . . 9
79	74, 77, 78	3eqtr4a 2682	. . . . . . . 8
80	79	adantl 482	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
81	73, 80	pm2.61dan 832	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
82	37	adantl 482	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
83		eleq1 2689	. . . . . . . . . 10
84		oveq2 6658	. . . . . . . . . . 11
85		oveq1 6657	. . . . . . . . . . 11
86	84, 85	oveq12d 6668	. . . . . . . . . 10
87	83, 86	ifbieq1d 4109	. . . . . . . . 9
88		ovex 6678	. . . . . . . . . 10
89		1ex 10035	. . . . . . . . . 10
90	88, 89	ifex 4156	. . . . . . . . 9
91	87, 34, 90	fvmpt 6282	. . . . . . . 8
92		oveq1 6657	. . . . . . . . . . 11
93	84, 92	oveq12d 6668	. . . . . . . . . 10
94	83, 93	ifbieq1d 4109	. . . . . . . . 9
95		ovex 6678	. . . . . . . . . 10
96	95, 89	ifex 4156	. . . . . . . . 9
97	94, 41, 96	fvmpt 6282	. . . . . . . 8
98	91, 97	oveq12d 6668	. . . . . . 7
99	82, 98	syl 17	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
100		oveq1 6657	. . . . . . . . . 10
101	84, 100	oveq12d 6668	. . . . . . . . 9
102	83, 101	ifbieq1d 4109	. . . . . . . 8
103		eqid 2622	. . . . . . . 8
104		ovex 6678	. . . . . . . . 9
105	104, 89	ifex 4156	. . . . . . . 8
106	102, 103, 105	fvmpt 6282	. . . . . . 7
107	82, 106	syl 17	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
108	81, 99, 107	3eqtr4rd 2667	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
109	16, 18, 20, 32, 40, 46, 108	seqcaopr 12838	. . . 4 $/\$ $/\$ $/\$ $/\$
110	33, 21, 22, 26, 27, 34	lgsdilem2 25058	. . . . 5 $/\$ $/\$ $/\$ $/\$
111	33, 22, 21, 27, 26, 41	lgsdilem2 25058	. . . . . 6 $/\$ $/\$ $/\$ $/\$
112	24, 25	mulcomd 10061	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
113	112	fveq2d 6195	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
114	113	fveq2d 6195	. . . . . 6 $/\$ $/\$ $/\$ $/\$
115	111, 114	eqtr4d 2659	. . . . 5 $/\$ $/\$ $/\$ $/\$
116	110, 115	oveq12d 6668	. . . 4 $/\$ $/\$ $/\$ $/\$
117	109, 116	eqtr4d 2659	. . 3 $/\$ $/\$ $/\$ $/\$
118	14, 117	oveq12d 6668	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
119	103	lgsval4 25042	. . 3 $/\$ $/\$ $/\$
120	33, 23, 28, 119	syl3anc 1326	. 2 $/\$ $/\$ $/\$ $/\$ $/\$
121	34	lgsval4 25042	. . . . 5 $/\$ $/\$ $/\$
122	33, 21, 26, 121	syl3anc 1326	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
123	41	lgsval4 25042	. . . . 5 $/\$ $/\$ $/\$
124	33, 22, 27, 123	syl3anc 1326	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
125	122, 124	oveq12d 6668	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
126		neg1cn 11124	. . . . . 6
127		ax-1cn 9994	. . . . . 6
128	126, 127	keepel 4155	. . . . 5 $/\$
129	128	a1i 11	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
130		nnabscl 14065	. . . . . . 7 $/\$
131	21, 26, 130	syl2anc 693	. . . . . 6 $/\$ $/\$ $/\$ $/\$
132	131, 31	syl6eleq 2711	. . . . 5 $/\$ $/\$ $/\$ $/\$
133		elfznn 12370	. . . . . . 7
134	36, 133, 38	syl2an 494	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
135	134	zcnd 11483	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
136		mulcl 10020	. . . . . 6 $/\$
137	136	adantl 482	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
138	132, 135, 137	seqcl 12821	. . . 4 $/\$ $/\$ $/\$ $/\$
139	126, 127	keepel 4155	. . . . 5 $/\$
140	139	a1i 11	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
141		nnabscl 14065	. . . . . . 7 $/\$
142	22, 27, 141	syl2anc 693	. . . . . 6 $/\$ $/\$ $/\$ $/\$
143	142, 31	syl6eleq 2711	. . . . 5 $/\$ $/\$ $/\$ $/\$
144		elfznn 12370	. . . . . . 7
145	43, 144, 44	syl2an 494	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
146	145	zcnd 11483	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
147	143, 146, 137	seqcl 12821	. . . 4 $/\$ $/\$ $/\$ $/\$
148	129, 138, 140, 147	mul4d 10248	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
149	125, 148	eqtrd 2656	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
150	118, 120, 149	3eqtr4d 2666	1 $/\$ $/\$ $/\$ $/\$

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4

wb 196 $/\$ 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

cc0 9936

c1 9937

caddc 9939

cmul 9941

clt 10074

cneg 10267

cn 11020

cn0 11292

cz 11377

cuz 11687

cfz 12326

cseq 12801

cexp 12860

cabs 13974

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-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: lgssq2 25063 lgsdinn0 25070 lgsquad2lem1 25109

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