Theorem lgsqr 22685

Description: The Legendre symbol for odd primes is 1 iff the number is not a multiple of the prime (in which case it is 0, see lgsne0 22672) and the number is a quadratic residue mod P (it is -u 1 for nonresidues by the process of elimination from lgsabs1 22673). Given our definition of the Legendre symbol, this theorem is equivalent to Euler's criterion. (Contributed by Mario Carneiro, 15-Jun-2015.)

Assertion

Ref	Expression
lgsqr	|- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( A /L P ) = 1 <-> ( -. P || A /\ E. x e. ZZ P || ( ( x ^ 2 ) - A ) ) ) )

Distinct variable groups:   x, A   x, P

Proof of Theorem lgsqr

Step	Hyp	Ref	Expression
1		eldifi 3478	. . . . . . . . . . 11 |- ( P e. ( Prime \ { 2 } ) -> P e. Prime )
2	1	adantl 466	. . . . . . . . . 10 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> P e. Prime )
3		prmz 13767	. . . . . . . . . 10 |- ( P e. Prime -> P e. ZZ )
4	2, 3	syl 16	. . . . . . . . 9 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> P e. ZZ )
5		simpl 457	. . . . . . . . 9 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> A e. ZZ )
6		gcdcom 13704	. . . . . . . . 9 |- ( ( P e. ZZ /\ A e. ZZ ) -> ( P gcd A ) = ( A gcd P ) )
7	4, 5, 6	syl2anc 661	. . . . . . . 8 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( P gcd A ) = ( A gcd P ) )
8	7	eqeq1d 2451	. . . . . . 7 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( P gcd A ) = 1 <-> ( A gcd P ) = 1 ) )
9		coprm 13786	. . . . . . . 8 |- ( ( P e. Prime /\ A e. ZZ ) -> ( -. P || A <-> ( P gcd A ) = 1 ) )
10	2, 5, 9	syl2anc 661	. . . . . . 7 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( -. P || A <-> ( P gcd A ) = 1 ) )
11		lgsne0 22672	. . . . . . . 8 |- ( ( A e. ZZ /\ P e. ZZ ) -> ( ( A /L P ) =/= 0 <-> ( A gcd P ) = 1 ) )
12	5, 4, 11	syl2anc 661	. . . . . . 7 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( A /L P ) =/= 0 <-> ( A gcd P ) = 1 ) )
13	8, 10, 12	3bitr4d 285	. . . . . 6 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( -. P || A <-> ( A /L P ) =/= 0 ) )
14	13	necon4bbid 2676	. . . . 5 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( P || A <-> ( A /L P ) = 0 ) )
15		0ne1 10389	. . . . . 6 |- 0 =/= 1
16		neeq1 2616	. . . . . 6 |- ( ( A /L P ) = 0 -> ( ( A /L P ) =/= 1 <-> 0 =/= 1 ) )
17	15, 16	mpbiri 233	. . . . 5 |- ( ( A /L P ) = 0 -> ( A /L P ) =/= 1 )
18	14, 17	syl6bi 228	. . . 4 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( P || A -> ( A /L P ) =/= 1 ) )
19	18	necon2bd 2660	. . 3 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( A /L P ) = 1 -> -. P || A ) )
20		lgsqrlem5 22684	. . . 4 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ ( A /L P ) = 1 ) -> E. x e. ZZ P || ( ( x ^ 2 ) - A ) )
21	20	3expia 1189	. . 3 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( A /L P ) = 1 -> E. x e. ZZ P || ( ( x ^ 2 ) - A ) ) )
22	19, 21	jcad 533	. 2 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( A /L P ) = 1 -> ( -. P || A /\ E. x e. ZZ P || ( ( x ^ 2 ) - A ) ) ) )
23		simprl 755	. . . . . . . 8 |- ( ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ -. P || A ) /\ ( x e. ZZ /\ P || ( ( x ^ 2 ) - A ) ) ) -> x e. ZZ )
24	23	zred 10747	. . . . . . 7 |- ( ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ -. P || A ) /\ ( x e. ZZ /\ P || ( ( x ^ 2 ) - A ) ) ) -> x e. RR )
25		absresq 12791	. . . . . . 7 |- ( x e. RR -> ( ( abs ` x ) ^ 2 ) = ( x ^ 2 ) )
26	24, 25	syl 16	. . . . . 6 |- ( ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ -. P || A ) /\ ( x e. ZZ /\ P || ( ( x ^ 2 ) - A ) ) ) -> ( ( abs ` x ) ^ 2 ) = ( x ^ 2 ) )
27	26	oveq1d 6106	. . . . 5 |- ( ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ -. P || A ) /\ ( x e. ZZ /\ P || ( ( x ^ 2 ) - A ) ) ) -> ( ( ( abs ` x ) ^ 2 ) /L P ) = ( ( x ^ 2 ) /L P ) )
28		simplr 754	. . . . . . . . . 10 |- ( ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ -. P || A ) /\ ( x e. ZZ /\ P || ( ( x ^ 2 ) - A ) ) ) -> -. P || A )
29	1	ad3antlr 730	. . . . . . . . . . . 12 |- ( ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ -. P || A ) /\ ( x e. ZZ /\ P || ( ( x ^ 2 ) - A ) ) ) -> P e. Prime )
30	29, 3	syl 16	. . . . . . . . . . 11 |- ( ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ -. P || A ) /\ ( x e. ZZ /\ P || ( ( x ^ 2 ) - A ) ) ) -> P e. ZZ )
31		zsqcl 11936	. . . . . . . . . . . 12 |- ( x e. ZZ -> ( x ^ 2 ) e. ZZ )
32	23, 31	syl 16	. . . . . . . . . . 11 |- ( ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ -. P || A ) /\ ( x e. ZZ /\ P || ( ( x ^ 2 ) - A ) ) ) -> ( x ^ 2 ) e. ZZ )
33		simplll 757	. . . . . . . . . . 11 |- ( ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ -. P || A ) /\ ( x e. ZZ /\ P || ( ( x ^ 2 ) - A ) ) ) -> A e. ZZ )
34		simprr 756	. . . . . . . . . . 11 |- ( ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ -. P || A ) /\ ( x e. ZZ /\ P || ( ( x ^ 2 ) - A ) ) ) -> P || ( ( x ^ 2 ) - A ) )
35		dvdssub2 13570	. . . . . . . . . . 11 |- ( ( ( P e. ZZ /\ ( x ^ 2 ) e. ZZ /\ A e. ZZ ) /\ P || ( ( x ^ 2 ) - A ) ) -> ( P || ( x ^ 2 ) <-> P || A ) )
36	30, 32, 33, 34, 35	syl31anc 1221	. . . . . . . . . 10 |- ( ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ -. P || A ) /\ ( x e. ZZ /\ P || ( ( x ^ 2 ) - A ) ) ) -> ( P || ( x ^ 2 ) <-> P || A ) )
37	28, 36	mtbird 301	. . . . . . . . 9 |- ( ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ -. P || A ) /\ ( x e. ZZ /\ P || ( ( x ^ 2 ) - A ) ) ) -> -. P || ( x ^ 2 ) )
38		2nn 10479	. . . . . . . . . . 11 |- 2 e. NN
39	38	a1i 11	. . . . . . . . . 10 |- ( ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ -. P || A ) /\ ( x e. ZZ /\ P || ( ( x ^ 2 ) - A ) ) ) -> 2 e. NN )
40		prmdvdsexp 13800	. . . . . . . . . 10 |- ( ( P e. Prime /\ x e. ZZ /\ 2 e. NN ) -> ( P || ( x ^ 2 ) <-> P || x ) )
41	29, 23, 39, 40	syl3anc 1218	. . . . . . . . 9 |- ( ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ -. P || A ) /\ ( x e. ZZ /\ P || ( ( x ^ 2 ) - A ) ) ) -> ( P || ( x ^ 2 ) <-> P || x ) )
42	37, 41	mtbid 300	. . . . . . . 8 |- ( ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ -. P || A ) /\ ( x e. ZZ /\ P || ( ( x ^ 2 ) - A ) ) ) -> -. P || x )
43		dvds0 13548	. . . . . . . . . . 11 |- ( P e. ZZ -> P || 0 )
44	30, 43	syl 16	. . . . . . . . . 10 |- ( ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ -. P || A ) /\ ( x e. ZZ /\ P || ( ( x ^ 2 ) - A ) ) ) -> P || 0 )
45		breq2 4296	. . . . . . . . . 10 |- ( x = 0 -> ( P || x <-> P || 0 ) )
46	44, 45	syl5ibrcom 222	. . . . . . . . 9 |- ( ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ -. P || A ) /\ ( x e. ZZ /\ P || ( ( x ^ 2 ) - A ) ) ) -> ( x = 0 -> P || x ) )
47	46	necon3bd 2645	. . . . . . . 8 |- ( ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ -. P || A ) /\ ( x e. ZZ /\ P || ( ( x ^ 2 ) - A ) ) ) -> ( -. P || x -> x =/= 0 ) )
48	42, 47	mpd 15	. . . . . . 7 |- ( ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ -. P || A ) /\ ( x e. ZZ /\ P || ( ( x ^ 2 ) - A ) ) ) -> x =/= 0 )
49		nnabscl 12813	. . . . . . 7 |- ( ( x e. ZZ /\ x =/= 0 ) -> ( abs ` x ) e. NN )
50	23, 48, 49	syl2anc 661	. . . . . 6 |- ( ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ -. P || A ) /\ ( x e. ZZ /\ P || ( ( x ^ 2 ) - A ) ) ) -> ( abs ` x ) e. NN )
51	50	nnzd 10746	. . . . . . . 8 |- ( ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ -. P || A ) /\ ( x e. ZZ /\ P || ( ( x ^ 2 ) - A ) ) ) -> ( abs ` x ) e. ZZ )
52		gcdcom 13704	. . . . . . . 8 |- ( ( ( abs ` x ) e. ZZ /\ P e. ZZ ) -> ( ( abs ` x ) gcd P ) = ( P gcd ( abs ` x ) ) )
53	51, 30, 52	syl2anc 661	. . . . . . 7 |- ( ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ -. P || A ) /\ ( x e. ZZ /\ P || ( ( x ^ 2 ) - A ) ) ) -> ( ( abs ` x ) gcd P ) = ( P gcd ( abs ` x ) ) )
54		dvdsabsb 13552	. . . . . . . . . 10 |- ( ( P e. ZZ /\ x e. ZZ ) -> ( P || x <-> P || ( abs ` x ) ) )
55	30, 23, 54	syl2anc 661	. . . . . . . . 9 |- ( ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ -. P || A ) /\ ( x e. ZZ /\ P || ( ( x ^ 2 ) - A ) ) ) -> ( P || x <-> P || ( abs ` x ) ) )
56	42, 55	mtbid 300	. . . . . . . 8 |- ( ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ -. P || A ) /\ ( x e. ZZ /\ P || ( ( x ^ 2 ) - A ) ) ) -> -. P || ( abs ` x ) )
57		coprm 13786	. . . . . . . . 9 |- ( ( P e. Prime /\ ( abs ` x ) e. ZZ ) -> ( -. P || ( abs ` x ) <-> ( P gcd ( abs ` x ) ) = 1 ) )
58	29, 51, 57	syl2anc 661	. . . . . . . 8 |- ( ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ -. P || A ) /\ ( x e. ZZ /\ P || ( ( x ^ 2 ) - A ) ) ) -> ( -. P || ( abs ` x ) <-> ( P gcd ( abs ` x ) ) = 1 ) )
59	56, 58	mpbid 210	. . . . . . 7 |- ( ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ -. P || A ) /\ ( x e. ZZ /\ P || ( ( x ^ 2 ) - A ) ) ) -> ( P gcd ( abs ` x ) ) = 1 )
60	53, 59	eqtrd 2475	. . . . . 6 |- ( ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ -. P || A ) /\ ( x e. ZZ /\ P || ( ( x ^ 2 ) - A ) ) ) -> ( ( abs ` x ) gcd P ) = 1 )
61		lgssq 22674	. . . . . 6 |- ( ( ( abs ` x ) e. NN /\ P e. ZZ /\ ( ( abs ` x ) gcd P ) = 1 ) -> ( ( ( abs ` x ) ^ 2 ) /L P ) = 1 )
62	50, 30, 60, 61	syl3anc 1218	. . . . 5 |- ( ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ -. P || A ) /\ ( x e. ZZ /\ P || ( ( x ^ 2 ) - A ) ) ) -> ( ( ( abs ` x ) ^ 2 ) /L P ) = 1 )
63		prmnn 13766	. . . . . . . . . 10 |- ( P e. Prime -> P e. NN )
64	29, 63	syl 16	. . . . . . . . 9 |- ( ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ -. P || A ) /\ ( x e. ZZ /\ P || ( ( x ^ 2 ) - A ) ) ) -> P e. NN )
65		moddvds 13542	. . . . . . . . 9 |- ( ( P e. NN /\ ( x ^ 2 ) e. ZZ /\ A e. ZZ ) -> ( ( ( x ^ 2 ) mod P ) = ( A mod P ) <-> P || ( ( x ^ 2 ) - A ) ) )
66	64, 32, 33, 65	syl3anc 1218	. . . . . . . 8 |- ( ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ -. P || A ) /\ ( x e. ZZ /\ P || ( ( x ^ 2 ) - A ) ) ) -> ( ( ( x ^ 2 ) mod P ) = ( A mod P ) <-> P || ( ( x ^ 2 ) - A ) ) )
67	34, 66	mpbird 232	. . . . . . 7 |- ( ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ -. P || A ) /\ ( x e. ZZ /\ P || ( ( x ^ 2 ) - A ) ) ) -> ( ( x ^ 2 ) mod P ) = ( A mod P ) )
68	67	oveq1d 6106	. . . . . 6 |- ( ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ -. P || A ) /\ ( x e. ZZ /\ P || ( ( x ^ 2 ) - A ) ) ) -> ( ( ( x ^ 2 ) mod P ) /L P ) = ( ( A mod P ) /L P ) )
69		eldifsni 4001	. . . . . . . . . 10 |- ( P e. ( Prime \ { 2 } ) -> P =/= 2 )
70	69	ad3antlr 730	. . . . . . . . 9 |- ( ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ -. P || A ) /\ ( x e. ZZ /\ P || ( ( x ^ 2 ) - A ) ) ) -> P =/= 2 )
71	70	necomd 2695	. . . . . . . 8 |- ( ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ -. P || A ) /\ ( x e. ZZ /\ P || ( ( x ^ 2 ) - A ) ) ) -> 2 =/= P )
72		2z 10678	. . . . . . . . . 10 |- 2 e. ZZ
73		uzid 10875	. . . . . . . . . 10 |- ( 2 e. ZZ -> 2 e. ( ZZ>= ` 2 ) )
74	72, 73	ax-mp 5	. . . . . . . . 9 |- 2 e. ( ZZ>= ` 2 )
75		dvdsprm 13785	. . . . . . . . . 10 |- ( ( 2 e. ( ZZ>= ` 2 ) /\ P e. Prime ) -> ( 2 || P <-> 2 = P ) )
76	75	necon3bbid 2642	. . . . . . . . 9 |- ( ( 2 e. ( ZZ>= ` 2 ) /\ P e. Prime ) -> ( -. 2 || P <-> 2 =/= P ) )
77	74, 29, 76	sylancr 663	. . . . . . . 8 |- ( ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ -. P || A ) /\ ( x e. ZZ /\ P || ( ( x ^ 2 ) - A ) ) ) -> ( -. 2 || P <-> 2 =/= P ) )
78	71, 77	mpbird 232	. . . . . . 7 |- ( ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ -. P || A ) /\ ( x e. ZZ /\ P || ( ( x ^ 2 ) - A ) ) ) -> -. 2 || P )
79		lgsmod 22660	. . . . . . 7 |- ( ( ( x ^ 2 ) e. ZZ /\ P e. NN /\ -. 2 || P ) -> ( ( ( x ^ 2 ) mod P ) /L P ) = ( ( x ^ 2 ) /L P ) )
80	32, 64, 78, 79	syl3anc 1218	. . . . . 6 |- ( ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ -. P || A ) /\ ( x e. ZZ /\ P || ( ( x ^ 2 ) - A ) ) ) -> ( ( ( x ^ 2 ) mod P ) /L P ) = ( ( x ^ 2 ) /L P ) )
81		lgsmod 22660	. . . . . . 7 |- ( ( A e. ZZ /\ P e. NN /\ -. 2 || P ) -> ( ( A mod P ) /L P ) = ( A /L P ) )
82	33, 64, 78, 81	syl3anc 1218	. . . . . 6 |- ( ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ -. P || A ) /\ ( x e. ZZ /\ P || ( ( x ^ 2 ) - A ) ) ) -> ( ( A mod P ) /L P ) = ( A /L P ) )
83	68, 80, 82	3eqtr3d 2483	. . . . 5 |- ( ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ -. P || A ) /\ ( x e. ZZ /\ P || ( ( x ^ 2 ) - A ) ) ) -> ( ( x ^ 2 ) /L P ) = ( A /L P ) )
84	27, 62, 83	3eqtr3rd 2484	. . . 4 |- ( ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ -. P || A ) /\ ( x e. ZZ /\ P || ( ( x ^ 2 ) - A ) ) ) -> ( A /L P ) = 1 )
85	84	rexlimdvaa 2842	. . 3 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ -. P || A ) -> ( E. x e. ZZ P || ( ( x ^ 2 ) - A ) -> ( A /L P ) = 1 ) )
86	85	expimpd 603	. 2 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( -. P || A /\ E. x e. ZZ P || ( ( x ^ 2 ) - A ) ) -> ( A /L P ) = 1 ) )
87	22, 86	impbid 191	1 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( A /L P ) = 1 <-> ( -. P || A /\ E. x e. ZZ P || ( ( x ^ 2 ) - A ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1369   e. wcel 1756   =/= wne 2606   E.wrex 2716   \ cdif 3325   {csn 3877   class class class wbr 4292   ` cfv 5418  (class class class)co 6091   RRcr 9281   0cc0 9282   1c1 9283   - cmin 9595   NNcn 10322   2c2 10371   ZZcz 10646   ZZ>=cuz 10861   mod cmo 11708   ^cexp 11865   abscabs 12723   || cdivides 13535   gcd cgcd 13690   Primecprime 13763   /Lclgs 22633

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-inf2 7847  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-pre-sup 9360  ax-addf 9361  ax-mulf 9362

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-iin 4174  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-se 4680  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-isom 5427  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-of 6320  df-ofr 6321  df-om 6477  df-1st 6577  df-2nd 6578  df-supp 6691  df-tpos 6745  df-recs 6832  df-rdg 6866  df-1o 6920  df-2o 6921  df-oadd 6924  df-er 7101  df-ec 7103  df-qs 7107  df-map 7216  df-pm 7217  df-ixp 7264  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-fsupp 7621  df-sup 7691  df-oi 7724  df-card 8109  df-cda 8337  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-2 10380  df-3 10381  df-4 10382  df-5 10383  df-6 10384  df-7 10385  df-8 10386  df-9 10387  df-10 10388  df-n0 10580  df-z 10647  df-dec 10756  df-uz 10862  df-q 10954  df-rp 10992  df-fz 11438  df-fzo 11549  df-fl 11642  df-mod 11709  df-seq 11807  df-exp 11866  df-hash 12104  df-cj 12588  df-re 12589  df-im 12590  df-sqr 12724  df-abs 12725  df-dvds 13536  df-gcd 13691  df-prm 13764  df-phi 13841  df-pc 13904  df-struct 14176  df-ndx 14177  df-slot 14178  df-base 14179  df-sets 14180  df-ress 14181  df-plusg 14251  df-mulr 14252  df-starv 14253  df-sca 14254  df-vsca 14255  df-ip 14256  df-tset 14257  df-ple 14258  df-ds 14260  df-unif 14261  df-hom 14262  df-cco 14263  df-0g 14380  df-gsum 14381  df-prds 14386  df-pws 14388  df-imas 14446  df-divs 14447  df-mre 14524  df-mrc 14525  df-acs 14527  df-mnd 15415  df-mhm 15464  df-submnd 15465  df-grp 15545  df-minusg 15546  df-sbg 15547  df-mulg 15548  df-subg 15678  df-nsg 15679  df-eqg 15680  df-ghm 15745  df-cntz 15835  df-cmn 16279  df-abl 16280  df-mgp 16592  df-ur 16604  df-srg 16608  df-rng 16647  df-cring 16648  df-oppr 16715  df-dvdsr 16733  df-unit 16734  df-invr 16764  df-dvr 16775  df-rnghom 16806  df-drng 16834  df-field 16835  df-subrg 16863  df-lmod 16950  df-lss 17014  df-lsp 17053  df-sra 17253  df-rgmod 17254  df-lidl 17255  df-rsp 17256  df-2idl 17314  df-nzr 17340  df-rlreg 17354  df-domn 17355  df-idom 17356  df-assa 17384  df-asp 17385  df-ascl 17386  df-psr 17423  df-mvr 17424  df-mpl 17425  df-opsr 17427  df-evls 17588  df-evl 17589  df-psr1 17636  df-vr1 17637  df-ply1 17638  df-coe1 17639  df-evl1 17751  df-cnfld 17819  df-zring 17884  df-zrh 17935  df-zn 17938  df-mdeg 21524  df-deg1 21525  df-mon1 21602  df-uc1p 21603  df-q1p 21604  df-r1p 21605  df-lgs 22634

This theorem is referenced by:  2sqlem11  22714  2sqblem  22716