Theorem lgsqr 22628

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 22615) and the number is a quadratic residue mod P (it is -u 1 for nonresidues by the process of elimination from lgsabs1 22616). 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 3475	. . . . . . . . . . 11 |- ( P e. ( Prime \ { 2 } ) -> P e. Prime )
2	1	adantl 463	. . . . . . . . . 10 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> P e. Prime )
3		prmz 13763	. . . . . . . . . 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 454	. . . . . . . . 9 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> A e. ZZ )
6		gcdcom 13700	. . . . . . . . 9 |- ( ( P e. ZZ /\ A e. ZZ ) -> ( P gcd A ) = ( A gcd P ) )
7	4, 5, 6	syl2anc 656	. . . . . . . 8 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( P gcd A ) = ( A gcd P ) )
8	7	eqeq1d 2449	. . . . . . 7 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( P gcd A ) = 1 <-> ( A gcd P ) = 1 ) )
9		coprm 13782	. . . . . . . 8 |- ( ( P e. Prime /\ A e. ZZ ) -> ( -. P || A <-> ( P gcd A ) = 1 ) )
10	2, 5, 9	syl2anc 656	. . . . . . 7 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( -. P || A <-> ( P gcd A ) = 1 ) )
11		lgsne0 22615	. . . . . . . 8 |- ( ( A e. ZZ /\ P e. ZZ ) -> ( ( A /L P ) =/= 0 <-> ( A gcd P ) = 1 ) )
12	5, 4, 11	syl2anc 656	. . . . . . 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 2674	. . . . 5 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( P || A <-> ( A /L P ) = 0 ) )
15		0ne1 10385	. . . . . 6 |- 0 =/= 1
16		neeq1 2614	. . . . . 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 2658	. . 3 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( A /L P ) = 1 -> -. P || A ) )
20		lgsqrlem5 22627	. . . 4 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ ( A /L P ) = 1 ) -> E. x e. ZZ P || ( ( x ^ 2 ) - A ) )
21	20	3expia 1184	. . 3 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( A /L P ) = 1 -> E. x e. ZZ P || ( ( x ^ 2 ) - A ) ) )
22	19, 21	jcad 530	. 2 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( A /L P ) = 1 -> ( -. P || A /\ E. x e. ZZ P || ( ( x ^ 2 ) - A ) ) ) )
23		simprl 750	. . . . . . . 8 |- ( ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ -. P || A ) /\ ( x e. ZZ /\ P || ( ( x ^ 2 ) - A ) ) ) -> x e. ZZ )
24	23	zred 10743	. . . . . . 7 |- ( ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ -. P || A ) /\ ( x e. ZZ /\ P || ( ( x ^ 2 ) - A ) ) ) -> x e. RR )
25		absresq 12787	. . . . . . 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 6105	. . . . 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 749	. . . . . . . . . 10 |- ( ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ -. P || A ) /\ ( x e. ZZ /\ P || ( ( x ^ 2 ) - A ) ) ) -> -. P || A )
29	1	ad3antlr 725	. . . . . . . . . . . 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 11932	. . . . . . . . . . . 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 752	. . . . . . . . . . 11 |- ( ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ -. P || A ) /\ ( x e. ZZ /\ P || ( ( x ^ 2 ) - A ) ) ) -> A e. ZZ )
34		simprr 751	. . . . . . . . . . 11 |- ( ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ -. P || A ) /\ ( x e. ZZ /\ P || ( ( x ^ 2 ) - A ) ) ) -> P || ( ( x ^ 2 ) - A ) )
35		dvdssub2 13566	. . . . . . . . . . 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 1216	. . . . . . . . . 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 10475	. . . . . . . . . . 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 13796	. . . . . . . . . 10 |- ( ( P e. Prime /\ x e. ZZ /\ 2 e. NN ) -> ( P || ( x ^ 2 ) <-> P || x ) )
41	29, 23, 39, 40	syl3anc 1213	. . . . . . . . 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 13544	. . . . . . . . . . 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 4293	. . . . . . . . . 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 2643	. . . . . . . 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 12809	. . . . . . 7 |- ( ( x e. ZZ /\ x =/= 0 ) -> ( abs ` x ) e. NN )
50	23, 48, 49	syl2anc 656	. . . . . 6 |- ( ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ -. P || A ) /\ ( x e. ZZ /\ P || ( ( x ^ 2 ) - A ) ) ) -> ( abs ` x ) e. NN )
51	50	nnzd 10742	. . . . . . . 8 |- ( ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ -. P || A ) /\ ( x e. ZZ /\ P || ( ( x ^ 2 ) - A ) ) ) -> ( abs ` x ) e. ZZ )
52		gcdcom 13700	. . . . . . . 8 |- ( ( ( abs ` x ) e. ZZ /\ P e. ZZ ) -> ( ( abs ` x ) gcd P ) = ( P gcd ( abs ` x ) ) )
53	51, 30, 52	syl2anc 656	. . . . . . 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 13548	. . . . . . . . . 10 |- ( ( P e. ZZ /\ x e. ZZ ) -> ( P || x <-> P || ( abs ` x ) ) )
55	30, 23, 54	syl2anc 656	. . . . . . . . 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 13782	. . . . . . . . 9 |- ( ( P e. Prime /\ ( abs ` x ) e. ZZ ) -> ( -. P || ( abs ` x ) <-> ( P gcd ( abs ` x ) ) = 1 ) )
58	29, 51, 57	syl2anc 656	. . . . . . . 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 2473	. . . . . 6 |- ( ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ -. P || A ) /\ ( x e. ZZ /\ P || ( ( x ^ 2 ) - A ) ) ) -> ( ( abs ` x ) gcd P ) = 1 )
61		lgssq 22617	. . . . . 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 1213	. . . . 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 13762	. . . . . . . . . 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 13538	. . . . . . . . 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 1213	. . . . . . . 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 6105	. . . . . 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 3998	. . . . . . . . . 10 |- ( P e. ( Prime \ { 2 } ) -> P =/= 2 )
70	69	ad3antlr 725	. . . . . . . . 9 |- ( ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ -. P || A ) /\ ( x e. ZZ /\ P || ( ( x ^ 2 ) - A ) ) ) -> P =/= 2 )
71	70	necomd 2693	. . . . . . . 8 |- ( ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ -. P || A ) /\ ( x e. ZZ /\ P || ( ( x ^ 2 ) - A ) ) ) -> 2 =/= P )
72		2z 10674	. . . . . . . . . 10 |- 2 e. ZZ
73		uzid 10871	. . . . . . . . . 10 |- ( 2 e. ZZ -> 2 e. ( ZZ>= ` 2 ) )
74	72, 73	ax-mp 5	. . . . . . . . 9 |- 2 e. ( ZZ>= ` 2 )
75		dvdsprm 13781	. . . . . . . . . 10 |- ( ( 2 e. ( ZZ>= ` 2 ) /\ P e. Prime ) -> ( 2 || P <-> 2 = P ) )
76	75	necon3bbid 2640	. . . . . . . . 9 |- ( ( 2 e. ( ZZ>= ` 2 ) /\ P e. Prime ) -> ( -. 2 || P <-> 2 =/= P ) )
77	74, 29, 76	sylancr 658	. . . . . . . 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 22603	. . . . . . 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 1213	. . . . . 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 22603	. . . . . . 7 |- ( ( A e. ZZ /\ P e. NN /\ -. 2 || P ) -> ( ( A mod P ) /L P ) = ( A /L P ) )
82	33, 64, 78, 81	syl3anc 1213	. . . . . 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 2481	. . . . 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 2482	. . . 4 |- ( ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ -. P || A ) /\ ( x e. ZZ /\ P || ( ( x ^ 2 ) - A ) ) ) -> ( A /L P ) = 1 )
85	84	rexlimdvaa 2840	. . 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 600	. 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 1364   e. wcel 1761   =/= wne 2604   E.wrex 2714   \ cdif 3322   {csn 3874   class class class wbr 4289   ` cfv 5415  (class class class)co 6090   RRcr 9277   0cc0 9278   1c1 9279   - cmin 9591   NNcn 10318   2c2 10367   ZZcz 10642   ZZ>=cuz 10857   mod cmo 11704   ^cexp 11861   abscabs 12719   || cdivides 13531   gcd cgcd 13686   Primecprime 13759   /Lclgs 22576

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356  ax-addf 9357  ax-mulf 9358

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-iin 4171  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-ofr 6320  df-om 6476  df-1st 6576  df-2nd 6577  df-supp 6690  df-tpos 6744  df-recs 6828  df-rdg 6862  df-1o 6916  df-2o 6917  df-oadd 6920  df-er 7097  df-ec 7099  df-qs 7103  df-map 7212  df-pm 7213  df-ixp 7260  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-fsupp 7617  df-sup 7687  df-oi 7720  df-card 8105  df-cda 8333  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  df-uz 10858  df-q 10950  df-rp 10988  df-fz 11434  df-fzo 11545  df-fl 11638  df-mod 11705  df-seq 11803  df-exp 11862  df-hash 12100  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-dvds 13532  df-gcd 13687  df-prm 13760  df-phi 13837  df-pc 13900  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-plusg 14247  df-mulr 14248  df-starv 14249  df-sca 14250  df-vsca 14251  df-ip 14252  df-tset 14253  df-ple 14254  df-ds 14256  df-unif 14257  df-hom 14258  df-cco 14259  df-0g 14376  df-gsum 14377  df-prds 14382  df-pws 14384  df-imas 14442  df-divs 14443  df-mre 14520  df-mrc 14521  df-acs 14523  df-mnd 15411  df-mhm 15460  df-submnd 15461  df-grp 15538  df-minusg 15539  df-sbg 15540  df-mulg 15541  df-subg 15671  df-nsg 15672  df-eqg 15673  df-ghm 15738  df-cntz 15828  df-cmn 16272  df-abl 16273  df-mgp 16582  df-ur 16594  df-rng 16637  df-cring 16638  df-oppr 16705  df-dvdsr 16723  df-unit 16724  df-invr 16754  df-dvr 16765  df-rnghom 16796  df-drng 16814  df-field 16815  df-subrg 16843  df-lmod 16930  df-lss 16992  df-lsp 17031  df-sra 17231  df-rgmod 17232  df-lidl 17233  df-rsp 17234  df-2idl 17292  df-nzr 17318  df-rlreg 17332  df-domn 17333  df-idom 17334  df-assa 17362  df-asp 17363  df-ascl 17364  df-psr 17407  df-mvr 17408  df-mpl 17409  df-evls 17410  df-evl 17411  df-opsr 17415  df-psr1 17589  df-vr1 17590  df-ply1 17591  df-evl1 17593  df-coe1 17594  df-cnfld 17719  df-zring 17784  df-zrh 17835  df-zn 17838  df-mdeg 21467  df-deg1 21468  df-mon1 21545  df-uc1p 21546  df-q1p 21547  df-r1p 21548  df-lgs 22577

This theorem is referenced by:  2sqlem11  22657  2sqblem  22659