Theorem lgsqr 23365

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 23352) and the number is a quadratic residue mod P (it is -u 1 for nonresidues by the process of elimination from lgsabs1 23353). 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 3626	. . . . . . . . . . 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 14079	. . . . . . . . . 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 14016	. . . . . . . . 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 2469	. . . . . . 7 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( P gcd A ) = 1 <-> ( A gcd P ) = 1 ) )
9		coprm 14099	. . . . . . . 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 23352	. . . . . . . 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 2720	. . . . 5 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( P || A <-> ( A /L P ) = 0 ) )
15		0ne1 10602	. . . . . 6 |- 0 =/= 1
16		neeq1 2748	. . . . . 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 2682	. . 3 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( A /L P ) = 1 -> -. P || A ) )
20		lgsqrlem5 23364	. . . 4 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ ( A /L P ) = 1 ) -> E. x e. ZZ P || ( ( x ^ 2 ) - A ) )
21	20	3expia 1198	. . 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 10965	. . . . . . 7 |- ( ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ -. P || A ) /\ ( x e. ZZ /\ P || ( ( x ^ 2 ) - A ) ) ) -> x e. RR )
25		absresq 13097	. . . . . . 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 6298	. . . . 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 12205	. . . . . . . . . . . 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 13881	. . . . . . . . . . 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 1231	. . . . . . . . . 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 10692	. . . . . . . . . . 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 14113	. . . . . . . . . 10 |- ( ( P e. Prime /\ x e. ZZ /\ 2 e. NN ) -> ( P || ( x ^ 2 ) <-> P || x ) )
41	29, 23, 39, 40	syl3anc 1228	. . . . . . . . 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 13859	. . . . . . . . . . 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 4451	. . . . . . . . . 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 2679	. . . . . . . 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 13120	. . . . . . 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 10964	. . . . . . . 8 |- ( ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ -. P || A ) /\ ( x e. ZZ /\ P || ( ( x ^ 2 ) - A ) ) ) -> ( abs ` x ) e. ZZ )
52		gcdcom 14016	. . . . . . . 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 13863	. . . . . . . . . 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 14099	. . . . . . . . 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 2508	. . . . . 6 |- ( ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ -. P || A ) /\ ( x e. ZZ /\ P || ( ( x ^ 2 ) - A ) ) ) -> ( ( abs ` x ) gcd P ) = 1 )
61		lgssq 23354	. . . . . 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 1228	. . . . 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 14078	. . . . . . . . . 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 13853	. . . . . . . . 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 1228	. . . . . . . 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 6298	. . . . . 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 4153	. . . . . . . . . 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 2738	. . . . . . . 8 |- ( ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ -. P || A ) /\ ( x e. ZZ /\ P || ( ( x ^ 2 ) - A ) ) ) -> 2 =/= P )
72		2z 10895	. . . . . . . . . 10 |- 2 e. ZZ
73		uzid 11095	. . . . . . . . . 10 |- ( 2 e. ZZ -> 2 e. ( ZZ>= ` 2 ) )
74	72, 73	ax-mp 5	. . . . . . . . 9 |- 2 e. ( ZZ>= ` 2 )
75		dvdsprm 14098	. . . . . . . . . 10 |- ( ( 2 e. ( ZZ>= ` 2 ) /\ P e. Prime ) -> ( 2 || P <-> 2 = P ) )
76	75	necon3bbid 2714	. . . . . . . . 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 23340	. . . . . . 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 1228	. . . . . 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 23340	. . . . . . 7 |- ( ( A e. ZZ /\ P e. NN /\ -. 2 || P ) -> ( ( A mod P ) /L P ) = ( A /L P ) )
82	33, 64, 78, 81	syl3anc 1228	. . . . . 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 2516	. . . . 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 2517	. . . 4 |- ( ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ -. P || A ) /\ ( x e. ZZ /\ P || ( ( x ^ 2 ) - A ) ) ) -> ( A /L P ) = 1 )
85	84	rexlimdvaa 2956	. . 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 1379   e. wcel 1767   =/= wne 2662   E.wrex 2815   \ cdif 3473   {csn 4027   class class class wbr 4447   ` cfv 5587  (class class class)co 6283   RRcr 9490   0cc0 9491   1c1 9492   - cmin 9804   NNcn 10535   2c2 10584   ZZcz 10863   ZZ>=cuz 11081   mod cmo 11963   ^cexp 12133   abscabs 13029   || cdivides 13846   gcd cgcd 14002   Primecprime 14075   /Lclgs 23313

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575  ax-inf2 8057  ax-cnex 9547  ax-resscn 9548  ax-1cn 9549  ax-icn 9550  ax-addcl 9551  ax-addrcl 9552  ax-mulcl 9553  ax-mulrcl 9554  ax-mulcom 9555  ax-addass 9556  ax-mulass 9557  ax-distr 9558  ax-i2m1 9559  ax-1ne0 9560  ax-1rid 9561  ax-rnegex 9562  ax-rrecex 9563  ax-cnre 9564  ax-pre-lttri 9565  ax-pre-lttrn 9566  ax-pre-ltadd 9567  ax-pre-mulgt0 9568  ax-pre-sup 9569  ax-addf 9570  ax-mulf 9571

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-isom 5596  df-riota 6244  df-ov 6286  df-oprab 6287  df-mpt2 6288  df-of 6523  df-ofr 6524  df-om 6680  df-1st 6784  df-2nd 6785  df-supp 6902  df-tpos 6955  df-recs 7042  df-rdg 7076  df-1o 7130  df-2o 7131  df-oadd 7134  df-er 7311  df-ec 7313  df-qs 7317  df-map 7422  df-pm 7423  df-ixp 7470  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-fsupp 7829  df-sup 7900  df-oi 7934  df-card 8319  df-cda 8547  df-pnf 9629  df-mnf 9630  df-xr 9631  df-ltxr 9632  df-le 9633  df-sub 9806  df-neg 9807  df-div 10206  df-nn 10536  df-2 10593  df-3 10594  df-4 10595  df-5 10596  df-6 10597  df-7 10598  df-8 10599  df-9 10600  df-10 10601  df-n0 10795  df-z 10864  df-dec 10976  df-uz 11082  df-q 11182  df-rp 11220  df-fz 11672  df-fzo 11792  df-fl 11896  df-mod 11964  df-seq 12075  df-exp 12134  df-hash 12373  df-cj 12894  df-re 12895  df-im 12896  df-sqrt 13030  df-abs 13031  df-dvds 13847  df-gcd 14003  df-prm 14076  df-phi 14154  df-pc 14219  df-struct 14491  df-ndx 14492  df-slot 14493  df-base 14494  df-sets 14495  df-ress 14496  df-plusg 14567  df-mulr 14568  df-starv 14569  df-sca 14570  df-vsca 14571  df-ip 14572  df-tset 14573  df-ple 14574  df-ds 14576  df-unif 14577  df-hom 14578  df-cco 14579  df-0g 14696  df-gsum 14697  df-prds 14702  df-pws 14704  df-imas 14762  df-divs 14763  df-mre 14840  df-mrc 14841  df-acs 14843  df-mnd 15731  df-mhm 15783  df-submnd 15784  df-grp 15864  df-minusg 15865  df-sbg 15866  df-mulg 15867  df-subg 16000  df-nsg 16001  df-eqg 16002  df-ghm 16067  df-cntz 16157  df-cmn 16603  df-abl 16604  df-mgp 16941  df-ur 16953  df-srg 16957  df-rng 16997  df-cring 16998  df-oppr 17068  df-dvdsr 17086  df-unit 17087  df-invr 17117  df-dvr 17128  df-rnghom 17160  df-drng 17193  df-field 17194  df-subrg 17222  df-lmod 17309  df-lss 17374  df-lsp 17413  df-sra 17613  df-rgmod 17614  df-lidl 17615  df-rsp 17616  df-2idl 17674  df-nzr 17700  df-rlreg 17718  df-domn 17719  df-idom 17720  df-assa 17748  df-asp 17749  df-ascl 17750  df-psr 17792  df-mvr 17793  df-mpl 17794  df-opsr 17796  df-evls 17958  df-evl 17959  df-psr1 18006  df-vr1 18007  df-ply1 18008  df-coe1 18009  df-evl1 18140  df-cnfld 18208  df-zring 18273  df-zrh 18324  df-zn 18327  df-mdeg 22204  df-deg1 22205  df-mon1 22282  df-uc1p 22283  df-q1p 22284  df-r1p 22285  df-lgs 23314

This theorem is referenced by:  2sqlem11  23394  2sqblem  23396