Theorem lgsqr 24353

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 24340) and the number is a quadratic residue mod P (it is -u 1 for nonresidues by the process of elimination from lgsabs1 24341). 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 3544	. . . . . . . . . . 11 |- ( P e. ( Prime \ { 2 } ) -> P e. Prime )
2	1	adantl 473	. . . . . . . . . 10 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> P e. Prime )
3		prmz 14705	. . . . . . . . . 10 |- ( P e. Prime -> P e. ZZ )
4	2, 3	syl 17	. . . . . . . . 9 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> P e. ZZ )
5		simpl 464	. . . . . . . . 9 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> A e. ZZ )
6		gcdcom 14563	. . . . . . . . 9 |- ( ( P e. ZZ /\ A e. ZZ ) -> ( P gcd A ) = ( A gcd P ) )
7	4, 5, 6	syl2anc 673	. . . . . . . 8 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( P gcd A ) = ( A gcd P ) )
8	7	eqeq1d 2473	. . . . . . 7 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( P gcd A ) = 1 <-> ( A gcd P ) = 1 ) )
9		coprm 14736	. . . . . . . 8 |- ( ( P e. Prime /\ A e. ZZ ) -> ( -. P || A <-> ( P gcd A ) = 1 ) )
10	2, 5, 9	syl2anc 673	. . . . . . 7 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( -. P || A <-> ( P gcd A ) = 1 ) )
11		lgsne0 24340	. . . . . . . 8 |- ( ( A e. ZZ /\ P e. ZZ ) -> ( ( A /L P ) =/= 0 <-> ( A gcd P ) = 1 ) )
12	5, 4, 11	syl2anc 673	. . . . . . 7 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( A /L P ) =/= 0 <-> ( A gcd P ) = 1 ) )
13	8, 10, 12	3bitr4d 293	. . . . . 6 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( -. P || A <-> ( A /L P ) =/= 0 ) )
14	13	necon4bbid 2684	. . . . 5 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( P || A <-> ( A /L P ) = 0 ) )
15		0ne1 10699	. . . . . 6 |- 0 =/= 1
16		neeq1 2705	. . . . . 6 |- ( ( A /L P ) = 0 -> ( ( A /L P ) =/= 1 <-> 0 =/= 1 ) )
17	15, 16	mpbiri 241	. . . . 5 |- ( ( A /L P ) = 0 -> ( A /L P ) =/= 1 )
18	14, 17	syl6bi 236	. . . 4 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( P || A -> ( A /L P ) =/= 1 ) )
19	18	necon2bd 2659	. . 3 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( A /L P ) = 1 -> -. P || A ) )
20		lgsqrlem5 24352	. . . 4 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ ( A /L P ) = 1 ) -> E. x e. ZZ P || ( ( x ^ 2 ) - A ) )
21	20	3expia 1233	. . 3 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( A /L P ) = 1 -> E. x e. ZZ P || ( ( x ^ 2 ) - A ) ) )
22	19, 21	jcad 542	. 2 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( A /L P ) = 1 -> ( -. P || A /\ E. x e. ZZ P || ( ( x ^ 2 ) - A ) ) ) )
23		simprl 772	. . . . . . . 8 |- ( ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ -. P || A ) /\ ( x e. ZZ /\ P || ( ( x ^ 2 ) - A ) ) ) -> x e. ZZ )
24	23	zred 11063	. . . . . . 7 |- ( ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ -. P || A ) /\ ( x e. ZZ /\ P || ( ( x ^ 2 ) - A ) ) ) -> x e. RR )
25		absresq 13442	. . . . . . 7 |- ( x e. RR -> ( ( abs ` x ) ^ 2 ) = ( x ^ 2 ) )
26	24, 25	syl 17	. . . . . 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 6323	. . . . 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 770	. . . . . . . . . 10 |- ( ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ -. P || A ) /\ ( x e. ZZ /\ P || ( ( x ^ 2 ) - A ) ) ) -> -. P || A )
29	1	ad3antlr 745	. . . . . . . . . . . 12 |- ( ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ -. P || A ) /\ ( x e. ZZ /\ P || ( ( x ^ 2 ) - A ) ) ) -> P e. Prime )
30	29, 3	syl 17	. . . . . . . . . . 11 |- ( ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ -. P || A ) /\ ( x e. ZZ /\ P || ( ( x ^ 2 ) - A ) ) ) -> P e. ZZ )
31		zsqcl 12383	. . . . . . . . . . . 12 |- ( x e. ZZ -> ( x ^ 2 ) e. ZZ )
32	23, 31	syl 17	. . . . . . . . . . 11 |- ( ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ -. P || A ) /\ ( x e. ZZ /\ P || ( ( x ^ 2 ) - A ) ) ) -> ( x ^ 2 ) e. ZZ )
33		simplll 776	. . . . . . . . . . 11 |- ( ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ -. P || A ) /\ ( x e. ZZ /\ P || ( ( x ^ 2 ) - A ) ) ) -> A e. ZZ )
34		simprr 774	. . . . . . . . . . 11 |- ( ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ -. P || A ) /\ ( x e. ZZ /\ P || ( ( x ^ 2 ) - A ) ) ) -> P || ( ( x ^ 2 ) - A ) )
35		dvdssub2 14419	. . . . . . . . . . 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 1295	. . . . . . . . . 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 308	. . . . . . . . 9 |- ( ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ -. P || A ) /\ ( x e. ZZ /\ P || ( ( x ^ 2 ) - A ) ) ) -> -. P || ( x ^ 2 ) )
38		2nn 10790	. . . . . . . . . . 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 14746	. . . . . . . . . 10 |- ( ( P e. Prime /\ x e. ZZ /\ 2 e. NN ) -> ( P || ( x ^ 2 ) <-> P || x ) )
41	29, 23, 39, 40	syl3anc 1292	. . . . . . . . 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 307	. . . . . . . 8 |- ( ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ -. P || A ) /\ ( x e. ZZ /\ P || ( ( x ^ 2 ) - A ) ) ) -> -. P || x )
43		dvds0 14395	. . . . . . . . . . 11 |- ( P e. ZZ -> P || 0 )
44	30, 43	syl 17	. . . . . . . . . 10 |- ( ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ -. P || A ) /\ ( x e. ZZ /\ P || ( ( x ^ 2 ) - A ) ) ) -> P || 0 )
45		breq2 4399	. . . . . . . . . 10 |- ( x = 0 -> ( P || x <-> P || 0 ) )
46	44, 45	syl5ibrcom 230	. . . . . . . . 9 |- ( ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ -. P || A ) /\ ( x e. ZZ /\ P || ( ( x ^ 2 ) - A ) ) ) -> ( x = 0 -> P || x ) )
47	46	necon3bd 2657	. . . . . . . 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 13465	. . . . . . 7 |- ( ( x e. ZZ /\ x =/= 0 ) -> ( abs ` x ) e. NN )
50	23, 48, 49	syl2anc 673	. . . . . 6 |- ( ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ -. P || A ) /\ ( x e. ZZ /\ P || ( ( x ^ 2 ) - A ) ) ) -> ( abs ` x ) e. NN )
51	50	nnzd 11062	. . . . . . . 8 |- ( ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ -. P || A ) /\ ( x e. ZZ /\ P || ( ( x ^ 2 ) - A ) ) ) -> ( abs ` x ) e. ZZ )
52		gcdcom 14563	. . . . . . . 8 |- ( ( ( abs ` x ) e. ZZ /\ P e. ZZ ) -> ( ( abs ` x ) gcd P ) = ( P gcd ( abs ` x ) ) )
53	51, 30, 52	syl2anc 673	. . . . . . 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 14399	. . . . . . . . . 10 |- ( ( P e. ZZ /\ x e. ZZ ) -> ( P || x <-> P || ( abs ` x ) ) )
55	30, 23, 54	syl2anc 673	. . . . . . . . 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 307	. . . . . . . 8 |- ( ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ -. P || A ) /\ ( x e. ZZ /\ P || ( ( x ^ 2 ) - A ) ) ) -> -. P || ( abs ` x ) )
57		coprm 14736	. . . . . . . . 9 |- ( ( P e. Prime /\ ( abs ` x ) e. ZZ ) -> ( -. P || ( abs ` x ) <-> ( P gcd ( abs ` x ) ) = 1 ) )
58	29, 51, 57	syl2anc 673	. . . . . . . 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 215	. . . . . . 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 2505	. . . . . 6 |- ( ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ -. P || A ) /\ ( x e. ZZ /\ P || ( ( x ^ 2 ) - A ) ) ) -> ( ( abs ` x ) gcd P ) = 1 )
61		lgssq 24342	. . . . . 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 1292	. . . . 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 14704	. . . . . . . . . 10 |- ( P e. Prime -> P e. NN )
64	29, 63	syl 17	. . . . . . . . 9 |- ( ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ -. P || A ) /\ ( x e. ZZ /\ P || ( ( x ^ 2 ) - A ) ) ) -> P e. NN )
65		moddvds 14389	. . . . . . . . 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 1292	. . . . . . . 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 240	. . . . . . 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 6323	. . . . . 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 4089	. . . . . . . . . 10 |- ( P e. ( Prime \ { 2 } ) -> P =/= 2 )
70	69	ad3antlr 745	. . . . . . . . 9 |- ( ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ -. P || A ) /\ ( x e. ZZ /\ P || ( ( x ^ 2 ) - A ) ) ) -> P =/= 2 )
71	70	necomd 2698	. . . . . . . 8 |- ( ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ -. P || A ) /\ ( x e. ZZ /\ P || ( ( x ^ 2 ) - A ) ) ) -> 2 =/= P )
72		2z 10993	. . . . . . . . . 10 |- 2 e. ZZ
73		uzid 11197	. . . . . . . . . 10 |- ( 2 e. ZZ -> 2 e. ( ZZ>= ` 2 ) )
74	72, 73	ax-mp 5	. . . . . . . . 9 |- 2 e. ( ZZ>= ` 2 )
75		dvdsprm 14726	. . . . . . . . . 10 |- ( ( 2 e. ( ZZ>= ` 2 ) /\ P e. Prime ) -> ( 2 || P <-> 2 = P ) )
76	75	necon3bbid 2680	. . . . . . . . 9 |- ( ( 2 e. ( ZZ>= ` 2 ) /\ P e. Prime ) -> ( -. 2 || P <-> 2 =/= P ) )
77	74, 29, 76	sylancr 676	. . . . . . . 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 240	. . . . . . 7 |- ( ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ -. P || A ) /\ ( x e. ZZ /\ P || ( ( x ^ 2 ) - A ) ) ) -> -. 2 || P )
79		lgsmod 24328	. . . . . . 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 1292	. . . . . 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 24328	. . . . . . 7 |- ( ( A e. ZZ /\ P e. NN /\ -. 2 || P ) -> ( ( A mod P ) /L P ) = ( A /L P ) )
82	33, 64, 78, 81	syl3anc 1292	. . . . . 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 2513	. . . . 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 2514	. . . 4 |- ( ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ -. P || A ) /\ ( x e. ZZ /\ P || ( ( x ^ 2 ) - A ) ) ) -> ( A /L P ) = 1 )
85	84	rexlimdvaa 2872	. . 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 614	. 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 195	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 189   /\ wa 376   = wceq 1452   e. wcel 1904   =/= wne 2641   E.wrex 2757   \ cdif 3387   {csn 3959   class class class wbr 4395   ` cfv 5589  (class class class)co 6308   RRcr 9556   0cc0 9557   1c1 9558   - cmin 9880   NNcn 10631   2c2 10681   ZZcz 10961   ZZ>=cuz 11182   mod cmo 12129   ^cexp 12310   abscabs 13374   || cdvds 14382   gcd cgcd 14547   Primecprime 14701   /Lclgs 24301

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635  ax-addf 9636  ax-mulf 9637

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-ofr 6551  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-tpos 6991  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-er 7381  df-ec 7383  df-qs 7387  df-map 7492  df-pm 7493  df-ixp 7541  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-q 11288  df-rp 11326  df-fz 11811  df-fzo 11943  df-fl 12061  df-mod 12130  df-seq 12252  df-exp 12311  df-hash 12554  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-dvds 14383  df-gcd 14548  df-prm 14702  df-phi 14793  df-pc 14866  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-starv 15283  df-sca 15284  df-vsca 15285  df-ip 15286  df-tset 15287  df-ple 15288  df-ds 15290  df-unif 15291  df-hom 15292  df-cco 15293  df-0g 15418  df-gsum 15419  df-prds 15424  df-pws 15426  df-imas 15485  df-qus 15487  df-mre 15570  df-mrc 15571  df-acs 15573  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-mhm 16660  df-submnd 16661  df-grp 16751  df-minusg 16752  df-sbg 16753  df-mulg 16754  df-subg 16892  df-nsg 16893  df-eqg 16894  df-ghm 16959  df-cntz 17049  df-cmn 17510  df-abl 17511  df-mgp 17802  df-ur 17814  df-srg 17818  df-ring 17860  df-cring 17861  df-oppr 17929  df-dvdsr 17947  df-unit 17948  df-invr 17978  df-dvr 17989  df-rnghom 18021  df-drng 18055  df-field 18056  df-subrg 18084  df-lmod 18171  df-lss 18234  df-lsp 18273  df-sra 18473  df-rgmod 18474  df-lidl 18475  df-rsp 18476  df-2idl 18533  df-nzr 18559  df-rlreg 18584  df-domn 18585  df-idom 18586  df-assa 18613  df-asp 18614  df-ascl 18615  df-psr 18657  df-mvr 18658  df-mpl 18659  df-opsr 18661  df-evls 18806  df-evl 18807  df-psr1 18850  df-vr1 18851  df-ply1 18852  df-coe1 18853  df-evl1 18982  df-cnfld 19048  df-zring 19117  df-zrh 19152  df-zn 19155  df-mdeg 23083  df-deg1 23084  df-mon1 23159  df-uc1p 23160  df-q1p 23161  df-r1p 23162  df-lgs 24302

This theorem is referenced by:  2sqlem11  24382  2sqblem  24384