Theorem lgsdir2lem4 25053

Description: Lemma for lgsdir2 25055. (Contributed by Mario Carneiro, 4-Feb-2015.)

Assertion

Ref	Expression
lgsdir2lem4	|- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) e. { 1 , 7 } ) -> ( ( ( A x. B ) mod 8 ) e. { 1 , 7 } <-> ( B mod 8 ) e. { 1 , 7 } ) )

Proof of Theorem lgsdir2lem4

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ovex 6678	. . 3 |- ( A mod 8 ) e. _V
2	1	elpr 4198	. 2 |- ( ( A mod 8 ) e. { 1 , 7 } <-> ( ( A mod 8 ) = 1 \/ ( A mod 8 ) = 7 ) )
3		zre 11381	. . . . . . 7 |- ( A e. ZZ -> A e. RR )
4	3	ad2antrr 762	. . . . . 6 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 1 ) -> A e. RR )
5		1red 10055	. . . . . 6 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 1 ) -> 1 e. RR )
6		simplr 792	. . . . . 6 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 1 ) -> B e. ZZ )
7		8re 11105	. . . . . . . 8 |- 8 e. RR
8		8pos 11121	. . . . . . . 8 |- 0 < 8
9	7, 8	elrpii 11835	. . . . . . 7 |- 8 e. RR+
10	9	a1i 11	. . . . . 6 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 1 ) -> 8 e. RR+ )
11		simpr 477	. . . . . . 7 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 1 ) -> ( A mod 8 ) = 1 )
12		lgsdir2lem1 25050	. . . . . . . . 9 |- ( ( ( 1 mod 8 ) = 1 /\ ( -u 1 mod 8 ) = 7 ) /\ ( ( 3 mod 8 ) = 3 /\ ( -u 3 mod 8 ) = 5 ) )
13	12	simpli 474	. . . . . . . 8 |- ( ( 1 mod 8 ) = 1 /\ ( -u 1 mod 8 ) = 7 )
14	13	simpli 474	. . . . . . 7 |- ( 1 mod 8 ) = 1
15	11, 14	syl6eqr 2674	. . . . . 6 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 1 ) -> ( A mod 8 ) = ( 1 mod 8 ) )
16		modmul1 12723	. . . . . 6 |- ( ( ( A e. RR /\ 1 e. RR ) /\ ( B e. ZZ /\ 8 e. RR+ ) /\ ( A mod 8 ) = ( 1 mod 8 ) ) -> ( ( A x. B ) mod 8 ) = ( ( 1 x. B ) mod 8 ) )
17	4, 5, 6, 10, 15, 16	syl221anc 1337	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 1 ) -> ( ( A x. B ) mod 8 ) = ( ( 1 x. B ) mod 8 ) )
18		zcn 11382	. . . . . . . 8 |- ( B e. ZZ -> B e. CC )
19	18	ad2antlr 763	. . . . . . 7 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 1 ) -> B e. CC )
20	19	mulid2d 10058	. . . . . 6 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 1 ) -> ( 1 x. B ) = B )
21	20	oveq1d 6665	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 1 ) -> ( ( 1 x. B ) mod 8 ) = ( B mod 8 ) )
22	17, 21	eqtrd 2656	. . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 1 ) -> ( ( A x. B ) mod 8 ) = ( B mod 8 ) )
23	22	eleq1d 2686	. . 3 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 1 ) -> ( ( ( A x. B ) mod 8 ) e. { 1 , 7 } <-> ( B mod 8 ) e. { 1 , 7 } ) )
24	3	ad2antrr 762	. . . . . . 7 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 7 ) -> A e. RR )
25		neg1rr 11125	. . . . . . . 8 |- -u 1 e. RR
26	25	a1i 11	. . . . . . 7 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 7 ) -> -u 1 e. RR )
27		simplr 792	. . . . . . 7 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 7 ) -> B e. ZZ )
28	9	a1i 11	. . . . . . 7 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 7 ) -> 8 e. RR+ )
29		simpr 477	. . . . . . . 8 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 7 ) -> ( A mod 8 ) = 7 )
30	13	simpri 478	. . . . . . . 8 |- ( -u 1 mod 8 ) = 7
31	29, 30	syl6eqr 2674	. . . . . . 7 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 7 ) -> ( A mod 8 ) = ( -u 1 mod 8 ) )
32		modmul1 12723	. . . . . . 7 |- ( ( ( A e. RR /\ -u 1 e. RR ) /\ ( B e. ZZ /\ 8 e. RR+ ) /\ ( A mod 8 ) = ( -u 1 mod 8 ) ) -> ( ( A x. B ) mod 8 ) = ( ( -u 1 x. B ) mod 8 ) )
33	24, 26, 27, 28, 31, 32	syl221anc 1337	. . . . . 6 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 7 ) -> ( ( A x. B ) mod 8 ) = ( ( -u 1 x. B ) mod 8 ) )
34	18	ad2antlr 763	. . . . . . . 8 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 7 ) -> B e. CC )
35	34	mulm1d 10482	. . . . . . 7 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 7 ) -> ( -u 1 x. B ) = -u B )
36	35	oveq1d 6665	. . . . . 6 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 7 ) -> ( ( -u 1 x. B ) mod 8 ) = ( -u B mod 8 ) )
37	33, 36	eqtrd 2656	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 7 ) -> ( ( A x. B ) mod 8 ) = ( -u B mod 8 ) )
38	37	eleq1d 2686	. . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 7 ) -> ( ( ( A x. B ) mod 8 ) e. { 1 , 7 } <-> ( -u B mod 8 ) e. { 1 , 7 } ) )
39		znegcl 11412	. . . . . . . 8 |- ( B e. ZZ -> -u B e. ZZ )
40		oveq1 6657	. . . . . . . . . . 11 |- ( x = -u B -> ( x mod 8 ) = ( -u B mod 8 ) )
41	40	eleq1d 2686	. . . . . . . . . 10 |- ( x = -u B -> ( ( x mod 8 ) e. { 1 , 7 } <-> ( -u B mod 8 ) e. { 1 , 7 } ) )
42		negeq 10273	. . . . . . . . . . . 12 |- ( x = -u B -> -u x = -u -u B )
43	42	oveq1d 6665	. . . . . . . . . . 11 |- ( x = -u B -> ( -u x mod 8 ) = ( -u -u B mod 8 ) )
44	43	eleq1d 2686	. . . . . . . . . 10 |- ( x = -u B -> ( ( -u x mod 8 ) e. { 1 , 7 } <-> ( -u -u B mod 8 ) e. { 1 , 7 } ) )
45	41, 44	imbi12d 334	. . . . . . . . 9 |- ( x = -u B -> ( ( ( x mod 8 ) e. { 1 , 7 } -> ( -u x mod 8 ) e. { 1 , 7 } ) <-> ( ( -u B mod 8 ) e. { 1 , 7 } -> ( -u -u B mod 8 ) e. { 1 , 7 } ) ) )
46		zcn 11382	. . . . . . . . . . . . . . . . 17 |- ( x e. ZZ -> x e. CC )
47		neg1cn 11124	. . . . . . . . . . . . . . . . . . 19 |- -u 1 e. CC
48		mulcom 10022	. . . . . . . . . . . . . . . . . . 19 |- ( ( x e. CC /\ -u 1 e. CC ) -> ( x x. -u 1 ) = ( -u 1 x. x ) )
49	47, 48	mpan2 707	. . . . . . . . . . . . . . . . . 18 |- ( x e. CC -> ( x x. -u 1 ) = ( -u 1 x. x ) )
50		mulm1 10471	. . . . . . . . . . . . . . . . . 18 |- ( x e. CC -> ( -u 1 x. x ) = -u x )
51	49, 50	eqtrd 2656	. . . . . . . . . . . . . . . . 17 |- ( x e. CC -> ( x x. -u 1 ) = -u x )
52	46, 51	syl 17	. . . . . . . . . . . . . . . 16 |- ( x e. ZZ -> ( x x. -u 1 ) = -u x )
53	52	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( x e. ZZ /\ ( x mod 8 ) = 1 ) -> ( x x. -u 1 ) = -u x )
54	53	oveq1d 6665	. . . . . . . . . . . . . 14 |- ( ( x e. ZZ /\ ( x mod 8 ) = 1 ) -> ( ( x x. -u 1 ) mod 8 ) = ( -u x mod 8 ) )
55		zre 11381	. . . . . . . . . . . . . . . 16 |- ( x e. ZZ -> x e. RR )
56	55	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( x e. ZZ /\ ( x mod 8 ) = 1 ) -> x e. RR )
57		1red 10055	. . . . . . . . . . . . . . 15 |- ( ( x e. ZZ /\ ( x mod 8 ) = 1 ) -> 1 e. RR )
58		neg1z 11413	. . . . . . . . . . . . . . . 16 |- -u 1 e. ZZ
59	58	a1i 11	. . . . . . . . . . . . . . 15 |- ( ( x e. ZZ /\ ( x mod 8 ) = 1 ) -> -u 1 e. ZZ )
60	9	a1i 11	. . . . . . . . . . . . . . 15 |- ( ( x e. ZZ /\ ( x mod 8 ) = 1 ) -> 8 e. RR+ )
61		simpr 477	. . . . . . . . . . . . . . . 16 |- ( ( x e. ZZ /\ ( x mod 8 ) = 1 ) -> ( x mod 8 ) = 1 )
62	61, 14	syl6eqr 2674	. . . . . . . . . . . . . . 15 |- ( ( x e. ZZ /\ ( x mod 8 ) = 1 ) -> ( x mod 8 ) = ( 1 mod 8 ) )
63		modmul1 12723	. . . . . . . . . . . . . . 15 |- ( ( ( x e. RR /\ 1 e. RR ) /\ ( -u 1 e. ZZ /\ 8 e. RR+ ) /\ ( x mod 8 ) = ( 1 mod 8 ) ) -> ( ( x x. -u 1 ) mod 8 ) = ( ( 1 x. -u 1 ) mod 8 ) )
64	56, 57, 59, 60, 62, 63	syl221anc 1337	. . . . . . . . . . . . . 14 |- ( ( x e. ZZ /\ ( x mod 8 ) = 1 ) -> ( ( x x. -u 1 ) mod 8 ) = ( ( 1 x. -u 1 ) mod 8 ) )
65	54, 64	eqtr3d 2658	. . . . . . . . . . . . 13 |- ( ( x e. ZZ /\ ( x mod 8 ) = 1 ) -> ( -u x mod 8 ) = ( ( 1 x. -u 1 ) mod 8 ) )
66	47	mulid2i 10043	. . . . . . . . . . . . . . 15 |- ( 1 x. -u 1 ) = -u 1
67	66	oveq1i 6660	. . . . . . . . . . . . . 14 |- ( ( 1 x. -u 1 ) mod 8 ) = ( -u 1 mod 8 )
68	67, 30	eqtri 2644	. . . . . . . . . . . . 13 |- ( ( 1 x. -u 1 ) mod 8 ) = 7
69	65, 68	syl6eq 2672	. . . . . . . . . . . 12 |- ( ( x e. ZZ /\ ( x mod 8 ) = 1 ) -> ( -u x mod 8 ) = 7 )
70	69	ex 450	. . . . . . . . . . 11 |- ( x e. ZZ -> ( ( x mod 8 ) = 1 -> ( -u x mod 8 ) = 7 ) )
71	52	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( x e. ZZ /\ ( x mod 8 ) = 7 ) -> ( x x. -u 1 ) = -u x )
72	71	oveq1d 6665	. . . . . . . . . . . . . 14 |- ( ( x e. ZZ /\ ( x mod 8 ) = 7 ) -> ( ( x x. -u 1 ) mod 8 ) = ( -u x mod 8 ) )
73	55	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( x e. ZZ /\ ( x mod 8 ) = 7 ) -> x e. RR )
74	25	a1i 11	. . . . . . . . . . . . . . 15 |- ( ( x e. ZZ /\ ( x mod 8 ) = 7 ) -> -u 1 e. RR )
75	58	a1i 11	. . . . . . . . . . . . . . 15 |- ( ( x e. ZZ /\ ( x mod 8 ) = 7 ) -> -u 1 e. ZZ )
76	9	a1i 11	. . . . . . . . . . . . . . 15 |- ( ( x e. ZZ /\ ( x mod 8 ) = 7 ) -> 8 e. RR+ )
77		simpr 477	. . . . . . . . . . . . . . . 16 |- ( ( x e. ZZ /\ ( x mod 8 ) = 7 ) -> ( x mod 8 ) = 7 )
78	77, 30	syl6eqr 2674	. . . . . . . . . . . . . . 15 |- ( ( x e. ZZ /\ ( x mod 8 ) = 7 ) -> ( x mod 8 ) = ( -u 1 mod 8 ) )
79		modmul1 12723	. . . . . . . . . . . . . . 15 |- ( ( ( x e. RR /\ -u 1 e. RR ) /\ ( -u 1 e. ZZ /\ 8 e. RR+ ) /\ ( x mod 8 ) = ( -u 1 mod 8 ) ) -> ( ( x x. -u 1 ) mod 8 ) = ( ( -u 1 x. -u 1 ) mod 8 ) )
80	73, 74, 75, 76, 78, 79	syl221anc 1337	. . . . . . . . . . . . . 14 |- ( ( x e. ZZ /\ ( x mod 8 ) = 7 ) -> ( ( x x. -u 1 ) mod 8 ) = ( ( -u 1 x. -u 1 ) mod 8 ) )
81	72, 80	eqtr3d 2658	. . . . . . . . . . . . 13 |- ( ( x e. ZZ /\ ( x mod 8 ) = 7 ) -> ( -u x mod 8 ) = ( ( -u 1 x. -u 1 ) mod 8 ) )
82		neg1mulneg1e1 11245	. . . . . . . . . . . . . . 15 |- ( -u 1 x. -u 1 ) = 1
83	82	oveq1i 6660	. . . . . . . . . . . . . 14 |- ( ( -u 1 x. -u 1 ) mod 8 ) = ( 1 mod 8 )
84	83, 14	eqtri 2644	. . . . . . . . . . . . 13 |- ( ( -u 1 x. -u 1 ) mod 8 ) = 1
85	81, 84	syl6eq 2672	. . . . . . . . . . . 12 |- ( ( x e. ZZ /\ ( x mod 8 ) = 7 ) -> ( -u x mod 8 ) = 1 )
86	85	ex 450	. . . . . . . . . . 11 |- ( x e. ZZ -> ( ( x mod 8 ) = 7 -> ( -u x mod 8 ) = 1 ) )
87	70, 86	orim12d 883	. . . . . . . . . 10 |- ( x e. ZZ -> ( ( ( x mod 8 ) = 1 \/ ( x mod 8 ) = 7 ) -> ( ( -u x mod 8 ) = 7 \/ ( -u x mod 8 ) = 1 ) ) )
88		ovex 6678	. . . . . . . . . . 11 |- ( x mod 8 ) e. _V
89	88	elpr 4198	. . . . . . . . . 10 |- ( ( x mod 8 ) e. { 1 , 7 } <-> ( ( x mod 8 ) = 1 \/ ( x mod 8 ) = 7 ) )
90		ovex 6678	. . . . . . . . . . . 12 |- ( -u x mod 8 ) e. _V
91	90	elpr 4198	. . . . . . . . . . 11 |- ( ( -u x mod 8 ) e. { 1 , 7 } <-> ( ( -u x mod 8 ) = 1 \/ ( -u x mod 8 ) = 7 ) )
92		orcom 402	. . . . . . . . . . 11 |- ( ( ( -u x mod 8 ) = 1 \/ ( -u x mod 8 ) = 7 ) <-> ( ( -u x mod 8 ) = 7 \/ ( -u x mod 8 ) = 1 ) )
93	91, 92	bitri 264	. . . . . . . . . 10 |- ( ( -u x mod 8 ) e. { 1 , 7 } <-> ( ( -u x mod 8 ) = 7 \/ ( -u x mod 8 ) = 1 ) )
94	87, 89, 93	3imtr4g 285	. . . . . . . . 9 |- ( x e. ZZ -> ( ( x mod 8 ) e. { 1 , 7 } -> ( -u x mod 8 ) e. { 1 , 7 } ) )
95	45, 94	vtoclga 3272	. . . . . . . 8 |- ( -u B e. ZZ -> ( ( -u B mod 8 ) e. { 1 , 7 } -> ( -u -u B mod 8 ) e. { 1 , 7 } ) )
96	39, 95	syl 17	. . . . . . 7 |- ( B e. ZZ -> ( ( -u B mod 8 ) e. { 1 , 7 } -> ( -u -u B mod 8 ) e. { 1 , 7 } ) )
97	18	negnegd 10383	. . . . . . . . 9 |- ( B e. ZZ -> -u -u B = B )
98	97	oveq1d 6665	. . . . . . . 8 |- ( B e. ZZ -> ( -u -u B mod 8 ) = ( B mod 8 ) )
99	98	eleq1d 2686	. . . . . . 7 |- ( B e. ZZ -> ( ( -u -u B mod 8 ) e. { 1 , 7 } <-> ( B mod 8 ) e. { 1 , 7 } ) )
100	96, 99	sylibd 229	. . . . . 6 |- ( B e. ZZ -> ( ( -u B mod 8 ) e. { 1 , 7 } -> ( B mod 8 ) e. { 1 , 7 } ) )
101		oveq1 6657	. . . . . . . . 9 |- ( x = B -> ( x mod 8 ) = ( B mod 8 ) )
102	101	eleq1d 2686	. . . . . . . 8 |- ( x = B -> ( ( x mod 8 ) e. { 1 , 7 } <-> ( B mod 8 ) e. { 1 , 7 } ) )
103		negeq 10273	. . . . . . . . . 10 |- ( x = B -> -u x = -u B )
104	103	oveq1d 6665	. . . . . . . . 9 |- ( x = B -> ( -u x mod 8 ) = ( -u B mod 8 ) )
105	104	eleq1d 2686	. . . . . . . 8 |- ( x = B -> ( ( -u x mod 8 ) e. { 1 , 7 } <-> ( -u B mod 8 ) e. { 1 , 7 } ) )
106	102, 105	imbi12d 334	. . . . . . 7 |- ( x = B -> ( ( ( x mod 8 ) e. { 1 , 7 } -> ( -u x mod 8 ) e. { 1 , 7 } ) <-> ( ( B mod 8 ) e. { 1 , 7 } -> ( -u B mod 8 ) e. { 1 , 7 } ) ) )
107	106, 94	vtoclga 3272	. . . . . 6 |- ( B e. ZZ -> ( ( B mod 8 ) e. { 1 , 7 } -> ( -u B mod 8 ) e. { 1 , 7 } ) )
108	100, 107	impbid 202	. . . . 5 |- ( B e. ZZ -> ( ( -u B mod 8 ) e. { 1 , 7 } <-> ( B mod 8 ) e. { 1 , 7 } ) )
109	108	ad2antlr 763	. . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 7 ) -> ( ( -u B mod 8 ) e. { 1 , 7 } <-> ( B mod 8 ) e. { 1 , 7 } ) )
110	38, 109	bitrd 268	. . 3 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 7 ) -> ( ( ( A x. B ) mod 8 ) e. { 1 , 7 } <-> ( B mod 8 ) e. { 1 , 7 } ) )
111	23, 110	jaodan 826	. 2 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) = 1 \/ ( A mod 8 ) = 7 ) ) -> ( ( ( A x. B ) mod 8 ) e. { 1 , 7 } <-> ( B mod 8 ) e. { 1 , 7 } ) )
112	2, 111	sylan2b 492	1 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) e. { 1 , 7 } ) -> ( ( ( A x. B ) mod 8 ) e. { 1 , 7 } <-> ( B mod 8 ) e. { 1 , 7 } ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   {cpr 4179  (class class class)co 6650   CCcc 9934   RRcr 9935   1c1 9937   x. cmul 9941   -ucneg 10267   3c3 11071   5c5 11073   7c7 11075   8c8 11076   ZZcz 11377   RR+crp 11832   mod cmo 12668

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-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-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-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  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-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fl 12593  df-mod 12669

This theorem is referenced by:  lgsdir2  25055