Theorem lgsdir2lem4 24191

Description: Lemma for lgsdir2 24193. (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 6272	. . 3 |- ( A mod 8 ) e. _V
2	1	elpr 3954	. 2 |- ( ( A mod 8 ) e. { 1 , 7 } <-> ( ( A mod 8 ) = 1 \/ ( A mod 8 ) = 7 ) )
3		zre 10887	. . . . . . 7 |- ( A e. ZZ -> A e. RR )
4	3	ad2antrr 730	. . . . . 6 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 1 ) -> A e. RR )
5		1red 9604	. . . . . 6 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 1 ) -> 1 e. RR )
6		simplr 760	. . . . . 6 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 1 ) -> B e. ZZ )
7		8re 10640	. . . . . . . 8 |- 8 e. RR
8		8pos 10656	. . . . . . . 8 |- 0 < 8
9	7, 8	elrpii 11251	. . . . . . 7 |- 8 e. RR+
10	9	a1i 11	. . . . . 6 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 1 ) -> 8 e. RR+ )
11		simpr 462	. . . . . . 7 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 1 ) -> ( A mod 8 ) = 1 )
12		lgsdir2lem1 24188	. . . . . . . . 9 |- ( ( ( 1 mod 8 ) = 1 /\ ( -u 1 mod 8 ) = 7 ) /\ ( ( 3 mod 8 ) = 3 /\ ( -u 3 mod 8 ) = 5 ) )
13	12	simpli 459	. . . . . . . 8 |- ( ( 1 mod 8 ) = 1 /\ ( -u 1 mod 8 ) = 7 )
14	13	simpli 459	. . . . . . 7 |- ( 1 mod 8 ) = 1
15	11, 14	syl6eqr 2475	. . . . . 6 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 1 ) -> ( A mod 8 ) = ( 1 mod 8 ) )
16		modmul1 12088	. . . . . 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 1275	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 1 ) -> ( ( A x. B ) mod 8 ) = ( ( 1 x. B ) mod 8 ) )
18		zcn 10888	. . . . . . . 8 |- ( B e. ZZ -> B e. CC )
19	18	ad2antlr 731	. . . . . . 7 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 1 ) -> B e. CC )
20	19	mulid2d 9607	. . . . . 6 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 1 ) -> ( 1 x. B ) = B )
21	20	oveq1d 6259	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 1 ) -> ( ( 1 x. B ) mod 8 ) = ( B mod 8 ) )
22	17, 21	eqtrd 2457	. . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 1 ) -> ( ( A x. B ) mod 8 ) = ( B mod 8 ) )
23	22	eleq1d 2485	. . 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 730	. . . . . . 7 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 7 ) -> A e. RR )
25		neg1rr 10660	. . . . . . . 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 760	. . . . . . 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 462	. . . . . . . 8 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 7 ) -> ( A mod 8 ) = 7 )
30	13	simpri 463	. . . . . . . 8 |- ( -u 1 mod 8 ) = 7
31	29, 30	syl6eqr 2475	. . . . . . 7 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 7 ) -> ( A mod 8 ) = ( -u 1 mod 8 ) )
32		modmul1 12088	. . . . . . 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 1275	. . . . . 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 731	. . . . . . . 8 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 7 ) -> B e. CC )
35	34	mulm1d 10016	. . . . . . 7 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 7 ) -> ( -u 1 x. B ) = -u B )
36	35	oveq1d 6259	. . . . . 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 2457	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 7 ) -> ( ( A x. B ) mod 8 ) = ( -u B mod 8 ) )
38	37	eleq1d 2485	. . . 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 10918	. . . . . . . 8 |- ( B e. ZZ -> -u B e. ZZ )
40		oveq1 6251	. . . . . . . . . . 11 |- ( x = -u B -> ( x mod 8 ) = ( -u B mod 8 ) )
41	40	eleq1d 2485	. . . . . . . . . 10 |- ( x = -u B -> ( ( x mod 8 ) e. { 1 , 7 } <-> ( -u B mod 8 ) e. { 1 , 7 } ) )
42		negeq 9813	. . . . . . . . . . . 12 |- ( x = -u B -> -u x = -u -u B )
43	42	oveq1d 6259	. . . . . . . . . . 11 |- ( x = -u B -> ( -u x mod 8 ) = ( -u -u B mod 8 ) )
44	43	eleq1d 2485	. . . . . . . . . 10 |- ( x = -u B -> ( ( -u x mod 8 ) e. { 1 , 7 } <-> ( -u -u B mod 8 ) e. { 1 , 7 } ) )
45	41, 44	imbi12d 321	. . . . . . . . 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 10888	. . . . . . . . . . . . . . . . 17 |- ( x e. ZZ -> x e. CC )
47		neg1cn 10659	. . . . . . . . . . . . . . . . . . 19 |- -u 1 e. CC
48		mulcom 9571	. . . . . . . . . . . . . . . . . . 19 |- ( ( x e. CC /\ -u 1 e. CC ) -> ( x x. -u 1 ) = ( -u 1 x. x ) )
49	47, 48	mpan2 675	. . . . . . . . . . . . . . . . . 18 |- ( x e. CC -> ( x x. -u 1 ) = ( -u 1 x. x ) )
50		mulm1 10006	. . . . . . . . . . . . . . . . . 18 |- ( x e. CC -> ( -u 1 x. x ) = -u x )
51	49, 50	eqtrd 2457	. . . . . . . . . . . . . . . . 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 466	. . . . . . . . . . . . . . 15 |- ( ( x e. ZZ /\ ( x mod 8 ) = 1 ) -> ( x x. -u 1 ) = -u x )
54	53	oveq1d 6259	. . . . . . . . . . . . . 14 |- ( ( x e. ZZ /\ ( x mod 8 ) = 1 ) -> ( ( x x. -u 1 ) mod 8 ) = ( -u x mod 8 ) )
55		zre 10887	. . . . . . . . . . . . . . . 16 |- ( x e. ZZ -> x e. RR )
56	55	adantr 466	. . . . . . . . . . . . . . 15 |- ( ( x e. ZZ /\ ( x mod 8 ) = 1 ) -> x e. RR )
57		1red 9604	. . . . . . . . . . . . . . 15 |- ( ( x e. ZZ /\ ( x mod 8 ) = 1 ) -> 1 e. RR )
58		neg1z 10919	. . . . . . . . . . . . . . . 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 462	. . . . . . . . . . . . . . . 16 |- ( ( x e. ZZ /\ ( x mod 8 ) = 1 ) -> ( x mod 8 ) = 1 )
62	61, 14	syl6eqr 2475	. . . . . . . . . . . . . . 15 |- ( ( x e. ZZ /\ ( x mod 8 ) = 1 ) -> ( x mod 8 ) = ( 1 mod 8 ) )
63		modmul1 12088	. . . . . . . . . . . . . . 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 1275	. . . . . . . . . . . . . 14 |- ( ( x e. ZZ /\ ( x mod 8 ) = 1 ) -> ( ( x x. -u 1 ) mod 8 ) = ( ( 1 x. -u 1 ) mod 8 ) )
65	54, 64	eqtr3d 2459	. . . . . . . . . . . . 13 |- ( ( x e. ZZ /\ ( x mod 8 ) = 1 ) -> ( -u x mod 8 ) = ( ( 1 x. -u 1 ) mod 8 ) )
66	47	mulid2i 9592	. . . . . . . . . . . . . . 15 |- ( 1 x. -u 1 ) = -u 1
67	66	oveq1i 6254	. . . . . . . . . . . . . 14 |- ( ( 1 x. -u 1 ) mod 8 ) = ( -u 1 mod 8 )
68	67, 30	eqtri 2445	. . . . . . . . . . . . 13 |- ( ( 1 x. -u 1 ) mod 8 ) = 7
69	65, 68	syl6eq 2473	. . . . . . . . . . . 12 |- ( ( x e. ZZ /\ ( x mod 8 ) = 1 ) -> ( -u x mod 8 ) = 7 )
70	69	ex 435	. . . . . . . . . . 11 |- ( x e. ZZ -> ( ( x mod 8 ) = 1 -> ( -u x mod 8 ) = 7 ) )
71	52	adantr 466	. . . . . . . . . . . . . . 15 |- ( ( x e. ZZ /\ ( x mod 8 ) = 7 ) -> ( x x. -u 1 ) = -u x )
72	71	oveq1d 6259	. . . . . . . . . . . . . 14 |- ( ( x e. ZZ /\ ( x mod 8 ) = 7 ) -> ( ( x x. -u 1 ) mod 8 ) = ( -u x mod 8 ) )
73	55	adantr 466	. . . . . . . . . . . . . . 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 462	. . . . . . . . . . . . . . . 16 |- ( ( x e. ZZ /\ ( x mod 8 ) = 7 ) -> ( x mod 8 ) = 7 )
78	77, 30	syl6eqr 2475	. . . . . . . . . . . . . . 15 |- ( ( x e. ZZ /\ ( x mod 8 ) = 7 ) -> ( x mod 8 ) = ( -u 1 mod 8 ) )
79		modmul1 12088	. . . . . . . . . . . . . . 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 1275	. . . . . . . . . . . . . 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 2459	. . . . . . . . . . . . 13 |- ( ( x e. ZZ /\ ( x mod 8 ) = 7 ) -> ( -u x mod 8 ) = ( ( -u 1 x. -u 1 ) mod 8 ) )
82		neg1mulneg1e1 10773	. . . . . . . . . . . . . . 15 |- ( -u 1 x. -u 1 ) = 1
83	82	oveq1i 6254	. . . . . . . . . . . . . 14 |- ( ( -u 1 x. -u 1 ) mod 8 ) = ( 1 mod 8 )
84	83, 14	eqtri 2445	. . . . . . . . . . . . 13 |- ( ( -u 1 x. -u 1 ) mod 8 ) = 1
85	81, 84	syl6eq 2473	. . . . . . . . . . . 12 |- ( ( x e. ZZ /\ ( x mod 8 ) = 7 ) -> ( -u x mod 8 ) = 1 )
86	85	ex 435	. . . . . . . . . . 11 |- ( x e. ZZ -> ( ( x mod 8 ) = 7 -> ( -u x mod 8 ) = 1 ) )
87	70, 86	orim12d 846	. . . . . . . . . 10 |- ( x e. ZZ -> ( ( ( x mod 8 ) = 1 \/ ( x mod 8 ) = 7 ) -> ( ( -u x mod 8 ) = 7 \/ ( -u x mod 8 ) = 1 ) ) )
88		ovex 6272	. . . . . . . . . . 11 |- ( x mod 8 ) e. _V
89	88	elpr 3954	. . . . . . . . . 10 |- ( ( x mod 8 ) e. { 1 , 7 } <-> ( ( x mod 8 ) = 1 \/ ( x mod 8 ) = 7 ) )
90		ovex 6272	. . . . . . . . . . . 12 |- ( -u x mod 8 ) e. _V
91	90	elpr 3954	. . . . . . . . . . 11 |- ( ( -u x mod 8 ) e. { 1 , 7 } <-> ( ( -u x mod 8 ) = 1 \/ ( -u x mod 8 ) = 7 ) )
92		orcom 388	. . . . . . . . . . 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 252	. . . . . . . . . 10 |- ( ( -u x mod 8 ) e. { 1 , 7 } <-> ( ( -u x mod 8 ) = 7 \/ ( -u x mod 8 ) = 1 ) )
94	87, 89, 93	3imtr4g 273	. . . . . . . . 9 |- ( x e. ZZ -> ( ( x mod 8 ) e. { 1 , 7 } -> ( -u x mod 8 ) e. { 1 , 7 } ) )
95	45, 94	vtoclga 3083	. . . . . . . 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 9923	. . . . . . . . 9 |- ( B e. ZZ -> -u -u B = B )
98	97	oveq1d 6259	. . . . . . . 8 |- ( B e. ZZ -> ( -u -u B mod 8 ) = ( B mod 8 ) )
99	98	eleq1d 2485	. . . . . . 7 |- ( B e. ZZ -> ( ( -u -u B mod 8 ) e. { 1 , 7 } <-> ( B mod 8 ) e. { 1 , 7 } ) )
100	96, 99	sylibd 217	. . . . . 6 |- ( B e. ZZ -> ( ( -u B mod 8 ) e. { 1 , 7 } -> ( B mod 8 ) e. { 1 , 7 } ) )
101		oveq1 6251	. . . . . . . . 9 |- ( x = B -> ( x mod 8 ) = ( B mod 8 ) )
102	101	eleq1d 2485	. . . . . . . 8 |- ( x = B -> ( ( x mod 8 ) e. { 1 , 7 } <-> ( B mod 8 ) e. { 1 , 7 } ) )
103		negeq 9813	. . . . . . . . . 10 |- ( x = B -> -u x = -u B )
104	103	oveq1d 6259	. . . . . . . . 9 |- ( x = B -> ( -u x mod 8 ) = ( -u B mod 8 ) )
105	104	eleq1d 2485	. . . . . . . 8 |- ( x = B -> ( ( -u x mod 8 ) e. { 1 , 7 } <-> ( -u B mod 8 ) e. { 1 , 7 } ) )
106	102, 105	imbi12d 321	. . . . . . 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 3083	. . . . . 6 |- ( B e. ZZ -> ( ( B mod 8 ) e. { 1 , 7 } -> ( -u B mod 8 ) e. { 1 , 7 } ) )
108	100, 107	impbid 193	. . . . 5 |- ( B e. ZZ -> ( ( -u B mod 8 ) e. { 1 , 7 } <-> ( B mod 8 ) e. { 1 , 7 } ) )
109	108	ad2antlr 731	. . . 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 256	. . 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 792	. 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 477	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 187   \/ wo 369   /\ wa 370   = wceq 1437   e. wcel 1872   {cpr 3938  (class class class)co 6244   CCcc 9483   RRcr 9484   1c1 9486   x. cmul 9490   -ucneg 9807   3c3 10606   5c5 10608   7c7 10610   8c8 10611   ZZcz 10883   RR+crp 11248   mod cmo 12041

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2058  ax-ext 2403  ax-sep 4484  ax-nul 4493  ax-pow 4540  ax-pr 4598  ax-un 6536  ax-cnex 9541  ax-resscn 9542  ax-1cn 9543  ax-icn 9544  ax-addcl 9545  ax-addrcl 9546  ax-mulcl 9547  ax-mulrcl 9548  ax-mulcom 9549  ax-addass 9550  ax-mulass 9551  ax-distr 9552  ax-i2m1 9553  ax-1ne0 9554  ax-1rid 9555  ax-rnegex 9556  ax-rrecex 9557  ax-cnre 9558  ax-pre-lttri 9559  ax-pre-lttrn 9560  ax-pre-ltadd 9561  ax-pre-mulgt0 9562  ax-pre-sup 9563

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2275  df-mo 2276  df-clab 2410  df-cleq 2416  df-clel 2419  df-nfc 2553  df-ne 2596  df-nel 2597  df-ral 2714  df-rex 2715  df-reu 2716  df-rmo 2717  df-rab 2718  df-v 3019  df-sbc 3238  df-csb 3334  df-dif 3377  df-un 3379  df-in 3381  df-ss 3388  df-pss 3390  df-nul 3700  df-if 3850  df-pw 3921  df-sn 3937  df-pr 3939  df-tp 3941  df-op 3943  df-uni 4158  df-iun 4239  df-br 4362  df-opab 4421  df-mpt 4422  df-tr 4457  df-eprel 4702  df-id 4706  df-po 4712  df-so 4713  df-fr 4750  df-we 4752  df-xp 4797  df-rel 4798  df-cnv 4799  df-co 4800  df-dm 4801  df-rn 4802  df-res 4803  df-ima 4804  df-pred 5337  df-ord 5383  df-on 5384  df-lim 5385  df-suc 5386  df-iota 5503  df-fun 5541  df-fn 5542  df-f 5543  df-f1 5544  df-fo 5545  df-f1o 5546  df-fv 5547  df-riota 6206  df-ov 6247  df-oprab 6248  df-mpt2 6249  df-om 6646  df-wrecs 6978  df-recs 7040  df-rdg 7078  df-er 7313  df-en 7520  df-dom 7521  df-sdom 7522  df-sup 7904  df-inf 7905  df-pnf 9623  df-mnf 9624  df-xr 9625  df-ltxr 9626  df-le 9627  df-sub 9808  df-neg 9809  df-div 10216  df-nn 10556  df-2 10614  df-3 10615  df-4 10616  df-5 10617  df-6 10618  df-7 10619  df-8 10620  df-n0 10816  df-z 10884  df-uz 11106  df-rp 11249  df-fl 11973  df-mod 12042

This theorem is referenced by:  lgsdir2  24193