Theorem lgsdir2lem4 23329

Description: Lemma for lgsdir2 23331. (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 6307	. . 3 |- ( A mod 8 ) e. _V
2	1	elpr 4045	. 2 |- ( ( A mod 8 ) e. { 1 , 7 } <-> ( ( A mod 8 ) = 1 \/ ( A mod 8 ) = 7 ) )
3		zre 10864	. . . . . . 7 |- ( A e. ZZ -> A e. RR )
4	3	ad2antrr 725	. . . . . 6 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 1 ) -> A e. RR )
5		1red 9607	. . . . . 6 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 1 ) -> 1 e. RR )
6		simplr 754	. . . . . 6 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 1 ) -> B e. ZZ )
7		8re 10616	. . . . . . . 8 |- 8 e. RR
8		8pos 10632	. . . . . . . 8 |- 0 < 8
9	7, 8	elrpii 11219	. . . . . . 7 |- 8 e. RR+
10	9	a1i 11	. . . . . 6 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 1 ) -> 8 e. RR+ )
11		simpr 461	. . . . . . 7 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 1 ) -> ( A mod 8 ) = 1 )
12		lgsdir2lem1 23326	. . . . . . . . 9 |- ( ( ( 1 mod 8 ) = 1 /\ ( -u 1 mod 8 ) = 7 ) /\ ( ( 3 mod 8 ) = 3 /\ ( -u 3 mod 8 ) = 5 ) )
13	12	simpli 458	. . . . . . . 8 |- ( ( 1 mod 8 ) = 1 /\ ( -u 1 mod 8 ) = 7 )
14	13	simpli 458	. . . . . . 7 |- ( 1 mod 8 ) = 1
15	11, 14	syl6eqr 2526	. . . . . 6 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 1 ) -> ( A mod 8 ) = ( 1 mod 8 ) )
16		modmul1 12004	. . . . . 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 1239	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 1 ) -> ( ( A x. B ) mod 8 ) = ( ( 1 x. B ) mod 8 ) )
18		zcn 10865	. . . . . . . 8 |- ( B e. ZZ -> B e. CC )
19	18	ad2antlr 726	. . . . . . 7 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 1 ) -> B e. CC )
20	19	mulid2d 9610	. . . . . 6 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 1 ) -> ( 1 x. B ) = B )
21	20	oveq1d 6297	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 1 ) -> ( ( 1 x. B ) mod 8 ) = ( B mod 8 ) )
22	17, 21	eqtrd 2508	. . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 1 ) -> ( ( A x. B ) mod 8 ) = ( B mod 8 ) )
23	22	eleq1d 2536	. . 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 725	. . . . . . 7 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 7 ) -> A e. RR )
25		neg1rr 10636	. . . . . . . 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 754	. . . . . . 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 461	. . . . . . . 8 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 7 ) -> ( A mod 8 ) = 7 )
30	13	simpri 462	. . . . . . . 8 |- ( -u 1 mod 8 ) = 7
31	29, 30	syl6eqr 2526	. . . . . . 7 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 7 ) -> ( A mod 8 ) = ( -u 1 mod 8 ) )
32		modmul1 12004	. . . . . . 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 1239	. . . . . 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 726	. . . . . . . 8 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 7 ) -> B e. CC )
35	34	mulm1d 10004	. . . . . . 7 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 7 ) -> ( -u 1 x. B ) = -u B )
36	35	oveq1d 6297	. . . . . 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 2508	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 7 ) -> ( ( A x. B ) mod 8 ) = ( -u B mod 8 ) )
38	37	eleq1d 2536	. . . 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 10894	. . . . . . . 8 |- ( B e. ZZ -> -u B e. ZZ )
40		oveq1 6289	. . . . . . . . . . 11 |- ( x = -u B -> ( x mod 8 ) = ( -u B mod 8 ) )
41	40	eleq1d 2536	. . . . . . . . . 10 |- ( x = -u B -> ( ( x mod 8 ) e. { 1 , 7 } <-> ( -u B mod 8 ) e. { 1 , 7 } ) )
42		negeq 9808	. . . . . . . . . . . 12 |- ( x = -u B -> -u x = -u -u B )
43	42	oveq1d 6297	. . . . . . . . . . 11 |- ( x = -u B -> ( -u x mod 8 ) = ( -u -u B mod 8 ) )
44	43	eleq1d 2536	. . . . . . . . . 10 |- ( x = -u B -> ( ( -u x mod 8 ) e. { 1 , 7 } <-> ( -u -u B mod 8 ) e. { 1 , 7 } ) )
45	41, 44	imbi12d 320	. . . . . . . . 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 10865	. . . . . . . . . . . . . . . . 17 |- ( x e. ZZ -> x e. CC )
47		neg1cn 10635	. . . . . . . . . . . . . . . . . . 19 |- -u 1 e. CC
48		mulcom 9574	. . . . . . . . . . . . . . . . . . 19 |- ( ( x e. CC /\ -u 1 e. CC ) -> ( x x. -u 1 ) = ( -u 1 x. x ) )
49	47, 48	mpan2 671	. . . . . . . . . . . . . . . . . 18 |- ( x e. CC -> ( x x. -u 1 ) = ( -u 1 x. x ) )
50		mulm1 9994	. . . . . . . . . . . . . . . . . 18 |- ( x e. CC -> ( -u 1 x. x ) = -u x )
51	49, 50	eqtrd 2508	. . . . . . . . . . . . . . . . 17 |- ( x e. CC -> ( x x. -u 1 ) = -u x )
52	46, 51	syl 16	. . . . . . . . . . . . . . . 16 |- ( x e. ZZ -> ( x x. -u 1 ) = -u x )
53	52	adantr 465	. . . . . . . . . . . . . . 15 |- ( ( x e. ZZ /\ ( x mod 8 ) = 1 ) -> ( x x. -u 1 ) = -u x )
54	53	oveq1d 6297	. . . . . . . . . . . . . 14 |- ( ( x e. ZZ /\ ( x mod 8 ) = 1 ) -> ( ( x x. -u 1 ) mod 8 ) = ( -u x mod 8 ) )
55		zre 10864	. . . . . . . . . . . . . . . 16 |- ( x e. ZZ -> x e. RR )
56	55	adantr 465	. . . . . . . . . . . . . . 15 |- ( ( x e. ZZ /\ ( x mod 8 ) = 1 ) -> x e. RR )
57		1red 9607	. . . . . . . . . . . . . . 15 |- ( ( x e. ZZ /\ ( x mod 8 ) = 1 ) -> 1 e. RR )
58		neg1z 10895	. . . . . . . . . . . . . . . 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 461	. . . . . . . . . . . . . . . 16 |- ( ( x e. ZZ /\ ( x mod 8 ) = 1 ) -> ( x mod 8 ) = 1 )
62	61, 14	syl6eqr 2526	. . . . . . . . . . . . . . 15 |- ( ( x e. ZZ /\ ( x mod 8 ) = 1 ) -> ( x mod 8 ) = ( 1 mod 8 ) )
63		modmul1 12004	. . . . . . . . . . . . . . 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 1239	. . . . . . . . . . . . . 14 |- ( ( x e. ZZ /\ ( x mod 8 ) = 1 ) -> ( ( x x. -u 1 ) mod 8 ) = ( ( 1 x. -u 1 ) mod 8 ) )
65	54, 64	eqtr3d 2510	. . . . . . . . . . . . 13 |- ( ( x e. ZZ /\ ( x mod 8 ) = 1 ) -> ( -u x mod 8 ) = ( ( 1 x. -u 1 ) mod 8 ) )
66	47	mulid2i 9595	. . . . . . . . . . . . . . 15 |- ( 1 x. -u 1 ) = -u 1
67	66	oveq1i 6292	. . . . . . . . . . . . . 14 |- ( ( 1 x. -u 1 ) mod 8 ) = ( -u 1 mod 8 )
68	67, 30	eqtri 2496	. . . . . . . . . . . . 13 |- ( ( 1 x. -u 1 ) mod 8 ) = 7
69	65, 68	syl6eq 2524	. . . . . . . . . . . 12 |- ( ( x e. ZZ /\ ( x mod 8 ) = 1 ) -> ( -u x mod 8 ) = 7 )
70	69	ex 434	. . . . . . . . . . 11 |- ( x e. ZZ -> ( ( x mod 8 ) = 1 -> ( -u x mod 8 ) = 7 ) )
71	52	adantr 465	. . . . . . . . . . . . . . 15 |- ( ( x e. ZZ /\ ( x mod 8 ) = 7 ) -> ( x x. -u 1 ) = -u x )
72	71	oveq1d 6297	. . . . . . . . . . . . . 14 |- ( ( x e. ZZ /\ ( x mod 8 ) = 7 ) -> ( ( x x. -u 1 ) mod 8 ) = ( -u x mod 8 ) )
73	55	adantr 465	. . . . . . . . . . . . . . 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 461	. . . . . . . . . . . . . . . 16 |- ( ( x e. ZZ /\ ( x mod 8 ) = 7 ) -> ( x mod 8 ) = 7 )
78	77, 30	syl6eqr 2526	. . . . . . . . . . . . . . 15 |- ( ( x e. ZZ /\ ( x mod 8 ) = 7 ) -> ( x mod 8 ) = ( -u 1 mod 8 ) )
79		modmul1 12004	. . . . . . . . . . . . . . 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 1239	. . . . . . . . . . . . . 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 2510	. . . . . . . . . . . . 13 |- ( ( x e. ZZ /\ ( x mod 8 ) = 7 ) -> ( -u x mod 8 ) = ( ( -u 1 x. -u 1 ) mod 8 ) )
82		neg1mulneg1e1 10749	. . . . . . . . . . . . . . 15 |- ( -u 1 x. -u 1 ) = 1
83	82	oveq1i 6292	. . . . . . . . . . . . . 14 |- ( ( -u 1 x. -u 1 ) mod 8 ) = ( 1 mod 8 )
84	83, 14	eqtri 2496	. . . . . . . . . . . . 13 |- ( ( -u 1 x. -u 1 ) mod 8 ) = 1
85	81, 84	syl6eq 2524	. . . . . . . . . . . 12 |- ( ( x e. ZZ /\ ( x mod 8 ) = 7 ) -> ( -u x mod 8 ) = 1 )
86	85	ex 434	. . . . . . . . . . 11 |- ( x e. ZZ -> ( ( x mod 8 ) = 7 -> ( -u x mod 8 ) = 1 ) )
87	70, 86	orim12d 836	. . . . . . . . . 10 |- ( x e. ZZ -> ( ( ( x mod 8 ) = 1 \/ ( x mod 8 ) = 7 ) -> ( ( -u x mod 8 ) = 7 \/ ( -u x mod 8 ) = 1 ) ) )
88		ovex 6307	. . . . . . . . . . 11 |- ( x mod 8 ) e. _V
89	88	elpr 4045	. . . . . . . . . 10 |- ( ( x mod 8 ) e. { 1 , 7 } <-> ( ( x mod 8 ) = 1 \/ ( x mod 8 ) = 7 ) )
90		ovex 6307	. . . . . . . . . . . 12 |- ( -u x mod 8 ) e. _V
91	90	elpr 4045	. . . . . . . . . . 11 |- ( ( -u x mod 8 ) e. { 1 , 7 } <-> ( ( -u x mod 8 ) = 1 \/ ( -u x mod 8 ) = 7 ) )
92		orcom 387	. . . . . . . . . . 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 249	. . . . . . . . . 10 |- ( ( -u x mod 8 ) e. { 1 , 7 } <-> ( ( -u x mod 8 ) = 7 \/ ( -u x mod 8 ) = 1 ) )
94	87, 89, 93	3imtr4g 270	. . . . . . . . 9 |- ( x e. ZZ -> ( ( x mod 8 ) e. { 1 , 7 } -> ( -u x mod 8 ) e. { 1 , 7 } ) )
95	45, 94	vtoclga 3177	. . . . . . . 8 |- ( -u B e. ZZ -> ( ( -u B mod 8 ) e. { 1 , 7 } -> ( -u -u B mod 8 ) e. { 1 , 7 } ) )
96	39, 95	syl 16	. . . . . . 7 |- ( B e. ZZ -> ( ( -u B mod 8 ) e. { 1 , 7 } -> ( -u -u B mod 8 ) e. { 1 , 7 } ) )
97	18	negnegd 9917	. . . . . . . . 9 |- ( B e. ZZ -> -u -u B = B )
98	97	oveq1d 6297	. . . . . . . 8 |- ( B e. ZZ -> ( -u -u B mod 8 ) = ( B mod 8 ) )
99	98	eleq1d 2536	. . . . . . 7 |- ( B e. ZZ -> ( ( -u -u B mod 8 ) e. { 1 , 7 } <-> ( B mod 8 ) e. { 1 , 7 } ) )
100	96, 99	sylibd 214	. . . . . 6 |- ( B e. ZZ -> ( ( -u B mod 8 ) e. { 1 , 7 } -> ( B mod 8 ) e. { 1 , 7 } ) )
101		oveq1 6289	. . . . . . . . 9 |- ( x = B -> ( x mod 8 ) = ( B mod 8 ) )
102	101	eleq1d 2536	. . . . . . . 8 |- ( x = B -> ( ( x mod 8 ) e. { 1 , 7 } <-> ( B mod 8 ) e. { 1 , 7 } ) )
103		negeq 9808	. . . . . . . . . 10 |- ( x = B -> -u x = -u B )
104	103	oveq1d 6297	. . . . . . . . 9 |- ( x = B -> ( -u x mod 8 ) = ( -u B mod 8 ) )
105	104	eleq1d 2536	. . . . . . . 8 |- ( x = B -> ( ( -u x mod 8 ) e. { 1 , 7 } <-> ( -u B mod 8 ) e. { 1 , 7 } ) )
106	102, 105	imbi12d 320	. . . . . . 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 3177	. . . . . 6 |- ( B e. ZZ -> ( ( B mod 8 ) e. { 1 , 7 } -> ( -u B mod 8 ) e. { 1 , 7 } ) )
108	100, 107	impbid 191	. . . . 5 |- ( B e. ZZ -> ( ( -u B mod 8 ) e. { 1 , 7 } <-> ( B mod 8 ) e. { 1 , 7 } ) )
109	108	ad2antlr 726	. . . 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 253	. . 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 783	. 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 475	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 184   \/ wo 368   /\ wa 369   = wceq 1379   e. wcel 1767   {cpr 4029  (class class class)co 6282   CCcc 9486   RRcr 9487   1c1 9489   x. cmul 9493   -ucneg 9802   3c3 10582   5c5 10584   7c7 10586   8c8 10587   ZZcz 10860   RR+crp 11216   mod cmo 11960

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-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566

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-iun 4327  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-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 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-recs 7039  df-rdg 7073  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-sup 7897  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-n0 10792  df-z 10861  df-uz 11079  df-rp 11217  df-fl 11893  df-mod 11961

This theorem is referenced by:  lgsdir2  23331