Theorem lgsdir2lem4 24310

Description: Lemma for lgsdir2 24312. (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 6348	. . 3 |- ( A mod 8 ) e. _V
2	1	elpr 3998	. 2 |- ( ( A mod 8 ) e. { 1 , 7 } <-> ( ( A mod 8 ) = 1 \/ ( A mod 8 ) = 7 ) )
3		zre 10975	. . . . . . 7 |- ( A e. ZZ -> A e. RR )
4	3	ad2antrr 737	. . . . . 6 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 1 ) -> A e. RR )
5		1red 9689	. . . . . 6 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 1 ) -> 1 e. RR )
6		simplr 767	. . . . . 6 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 1 ) -> B e. ZZ )
7		8re 10727	. . . . . . . 8 |- 8 e. RR
8		8pos 10743	. . . . . . . 8 |- 0 < 8
9	7, 8	elrpii 11339	. . . . . . 7 |- 8 e. RR+
10	9	a1i 11	. . . . . 6 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 1 ) -> 8 e. RR+ )
11		simpr 467	. . . . . . 7 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 1 ) -> ( A mod 8 ) = 1 )
12		lgsdir2lem1 24307	. . . . . . . . 9 |- ( ( ( 1 mod 8 ) = 1 /\ ( -u 1 mod 8 ) = 7 ) /\ ( ( 3 mod 8 ) = 3 /\ ( -u 3 mod 8 ) = 5 ) )
13	12	simpli 464	. . . . . . . 8 |- ( ( 1 mod 8 ) = 1 /\ ( -u 1 mod 8 ) = 7 )
14	13	simpli 464	. . . . . . 7 |- ( 1 mod 8 ) = 1
15	11, 14	syl6eqr 2514	. . . . . 6 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 1 ) -> ( A mod 8 ) = ( 1 mod 8 ) )
16		modmul1 12181	. . . . . 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 1287	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 1 ) -> ( ( A x. B ) mod 8 ) = ( ( 1 x. B ) mod 8 ) )
18		zcn 10976	. . . . . . . 8 |- ( B e. ZZ -> B e. CC )
19	18	ad2antlr 738	. . . . . . 7 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 1 ) -> B e. CC )
20	19	mulid2d 9692	. . . . . 6 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 1 ) -> ( 1 x. B ) = B )
21	20	oveq1d 6335	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 1 ) -> ( ( 1 x. B ) mod 8 ) = ( B mod 8 ) )
22	17, 21	eqtrd 2496	. . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 1 ) -> ( ( A x. B ) mod 8 ) = ( B mod 8 ) )
23	22	eleq1d 2524	. . 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 737	. . . . . . 7 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 7 ) -> A e. RR )
25		neg1rr 10747	. . . . . . . 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 767	. . . . . . 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 467	. . . . . . . 8 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 7 ) -> ( A mod 8 ) = 7 )
30	13	simpri 468	. . . . . . . 8 |- ( -u 1 mod 8 ) = 7
31	29, 30	syl6eqr 2514	. . . . . . 7 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 7 ) -> ( A mod 8 ) = ( -u 1 mod 8 ) )
32		modmul1 12181	. . . . . . 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 1287	. . . . . 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 738	. . . . . . . 8 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 7 ) -> B e. CC )
35	34	mulm1d 10103	. . . . . . 7 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 7 ) -> ( -u 1 x. B ) = -u B )
36	35	oveq1d 6335	. . . . . 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 2496	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 7 ) -> ( ( A x. B ) mod 8 ) = ( -u B mod 8 ) )
38	37	eleq1d 2524	. . . 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 11006	. . . . . . . 8 |- ( B e. ZZ -> -u B e. ZZ )
40		oveq1 6327	. . . . . . . . . . 11 |- ( x = -u B -> ( x mod 8 ) = ( -u B mod 8 ) )
41	40	eleq1d 2524	. . . . . . . . . 10 |- ( x = -u B -> ( ( x mod 8 ) e. { 1 , 7 } <-> ( -u B mod 8 ) e. { 1 , 7 } ) )
42		negeq 9898	. . . . . . . . . . . 12 |- ( x = -u B -> -u x = -u -u B )
43	42	oveq1d 6335	. . . . . . . . . . 11 |- ( x = -u B -> ( -u x mod 8 ) = ( -u -u B mod 8 ) )
44	43	eleq1d 2524	. . . . . . . . . 10 |- ( x = -u B -> ( ( -u x mod 8 ) e. { 1 , 7 } <-> ( -u -u B mod 8 ) e. { 1 , 7 } ) )
45	41, 44	imbi12d 326	. . . . . . . . 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 10976	. . . . . . . . . . . . . . . . 17 |- ( x e. ZZ -> x e. CC )
47		neg1cn 10746	. . . . . . . . . . . . . . . . . . 19 |- -u 1 e. CC
48		mulcom 9656	. . . . . . . . . . . . . . . . . . 19 |- ( ( x e. CC /\ -u 1 e. CC ) -> ( x x. -u 1 ) = ( -u 1 x. x ) )
49	47, 48	mpan2 682	. . . . . . . . . . . . . . . . . 18 |- ( x e. CC -> ( x x. -u 1 ) = ( -u 1 x. x ) )
50		mulm1 10093	. . . . . . . . . . . . . . . . . 18 |- ( x e. CC -> ( -u 1 x. x ) = -u x )
51	49, 50	eqtrd 2496	. . . . . . . . . . . . . . . . 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 471	. . . . . . . . . . . . . . 15 |- ( ( x e. ZZ /\ ( x mod 8 ) = 1 ) -> ( x x. -u 1 ) = -u x )
54	53	oveq1d 6335	. . . . . . . . . . . . . 14 |- ( ( x e. ZZ /\ ( x mod 8 ) = 1 ) -> ( ( x x. -u 1 ) mod 8 ) = ( -u x mod 8 ) )
55		zre 10975	. . . . . . . . . . . . . . . 16 |- ( x e. ZZ -> x e. RR )
56	55	adantr 471	. . . . . . . . . . . . . . 15 |- ( ( x e. ZZ /\ ( x mod 8 ) = 1 ) -> x e. RR )
57		1red 9689	. . . . . . . . . . . . . . 15 |- ( ( x e. ZZ /\ ( x mod 8 ) = 1 ) -> 1 e. RR )
58		neg1z 11007	. . . . . . . . . . . . . . . 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 467	. . . . . . . . . . . . . . . 16 |- ( ( x e. ZZ /\ ( x mod 8 ) = 1 ) -> ( x mod 8 ) = 1 )
62	61, 14	syl6eqr 2514	. . . . . . . . . . . . . . 15 |- ( ( x e. ZZ /\ ( x mod 8 ) = 1 ) -> ( x mod 8 ) = ( 1 mod 8 ) )
63		modmul1 12181	. . . . . . . . . . . . . . 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 1287	. . . . . . . . . . . . . 14 |- ( ( x e. ZZ /\ ( x mod 8 ) = 1 ) -> ( ( x x. -u 1 ) mod 8 ) = ( ( 1 x. -u 1 ) mod 8 ) )
65	54, 64	eqtr3d 2498	. . . . . . . . . . . . 13 |- ( ( x e. ZZ /\ ( x mod 8 ) = 1 ) -> ( -u x mod 8 ) = ( ( 1 x. -u 1 ) mod 8 ) )
66	47	mulid2i 9677	. . . . . . . . . . . . . . 15 |- ( 1 x. -u 1 ) = -u 1
67	66	oveq1i 6330	. . . . . . . . . . . . . 14 |- ( ( 1 x. -u 1 ) mod 8 ) = ( -u 1 mod 8 )
68	67, 30	eqtri 2484	. . . . . . . . . . . . 13 |- ( ( 1 x. -u 1 ) mod 8 ) = 7
69	65, 68	syl6eq 2512	. . . . . . . . . . . 12 |- ( ( x e. ZZ /\ ( x mod 8 ) = 1 ) -> ( -u x mod 8 ) = 7 )
70	69	ex 440	. . . . . . . . . . 11 |- ( x e. ZZ -> ( ( x mod 8 ) = 1 -> ( -u x mod 8 ) = 7 ) )
71	52	adantr 471	. . . . . . . . . . . . . . 15 |- ( ( x e. ZZ /\ ( x mod 8 ) = 7 ) -> ( x x. -u 1 ) = -u x )
72	71	oveq1d 6335	. . . . . . . . . . . . . 14 |- ( ( x e. ZZ /\ ( x mod 8 ) = 7 ) -> ( ( x x. -u 1 ) mod 8 ) = ( -u x mod 8 ) )
73	55	adantr 471	. . . . . . . . . . . . . . 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 467	. . . . . . . . . . . . . . . 16 |- ( ( x e. ZZ /\ ( x mod 8 ) = 7 ) -> ( x mod 8 ) = 7 )
78	77, 30	syl6eqr 2514	. . . . . . . . . . . . . . 15 |- ( ( x e. ZZ /\ ( x mod 8 ) = 7 ) -> ( x mod 8 ) = ( -u 1 mod 8 ) )
79		modmul1 12181	. . . . . . . . . . . . . . 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 1287	. . . . . . . . . . . . . 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 2498	. . . . . . . . . . . . 13 |- ( ( x e. ZZ /\ ( x mod 8 ) = 7 ) -> ( -u x mod 8 ) = ( ( -u 1 x. -u 1 ) mod 8 ) )
82		neg1mulneg1e1 10861	. . . . . . . . . . . . . . 15 |- ( -u 1 x. -u 1 ) = 1
83	82	oveq1i 6330	. . . . . . . . . . . . . 14 |- ( ( -u 1 x. -u 1 ) mod 8 ) = ( 1 mod 8 )
84	83, 14	eqtri 2484	. . . . . . . . . . . . 13 |- ( ( -u 1 x. -u 1 ) mod 8 ) = 1
85	81, 84	syl6eq 2512	. . . . . . . . . . . 12 |- ( ( x e. ZZ /\ ( x mod 8 ) = 7 ) -> ( -u x mod 8 ) = 1 )
86	85	ex 440	. . . . . . . . . . 11 |- ( x e. ZZ -> ( ( x mod 8 ) = 7 -> ( -u x mod 8 ) = 1 ) )
87	70, 86	orim12d 854	. . . . . . . . . 10 |- ( x e. ZZ -> ( ( ( x mod 8 ) = 1 \/ ( x mod 8 ) = 7 ) -> ( ( -u x mod 8 ) = 7 \/ ( -u x mod 8 ) = 1 ) ) )
88		ovex 6348	. . . . . . . . . . 11 |- ( x mod 8 ) e. _V
89	88	elpr 3998	. . . . . . . . . 10 |- ( ( x mod 8 ) e. { 1 , 7 } <-> ( ( x mod 8 ) = 1 \/ ( x mod 8 ) = 7 ) )
90		ovex 6348	. . . . . . . . . . . 12 |- ( -u x mod 8 ) e. _V
91	90	elpr 3998	. . . . . . . . . . 11 |- ( ( -u x mod 8 ) e. { 1 , 7 } <-> ( ( -u x mod 8 ) = 1 \/ ( -u x mod 8 ) = 7 ) )
92		orcom 393	. . . . . . . . . . 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 257	. . . . . . . . . 10 |- ( ( -u x mod 8 ) e. { 1 , 7 } <-> ( ( -u x mod 8 ) = 7 \/ ( -u x mod 8 ) = 1 ) )
94	87, 89, 93	3imtr4g 278	. . . . . . . . 9 |- ( x e. ZZ -> ( ( x mod 8 ) e. { 1 , 7 } -> ( -u x mod 8 ) e. { 1 , 7 } ) )
95	45, 94	vtoclga 3125	. . . . . . . 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 10008	. . . . . . . . 9 |- ( B e. ZZ -> -u -u B = B )
98	97	oveq1d 6335	. . . . . . . 8 |- ( B e. ZZ -> ( -u -u B mod 8 ) = ( B mod 8 ) )
99	98	eleq1d 2524	. . . . . . 7 |- ( B e. ZZ -> ( ( -u -u B mod 8 ) e. { 1 , 7 } <-> ( B mod 8 ) e. { 1 , 7 } ) )
100	96, 99	sylibd 222	. . . . . 6 |- ( B e. ZZ -> ( ( -u B mod 8 ) e. { 1 , 7 } -> ( B mod 8 ) e. { 1 , 7 } ) )
101		oveq1 6327	. . . . . . . . 9 |- ( x = B -> ( x mod 8 ) = ( B mod 8 ) )
102	101	eleq1d 2524	. . . . . . . 8 |- ( x = B -> ( ( x mod 8 ) e. { 1 , 7 } <-> ( B mod 8 ) e. { 1 , 7 } ) )
103		negeq 9898	. . . . . . . . . 10 |- ( x = B -> -u x = -u B )
104	103	oveq1d 6335	. . . . . . . . 9 |- ( x = B -> ( -u x mod 8 ) = ( -u B mod 8 ) )
105	104	eleq1d 2524	. . . . . . . 8 |- ( x = B -> ( ( -u x mod 8 ) e. { 1 , 7 } <-> ( -u B mod 8 ) e. { 1 , 7 } ) )
106	102, 105	imbi12d 326	. . . . . . 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 3125	. . . . . 6 |- ( B e. ZZ -> ( ( B mod 8 ) e. { 1 , 7 } -> ( -u B mod 8 ) e. { 1 , 7 } ) )
108	100, 107	impbid 195	. . . . 5 |- ( B e. ZZ -> ( ( -u B mod 8 ) e. { 1 , 7 } <-> ( B mod 8 ) e. { 1 , 7 } ) )
109	108	ad2antlr 738	. . . 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 261	. . 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 799	. 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 482	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 189   \/ wo 374   /\ wa 375   = wceq 1455   e. wcel 1898   {cpr 3982  (class class class)co 6320   CCcc 9568   RRcr 9569   1c1 9571   x. cmul 9575   -ucneg 9892   3c3 10693   5c5 10695   7c7 10697   8c8 10698   ZZcz 10971   RR+crp 11336   mod cmo 12134

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4541  ax-nul 4550  ax-pow 4598  ax-pr 4656  ax-un 6615  ax-cnex 9626  ax-resscn 9627  ax-1cn 9628  ax-icn 9629  ax-addcl 9630  ax-addrcl 9631  ax-mulcl 9632  ax-mulrcl 9633  ax-mulcom 9634  ax-addass 9635  ax-mulass 9636  ax-distr 9637  ax-i2m1 9638  ax-1ne0 9639  ax-1rid 9640  ax-rnegex 9641  ax-rrecex 9642  ax-cnre 9643  ax-pre-lttri 9644  ax-pre-lttrn 9645  ax-pre-ltadd 9646  ax-pre-mulgt0 9647  ax-pre-sup 9648

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-nel 2636  df-ral 2754  df-rex 2755  df-reu 2756  df-rmo 2757  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-iun 4294  df-br 4419  df-opab 4478  df-mpt 4479  df-tr 4514  df-eprel 4767  df-id 4771  df-po 4777  df-so 4778  df-fr 4815  df-we 4817  df-xp 4862  df-rel 4863  df-cnv 4864  df-co 4865  df-dm 4866  df-rn 4867  df-res 4868  df-ima 4869  df-pred 5403  df-ord 5449  df-on 5450  df-lim 5451  df-suc 5452  df-iota 5569  df-fun 5607  df-fn 5608  df-f 5609  df-f1 5610  df-fo 5611  df-f1o 5612  df-fv 5613  df-riota 6282  df-ov 6323  df-oprab 6324  df-mpt2 6325  df-om 6725  df-wrecs 7059  df-recs 7121  df-rdg 7159  df-er 7394  df-en 7601  df-dom 7602  df-sdom 7603  df-sup 7987  df-inf 7988  df-pnf 9708  df-mnf 9709  df-xr 9710  df-ltxr 9711  df-le 9712  df-sub 9893  df-neg 9894  df-div 10303  df-nn 10643  df-2 10701  df-3 10702  df-4 10703  df-5 10704  df-6 10705  df-7 10706  df-8 10707  df-n0 10904  df-z 10972  df-uz 11194  df-rp 11337  df-fl 12066  df-mod 12135

This theorem is referenced by:  lgsdir2  24312