Theorem lgsdir2lem4 21063

Description: Lemma for lgsdir2 21065. (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 6065	. . 3 |- ( A mod 8 ) e. _V
2	1	elpr 3792	. 2 |- ( ( A mod 8 ) e. { 1 , 7 } <-> ( ( A mod 8 ) = 1 \/ ( A mod 8 ) = 7 ) )
3		zre 10242	. . . . . . 7 |- ( A e. ZZ -> A e. RR )
4	3	ad2antrr 707	. . . . . 6 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 1 ) -> A e. RR )
5		1re 9046	. . . . . . 7 |- 1 e. RR
6	5	a1i 11	. . . . . 6 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 1 ) -> 1 e. RR )
7		simplr 732	. . . . . 6 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 1 ) -> B e. ZZ )
8		8re 10034	. . . . . . . 8 |- 8 e. RR
9		8pos 10046	. . . . . . . 8 |- 0 < 8
10	8, 9	elrpii 10571	. . . . . . 7 |- 8 e. RR+
11	10	a1i 11	. . . . . 6 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 1 ) -> 8 e. RR+ )
12		simpr 448	. . . . . . 7 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 1 ) -> ( A mod 8 ) = 1 )
13		lgsdir2lem1 21060	. . . . . . . . 9 |- ( ( ( 1 mod 8 ) = 1 /\ ( -u 1 mod 8 ) = 7 ) /\ ( ( 3 mod 8 ) = 3 /\ ( -u 3 mod 8 ) = 5 ) )
14	13	simpli 445	. . . . . . . 8 |- ( ( 1 mod 8 ) = 1 /\ ( -u 1 mod 8 ) = 7 )
15	14	simpli 445	. . . . . . 7 |- ( 1 mod 8 ) = 1
16	12, 15	syl6eqr 2454	. . . . . 6 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 1 ) -> ( A mod 8 ) = ( 1 mod 8 ) )
17		modmul1 11234	. . . . . 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 ) )
18	4, 6, 7, 11, 16, 17	syl221anc 1195	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 1 ) -> ( ( A x. B ) mod 8 ) = ( ( 1 x. B ) mod 8 ) )
19		zcn 10243	. . . . . . . 8 |- ( B e. ZZ -> B e. CC )
20	19	ad2antlr 708	. . . . . . 7 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 1 ) -> B e. CC )
21	20	mulid2d 9062	. . . . . 6 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 1 ) -> ( 1 x. B ) = B )
22	21	oveq1d 6055	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 1 ) -> ( ( 1 x. B ) mod 8 ) = ( B mod 8 ) )
23	18, 22	eqtrd 2436	. . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 1 ) -> ( ( A x. B ) mod 8 ) = ( B mod 8 ) )
24	23	eleq1d 2470	. . 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 } ) )
25	3	ad2antrr 707	. . . . . . 7 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 7 ) -> A e. RR )
26	5	renegcli 9318	. . . . . . . 8 |- -u 1 e. RR
27	26	a1i 11	. . . . . . 7 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 7 ) -> -u 1 e. RR )
28		simplr 732	. . . . . . 7 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 7 ) -> B e. ZZ )
29	10	a1i 11	. . . . . . 7 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 7 ) -> 8 e. RR+ )
30		simpr 448	. . . . . . . 8 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 7 ) -> ( A mod 8 ) = 7 )
31	14	simpri 449	. . . . . . . 8 |- ( -u 1 mod 8 ) = 7
32	30, 31	syl6eqr 2454	. . . . . . 7 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 7 ) -> ( A mod 8 ) = ( -u 1 mod 8 ) )
33		modmul1 11234	. . . . . . 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 ) )
34	25, 27, 28, 29, 32, 33	syl221anc 1195	. . . . . 6 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 7 ) -> ( ( A x. B ) mod 8 ) = ( ( -u 1 x. B ) mod 8 ) )
35	19	ad2antlr 708	. . . . . . . 8 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 7 ) -> B e. CC )
36	35	mulm1d 9441	. . . . . . 7 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 7 ) -> ( -u 1 x. B ) = -u B )
37	36	oveq1d 6055	. . . . . 6 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 7 ) -> ( ( -u 1 x. B ) mod 8 ) = ( -u B mod 8 ) )
38	34, 37	eqtrd 2436	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 7 ) -> ( ( A x. B ) mod 8 ) = ( -u B mod 8 ) )
39	38	eleq1d 2470	. . . 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 } ) )
40		znegcl 10269	. . . . . . . 8 |- ( B e. ZZ -> -u B e. ZZ )
41		oveq1 6047	. . . . . . . . . . 11 |- ( x = -u B -> ( x mod 8 ) = ( -u B mod 8 ) )
42	41	eleq1d 2470	. . . . . . . . . 10 |- ( x = -u B -> ( ( x mod 8 ) e. { 1 , 7 } <-> ( -u B mod 8 ) e. { 1 , 7 } ) )
43		negeq 9254	. . . . . . . . . . . 12 |- ( x = -u B -> -u x = -u -u B )
44	43	oveq1d 6055	. . . . . . . . . . 11 |- ( x = -u B -> ( -u x mod 8 ) = ( -u -u B mod 8 ) )
45	44	eleq1d 2470	. . . . . . . . . 10 |- ( x = -u B -> ( ( -u x mod 8 ) e. { 1 , 7 } <-> ( -u -u B mod 8 ) e. { 1 , 7 } ) )
46	42, 45	imbi12d 312	. . . . . . . . 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 } ) ) )
47		zcn 10243	. . . . . . . . . . . . . . . . 17 |- ( x e. ZZ -> x e. CC )
48		neg1cn 10023	. . . . . . . . . . . . . . . . . . 19 |- -u 1 e. CC
49		mulcom 9032	. . . . . . . . . . . . . . . . . . 19 |- ( ( x e. CC /\ -u 1 e. CC ) -> ( x x. -u 1 ) = ( -u 1 x. x ) )
50	48, 49	mpan2 653	. . . . . . . . . . . . . . . . . 18 |- ( x e. CC -> ( x x. -u 1 ) = ( -u 1 x. x ) )
51		mulm1 9431	. . . . . . . . . . . . . . . . . 18 |- ( x e. CC -> ( -u 1 x. x ) = -u x )
52	50, 51	eqtrd 2436	. . . . . . . . . . . . . . . . 17 |- ( x e. CC -> ( x x. -u 1 ) = -u x )
53	47, 52	syl 16	. . . . . . . . . . . . . . . 16 |- ( x e. ZZ -> ( x x. -u 1 ) = -u x )
54	53	adantr 452	. . . . . . . . . . . . . . 15 |- ( ( x e. ZZ /\ ( x mod 8 ) = 1 ) -> ( x x. -u 1 ) = -u x )
55	54	oveq1d 6055	. . . . . . . . . . . . . 14 |- ( ( x e. ZZ /\ ( x mod 8 ) = 1 ) -> ( ( x x. -u 1 ) mod 8 ) = ( -u x mod 8 ) )
56		zre 10242	. . . . . . . . . . . . . . . 16 |- ( x e. ZZ -> x e. RR )
57	56	adantr 452	. . . . . . . . . . . . . . 15 |- ( ( x e. ZZ /\ ( x mod 8 ) = 1 ) -> x e. RR )
58	5	a1i 11	. . . . . . . . . . . . . . 15 |- ( ( x e. ZZ /\ ( x mod 8 ) = 1 ) -> 1 e. RR )
59		1z 10267	. . . . . . . . . . . . . . . . 17 |- 1 e. ZZ
60		znegcl 10269	. . . . . . . . . . . . . . . . 17 |- ( 1 e. ZZ -> -u 1 e. ZZ )
61	59, 60	ax-mp 8	. . . . . . . . . . . . . . . 16 |- -u 1 e. ZZ
62	61	a1i 11	. . . . . . . . . . . . . . 15 |- ( ( x e. ZZ /\ ( x mod 8 ) = 1 ) -> -u 1 e. ZZ )
63	10	a1i 11	. . . . . . . . . . . . . . 15 |- ( ( x e. ZZ /\ ( x mod 8 ) = 1 ) -> 8 e. RR+ )
64		simpr 448	. . . . . . . . . . . . . . . 16 |- ( ( x e. ZZ /\ ( x mod 8 ) = 1 ) -> ( x mod 8 ) = 1 )
65	64, 15	syl6eqr 2454	. . . . . . . . . . . . . . 15 |- ( ( x e. ZZ /\ ( x mod 8 ) = 1 ) -> ( x mod 8 ) = ( 1 mod 8 ) )
66		modmul1 11234	. . . . . . . . . . . . . . 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 ) )
67	57, 58, 62, 63, 65, 66	syl221anc 1195	. . . . . . . . . . . . . 14 |- ( ( x e. ZZ /\ ( x mod 8 ) = 1 ) -> ( ( x x. -u 1 ) mod 8 ) = ( ( 1 x. -u 1 ) mod 8 ) )
68	55, 67	eqtr3d 2438	. . . . . . . . . . . . 13 |- ( ( x e. ZZ /\ ( x mod 8 ) = 1 ) -> ( -u x mod 8 ) = ( ( 1 x. -u 1 ) mod 8 ) )
69	48	mulid2i 9049	. . . . . . . . . . . . . . 15 |- ( 1 x. -u 1 ) = -u 1
70	69	oveq1i 6050	. . . . . . . . . . . . . 14 |- ( ( 1 x. -u 1 ) mod 8 ) = ( -u 1 mod 8 )
71	70, 31	eqtri 2424	. . . . . . . . . . . . 13 |- ( ( 1 x. -u 1 ) mod 8 ) = 7
72	68, 71	syl6eq 2452	. . . . . . . . . . . 12 |- ( ( x e. ZZ /\ ( x mod 8 ) = 1 ) -> ( -u x mod 8 ) = 7 )
73	72	ex 424	. . . . . . . . . . 11 |- ( x e. ZZ -> ( ( x mod 8 ) = 1 -> ( -u x mod 8 ) = 7 ) )
74	53	adantr 452	. . . . . . . . . . . . . . 15 |- ( ( x e. ZZ /\ ( x mod 8 ) = 7 ) -> ( x x. -u 1 ) = -u x )
75	74	oveq1d 6055	. . . . . . . . . . . . . 14 |- ( ( x e. ZZ /\ ( x mod 8 ) = 7 ) -> ( ( x x. -u 1 ) mod 8 ) = ( -u x mod 8 ) )
76	56	adantr 452	. . . . . . . . . . . . . . 15 |- ( ( x e. ZZ /\ ( x mod 8 ) = 7 ) -> x e. RR )
77	26	a1i 11	. . . . . . . . . . . . . . 15 |- ( ( x e. ZZ /\ ( x mod 8 ) = 7 ) -> -u 1 e. RR )
78	61	a1i 11	. . . . . . . . . . . . . . 15 |- ( ( x e. ZZ /\ ( x mod 8 ) = 7 ) -> -u 1 e. ZZ )
79	10	a1i 11	. . . . . . . . . . . . . . 15 |- ( ( x e. ZZ /\ ( x mod 8 ) = 7 ) -> 8 e. RR+ )
80		simpr 448	. . . . . . . . . . . . . . . 16 |- ( ( x e. ZZ /\ ( x mod 8 ) = 7 ) -> ( x mod 8 ) = 7 )
81	80, 31	syl6eqr 2454	. . . . . . . . . . . . . . 15 |- ( ( x e. ZZ /\ ( x mod 8 ) = 7 ) -> ( x mod 8 ) = ( -u 1 mod 8 ) )
82		modmul1 11234	. . . . . . . . . . . . . . 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 ) )
83	76, 77, 78, 79, 81, 82	syl221anc 1195	. . . . . . . . . . . . . 14 |- ( ( x e. ZZ /\ ( x mod 8 ) = 7 ) -> ( ( x x. -u 1 ) mod 8 ) = ( ( -u 1 x. -u 1 ) mod 8 ) )
84	75, 83	eqtr3d 2438	. . . . . . . . . . . . 13 |- ( ( x e. ZZ /\ ( x mod 8 ) = 7 ) -> ( -u x mod 8 ) = ( ( -u 1 x. -u 1 ) mod 8 ) )
85	48	mulm1i 9434	. . . . . . . . . . . . . . . 16 |- ( -u 1 x. -u 1 ) = -u -u 1
86		ax-1cn 9004	. . . . . . . . . . . . . . . . 17 |- 1 e. CC
87	86	negnegi 9326	. . . . . . . . . . . . . . . 16 |- -u -u 1 = 1
88	85, 87	eqtri 2424	. . . . . . . . . . . . . . 15 |- ( -u 1 x. -u 1 ) = 1
89	88	oveq1i 6050	. . . . . . . . . . . . . 14 |- ( ( -u 1 x. -u 1 ) mod 8 ) = ( 1 mod 8 )
90	89, 15	eqtri 2424	. . . . . . . . . . . . 13 |- ( ( -u 1 x. -u 1 ) mod 8 ) = 1
91	84, 90	syl6eq 2452	. . . . . . . . . . . 12 |- ( ( x e. ZZ /\ ( x mod 8 ) = 7 ) -> ( -u x mod 8 ) = 1 )
92	91	ex 424	. . . . . . . . . . 11 |- ( x e. ZZ -> ( ( x mod 8 ) = 7 -> ( -u x mod 8 ) = 1 ) )
93	73, 92	orim12d 812	. . . . . . . . . 10 |- ( x e. ZZ -> ( ( ( x mod 8 ) = 1 \/ ( x mod 8 ) = 7 ) -> ( ( -u x mod 8 ) = 7 \/ ( -u x mod 8 ) = 1 ) ) )
94		ovex 6065	. . . . . . . . . . 11 |- ( x mod 8 ) e. _V
95	94	elpr 3792	. . . . . . . . . 10 |- ( ( x mod 8 ) e. { 1 , 7 } <-> ( ( x mod 8 ) = 1 \/ ( x mod 8 ) = 7 ) )
96		ovex 6065	. . . . . . . . . . . 12 |- ( -u x mod 8 ) e. _V
97	96	elpr 3792	. . . . . . . . . . 11 |- ( ( -u x mod 8 ) e. { 1 , 7 } <-> ( ( -u x mod 8 ) = 1 \/ ( -u x mod 8 ) = 7 ) )
98		orcom 377	. . . . . . . . . . 11 |- ( ( ( -u x mod 8 ) = 1 \/ ( -u x mod 8 ) = 7 ) <-> ( ( -u x mod 8 ) = 7 \/ ( -u x mod 8 ) = 1 ) )
99	97, 98	bitri 241	. . . . . . . . . 10 |- ( ( -u x mod 8 ) e. { 1 , 7 } <-> ( ( -u x mod 8 ) = 7 \/ ( -u x mod 8 ) = 1 ) )
100	93, 95, 99	3imtr4g 262	. . . . . . . . 9 |- ( x e. ZZ -> ( ( x mod 8 ) e. { 1 , 7 } -> ( -u x mod 8 ) e. { 1 , 7 } ) )
101	46, 100	vtoclga 2977	. . . . . . . 8 |- ( -u B e. ZZ -> ( ( -u B mod 8 ) e. { 1 , 7 } -> ( -u -u B mod 8 ) e. { 1 , 7 } ) )
102	40, 101	syl 16	. . . . . . 7 |- ( B e. ZZ -> ( ( -u B mod 8 ) e. { 1 , 7 } -> ( -u -u B mod 8 ) e. { 1 , 7 } ) )
103	19	negnegd 9358	. . . . . . . . 9 |- ( B e. ZZ -> -u -u B = B )
104	103	oveq1d 6055	. . . . . . . 8 |- ( B e. ZZ -> ( -u -u B mod 8 ) = ( B mod 8 ) )
105	104	eleq1d 2470	. . . . . . 7 |- ( B e. ZZ -> ( ( -u -u B mod 8 ) e. { 1 , 7 } <-> ( B mod 8 ) e. { 1 , 7 } ) )
106	102, 105	sylibd 206	. . . . . 6 |- ( B e. ZZ -> ( ( -u B mod 8 ) e. { 1 , 7 } -> ( B mod 8 ) e. { 1 , 7 } ) )
107		oveq1 6047	. . . . . . . . 9 |- ( x = B -> ( x mod 8 ) = ( B mod 8 ) )
108	107	eleq1d 2470	. . . . . . . 8 |- ( x = B -> ( ( x mod 8 ) e. { 1 , 7 } <-> ( B mod 8 ) e. { 1 , 7 } ) )
109		negeq 9254	. . . . . . . . . 10 |- ( x = B -> -u x = -u B )
110	109	oveq1d 6055	. . . . . . . . 9 |- ( x = B -> ( -u x mod 8 ) = ( -u B mod 8 ) )
111	110	eleq1d 2470	. . . . . . . 8 |- ( x = B -> ( ( -u x mod 8 ) e. { 1 , 7 } <-> ( -u B mod 8 ) e. { 1 , 7 } ) )
112	108, 111	imbi12d 312	. . . . . . 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 } ) ) )
113	112, 100	vtoclga 2977	. . . . . 6 |- ( B e. ZZ -> ( ( B mod 8 ) e. { 1 , 7 } -> ( -u B mod 8 ) e. { 1 , 7 } ) )
114	106, 113	impbid 184	. . . . 5 |- ( B e. ZZ -> ( ( -u B mod 8 ) e. { 1 , 7 } <-> ( B mod 8 ) e. { 1 , 7 } ) )
115	114	ad2antlr 708	. . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) = 7 ) -> ( ( -u B mod 8 ) e. { 1 , 7 } <-> ( B mod 8 ) e. { 1 , 7 } ) )
116	39, 115	bitrd 245	. . 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 } ) )
117	24, 116	jaodan 761	. 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 } ) )
118	2, 117	sylan2b 462	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 177   \/ wo 358   /\ wa 359   = wceq 1649   e. wcel 1721   {cpr 3775  (class class class)co 6040   CCcc 8944   RRcr 8945   1c1 8947   x. cmul 8951   -ucneg 9248   3c3 10006   5c5 10008   7c7 10010   8c8 10011   ZZcz 10238   RR+crp 10568   mod cmo 11205

This theorem is referenced by:  lgsdir2  21065

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-sup 7404  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-fl 11157  df-mod 11206