Theorem signswch 29443

Description: The zero-skipping operation changes value when the operands change signs. (Contributed by Thierry Arnoux, 9-Oct-2018.)

Hypotheses

Ref	Expression
signsw.p	|- .+^ = ( a e. { -u 1 , 0 , 1 } , b e. { -u 1 , 0 , 1 } |-> if ( b = 0 , a , b ) )
signsw.w	|- W = { <. ( Base ` ndx ) , { -u 1 , 0 , 1 } >. , <. ( +g ` ndx ) , .+^ >. }

Assertion

Ref	Expression
signswch	|- ( ( X e. { -u 1 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) -> ( ( X .+^ Y ) =/= X <-> ( X x. Y ) < 0 ) )

Distinct variable groups:   a, b, X   Y, a, b

Allowed substitution hints:   .+^ ( a, b)   W( a, b)

Proof of Theorem signswch

Step	Hyp	Ref	Expression
1		df-pr 3970	. . . . . 6 |- { -u 1 , 1 } = ( { -u 1 } u. { 1 } )
2		snsstp1 4122	. . . . . . 7 |- { -u 1 } C_ { -u 1 , 0 , 1 }
3		snsstp3 4124	. . . . . . 7 |- { 1 } C_ { -u 1 , 0 , 1 }
4	2, 3	unssi 3608	. . . . . 6 |- ( { -u 1 } u. { 1 } ) C_ { -u 1 , 0 , 1 }
5	1, 4	eqsstri 3461	. . . . 5 |- { -u 1 , 1 } C_ { -u 1 , 0 , 1 }
6	5	sseli 3427	. . . 4 |- ( X e. { -u 1 , 1 } -> X e. { -u 1 , 0 , 1 } )
7		signsw.p	. . . . 5 |- .+^ = ( a e. { -u 1 , 0 , 1 } , b e. { -u 1 , 0 , 1 } |-> if ( b = 0 , a , b ) )
8	7	signspval 29434	. . . 4 |- ( ( X e. { -u 1 , 0 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) -> ( X .+^ Y ) = if ( Y = 0 , X , Y ) )
9	6, 8	sylan 474	. . 3 |- ( ( X e. { -u 1 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) -> ( X .+^ Y ) = if ( Y = 0 , X , Y ) )
10	9	neeq1d 2682	. 2 |- ( ( X e. { -u 1 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) -> ( ( X .+^ Y ) =/= X <-> if ( Y = 0 , X , Y ) =/= X ) )
11		neeq1 2685	. . . 4 |- ( X = if ( Y = 0 , X , Y ) -> ( X =/= X <-> if ( Y = 0 , X , Y ) =/= X ) )
12	11	bibi1d 321	. . 3 |- ( X = if ( Y = 0 , X , Y ) -> ( ( X =/= X <-> ( X x. Y ) < 0 ) <-> ( if ( Y = 0 , X , Y ) =/= X <-> ( X x. Y ) < 0 ) ) )
13		neeq1 2685	. . . 4 |- ( Y = if ( Y = 0 , X , Y ) -> ( Y =/= X <-> if ( Y = 0 , X , Y ) =/= X ) )
14	13	bibi1d 321	. . 3 |- ( Y = if ( Y = 0 , X , Y ) -> ( ( Y =/= X <-> ( X x. Y ) < 0 ) <-> ( if ( Y = 0 , X , Y ) =/= X <-> ( X x. Y ) < 0 ) ) )
15		neirr 2631	. . . . 5 |- -. X =/= X
16	15	a1i 11	. . . 4 |- ( ( ( X e. { -u 1 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) /\ Y = 0 ) -> -. X =/= X )
17		0re 9640	. . . . . 6 |- 0 e. RR
18	17	ltnri 9740	. . . . 5 |- -. 0 < 0
19		simpr 463	. . . . . . . 8 |- ( ( ( X e. { -u 1 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) /\ Y = 0 ) -> Y = 0 )
20	19	oveq2d 6304	. . . . . . 7 |- ( ( ( X e. { -u 1 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) /\ Y = 0 ) -> ( X x. Y ) = ( X x. 0 ) )
21		neg1cn 10710	. . . . . . . . . 10 |- -u 1 e. CC
22		ax-1cn 9594	. . . . . . . . . 10 |- 1 e. CC
23		prssi 4127	. . . . . . . . . 10 |- ( ( -u 1 e. CC /\ 1 e. CC ) -> { -u 1 , 1 } C_ CC )
24	21, 22, 23	mp2an 677	. . . . . . . . 9 |- { -u 1 , 1 } C_ CC
25		simpll 759	. . . . . . . . 9 |- ( ( ( X e. { -u 1 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) /\ Y = 0 ) -> X e. { -u 1 , 1 } )
26	24, 25	sseldi 3429	. . . . . . . 8 |- ( ( ( X e. { -u 1 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) /\ Y = 0 ) -> X e. CC )
27	26	mul01d 9829	. . . . . . 7 |- ( ( ( X e. { -u 1 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) /\ Y = 0 ) -> ( X x. 0 ) = 0 )
28	20, 27	eqtrd 2484	. . . . . 6 |- ( ( ( X e. { -u 1 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) /\ Y = 0 ) -> ( X x. Y ) = 0 )
29	28	breq1d 4411	. . . . 5 |- ( ( ( X e. { -u 1 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) /\ Y = 0 ) -> ( ( X x. Y ) < 0 <-> 0 < 0 ) )
30	18, 29	mtbiri 305	. . . 4 |- ( ( ( X e. { -u 1 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) /\ Y = 0 ) -> -. ( X x. Y ) < 0 )
31	16, 30	2falsed 353	. . 3 |- ( ( ( X e. { -u 1 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) /\ Y = 0 ) -> ( X =/= X <-> ( X x. Y ) < 0 ) )
32		simplr 761	. . . . . . . 8 |- ( ( ( X e. { -u 1 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) /\ -. Y = 0 ) -> Y e. { -u 1 , 0 , 1 } )
33		tpcomb 4068	. . . . . . . 8 |- { -u 1 , 0 , 1 } = { -u 1 , 1 , 0 }
34	32, 33	syl6eleq 2538	. . . . . . 7 |- ( ( ( X e. { -u 1 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) /\ -. Y = 0 ) -> Y e. { -u 1 , 1 , 0 } )
35		simpr 463	. . . . . . . 8 |- ( ( ( X e. { -u 1 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) /\ -. Y = 0 ) -> -. Y = 0 )
36	35	neqned 2630	. . . . . . 7 |- ( ( ( X e. { -u 1 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) /\ -. Y = 0 ) -> Y =/= 0 )
37	34, 36	jca 535	. . . . . 6 |- ( ( ( X e. { -u 1 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) /\ -. Y = 0 ) -> ( Y e. { -u 1 , 1 , 0 } /\ Y =/= 0 ) )
38		eldifsn 4096	. . . . . . 7 |- ( Y e. ( { -u 1 , 1 , 0 } \ { 0 } ) <-> ( Y e. { -u 1 , 1 , 0 } /\ Y =/= 0 ) )
39		neg1ne0 10712	. . . . . . . . 9 |- -u 1 =/= 0
40		ax-1ne0 9605	. . . . . . . . 9 |- 1 =/= 0
41		diftpsn3 4109	. . . . . . . . 9 |- ( ( -u 1 =/= 0 /\ 1 =/= 0 ) -> ( { -u 1 , 1 , 0 } \ { 0 } ) = { -u 1 , 1 } )
42	39, 40, 41	mp2an 677	. . . . . . . 8 |- ( { -u 1 , 1 , 0 } \ { 0 } ) = { -u 1 , 1 }
43	42	eleq2i 2520	. . . . . . 7 |- ( Y e. ( { -u 1 , 1 , 0 } \ { 0 } ) <-> Y e. { -u 1 , 1 } )
44	38, 43	bitr3i 255	. . . . . 6 |- ( ( Y e. { -u 1 , 1 , 0 } /\ Y =/= 0 ) <-> Y e. { -u 1 , 1 } )
45	37, 44	sylib 200	. . . . 5 |- ( ( ( X e. { -u 1 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) /\ -. Y = 0 ) -> Y e. { -u 1 , 1 } )
46		neirr 2631	. . . . . . . . . . 11 |- -. -u 1 =/= -u 1
47		0le1 10134	. . . . . . . . . . . . 13 |- 0 <_ 1
48		1re 9639	. . . . . . . . . . . . . 14 |- 1 e. RR
49	17, 48	lenlti 9751	. . . . . . . . . . . . 13 |- ( 0 <_ 1 <-> -. 1 < 0 )
50	47, 49	mpbi 212	. . . . . . . . . . . 12 |- -. 1 < 0
51		neg1mulneg1e1 10824	. . . . . . . . . . . . 13 |- ( -u 1 x. -u 1 ) = 1
52	51	breq1i 4408	. . . . . . . . . . . 12 |- ( ( -u 1 x. -u 1 ) < 0 <-> 1 < 0 )
53	50, 52	mtbir 301	. . . . . . . . . . 11 |- -. ( -u 1 x. -u 1 ) < 0
54	46, 53	2false 352	. . . . . . . . . 10 |- ( -u 1 =/= -u 1 <-> ( -u 1 x. -u 1 ) < 0 )
55		neeq1 2685	. . . . . . . . . . 11 |- ( Y = -u 1 -> ( Y =/= -u 1 <-> -u 1 =/= -u 1 ) )
56		oveq2 6296	. . . . . . . . . . . 12 |- ( Y = -u 1 -> ( -u 1 x. Y ) = ( -u 1 x. -u 1 ) )
57	56	breq1d 4411	. . . . . . . . . . 11 |- ( Y = -u 1 -> ( ( -u 1 x. Y ) < 0 <-> ( -u 1 x. -u 1 ) < 0 ) )
58	55, 57	bibi12d 323	. . . . . . . . . 10 |- ( Y = -u 1 -> ( ( Y =/= -u 1 <-> ( -u 1 x. Y ) < 0 ) <-> ( -u 1 =/= -u 1 <-> ( -u 1 x. -u 1 ) < 0 ) ) )
59	54, 58	mpbiri 237	. . . . . . . . 9 |- ( Y = -u 1 -> ( Y =/= -u 1 <-> ( -u 1 x. Y ) < 0 ) )
60	59	adantl 468	. . . . . . . 8 |- ( ( Y e. { -u 1 , 1 } /\ Y = -u 1 ) -> ( Y =/= -u 1 <-> ( -u 1 x. Y ) < 0 ) )
61		neg1rr 10711	. . . . . . . . . . . 12 |- -u 1 e. RR
62		neg1lt0 10713	. . . . . . . . . . . . 13 |- -u 1 < 0
63		0lt1 10133	. . . . . . . . . . . . 13 |- 0 < 1
64	61, 17, 48	lttri 9757	. . . . . . . . . . . . 13 |- ( ( -u 1 < 0 /\ 0 < 1 ) -> -u 1 < 1 )
65	62, 63, 64	mp2an 677	. . . . . . . . . . . 12 |- -u 1 < 1
66	61, 65	gtneii 9743	. . . . . . . . . . 11 |- 1 =/= -u 1
67	21	mulid1i 9642	. . . . . . . . . . . 12 |- ( -u 1 x. 1 ) = -u 1
68	67, 62	eqbrtri 4421	. . . . . . . . . . 11 |- ( -u 1 x. 1 ) < 0
69	66, 68	2th 243	. . . . . . . . . 10 |- ( 1 =/= -u 1 <-> ( -u 1 x. 1 ) < 0 )
70		neeq1 2685	. . . . . . . . . . 11 |- ( Y = 1 -> ( Y =/= -u 1 <-> 1 =/= -u 1 ) )
71		oveq2 6296	. . . . . . . . . . . 12 |- ( Y = 1 -> ( -u 1 x. Y ) = ( -u 1 x. 1 ) )
72	71	breq1d 4411	. . . . . . . . . . 11 |- ( Y = 1 -> ( ( -u 1 x. Y ) < 0 <-> ( -u 1 x. 1 ) < 0 ) )
73	70, 72	bibi12d 323	. . . . . . . . . 10 |- ( Y = 1 -> ( ( Y =/= -u 1 <-> ( -u 1 x. Y ) < 0 ) <-> ( 1 =/= -u 1 <-> ( -u 1 x. 1 ) < 0 ) ) )
74	69, 73	mpbiri 237	. . . . . . . . 9 |- ( Y = 1 -> ( Y =/= -u 1 <-> ( -u 1 x. Y ) < 0 ) )
75	74	adantl 468	. . . . . . . 8 |- ( ( Y e. { -u 1 , 1 } /\ Y = 1 ) -> ( Y =/= -u 1 <-> ( -u 1 x. Y ) < 0 ) )
76		elpri 3984	. . . . . . . 8 |- ( Y e. { -u 1 , 1 } -> ( Y = -u 1 \/ Y = 1 ) )
77	60, 75, 76	mpjaodan 794	. . . . . . 7 |- ( Y e. { -u 1 , 1 } -> ( Y =/= -u 1 <-> ( -u 1 x. Y ) < 0 ) )
78	77	adantr 467	. . . . . 6 |- ( ( Y e. { -u 1 , 1 } /\ X = -u 1 ) -> ( Y =/= -u 1 <-> ( -u 1 x. Y ) < 0 ) )
79		neeq2 2686	. . . . . . . 8 |- ( X = -u 1 -> ( Y =/= X <-> Y =/= -u 1 ) )
80		oveq1 6295	. . . . . . . . 9 |- ( X = -u 1 -> ( X x. Y ) = ( -u 1 x. Y ) )
81	80	breq1d 4411	. . . . . . . 8 |- ( X = -u 1 -> ( ( X x. Y ) < 0 <-> ( -u 1 x. Y ) < 0 ) )
82	79, 81	bibi12d 323	. . . . . . 7 |- ( X = -u 1 -> ( ( Y =/= X <-> ( X x. Y ) < 0 ) <-> ( Y =/= -u 1 <-> ( -u 1 x. Y ) < 0 ) ) )
83	82	adantl 468	. . . . . 6 |- ( ( Y e. { -u 1 , 1 } /\ X = -u 1 ) -> ( ( Y =/= X <-> ( X x. Y ) < 0 ) <-> ( Y =/= -u 1 <-> ( -u 1 x. Y ) < 0 ) ) )
84	78, 83	mpbird 236	. . . . 5 |- ( ( Y e. { -u 1 , 1 } /\ X = -u 1 ) -> ( Y =/= X <-> ( X x. Y ) < 0 ) )
85	45, 84	sylan 474	. . . 4 |- ( ( ( ( X e. { -u 1 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) /\ -. Y = 0 ) /\ X = -u 1 ) -> ( Y =/= X <-> ( X x. Y ) < 0 ) )
86	66	necomi 2677	. . . . . . . . . . 11 |- -u 1 =/= 1
87	21, 22	mulcomi 9646	. . . . . . . . . . . . 13 |- ( -u 1 x. 1 ) = ( 1 x. -u 1 )
88	87	breq1i 4408	. . . . . . . . . . . 12 |- ( ( -u 1 x. 1 ) < 0 <-> ( 1 x. -u 1 ) < 0 )
89	68, 88	mpbi 212	. . . . . . . . . . 11 |- ( 1 x. -u 1 ) < 0
90	86, 89	2th 243	. . . . . . . . . 10 |- ( -u 1 =/= 1 <-> ( 1 x. -u 1 ) < 0 )
91		neeq1 2685	. . . . . . . . . . 11 |- ( Y = -u 1 -> ( Y =/= 1 <-> -u 1 =/= 1 ) )
92		oveq2 6296	. . . . . . . . . . . 12 |- ( Y = -u 1 -> ( 1 x. Y ) = ( 1 x. -u 1 ) )
93	92	breq1d 4411	. . . . . . . . . . 11 |- ( Y = -u 1 -> ( ( 1 x. Y ) < 0 <-> ( 1 x. -u 1 ) < 0 ) )
94	91, 93	bibi12d 323	. . . . . . . . . 10 |- ( Y = -u 1 -> ( ( Y =/= 1 <-> ( 1 x. Y ) < 0 ) <-> ( -u 1 =/= 1 <-> ( 1 x. -u 1 ) < 0 ) ) )
95	90, 94	mpbiri 237	. . . . . . . . 9 |- ( Y = -u 1 -> ( Y =/= 1 <-> ( 1 x. Y ) < 0 ) )
96	95	adantl 468	. . . . . . . 8 |- ( ( Y e. { -u 1 , 1 } /\ Y = -u 1 ) -> ( Y =/= 1 <-> ( 1 x. Y ) < 0 ) )
97		neirr 2631	. . . . . . . . . . 11 |- -. 1 =/= 1
98	22	mulid1i 9642	. . . . . . . . . . . . 13 |- ( 1 x. 1 ) = 1
99	98	breq1i 4408	. . . . . . . . . . . 12 |- ( ( 1 x. 1 ) < 0 <-> 1 < 0 )
100	50, 99	mtbir 301	. . . . . . . . . . 11 |- -. ( 1 x. 1 ) < 0
101	97, 100	2false 352	. . . . . . . . . 10 |- ( 1 =/= 1 <-> ( 1 x. 1 ) < 0 )
102		neeq1 2685	. . . . . . . . . . 11 |- ( Y = 1 -> ( Y =/= 1 <-> 1 =/= 1 ) )
103		oveq2 6296	. . . . . . . . . . . 12 |- ( Y = 1 -> ( 1 x. Y ) = ( 1 x. 1 ) )
104	103	breq1d 4411	. . . . . . . . . . 11 |- ( Y = 1 -> ( ( 1 x. Y ) < 0 <-> ( 1 x. 1 ) < 0 ) )
105	102, 104	bibi12d 323	. . . . . . . . . 10 |- ( Y = 1 -> ( ( Y =/= 1 <-> ( 1 x. Y ) < 0 ) <-> ( 1 =/= 1 <-> ( 1 x. 1 ) < 0 ) ) )
106	101, 105	mpbiri 237	. . . . . . . . 9 |- ( Y = 1 -> ( Y =/= 1 <-> ( 1 x. Y ) < 0 ) )
107	106	adantl 468	. . . . . . . 8 |- ( ( Y e. { -u 1 , 1 } /\ Y = 1 ) -> ( Y =/= 1 <-> ( 1 x. Y ) < 0 ) )
108	96, 107, 76	mpjaodan 794	. . . . . . 7 |- ( Y e. { -u 1 , 1 } -> ( Y =/= 1 <-> ( 1 x. Y ) < 0 ) )
109	108	adantr 467	. . . . . 6 |- ( ( Y e. { -u 1 , 1 } /\ X = 1 ) -> ( Y =/= 1 <-> ( 1 x. Y ) < 0 ) )
110		neeq2 2686	. . . . . . . 8 |- ( X = 1 -> ( Y =/= X <-> Y =/= 1 ) )
111		oveq1 6295	. . . . . . . . 9 |- ( X = 1 -> ( X x. Y ) = ( 1 x. Y ) )
112	111	breq1d 4411	. . . . . . . 8 |- ( X = 1 -> ( ( X x. Y ) < 0 <-> ( 1 x. Y ) < 0 ) )
113	110, 112	bibi12d 323	. . . . . . 7 |- ( X = 1 -> ( ( Y =/= X <-> ( X x. Y ) < 0 ) <-> ( Y =/= 1 <-> ( 1 x. Y ) < 0 ) ) )
114	113	adantl 468	. . . . . 6 |- ( ( Y e. { -u 1 , 1 } /\ X = 1 ) -> ( ( Y =/= X <-> ( X x. Y ) < 0 ) <-> ( Y =/= 1 <-> ( 1 x. Y ) < 0 ) ) )
115	109, 114	mpbird 236	. . . . 5 |- ( ( Y e. { -u 1 , 1 } /\ X = 1 ) -> ( Y =/= X <-> ( X x. Y ) < 0 ) )
116	45, 115	sylan 474	. . . 4 |- ( ( ( ( X e. { -u 1 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) /\ -. Y = 0 ) /\ X = 1 ) -> ( Y =/= X <-> ( X x. Y ) < 0 ) )
117		elpri 3984	. . . . 5 |- ( X e. { -u 1 , 1 } -> ( X = -u 1 \/ X = 1 ) )
118	117	ad2antrr 731	. . . 4 |- ( ( ( X e. { -u 1 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) /\ -. Y = 0 ) -> ( X = -u 1 \/ X = 1 ) )
119	85, 116, 118	mpjaodan 794	. . 3 |- ( ( ( X e. { -u 1 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) /\ -. Y = 0 ) -> ( Y =/= X <-> ( X x. Y ) < 0 ) )
120	12, 14, 31, 119	ifbothda 3915	. 2 |- ( ( X e. { -u 1 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) -> ( if ( Y = 0 , X , Y ) =/= X <-> ( X x. Y ) < 0 ) )
121	10, 120	bitrd 257	1 |- ( ( X e. { -u 1 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) -> ( ( X .+^ Y ) =/= X <-> ( X x. Y ) < 0 ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 188   \/ wo 370   /\ wa 371   = wceq 1443   e. wcel 1886   =/= wne 2621   \ cdif 3400   u. cun 3401   C_ wss 3403   ifcif 3880   {csn 3967   {cpr 3969   {ctp 3971   <.cop 3973   class class class wbr 4401   ` cfv 5581  (class class class)co 6288   |-> cmpt2 6290   CCcc 9534   0cc0 9536   1c1 9537   x. cmul 9541   < clt 9672   <_ cle 9673   -ucneg 9858   ndxcnx 15111   Basecbs 15114   +g cplusg 15183

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-br 4402  df-opab 4461  df-mpt 4462  df-id 4748  df-po 4754  df-so 4755  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-er 7360  df-en 7567  df-dom 7568  df-sdom 7569  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860

This theorem is referenced by:  signsvfn  29464