Theorem signswch 30638

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 4180	. . . . . 6 |- { -u 1 , 1 } = ( { -u 1 } u. { 1 } )
2		snsstp1 4347	. . . . . . 7 |- { -u 1 } C_ { -u 1 , 0 , 1 }
3		snsstp3 4349	. . . . . . 7 |- { 1 } C_ { -u 1 , 0 , 1 }
4	2, 3	unssi 3788	. . . . . 6 |- ( { -u 1 } u. { 1 } ) C_ { -u 1 , 0 , 1 }
5	1, 4	eqsstri 3635	. . . . 5 |- { -u 1 , 1 } C_ { -u 1 , 0 , 1 }
6	5	sseli 3599	. . . 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 30629	. . . 4 |- ( ( X e. { -u 1 , 0 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) -> ( X .+^ Y ) = if ( Y = 0 , X , Y ) )
9	6, 8	sylan 488	. . 3 |- ( ( X e. { -u 1 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) -> ( X .+^ Y ) = if ( Y = 0 , X , Y ) )
10	9	neeq1d 2853	. 2 |- ( ( X e. { -u 1 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) -> ( ( X .+^ Y ) =/= X <-> if ( Y = 0 , X , Y ) =/= X ) )
11		neeq1 2856	. . . 4 |- ( X = if ( Y = 0 , X , Y ) -> ( X =/= X <-> if ( Y = 0 , X , Y ) =/= X ) )
12	11	bibi1d 333	. . 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 2856	. . . 4 |- ( Y = if ( Y = 0 , X , Y ) -> ( Y =/= X <-> if ( Y = 0 , X , Y ) =/= X ) )
14	13	bibi1d 333	. . 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 2803	. . . . 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 10040	. . . . . 6 |- 0 e. RR
18	17	ltnri 10146	. . . . 5 |- -. 0 < 0
19		simpr 477	. . . . . . . 8 |- ( ( ( X e. { -u 1 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) /\ Y = 0 ) -> Y = 0 )
20	19	oveq2d 6666	. . . . . . 7 |- ( ( ( X e. { -u 1 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) /\ Y = 0 ) -> ( X x. Y ) = ( X x. 0 ) )
21		neg1cn 11124	. . . . . . . . . 10 |- -u 1 e. CC
22		ax-1cn 9994	. . . . . . . . . 10 |- 1 e. CC
23		prssi 4353	. . . . . . . . . 10 |- ( ( -u 1 e. CC /\ 1 e. CC ) -> { -u 1 , 1 } C_ CC )
24	21, 22, 23	mp2an 708	. . . . . . . . 9 |- { -u 1 , 1 } C_ CC
25		simpll 790	. . . . . . . . 9 |- ( ( ( X e. { -u 1 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) /\ Y = 0 ) -> X e. { -u 1 , 1 } )
26	24, 25	sseldi 3601	. . . . . . . 8 |- ( ( ( X e. { -u 1 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) /\ Y = 0 ) -> X e. CC )
27	26	mul01d 10235	. . . . . . 7 |- ( ( ( X e. { -u 1 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) /\ Y = 0 ) -> ( X x. 0 ) = 0 )
28	20, 27	eqtrd 2656	. . . . . 6 |- ( ( ( X e. { -u 1 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) /\ Y = 0 ) -> ( X x. Y ) = 0 )
29	28	breq1d 4663	. . . . 5 |- ( ( ( X e. { -u 1 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) /\ Y = 0 ) -> ( ( X x. Y ) < 0 <-> 0 < 0 ) )
30	18, 29	mtbiri 317	. . . 4 |- ( ( ( X e. { -u 1 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) /\ Y = 0 ) -> -. ( X x. Y ) < 0 )
31	16, 30	2falsed 366	. . 3 |- ( ( ( X e. { -u 1 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) /\ Y = 0 ) -> ( X =/= X <-> ( X x. Y ) < 0 ) )
32		simplr 792	. . . . . . . 8 |- ( ( ( X e. { -u 1 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) /\ -. Y = 0 ) -> Y e. { -u 1 , 0 , 1 } )
33		tpcomb 4286	. . . . . . . 8 |- { -u 1 , 0 , 1 } = { -u 1 , 1 , 0 }
34	32, 33	syl6eleq 2711	. . . . . . 7 |- ( ( ( X e. { -u 1 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) /\ -. Y = 0 ) -> Y e. { -u 1 , 1 , 0 } )
35		simpr 477	. . . . . . . 8 |- ( ( ( X e. { -u 1 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) /\ -. Y = 0 ) -> -. Y = 0 )
36	35	neqned 2801	. . . . . . 7 |- ( ( ( X e. { -u 1 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) /\ -. Y = 0 ) -> Y =/= 0 )
37	34, 36	jca 554	. . . . . 6 |- ( ( ( X e. { -u 1 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) /\ -. Y = 0 ) -> ( Y e. { -u 1 , 1 , 0 } /\ Y =/= 0 ) )
38		eldifsn 4317	. . . . . . 7 |- ( Y e. ( { -u 1 , 1 , 0 } \ { 0 } ) <-> ( Y e. { -u 1 , 1 , 0 } /\ Y =/= 0 ) )
39		neg1ne0 11126	. . . . . . . . 9 |- -u 1 =/= 0
40		ax-1ne0 10005	. . . . . . . . 9 |- 1 =/= 0
41		diftpsn3 4332	. . . . . . . . 9 |- ( ( -u 1 =/= 0 /\ 1 =/= 0 ) -> ( { -u 1 , 1 , 0 } \ { 0 } ) = { -u 1 , 1 } )
42	39, 40, 41	mp2an 708	. . . . . . . 8 |- ( { -u 1 , 1 , 0 } \ { 0 } ) = { -u 1 , 1 }
43	42	eleq2i 2693	. . . . . . 7 |- ( Y e. ( { -u 1 , 1 , 0 } \ { 0 } ) <-> Y e. { -u 1 , 1 } )
44	38, 43	bitr3i 266	. . . . . 6 |- ( ( Y e. { -u 1 , 1 , 0 } /\ Y =/= 0 ) <-> Y e. { -u 1 , 1 } )
45	37, 44	sylib 208	. . . . 5 |- ( ( ( X e. { -u 1 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) /\ -. Y = 0 ) -> Y e. { -u 1 , 1 } )
46		neirr 2803	. . . . . . . . . . 11 |- -. -u 1 =/= -u 1
47		0le1 10551	. . . . . . . . . . . . 13 |- 0 <_ 1
48		1re 10039	. . . . . . . . . . . . . 14 |- 1 e. RR
49	17, 48	lenlti 10157	. . . . . . . . . . . . 13 |- ( 0 <_ 1 <-> -. 1 < 0 )
50	47, 49	mpbi 220	. . . . . . . . . . . 12 |- -. 1 < 0
51		neg1mulneg1e1 11245	. . . . . . . . . . . . 13 |- ( -u 1 x. -u 1 ) = 1
52	51	breq1i 4660	. . . . . . . . . . . 12 |- ( ( -u 1 x. -u 1 ) < 0 <-> 1 < 0 )
53	50, 52	mtbir 313	. . . . . . . . . . 11 |- -. ( -u 1 x. -u 1 ) < 0
54	46, 53	2false 365	. . . . . . . . . 10 |- ( -u 1 =/= -u 1 <-> ( -u 1 x. -u 1 ) < 0 )
55		neeq1 2856	. . . . . . . . . . 11 |- ( Y = -u 1 -> ( Y =/= -u 1 <-> -u 1 =/= -u 1 ) )
56		oveq2 6658	. . . . . . . . . . . 12 |- ( Y = -u 1 -> ( -u 1 x. Y ) = ( -u 1 x. -u 1 ) )
57	56	breq1d 4663	. . . . . . . . . . 11 |- ( Y = -u 1 -> ( ( -u 1 x. Y ) < 0 <-> ( -u 1 x. -u 1 ) < 0 ) )
58	55, 57	bibi12d 335	. . . . . . . . . 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 248	. . . . . . . . 9 |- ( Y = -u 1 -> ( Y =/= -u 1 <-> ( -u 1 x. Y ) < 0 ) )
60	59	adantl 482	. . . . . . . 8 |- ( ( Y e. { -u 1 , 1 } /\ Y = -u 1 ) -> ( Y =/= -u 1 <-> ( -u 1 x. Y ) < 0 ) )
61		neg1rr 11125	. . . . . . . . . . . 12 |- -u 1 e. RR
62		neg1lt0 11127	. . . . . . . . . . . . 13 |- -u 1 < 0
63		0lt1 10550	. . . . . . . . . . . . 13 |- 0 < 1
64	61, 17, 48	lttri 10163	. . . . . . . . . . . . 13 |- ( ( -u 1 < 0 /\ 0 < 1 ) -> -u 1 < 1 )
65	62, 63, 64	mp2an 708	. . . . . . . . . . . 12 |- -u 1 < 1
66	61, 65	gtneii 10149	. . . . . . . . . . 11 |- 1 =/= -u 1
67	21	mulid1i 10042	. . . . . . . . . . . 12 |- ( -u 1 x. 1 ) = -u 1
68	67, 62	eqbrtri 4674	. . . . . . . . . . 11 |- ( -u 1 x. 1 ) < 0
69	66, 68	2th 254	. . . . . . . . . 10 |- ( 1 =/= -u 1 <-> ( -u 1 x. 1 ) < 0 )
70		neeq1 2856	. . . . . . . . . . 11 |- ( Y = 1 -> ( Y =/= -u 1 <-> 1 =/= -u 1 ) )
71		oveq2 6658	. . . . . . . . . . . 12 |- ( Y = 1 -> ( -u 1 x. Y ) = ( -u 1 x. 1 ) )
72	71	breq1d 4663	. . . . . . . . . . 11 |- ( Y = 1 -> ( ( -u 1 x. Y ) < 0 <-> ( -u 1 x. 1 ) < 0 ) )
73	70, 72	bibi12d 335	. . . . . . . . . 10 |- ( Y = 1 -> ( ( Y =/= -u 1 <-> ( -u 1 x. Y ) < 0 ) <-> ( 1 =/= -u 1 <-> ( -u 1 x. 1 ) < 0 ) ) )
74	69, 73	mpbiri 248	. . . . . . . . 9 |- ( Y = 1 -> ( Y =/= -u 1 <-> ( -u 1 x. Y ) < 0 ) )
75	74	adantl 482	. . . . . . . 8 |- ( ( Y e. { -u 1 , 1 } /\ Y = 1 ) -> ( Y =/= -u 1 <-> ( -u 1 x. Y ) < 0 ) )
76		elpri 4197	. . . . . . . 8 |- ( Y e. { -u 1 , 1 } -> ( Y = -u 1 \/ Y = 1 ) )
77	60, 75, 76	mpjaodan 827	. . . . . . 7 |- ( Y e. { -u 1 , 1 } -> ( Y =/= -u 1 <-> ( -u 1 x. Y ) < 0 ) )
78	77	adantr 481	. . . . . 6 |- ( ( Y e. { -u 1 , 1 } /\ X = -u 1 ) -> ( Y =/= -u 1 <-> ( -u 1 x. Y ) < 0 ) )
79		neeq2 2857	. . . . . . . 8 |- ( X = -u 1 -> ( Y =/= X <-> Y =/= -u 1 ) )
80		oveq1 6657	. . . . . . . . 9 |- ( X = -u 1 -> ( X x. Y ) = ( -u 1 x. Y ) )
81	80	breq1d 4663	. . . . . . . 8 |- ( X = -u 1 -> ( ( X x. Y ) < 0 <-> ( -u 1 x. Y ) < 0 ) )
82	79, 81	bibi12d 335	. . . . . . 7 |- ( X = -u 1 -> ( ( Y =/= X <-> ( X x. Y ) < 0 ) <-> ( Y =/= -u 1 <-> ( -u 1 x. Y ) < 0 ) ) )
83	82	adantl 482	. . . . . 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 247	. . . . 5 |- ( ( Y e. { -u 1 , 1 } /\ X = -u 1 ) -> ( Y =/= X <-> ( X x. Y ) < 0 ) )
85	45, 84	sylan 488	. . . 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 2848	. . . . . . . . . . 11 |- -u 1 =/= 1
87	21, 22	mulcomi 10046	. . . . . . . . . . . . 13 |- ( -u 1 x. 1 ) = ( 1 x. -u 1 )
88	87	breq1i 4660	. . . . . . . . . . . 12 |- ( ( -u 1 x. 1 ) < 0 <-> ( 1 x. -u 1 ) < 0 )
89	68, 88	mpbi 220	. . . . . . . . . . 11 |- ( 1 x. -u 1 ) < 0
90	86, 89	2th 254	. . . . . . . . . 10 |- ( -u 1 =/= 1 <-> ( 1 x. -u 1 ) < 0 )
91		neeq1 2856	. . . . . . . . . . 11 |- ( Y = -u 1 -> ( Y =/= 1 <-> -u 1 =/= 1 ) )
92		oveq2 6658	. . . . . . . . . . . 12 |- ( Y = -u 1 -> ( 1 x. Y ) = ( 1 x. -u 1 ) )
93	92	breq1d 4663	. . . . . . . . . . 11 |- ( Y = -u 1 -> ( ( 1 x. Y ) < 0 <-> ( 1 x. -u 1 ) < 0 ) )
94	91, 93	bibi12d 335	. . . . . . . . . 10 |- ( Y = -u 1 -> ( ( Y =/= 1 <-> ( 1 x. Y ) < 0 ) <-> ( -u 1 =/= 1 <-> ( 1 x. -u 1 ) < 0 ) ) )
95	90, 94	mpbiri 248	. . . . . . . . 9 |- ( Y = -u 1 -> ( Y =/= 1 <-> ( 1 x. Y ) < 0 ) )
96	95	adantl 482	. . . . . . . 8 |- ( ( Y e. { -u 1 , 1 } /\ Y = -u 1 ) -> ( Y =/= 1 <-> ( 1 x. Y ) < 0 ) )
97		neirr 2803	. . . . . . . . . . 11 |- -. 1 =/= 1
98	22	mulid1i 10042	. . . . . . . . . . . . 13 |- ( 1 x. 1 ) = 1
99	98	breq1i 4660	. . . . . . . . . . . 12 |- ( ( 1 x. 1 ) < 0 <-> 1 < 0 )
100	50, 99	mtbir 313	. . . . . . . . . . 11 |- -. ( 1 x. 1 ) < 0
101	97, 100	2false 365	. . . . . . . . . 10 |- ( 1 =/= 1 <-> ( 1 x. 1 ) < 0 )
102		neeq1 2856	. . . . . . . . . . 11 |- ( Y = 1 -> ( Y =/= 1 <-> 1 =/= 1 ) )
103		oveq2 6658	. . . . . . . . . . . 12 |- ( Y = 1 -> ( 1 x. Y ) = ( 1 x. 1 ) )
104	103	breq1d 4663	. . . . . . . . . . 11 |- ( Y = 1 -> ( ( 1 x. Y ) < 0 <-> ( 1 x. 1 ) < 0 ) )
105	102, 104	bibi12d 335	. . . . . . . . . 10 |- ( Y = 1 -> ( ( Y =/= 1 <-> ( 1 x. Y ) < 0 ) <-> ( 1 =/= 1 <-> ( 1 x. 1 ) < 0 ) ) )
106	101, 105	mpbiri 248	. . . . . . . . 9 |- ( Y = 1 -> ( Y =/= 1 <-> ( 1 x. Y ) < 0 ) )
107	106	adantl 482	. . . . . . . 8 |- ( ( Y e. { -u 1 , 1 } /\ Y = 1 ) -> ( Y =/= 1 <-> ( 1 x. Y ) < 0 ) )
108	96, 107, 76	mpjaodan 827	. . . . . . 7 |- ( Y e. { -u 1 , 1 } -> ( Y =/= 1 <-> ( 1 x. Y ) < 0 ) )
109	108	adantr 481	. . . . . 6 |- ( ( Y e. { -u 1 , 1 } /\ X = 1 ) -> ( Y =/= 1 <-> ( 1 x. Y ) < 0 ) )
110		neeq2 2857	. . . . . . . 8 |- ( X = 1 -> ( Y =/= X <-> Y =/= 1 ) )
111		oveq1 6657	. . . . . . . . 9 |- ( X = 1 -> ( X x. Y ) = ( 1 x. Y ) )
112	111	breq1d 4663	. . . . . . . 8 |- ( X = 1 -> ( ( X x. Y ) < 0 <-> ( 1 x. Y ) < 0 ) )
113	110, 112	bibi12d 335	. . . . . . 7 |- ( X = 1 -> ( ( Y =/= X <-> ( X x. Y ) < 0 ) <-> ( Y =/= 1 <-> ( 1 x. Y ) < 0 ) ) )
114	113	adantl 482	. . . . . 6 |- ( ( Y e. { -u 1 , 1 } /\ X = 1 ) -> ( ( Y =/= X <-> ( X x. Y ) < 0 ) <-> ( Y =/= 1 <-> ( 1 x. Y ) < 0 ) ) )
115	109, 114	mpbird 247	. . . . 5 |- ( ( Y e. { -u 1 , 1 } /\ X = 1 ) -> ( Y =/= X <-> ( X x. Y ) < 0 ) )
116	45, 115	sylan 488	. . . 4 |- ( ( ( ( X e. { -u 1 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) /\ -. Y = 0 ) /\ X = 1 ) -> ( Y =/= X <-> ( X x. Y ) < 0 ) )
117		elpri 4197	. . . . 5 |- ( X e. { -u 1 , 1 } -> ( X = -u 1 \/ X = 1 ) )
118	117	ad2antrr 762	. . . 4 |- ( ( ( X e. { -u 1 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) /\ -. Y = 0 ) -> ( X = -u 1 \/ X = 1 ) )
119	85, 116, 118	mpjaodan 827	. . 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 4123	. 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 268	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 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   \ cdif 3571   u. cun 3572   C_ wss 3574   ifcif 4086   {csn 4177   {cpr 4179   {ctp 4181   <.cop 4183   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   CCcc 9934   0cc0 9936   1c1 9937   x. cmul 9941   < clt 10074   <_ cle 10075   -ucneg 10267   ndxcnx 15854   Basecbs 15857   +g cplusg 15941

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269

This theorem is referenced by:  signsvfn  30659