Theorem signswch 29522

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 3962	. . . . . 6 |- { -u 1 , 1 } = ( { -u 1 } u. { 1 } )
2		snsstp1 4114	. . . . . . 7 |- { -u 1 } C_ { -u 1 , 0 , 1 }
3		snsstp3 4116	. . . . . . 7 |- { 1 } C_ { -u 1 , 0 , 1 }
4	2, 3	unssi 3600	. . . . . 6 |- ( { -u 1 } u. { 1 } ) C_ { -u 1 , 0 , 1 }
5	1, 4	eqsstri 3448	. . . . 5 |- { -u 1 , 1 } C_ { -u 1 , 0 , 1 }
6	5	sseli 3414	. . . 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 29513	. . . 4 |- ( ( X e. { -u 1 , 0 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) -> ( X .+^ Y ) = if ( Y = 0 , X , Y ) )
9	6, 8	sylan 479	. . 3 |- ( ( X e. { -u 1 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) -> ( X .+^ Y ) = if ( Y = 0 , X , Y ) )
10	9	neeq1d 2702	. 2 |- ( ( X e. { -u 1 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) -> ( ( X .+^ Y ) =/= X <-> if ( Y = 0 , X , Y ) =/= X ) )
11		neeq1 2705	. . . 4 |- ( X = if ( Y = 0 , X , Y ) -> ( X =/= X <-> if ( Y = 0 , X , Y ) =/= X ) )
12	11	bibi1d 326	. . 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 2705	. . . 4 |- ( Y = if ( Y = 0 , X , Y ) -> ( Y =/= X <-> if ( Y = 0 , X , Y ) =/= X ) )
14	13	bibi1d 326	. . 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 2652	. . . . 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 9661	. . . . . 6 |- 0 e. RR
18	17	ltnri 9761	. . . . 5 |- -. 0 < 0
19		simpr 468	. . . . . . . 8 |- ( ( ( X e. { -u 1 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) /\ Y = 0 ) -> Y = 0 )
20	19	oveq2d 6324	. . . . . . 7 |- ( ( ( X e. { -u 1 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) /\ Y = 0 ) -> ( X x. Y ) = ( X x. 0 ) )
21		neg1cn 10735	. . . . . . . . . 10 |- -u 1 e. CC
22		ax-1cn 9615	. . . . . . . . . 10 |- 1 e. CC
23		prssi 4119	. . . . . . . . . 10 |- ( ( -u 1 e. CC /\ 1 e. CC ) -> { -u 1 , 1 } C_ CC )
24	21, 22, 23	mp2an 686	. . . . . . . . 9 |- { -u 1 , 1 } C_ CC
25		simpll 768	. . . . . . . . 9 |- ( ( ( X e. { -u 1 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) /\ Y = 0 ) -> X e. { -u 1 , 1 } )
26	24, 25	sseldi 3416	. . . . . . . 8 |- ( ( ( X e. { -u 1 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) /\ Y = 0 ) -> X e. CC )
27	26	mul01d 9850	. . . . . . 7 |- ( ( ( X e. { -u 1 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) /\ Y = 0 ) -> ( X x. 0 ) = 0 )
28	20, 27	eqtrd 2505	. . . . . 6 |- ( ( ( X e. { -u 1 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) /\ Y = 0 ) -> ( X x. Y ) = 0 )
29	28	breq1d 4405	. . . . 5 |- ( ( ( X e. { -u 1 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) /\ Y = 0 ) -> ( ( X x. Y ) < 0 <-> 0 < 0 ) )
30	18, 29	mtbiri 310	. . . 4 |- ( ( ( X e. { -u 1 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) /\ Y = 0 ) -> -. ( X x. Y ) < 0 )
31	16, 30	2falsed 358	. . 3 |- ( ( ( X e. { -u 1 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) /\ Y = 0 ) -> ( X =/= X <-> ( X x. Y ) < 0 ) )
32		simplr 770	. . . . . . . 8 |- ( ( ( X e. { -u 1 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) /\ -. Y = 0 ) -> Y e. { -u 1 , 0 , 1 } )
33		tpcomb 4060	. . . . . . . 8 |- { -u 1 , 0 , 1 } = { -u 1 , 1 , 0 }
34	32, 33	syl6eleq 2559	. . . . . . 7 |- ( ( ( X e. { -u 1 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) /\ -. Y = 0 ) -> Y e. { -u 1 , 1 , 0 } )
35		simpr 468	. . . . . . . 8 |- ( ( ( X e. { -u 1 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) /\ -. Y = 0 ) -> -. Y = 0 )
36	35	neqned 2650	. . . . . . 7 |- ( ( ( X e. { -u 1 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) /\ -. Y = 0 ) -> Y =/= 0 )
37	34, 36	jca 541	. . . . . 6 |- ( ( ( X e. { -u 1 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) /\ -. Y = 0 ) -> ( Y e. { -u 1 , 1 , 0 } /\ Y =/= 0 ) )
38		eldifsn 4088	. . . . . . 7 |- ( Y e. ( { -u 1 , 1 , 0 } \ { 0 } ) <-> ( Y e. { -u 1 , 1 , 0 } /\ Y =/= 0 ) )
39		neg1ne0 10737	. . . . . . . . 9 |- -u 1 =/= 0
40		ax-1ne0 9626	. . . . . . . . 9 |- 1 =/= 0
41		diftpsn3 4101	. . . . . . . . 9 |- ( ( -u 1 =/= 0 /\ 1 =/= 0 ) -> ( { -u 1 , 1 , 0 } \ { 0 } ) = { -u 1 , 1 } )
42	39, 40, 41	mp2an 686	. . . . . . . 8 |- ( { -u 1 , 1 , 0 } \ { 0 } ) = { -u 1 , 1 }
43	42	eleq2i 2541	. . . . . . 7 |- ( Y e. ( { -u 1 , 1 , 0 } \ { 0 } ) <-> Y e. { -u 1 , 1 } )
44	38, 43	bitr3i 259	. . . . . 6 |- ( ( Y e. { -u 1 , 1 , 0 } /\ Y =/= 0 ) <-> Y e. { -u 1 , 1 } )
45	37, 44	sylib 201	. . . . 5 |- ( ( ( X e. { -u 1 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) /\ -. Y = 0 ) -> Y e. { -u 1 , 1 } )
46		neirr 2652	. . . . . . . . . . 11 |- -. -u 1 =/= -u 1
47		0le1 10158	. . . . . . . . . . . . 13 |- 0 <_ 1
48		1re 9660	. . . . . . . . . . . . . 14 |- 1 e. RR
49	17, 48	lenlti 9772	. . . . . . . . . . . . 13 |- ( 0 <_ 1 <-> -. 1 < 0 )
50	47, 49	mpbi 213	. . . . . . . . . . . 12 |- -. 1 < 0
51		neg1mulneg1e1 10850	. . . . . . . . . . . . 13 |- ( -u 1 x. -u 1 ) = 1
52	51	breq1i 4402	. . . . . . . . . . . 12 |- ( ( -u 1 x. -u 1 ) < 0 <-> 1 < 0 )
53	50, 52	mtbir 306	. . . . . . . . . . 11 |- -. ( -u 1 x. -u 1 ) < 0
54	46, 53	2false 357	. . . . . . . . . 10 |- ( -u 1 =/= -u 1 <-> ( -u 1 x. -u 1 ) < 0 )
55		neeq1 2705	. . . . . . . . . . 11 |- ( Y = -u 1 -> ( Y =/= -u 1 <-> -u 1 =/= -u 1 ) )
56		oveq2 6316	. . . . . . . . . . . 12 |- ( Y = -u 1 -> ( -u 1 x. Y ) = ( -u 1 x. -u 1 ) )
57	56	breq1d 4405	. . . . . . . . . . 11 |- ( Y = -u 1 -> ( ( -u 1 x. Y ) < 0 <-> ( -u 1 x. -u 1 ) < 0 ) )
58	55, 57	bibi12d 328	. . . . . . . . . 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 241	. . . . . . . . 9 |- ( Y = -u 1 -> ( Y =/= -u 1 <-> ( -u 1 x. Y ) < 0 ) )
60	59	adantl 473	. . . . . . . 8 |- ( ( Y e. { -u 1 , 1 } /\ Y = -u 1 ) -> ( Y =/= -u 1 <-> ( -u 1 x. Y ) < 0 ) )
61		neg1rr 10736	. . . . . . . . . . . 12 |- -u 1 e. RR
62		neg1lt0 10738	. . . . . . . . . . . . 13 |- -u 1 < 0
63		0lt1 10157	. . . . . . . . . . . . 13 |- 0 < 1
64	61, 17, 48	lttri 9778	. . . . . . . . . . . . 13 |- ( ( -u 1 < 0 /\ 0 < 1 ) -> -u 1 < 1 )
65	62, 63, 64	mp2an 686	. . . . . . . . . . . 12 |- -u 1 < 1
66	61, 65	gtneii 9764	. . . . . . . . . . 11 |- 1 =/= -u 1
67	21	mulid1i 9663	. . . . . . . . . . . 12 |- ( -u 1 x. 1 ) = -u 1
68	67, 62	eqbrtri 4415	. . . . . . . . . . 11 |- ( -u 1 x. 1 ) < 0
69	66, 68	2th 247	. . . . . . . . . 10 |- ( 1 =/= -u 1 <-> ( -u 1 x. 1 ) < 0 )
70		neeq1 2705	. . . . . . . . . . 11 |- ( Y = 1 -> ( Y =/= -u 1 <-> 1 =/= -u 1 ) )
71		oveq2 6316	. . . . . . . . . . . 12 |- ( Y = 1 -> ( -u 1 x. Y ) = ( -u 1 x. 1 ) )
72	71	breq1d 4405	. . . . . . . . . . 11 |- ( Y = 1 -> ( ( -u 1 x. Y ) < 0 <-> ( -u 1 x. 1 ) < 0 ) )
73	70, 72	bibi12d 328	. . . . . . . . . 10 |- ( Y = 1 -> ( ( Y =/= -u 1 <-> ( -u 1 x. Y ) < 0 ) <-> ( 1 =/= -u 1 <-> ( -u 1 x. 1 ) < 0 ) ) )
74	69, 73	mpbiri 241	. . . . . . . . 9 |- ( Y = 1 -> ( Y =/= -u 1 <-> ( -u 1 x. Y ) < 0 ) )
75	74	adantl 473	. . . . . . . 8 |- ( ( Y e. { -u 1 , 1 } /\ Y = 1 ) -> ( Y =/= -u 1 <-> ( -u 1 x. Y ) < 0 ) )
76		elpri 3976	. . . . . . . 8 |- ( Y e. { -u 1 , 1 } -> ( Y = -u 1 \/ Y = 1 ) )
77	60, 75, 76	mpjaodan 803	. . . . . . 7 |- ( Y e. { -u 1 , 1 } -> ( Y =/= -u 1 <-> ( -u 1 x. Y ) < 0 ) )
78	77	adantr 472	. . . . . 6 |- ( ( Y e. { -u 1 , 1 } /\ X = -u 1 ) -> ( Y =/= -u 1 <-> ( -u 1 x. Y ) < 0 ) )
79		neeq2 2706	. . . . . . . 8 |- ( X = -u 1 -> ( Y =/= X <-> Y =/= -u 1 ) )
80		oveq1 6315	. . . . . . . . 9 |- ( X = -u 1 -> ( X x. Y ) = ( -u 1 x. Y ) )
81	80	breq1d 4405	. . . . . . . 8 |- ( X = -u 1 -> ( ( X x. Y ) < 0 <-> ( -u 1 x. Y ) < 0 ) )
82	79, 81	bibi12d 328	. . . . . . 7 |- ( X = -u 1 -> ( ( Y =/= X <-> ( X x. Y ) < 0 ) <-> ( Y =/= -u 1 <-> ( -u 1 x. Y ) < 0 ) ) )
83	82	adantl 473	. . . . . 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 240	. . . . 5 |- ( ( Y e. { -u 1 , 1 } /\ X = -u 1 ) -> ( Y =/= X <-> ( X x. Y ) < 0 ) )
85	45, 84	sylan 479	. . . 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 2697	. . . . . . . . . . 11 |- -u 1 =/= 1
87	21, 22	mulcomi 9667	. . . . . . . . . . . . 13 |- ( -u 1 x. 1 ) = ( 1 x. -u 1 )
88	87	breq1i 4402	. . . . . . . . . . . 12 |- ( ( -u 1 x. 1 ) < 0 <-> ( 1 x. -u 1 ) < 0 )
89	68, 88	mpbi 213	. . . . . . . . . . 11 |- ( 1 x. -u 1 ) < 0
90	86, 89	2th 247	. . . . . . . . . 10 |- ( -u 1 =/= 1 <-> ( 1 x. -u 1 ) < 0 )
91		neeq1 2705	. . . . . . . . . . 11 |- ( Y = -u 1 -> ( Y =/= 1 <-> -u 1 =/= 1 ) )
92		oveq2 6316	. . . . . . . . . . . 12 |- ( Y = -u 1 -> ( 1 x. Y ) = ( 1 x. -u 1 ) )
93	92	breq1d 4405	. . . . . . . . . . 11 |- ( Y = -u 1 -> ( ( 1 x. Y ) < 0 <-> ( 1 x. -u 1 ) < 0 ) )
94	91, 93	bibi12d 328	. . . . . . . . . 10 |- ( Y = -u 1 -> ( ( Y =/= 1 <-> ( 1 x. Y ) < 0 ) <-> ( -u 1 =/= 1 <-> ( 1 x. -u 1 ) < 0 ) ) )
95	90, 94	mpbiri 241	. . . . . . . . 9 |- ( Y = -u 1 -> ( Y =/= 1 <-> ( 1 x. Y ) < 0 ) )
96	95	adantl 473	. . . . . . . 8 |- ( ( Y e. { -u 1 , 1 } /\ Y = -u 1 ) -> ( Y =/= 1 <-> ( 1 x. Y ) < 0 ) )
97		neirr 2652	. . . . . . . . . . 11 |- -. 1 =/= 1
98	22	mulid1i 9663	. . . . . . . . . . . . 13 |- ( 1 x. 1 ) = 1
99	98	breq1i 4402	. . . . . . . . . . . 12 |- ( ( 1 x. 1 ) < 0 <-> 1 < 0 )
100	50, 99	mtbir 306	. . . . . . . . . . 11 |- -. ( 1 x. 1 ) < 0
101	97, 100	2false 357	. . . . . . . . . 10 |- ( 1 =/= 1 <-> ( 1 x. 1 ) < 0 )
102		neeq1 2705	. . . . . . . . . . 11 |- ( Y = 1 -> ( Y =/= 1 <-> 1 =/= 1 ) )
103		oveq2 6316	. . . . . . . . . . . 12 |- ( Y = 1 -> ( 1 x. Y ) = ( 1 x. 1 ) )
104	103	breq1d 4405	. . . . . . . . . . 11 |- ( Y = 1 -> ( ( 1 x. Y ) < 0 <-> ( 1 x. 1 ) < 0 ) )
105	102, 104	bibi12d 328	. . . . . . . . . 10 |- ( Y = 1 -> ( ( Y =/= 1 <-> ( 1 x. Y ) < 0 ) <-> ( 1 =/= 1 <-> ( 1 x. 1 ) < 0 ) ) )
106	101, 105	mpbiri 241	. . . . . . . . 9 |- ( Y = 1 -> ( Y =/= 1 <-> ( 1 x. Y ) < 0 ) )
107	106	adantl 473	. . . . . . . 8 |- ( ( Y e. { -u 1 , 1 } /\ Y = 1 ) -> ( Y =/= 1 <-> ( 1 x. Y ) < 0 ) )
108	96, 107, 76	mpjaodan 803	. . . . . . 7 |- ( Y e. { -u 1 , 1 } -> ( Y =/= 1 <-> ( 1 x. Y ) < 0 ) )
109	108	adantr 472	. . . . . 6 |- ( ( Y e. { -u 1 , 1 } /\ X = 1 ) -> ( Y =/= 1 <-> ( 1 x. Y ) < 0 ) )
110		neeq2 2706	. . . . . . . 8 |- ( X = 1 -> ( Y =/= X <-> Y =/= 1 ) )
111		oveq1 6315	. . . . . . . . 9 |- ( X = 1 -> ( X x. Y ) = ( 1 x. Y ) )
112	111	breq1d 4405	. . . . . . . 8 |- ( X = 1 -> ( ( X x. Y ) < 0 <-> ( 1 x. Y ) < 0 ) )
113	110, 112	bibi12d 328	. . . . . . 7 |- ( X = 1 -> ( ( Y =/= X <-> ( X x. Y ) < 0 ) <-> ( Y =/= 1 <-> ( 1 x. Y ) < 0 ) ) )
114	113	adantl 473	. . . . . 6 |- ( ( Y e. { -u 1 , 1 } /\ X = 1 ) -> ( ( Y =/= X <-> ( X x. Y ) < 0 ) <-> ( Y =/= 1 <-> ( 1 x. Y ) < 0 ) ) )
115	109, 114	mpbird 240	. . . . 5 |- ( ( Y e. { -u 1 , 1 } /\ X = 1 ) -> ( Y =/= X <-> ( X x. Y ) < 0 ) )
116	45, 115	sylan 479	. . . 4 |- ( ( ( ( X e. { -u 1 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) /\ -. Y = 0 ) /\ X = 1 ) -> ( Y =/= X <-> ( X x. Y ) < 0 ) )
117		elpri 3976	. . . . 5 |- ( X e. { -u 1 , 1 } -> ( X = -u 1 \/ X = 1 ) )
118	117	ad2antrr 740	. . . 4 |- ( ( ( X e. { -u 1 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) /\ -. Y = 0 ) -> ( X = -u 1 \/ X = 1 ) )
119	85, 116, 118	mpjaodan 803	. . 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 3907	. 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 261	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 189   \/ wo 375   /\ wa 376   = wceq 1452   e. wcel 1904   =/= wne 2641   \ cdif 3387   u. cun 3388   C_ wss 3390   ifcif 3872   {csn 3959   {cpr 3961   {ctp 3963   <.cop 3965   class class class wbr 4395   ` cfv 5589  (class class class)co 6308   |-> cmpt2 6310   CCcc 9555   0cc0 9557   1c1 9558   x. cmul 9562   < clt 9693   <_ cle 9694   -ucneg 9881   ndxcnx 15196   Basecbs 15199   +g cplusg 15268

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-po 4760  df-so 4761  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883

This theorem is referenced by:  signsvfn  29543