Theorem signswch 28155

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 4030	. . . . . . 7 |- { -u 1 , 1 } = ( { -u 1 } u. { 1 } )
2		snsstp1 4178	. . . . . . . 8 |- { -u 1 } C_ { -u 1 , 0 , 1 }
3		snsstp3 4180	. . . . . . . 8 |- { 1 } C_ { -u 1 , 0 , 1 }
4	2, 3	unssi 3679	. . . . . . 7 |- ( { -u 1 } u. { 1 } ) C_ { -u 1 , 0 , 1 }
5	1, 4	eqsstri 3534	. . . . . 6 |- { -u 1 , 1 } C_ { -u 1 , 0 , 1 }
6	5	sseli 3500	. . . . 5 |- ( X e. { -u 1 , 1 } -> X e. { -u 1 , 0 , 1 } )
7	6	adantr 465	. . . 4 |- ( ( X e. { -u 1 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) -> X e. { -u 1 , 0 , 1 } )
8		signsw.p	. . . . 5 |- .+^ = ( a e. { -u 1 , 0 , 1 } , b e. { -u 1 , 0 , 1 } |-> if ( b = 0 , a , b ) )
9	8	signspval 28146	. . . 4 |- ( ( X e. { -u 1 , 0 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) -> ( X .+^ Y ) = if ( Y = 0 , X , Y ) )
10	7, 9	sylancom 667	. . 3 |- ( ( X e. { -u 1 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) -> ( X .+^ Y ) = if ( Y = 0 , X , Y ) )
11	10	neeq1d 2744	. 2 |- ( ( X e. { -u 1 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) -> ( ( X .+^ Y ) =/= X <-> if ( Y = 0 , X , Y ) =/= X ) )
12		neeq1 2748	. . . 4 |- ( X = if ( Y = 0 , X , Y ) -> ( X =/= X <-> if ( Y = 0 , X , Y ) =/= X ) )
13	12	bibi1d 319	. . 3 |- ( X = if ( Y = 0 , X , Y ) -> ( ( X =/= X <-> ( X x. Y ) < 0 ) <-> ( if ( Y = 0 , X , Y ) =/= X <-> ( X x. Y ) < 0 ) ) )
14		neeq1 2748	. . . 4 |- ( Y = if ( Y = 0 , X , Y ) -> ( Y =/= X <-> if ( Y = 0 , X , Y ) =/= X ) )
15	14	bibi1d 319	. . 3 |- ( Y = if ( Y = 0 , X , Y ) -> ( ( Y =/= X <-> ( X x. Y ) < 0 ) <-> ( if ( Y = 0 , X , Y ) =/= X <-> ( X x. Y ) < 0 ) ) )
16		neirr 2671	. . . . 5 |- -. X =/= X
17	16	a1i 11	. . . 4 |- ( ( ( X e. { -u 1 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) /\ Y = 0 ) -> -. X =/= X )
18		0re 9592	. . . . . 6 |- 0 e. RR
19	18	ltnri 9689	. . . . 5 |- -. 0 < 0
20		simpr 461	. . . . . . . 8 |- ( ( ( X e. { -u 1 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) /\ Y = 0 ) -> Y = 0 )
21	20	oveq2d 6298	. . . . . . 7 |- ( ( ( X e. { -u 1 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) /\ Y = 0 ) -> ( X x. Y ) = ( X x. 0 ) )
22		neg1cn 10635	. . . . . . . . . 10 |- -u 1 e. CC
23		ax-1cn 9546	. . . . . . . . . 10 |- 1 e. CC
24		prssi 4183	. . . . . . . . . 10 |- ( ( -u 1 e. CC /\ 1 e. CC ) -> { -u 1 , 1 } C_ CC )
25	22, 23, 24	mp2an 672	. . . . . . . . 9 |- { -u 1 , 1 } C_ CC
26		simpll 753	. . . . . . . . 9 |- ( ( ( X e. { -u 1 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) /\ Y = 0 ) -> X e. { -u 1 , 1 } )
27	25, 26	sseldi 3502	. . . . . . . 8 |- ( ( ( X e. { -u 1 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) /\ Y = 0 ) -> X e. CC )
28	27	mul01d 9774	. . . . . . 7 |- ( ( ( X e. { -u 1 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) /\ Y = 0 ) -> ( X x. 0 ) = 0 )
29	21, 28	eqtrd 2508	. . . . . 6 |- ( ( ( X e. { -u 1 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) /\ Y = 0 ) -> ( X x. Y ) = 0 )
30	29	breq1d 4457	. . . . 5 |- ( ( ( X e. { -u 1 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) /\ Y = 0 ) -> ( ( X x. Y ) < 0 <-> 0 < 0 ) )
31	19, 30	mtbiri 303	. . . 4 |- ( ( ( X e. { -u 1 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) /\ Y = 0 ) -> -. ( X x. Y ) < 0 )
32	17, 31	2falsed 351	. . 3 |- ( ( ( X e. { -u 1 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) /\ Y = 0 ) -> ( X =/= X <-> ( X x. Y ) < 0 ) )
33		simplr 754	. . . . . . . . 9 |- ( ( ( X e. { -u 1 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) /\ -. Y = 0 ) -> Y e. { -u 1 , 0 , 1 } )
34		tpcomb 4124	. . . . . . . . 9 |- { -u 1 , 0 , 1 } = { -u 1 , 1 , 0 }
35	33, 34	syl6eleq 2565	. . . . . . . 8 |- ( ( ( X e. { -u 1 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) /\ -. Y = 0 ) -> Y e. { -u 1 , 1 , 0 } )
36		simpr 461	. . . . . . . . 9 |- ( ( ( X e. { -u 1 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) /\ -. Y = 0 ) -> -. Y = 0 )
37	36	neqned 2670	. . . . . . . 8 |- ( ( ( X e. { -u 1 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) /\ -. Y = 0 ) -> Y =/= 0 )
38	35, 37	jca 532	. . . . . . 7 |- ( ( ( X e. { -u 1 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) /\ -. Y = 0 ) -> ( Y e. { -u 1 , 1 , 0 } /\ Y =/= 0 ) )
39		eldifsn 4152	. . . . . . . 8 |- ( Y e. ( { -u 1 , 1 , 0 } \ { 0 } ) <-> ( Y e. { -u 1 , 1 , 0 } /\ Y =/= 0 ) )
40		neg1ne0 10637	. . . . . . . . . 10 |- -u 1 =/= 0
41		ax-1ne0 9557	. . . . . . . . . 10 |- 1 =/= 0
42		diftpsn3 4165	. . . . . . . . . 10 |- ( ( -u 1 =/= 0 /\ 1 =/= 0 ) -> ( { -u 1 , 1 , 0 } \ { 0 } ) = { -u 1 , 1 } )
43	40, 41, 42	mp2an 672	. . . . . . . . 9 |- ( { -u 1 , 1 , 0 } \ { 0 } ) = { -u 1 , 1 }
44	43	eleq2i 2545	. . . . . . . 8 |- ( Y e. ( { -u 1 , 1 , 0 } \ { 0 } ) <-> Y e. { -u 1 , 1 } )
45	39, 44	bitr3i 251	. . . . . . 7 |- ( ( Y e. { -u 1 , 1 , 0 } /\ Y =/= 0 ) <-> Y e. { -u 1 , 1 } )
46	38, 45	sylib 196	. . . . . 6 |- ( ( ( X e. { -u 1 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) /\ -. Y = 0 ) -> Y e. { -u 1 , 1 } )
47	46	adantr 465	. . . . 5 |- ( ( ( ( X e. { -u 1 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) /\ -. Y = 0 ) /\ X = -u 1 ) -> Y e. { -u 1 , 1 } )
48		neirr 2671	. . . . . . . . . . 11 |- -. -u 1 =/= -u 1
49		0le1 10072	. . . . . . . . . . . . 13 |- 0 <_ 1
50		1re 9591	. . . . . . . . . . . . . 14 |- 1 e. RR
51	18, 50	lenlti 9700	. . . . . . . . . . . . 13 |- ( 0 <_ 1 <-> -. 1 < 0 )
52	49, 51	mpbi 208	. . . . . . . . . . . 12 |- -. 1 < 0
53		neg1mulneg1e1 10749	. . . . . . . . . . . . 13 |- ( -u 1 x. -u 1 ) = 1
54	53	breq1i 4454	. . . . . . . . . . . 12 |- ( ( -u 1 x. -u 1 ) < 0 <-> 1 < 0 )
55	52, 54	mtbir 299	. . . . . . . . . . 11 |- -. ( -u 1 x. -u 1 ) < 0
56	48, 55	2false 350	. . . . . . . . . 10 |- ( -u 1 =/= -u 1 <-> ( -u 1 x. -u 1 ) < 0 )
57		neeq1 2748	. . . . . . . . . . 11 |- ( Y = -u 1 -> ( Y =/= -u 1 <-> -u 1 =/= -u 1 ) )
58		oveq2 6290	. . . . . . . . . . . 12 |- ( Y = -u 1 -> ( -u 1 x. Y ) = ( -u 1 x. -u 1 ) )
59	58	breq1d 4457	. . . . . . . . . . 11 |- ( Y = -u 1 -> ( ( -u 1 x. Y ) < 0 <-> ( -u 1 x. -u 1 ) < 0 ) )
60	57, 59	bibi12d 321	. . . . . . . . . 10 |- ( Y = -u 1 -> ( ( Y =/= -u 1 <-> ( -u 1 x. Y ) < 0 ) <-> ( -u 1 =/= -u 1 <-> ( -u 1 x. -u 1 ) < 0 ) ) )
61	56, 60	mpbiri 233	. . . . . . . . 9 |- ( Y = -u 1 -> ( Y =/= -u 1 <-> ( -u 1 x. Y ) < 0 ) )
62	61	adantl 466	. . . . . . . 8 |- ( ( Y e. { -u 1 , 1 } /\ Y = -u 1 ) -> ( Y =/= -u 1 <-> ( -u 1 x. Y ) < 0 ) )
63		neg1lt0 10638	. . . . . . . . . . . . 13 |- -u 1 < 0
64		0lt1 10071	. . . . . . . . . . . . 13 |- 0 < 1
65		neg1rr 10636	. . . . . . . . . . . . . 14 |- -u 1 e. RR
66	65, 18, 50	lttri 9706	. . . . . . . . . . . . 13 |- ( ( -u 1 < 0 /\ 0 < 1 ) -> -u 1 < 1 )
67	63, 64, 66	mp2an 672	. . . . . . . . . . . 12 |- -u 1 < 1
68	65, 50	ltnei 9704	. . . . . . . . . . . 12 |- ( -u 1 < 1 -> 1 =/= -u 1 )
69	67, 68	ax-mp 5	. . . . . . . . . . 11 |- 1 =/= -u 1
70	22	mulid1i 9594	. . . . . . . . . . . 12 |- ( -u 1 x. 1 ) = -u 1
71	70, 63	eqbrtri 4466	. . . . . . . . . . 11 |- ( -u 1 x. 1 ) < 0
72	69, 71	2th 239	. . . . . . . . . 10 |- ( 1 =/= -u 1 <-> ( -u 1 x. 1 ) < 0 )
73		neeq1 2748	. . . . . . . . . . 11 |- ( Y = 1 -> ( Y =/= -u 1 <-> 1 =/= -u 1 ) )
74		oveq2 6290	. . . . . . . . . . . 12 |- ( Y = 1 -> ( -u 1 x. Y ) = ( -u 1 x. 1 ) )
75	74	breq1d 4457	. . . . . . . . . . 11 |- ( Y = 1 -> ( ( -u 1 x. Y ) < 0 <-> ( -u 1 x. 1 ) < 0 ) )
76	73, 75	bibi12d 321	. . . . . . . . . 10 |- ( Y = 1 -> ( ( Y =/= -u 1 <-> ( -u 1 x. Y ) < 0 ) <-> ( 1 =/= -u 1 <-> ( -u 1 x. 1 ) < 0 ) ) )
77	72, 76	mpbiri 233	. . . . . . . . 9 |- ( Y = 1 -> ( Y =/= -u 1 <-> ( -u 1 x. Y ) < 0 ) )
78	77	adantl 466	. . . . . . . 8 |- ( ( Y e. { -u 1 , 1 } /\ Y = 1 ) -> ( Y =/= -u 1 <-> ( -u 1 x. Y ) < 0 ) )
79		elpri 4047	. . . . . . . 8 |- ( Y e. { -u 1 , 1 } -> ( Y = -u 1 \/ Y = 1 ) )
80	62, 78, 79	mpjaodan 784	. . . . . . 7 |- ( Y e. { -u 1 , 1 } -> ( Y =/= -u 1 <-> ( -u 1 x. Y ) < 0 ) )
81	80	adantr 465	. . . . . 6 |- ( ( Y e. { -u 1 , 1 } /\ X = -u 1 ) -> ( Y =/= -u 1 <-> ( -u 1 x. Y ) < 0 ) )
82		neeq2 2750	. . . . . . . 8 |- ( X = -u 1 -> ( Y =/= X <-> Y =/= -u 1 ) )
83		oveq1 6289	. . . . . . . . 9 |- ( X = -u 1 -> ( X x. Y ) = ( -u 1 x. Y ) )
84	83	breq1d 4457	. . . . . . . 8 |- ( X = -u 1 -> ( ( X x. Y ) < 0 <-> ( -u 1 x. Y ) < 0 ) )
85	82, 84	bibi12d 321	. . . . . . 7 |- ( X = -u 1 -> ( ( Y =/= X <-> ( X x. Y ) < 0 ) <-> ( Y =/= -u 1 <-> ( -u 1 x. Y ) < 0 ) ) )
86	85	adantl 466	. . . . . 6 |- ( ( Y e. { -u 1 , 1 } /\ X = -u 1 ) -> ( ( Y =/= X <-> ( X x. Y ) < 0 ) <-> ( Y =/= -u 1 <-> ( -u 1 x. Y ) < 0 ) ) )
87	81, 86	mpbird 232	. . . . 5 |- ( ( Y e. { -u 1 , 1 } /\ X = -u 1 ) -> ( Y =/= X <-> ( X x. Y ) < 0 ) )
88	47, 87	sylancom 667	. . . 4 |- ( ( ( ( X e. { -u 1 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) /\ -. Y = 0 ) /\ X = -u 1 ) -> ( Y =/= X <-> ( X x. Y ) < 0 ) )
89	46	adantr 465	. . . . 5 |- ( ( ( ( X e. { -u 1 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) /\ -. Y = 0 ) /\ X = 1 ) -> Y e. { -u 1 , 1 } )
90	69	necomi 2737	. . . . . . . . . . 11 |- -u 1 =/= 1
91	22, 23	mulcomi 9598	. . . . . . . . . . . . 13 |- ( -u 1 x. 1 ) = ( 1 x. -u 1 )
92	91	breq1i 4454	. . . . . . . . . . . 12 |- ( ( -u 1 x. 1 ) < 0 <-> ( 1 x. -u 1 ) < 0 )
93	71, 92	mpbi 208	. . . . . . . . . . 11 |- ( 1 x. -u 1 ) < 0
94	90, 93	2th 239	. . . . . . . . . 10 |- ( -u 1 =/= 1 <-> ( 1 x. -u 1 ) < 0 )
95		neeq1 2748	. . . . . . . . . . 11 |- ( Y = -u 1 -> ( Y =/= 1 <-> -u 1 =/= 1 ) )
96		oveq2 6290	. . . . . . . . . . . 12 |- ( Y = -u 1 -> ( 1 x. Y ) = ( 1 x. -u 1 ) )
97	96	breq1d 4457	. . . . . . . . . . 11 |- ( Y = -u 1 -> ( ( 1 x. Y ) < 0 <-> ( 1 x. -u 1 ) < 0 ) )
98	95, 97	bibi12d 321	. . . . . . . . . 10 |- ( Y = -u 1 -> ( ( Y =/= 1 <-> ( 1 x. Y ) < 0 ) <-> ( -u 1 =/= 1 <-> ( 1 x. -u 1 ) < 0 ) ) )
99	94, 98	mpbiri 233	. . . . . . . . 9 |- ( Y = -u 1 -> ( Y =/= 1 <-> ( 1 x. Y ) < 0 ) )
100	99	adantl 466	. . . . . . . 8 |- ( ( Y e. { -u 1 , 1 } /\ Y = -u 1 ) -> ( Y =/= 1 <-> ( 1 x. Y ) < 0 ) )
101		neirr 2671	. . . . . . . . . . 11 |- -. 1 =/= 1
102	23	mulid1i 9594	. . . . . . . . . . . . 13 |- ( 1 x. 1 ) = 1
103	102	breq1i 4454	. . . . . . . . . . . 12 |- ( ( 1 x. 1 ) < 0 <-> 1 < 0 )
104	52, 103	mtbir 299	. . . . . . . . . . 11 |- -. ( 1 x. 1 ) < 0
105	101, 104	2false 350	. . . . . . . . . 10 |- ( 1 =/= 1 <-> ( 1 x. 1 ) < 0 )
106		neeq1 2748	. . . . . . . . . . 11 |- ( Y = 1 -> ( Y =/= 1 <-> 1 =/= 1 ) )
107		oveq2 6290	. . . . . . . . . . . 12 |- ( Y = 1 -> ( 1 x. Y ) = ( 1 x. 1 ) )
108	107	breq1d 4457	. . . . . . . . . . 11 |- ( Y = 1 -> ( ( 1 x. Y ) < 0 <-> ( 1 x. 1 ) < 0 ) )
109	106, 108	bibi12d 321	. . . . . . . . . 10 |- ( Y = 1 -> ( ( Y =/= 1 <-> ( 1 x. Y ) < 0 ) <-> ( 1 =/= 1 <-> ( 1 x. 1 ) < 0 ) ) )
110	105, 109	mpbiri 233	. . . . . . . . 9 |- ( Y = 1 -> ( Y =/= 1 <-> ( 1 x. Y ) < 0 ) )
111	110	adantl 466	. . . . . . . 8 |- ( ( Y e. { -u 1 , 1 } /\ Y = 1 ) -> ( Y =/= 1 <-> ( 1 x. Y ) < 0 ) )
112	100, 111, 79	mpjaodan 784	. . . . . . 7 |- ( Y e. { -u 1 , 1 } -> ( Y =/= 1 <-> ( 1 x. Y ) < 0 ) )
113	112	adantr 465	. . . . . 6 |- ( ( Y e. { -u 1 , 1 } /\ X = 1 ) -> ( Y =/= 1 <-> ( 1 x. Y ) < 0 ) )
114		neeq2 2750	. . . . . . . 8 |- ( X = 1 -> ( Y =/= X <-> Y =/= 1 ) )
115		oveq1 6289	. . . . . . . . 9 |- ( X = 1 -> ( X x. Y ) = ( 1 x. Y ) )
116	115	breq1d 4457	. . . . . . . 8 |- ( X = 1 -> ( ( X x. Y ) < 0 <-> ( 1 x. Y ) < 0 ) )
117	114, 116	bibi12d 321	. . . . . . 7 |- ( X = 1 -> ( ( Y =/= X <-> ( X x. Y ) < 0 ) <-> ( Y =/= 1 <-> ( 1 x. Y ) < 0 ) ) )
118	117	adantl 466	. . . . . 6 |- ( ( Y e. { -u 1 , 1 } /\ X = 1 ) -> ( ( Y =/= X <-> ( X x. Y ) < 0 ) <-> ( Y =/= 1 <-> ( 1 x. Y ) < 0 ) ) )
119	113, 118	mpbird 232	. . . . 5 |- ( ( Y e. { -u 1 , 1 } /\ X = 1 ) -> ( Y =/= X <-> ( X x. Y ) < 0 ) )
120	89, 119	sylancom 667	. . . 4 |- ( ( ( ( X e. { -u 1 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) /\ -. Y = 0 ) /\ X = 1 ) -> ( Y =/= X <-> ( X x. Y ) < 0 ) )
121		elpri 4047	. . . . 5 |- ( X e. { -u 1 , 1 } -> ( X = -u 1 \/ X = 1 ) )
122	121	ad2antrr 725	. . . 4 |- ( ( ( X e. { -u 1 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) /\ -. Y = 0 ) -> ( X = -u 1 \/ X = 1 ) )
123	88, 120, 122	mpjaodan 784	. . 3 |- ( ( ( X e. { -u 1 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) /\ -. Y = 0 ) -> ( Y =/= X <-> ( X x. Y ) < 0 ) )
124	13, 15, 32, 123	ifbothda 3974	. 2 |- ( ( X e. { -u 1 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) -> ( if ( Y = 0 , X , Y ) =/= X <-> ( X x. Y ) < 0 ) )
125	11, 124	bitrd 253	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 184   \/ wo 368   /\ wa 369   = wceq 1379   e. wcel 1767   =/= wne 2662   \ cdif 3473   u. cun 3474   C_ wss 3476   ifcif 3939   {csn 4027   {cpr 4029   {ctp 4031   <.cop 4033   class class class wbr 4447   ` cfv 5586  (class class class)co 6282   |-> cmpt2 6284   CCcc 9486   0cc0 9488   1c1 9489   x. cmul 9493   < clt 9624   <_ cle 9625   -ucneg 9802   ndxcnx 14480   Basecbs 14483   +g cplusg 14548

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804

This theorem is referenced by:  signsvfn  28176