Theorem signswch 27098

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 3980	. . . . . . 7 |- { -u 1 , 1 } = ( { -u 1 } u. { 1 } )
2		snsstp1 4124	. . . . . . . 8 |- { -u 1 } C_ { -u 1 , 0 , 1 }
3		snsstp3 4126	. . . . . . . 8 |- { 1 } C_ { -u 1 , 0 , 1 }
4	2, 3	unssi 3631	. . . . . . 7 |- ( { -u 1 } u. { 1 } ) C_ { -u 1 , 0 , 1 }
5	1, 4	eqsstri 3486	. . . . . 6 |- { -u 1 , 1 } C_ { -u 1 , 0 , 1 }
6	5	sseli 3452	. . . . 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 27089	. . . 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 2725	. 2 |- ( ( X e. { -u 1 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) -> ( ( X .+^ Y ) =/= X <-> if ( Y = 0 , X , Y ) =/= X ) )
12		neeq1 2729	. . . 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 2729	. . . 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 2653	. . . . 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 9489	. . . . . 6 |- 0 e. RR
19	18	ltnri 9586	. . . . 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 6208	. . . . . . 7 |- ( ( ( X e. { -u 1 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) /\ Y = 0 ) -> ( X x. Y ) = ( X x. 0 ) )
22		neg1cn 10528	. . . . . . . . . 10 |- -u 1 e. CC
23		ax-1cn 9443	. . . . . . . . . 10 |- 1 e. CC
24		prssi 4129	. . . . . . . . . 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 3454	. . . . . . . 8 |- ( ( ( X e. { -u 1 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) /\ Y = 0 ) -> X e. CC )
28	27	mul01d 9671	. . . . . . 7 |- ( ( ( X e. { -u 1 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) /\ Y = 0 ) -> ( X x. 0 ) = 0 )
29	21, 28	eqtrd 2492	. . . . . 6 |- ( ( ( X e. { -u 1 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) /\ Y = 0 ) -> ( X x. Y ) = 0 )
30	29	breq1d 4402	. . . . 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 4072	. . . . . . . . 9 |- { -u 1 , 0 , 1 } = { -u 1 , 1 , 0 }
35	33, 34	syl6eleq 2549	. . . . . . . 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	neneqad 2652	. . . . . . . 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 4100	. . . . . . . 8 |- ( Y e. ( { -u 1 , 1 , 0 } \ { 0 } ) <-> ( Y e. { -u 1 , 1 , 0 } /\ Y =/= 0 ) )
40		neg1ne0 10530	. . . . . . . . . 10 |- -u 1 =/= 0
41		ax-1ne0 9454	. . . . . . . . . 10 |- 1 =/= 0
42		diftpsn3 4112	. . . . . . . . . 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 2529	. . . . . . . 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 2653	. . . . . . . . . . 11 |- -. -u 1 =/= -u 1
49		0le1 9966	. . . . . . . . . . . . 13 |- 0 <_ 1
50		1re 9488	. . . . . . . . . . . . . 14 |- 1 e. RR
51	18, 50	lenlti 9597	. . . . . . . . . . . . 13 |- ( 0 <_ 1 <-> -. 1 < 0 )
52	49, 51	mpbi 208	. . . . . . . . . . . 12 |- -. 1 < 0
53		neg1mulneg1e1 10642	. . . . . . . . . . . . 13 |- ( -u 1 x. -u 1 ) = 1
54	53	breq1i 4399	. . . . . . . . . . . 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 2729	. . . . . . . . . . 11 |- ( Y = -u 1 -> ( Y =/= -u 1 <-> -u 1 =/= -u 1 ) )
58		oveq2 6200	. . . . . . . . . . . 12 |- ( Y = -u 1 -> ( -u 1 x. Y ) = ( -u 1 x. -u 1 ) )
59	58	breq1d 4402	. . . . . . . . . . 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 10531	. . . . . . . . . . . . 13 |- -u 1 < 0
64		0lt1 9965	. . . . . . . . . . . . 13 |- 0 < 1
65		neg1rr 10529	. . . . . . . . . . . . . 14 |- -u 1 e. RR
66	65, 18, 50	lttri 9603	. . . . . . . . . . . . 13 |- ( ( -u 1 < 0 /\ 0 < 1 ) -> -u 1 < 1 )
67	63, 64, 66	mp2an 672	. . . . . . . . . . . 12 |- -u 1 < 1
68	65, 50	ltnei 9601	. . . . . . . . . . . 12 |- ( -u 1 < 1 -> 1 =/= -u 1 )
69	67, 68	ax-mp 5	. . . . . . . . . . 11 |- 1 =/= -u 1
70	22	mulid1i 9491	. . . . . . . . . . . 12 |- ( -u 1 x. 1 ) = -u 1
71	70, 63	eqbrtri 4411	. . . . . . . . . . 11 |- ( -u 1 x. 1 ) < 0
72	69, 71	2th 239	. . . . . . . . . 10 |- ( 1 =/= -u 1 <-> ( -u 1 x. 1 ) < 0 )
73		neeq1 2729	. . . . . . . . . . 11 |- ( Y = 1 -> ( Y =/= -u 1 <-> 1 =/= -u 1 ) )
74		oveq2 6200	. . . . . . . . . . . 12 |- ( Y = 1 -> ( -u 1 x. Y ) = ( -u 1 x. 1 ) )
75	74	breq1d 4402	. . . . . . . . . . 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 3997	. . . . . . . 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 2731	. . . . . . . 8 |- ( X = -u 1 -> ( Y =/= X <-> Y =/= -u 1 ) )
83		oveq1 6199	. . . . . . . . 9 |- ( X = -u 1 -> ( X x. Y ) = ( -u 1 x. Y ) )
84	83	breq1d 4402	. . . . . . . 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 2718	. . . . . . . . . . 11 |- -u 1 =/= 1
91	22, 23	mulcomi 9495	. . . . . . . . . . . . 13 |- ( -u 1 x. 1 ) = ( 1 x. -u 1 )
92	91	breq1i 4399	. . . . . . . . . . . 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 2729	. . . . . . . . . . 11 |- ( Y = -u 1 -> ( Y =/= 1 <-> -u 1 =/= 1 ) )
96		oveq2 6200	. . . . . . . . . . . 12 |- ( Y = -u 1 -> ( 1 x. Y ) = ( 1 x. -u 1 ) )
97	96	breq1d 4402	. . . . . . . . . . 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 2653	. . . . . . . . . . 11 |- -. 1 =/= 1
102	23	mulid1i 9491	. . . . . . . . . . . . 13 |- ( 1 x. 1 ) = 1
103	102	breq1i 4399	. . . . . . . . . . . 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 2729	. . . . . . . . . . 11 |- ( Y = 1 -> ( Y =/= 1 <-> 1 =/= 1 ) )
107		oveq2 6200	. . . . . . . . . . . 12 |- ( Y = 1 -> ( 1 x. Y ) = ( 1 x. 1 ) )
108	107	breq1d 4402	. . . . . . . . . . 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 2731	. . . . . . . 8 |- ( X = 1 -> ( Y =/= X <-> Y =/= 1 ) )
115		oveq1 6199	. . . . . . . . 9 |- ( X = 1 -> ( X x. Y ) = ( 1 x. Y ) )
116	115	breq1d 4402	. . . . . . . 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 3997	. . . . 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 3924	. 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 1370   e. wcel 1758   =/= wne 2644   \ cdif 3425   u. cun 3426   C_ wss 3428   ifcif 3891   {csn 3977   {cpr 3979   {ctp 3981   <.cop 3983   class class class wbr 4392   ` cfv 5518  (class class class)co 6192   |-> cmpt2 6194   CCcc 9383   0cc0 9385   1c1 9386   x. cmul 9390   < clt 9521   <_ cle 9522   -ucneg 9699   ndxcnx 14275   Basecbs 14278   +g cplusg 14342

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-resscn 9442  ax-1cn 9443  ax-icn 9444  ax-addcl 9445  ax-addrcl 9446  ax-mulcl 9447  ax-mulrcl 9448  ax-mulcom 9449  ax-addass 9450  ax-mulass 9451  ax-distr 9452  ax-i2m1 9453  ax-1ne0 9454  ax-1rid 9455  ax-rnegex 9456  ax-rrecex 9457  ax-cnre 9458  ax-pre-lttri 9459  ax-pre-lttrn 9460  ax-pre-ltadd 9461  ax-pre-mulgt0 9462

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-po 4741  df-so 4742  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-er 7203  df-en 7413  df-dom 7414  df-sdom 7415  df-pnf 9523  df-mnf 9524  df-xr 9525  df-ltxr 9526  df-le 9527  df-sub 9700  df-neg 9701

This theorem is referenced by:  signsvfn  27119