Theorem signswmnd 30634

Description: W is a monoid structure on { -u 1 , 0 , 1 } which operation retains the right side, but skips zeroes. This will be used for skipping zeroes when counting sign changes. (Contributed by Thierry Arnoux, 9-Sep-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
signswmnd	|- W e. Mnd

Distinct variable group:   a, b, .+^

Allowed substitution hints:   W( a, b)

Proof of Theorem signswmnd

Dummy variables u e v w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		signsw.p	. . . . . 6 |- .+^ = ( a e. { -u 1 , 0 , 1 } , b e. { -u 1 , 0 , 1 } |-> if ( b = 0 , a , b ) )
2	1	signspval 30629	. . . . 5 |- ( ( u e. { -u 1 , 0 , 1 } /\ v e. { -u 1 , 0 , 1 } ) -> ( u .+^ v ) = if ( v = 0 , u , v ) )
3		ifcl 4130	. . . . 5 |- ( ( u e. { -u 1 , 0 , 1 } /\ v e. { -u 1 , 0 , 1 } ) -> if ( v = 0 , u , v ) e. { -u 1 , 0 , 1 } )
4	2, 3	eqeltrd 2701	. . . 4 |- ( ( u e. { -u 1 , 0 , 1 } /\ v e. { -u 1 , 0 , 1 } ) -> ( u .+^ v ) e. { -u 1 , 0 , 1 } )
5	1	signspval 30629	. . . . . . . . . . . . 13 |- ( ( ( u .+^ v ) e. { -u 1 , 0 , 1 } /\ w e. { -u 1 , 0 , 1 } ) -> ( ( u .+^ v ) .+^ w ) = if ( w = 0 , ( u .+^ v ) , w ) )
6	4, 5	stoic3 1701	. . . . . . . . . . . 12 |- ( ( u e. { -u 1 , 0 , 1 } /\ v e. { -u 1 , 0 , 1 } /\ w e. { -u 1 , 0 , 1 } ) -> ( ( u .+^ v ) .+^ w ) = if ( w = 0 , ( u .+^ v ) , w ) )
7		iftrue 4092	. . . . . . . . . . . 12 |- ( w = 0 -> if ( w = 0 , ( u .+^ v ) , w ) = ( u .+^ v ) )
8	6, 7	sylan9eq 2676	. . . . . . . . . . 11 |- ( ( ( u e. { -u 1 , 0 , 1 } /\ v e. { -u 1 , 0 , 1 } /\ w e. { -u 1 , 0 , 1 } ) /\ w = 0 ) -> ( ( u .+^ v ) .+^ w ) = ( u .+^ v ) )
9	8	adantr 481	. . . . . . . . . 10 |- ( ( ( ( u e. { -u 1 , 0 , 1 } /\ v e. { -u 1 , 0 , 1 } /\ w e. { -u 1 , 0 , 1 } ) /\ w = 0 ) /\ v = 0 ) -> ( ( u .+^ v ) .+^ w ) = ( u .+^ v ) )
10	2	3adant3 1081	. . . . . . . . . . 11 |- ( ( u e. { -u 1 , 0 , 1 } /\ v e. { -u 1 , 0 , 1 } /\ w e. { -u 1 , 0 , 1 } ) -> ( u .+^ v ) = if ( v = 0 , u , v ) )
11	10	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( ( u e. { -u 1 , 0 , 1 } /\ v e. { -u 1 , 0 , 1 } /\ w e. { -u 1 , 0 , 1 } ) /\ w = 0 ) /\ v = 0 ) -> ( u .+^ v ) = if ( v = 0 , u , v ) )
12		iftrue 4092	. . . . . . . . . . 11 |- ( v = 0 -> if ( v = 0 , u , v ) = u )
13	12	adantl 482	. . . . . . . . . 10 |- ( ( ( ( u e. { -u 1 , 0 , 1 } /\ v e. { -u 1 , 0 , 1 } /\ w e. { -u 1 , 0 , 1 } ) /\ w = 0 ) /\ v = 0 ) -> if ( v = 0 , u , v ) = u )
14	9, 11, 13	3eqtrd 2660	. . . . . . . . 9 |- ( ( ( ( u e. { -u 1 , 0 , 1 } /\ v e. { -u 1 , 0 , 1 } /\ w e. { -u 1 , 0 , 1 } ) /\ w = 0 ) /\ v = 0 ) -> ( ( u .+^ v ) .+^ w ) = u )
15		simp1 1061	. . . . . . . . . . . 12 |- ( ( u e. { -u 1 , 0 , 1 } /\ v e. { -u 1 , 0 , 1 } /\ w e. { -u 1 , 0 , 1 } ) -> u e. { -u 1 , 0 , 1 } )
16	1	signspval 30629	. . . . . . . . . . . . . 14 |- ( ( v e. { -u 1 , 0 , 1 } /\ w e. { -u 1 , 0 , 1 } ) -> ( v .+^ w ) = if ( w = 0 , v , w ) )
17	16	3adant1 1079	. . . . . . . . . . . . 13 |- ( ( u e. { -u 1 , 0 , 1 } /\ v e. { -u 1 , 0 , 1 } /\ w e. { -u 1 , 0 , 1 } ) -> ( v .+^ w ) = if ( w = 0 , v , w ) )
18		simpl2 1065	. . . . . . . . . . . . . 14 |- ( ( ( u e. { -u 1 , 0 , 1 } /\ v e. { -u 1 , 0 , 1 } /\ w e. { -u 1 , 0 , 1 } ) /\ w = 0 ) -> v e. { -u 1 , 0 , 1 } )
19		simpl3 1066	. . . . . . . . . . . . . 14 |- ( ( ( u e. { -u 1 , 0 , 1 } /\ v e. { -u 1 , 0 , 1 } /\ w e. { -u 1 , 0 , 1 } ) /\ -. w = 0 ) -> w e. { -u 1 , 0 , 1 } )
20	18, 19	ifclda 4120	. . . . . . . . . . . . 13 |- ( ( u e. { -u 1 , 0 , 1 } /\ v e. { -u 1 , 0 , 1 } /\ w e. { -u 1 , 0 , 1 } ) -> if ( w = 0 , v , w ) e. { -u 1 , 0 , 1 } )
21	17, 20	eqeltrd 2701	. . . . . . . . . . . 12 |- ( ( u e. { -u 1 , 0 , 1 } /\ v e. { -u 1 , 0 , 1 } /\ w e. { -u 1 , 0 , 1 } ) -> ( v .+^ w ) e. { -u 1 , 0 , 1 } )
22	1	signspval 30629	. . . . . . . . . . . 12 |- ( ( u e. { -u 1 , 0 , 1 } /\ ( v .+^ w ) e. { -u 1 , 0 , 1 } ) -> ( u .+^ ( v .+^ w ) ) = if ( ( v .+^ w ) = 0 , u , ( v .+^ w ) ) )
23	15, 21, 22	syl2anc 693	. . . . . . . . . . 11 |- ( ( u e. { -u 1 , 0 , 1 } /\ v e. { -u 1 , 0 , 1 } /\ w e. { -u 1 , 0 , 1 } ) -> ( u .+^ ( v .+^ w ) ) = if ( ( v .+^ w ) = 0 , u , ( v .+^ w ) ) )
24	23	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( ( u e. { -u 1 , 0 , 1 } /\ v e. { -u 1 , 0 , 1 } /\ w e. { -u 1 , 0 , 1 } ) /\ w = 0 ) /\ v = 0 ) -> ( u .+^ ( v .+^ w ) ) = if ( ( v .+^ w ) = 0 , u , ( v .+^ w ) ) )
25		iftrue 4092	. . . . . . . . . . . . 13 |- ( w = 0 -> if ( w = 0 , v , w ) = v )
26	17, 25	sylan9eq 2676	. . . . . . . . . . . 12 |- ( ( ( u e. { -u 1 , 0 , 1 } /\ v e. { -u 1 , 0 , 1 } /\ w e. { -u 1 , 0 , 1 } ) /\ w = 0 ) -> ( v .+^ w ) = v )
27		id 22	. . . . . . . . . . . 12 |- ( v = 0 -> v = 0 )
28	26, 27	sylan9eq 2676	. . . . . . . . . . 11 |- ( ( ( ( u e. { -u 1 , 0 , 1 } /\ v e. { -u 1 , 0 , 1 } /\ w e. { -u 1 , 0 , 1 } ) /\ w = 0 ) /\ v = 0 ) -> ( v .+^ w ) = 0 )
29	28	iftrued 4094	. . . . . . . . . 10 |- ( ( ( ( u e. { -u 1 , 0 , 1 } /\ v e. { -u 1 , 0 , 1 } /\ w e. { -u 1 , 0 , 1 } ) /\ w = 0 ) /\ v = 0 ) -> if ( ( v .+^ w ) = 0 , u , ( v .+^ w ) ) = u )
30	24, 29	eqtrd 2656	. . . . . . . . 9 |- ( ( ( ( u e. { -u 1 , 0 , 1 } /\ v e. { -u 1 , 0 , 1 } /\ w e. { -u 1 , 0 , 1 } ) /\ w = 0 ) /\ v = 0 ) -> ( u .+^ ( v .+^ w ) ) = u )
31	14, 30	eqtr4d 2659	. . . . . . . 8 |- ( ( ( ( u e. { -u 1 , 0 , 1 } /\ v e. { -u 1 , 0 , 1 } /\ w e. { -u 1 , 0 , 1 } ) /\ w = 0 ) /\ v = 0 ) -> ( ( u .+^ v ) .+^ w ) = ( u .+^ ( v .+^ w ) ) )
32	6	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( ( u e. { -u 1 , 0 , 1 } /\ v e. { -u 1 , 0 , 1 } /\ w e. { -u 1 , 0 , 1 } ) /\ w = 0 ) /\ -. v = 0 ) -> ( ( u .+^ v ) .+^ w ) = if ( w = 0 , ( u .+^ v ) , w ) )
33	7	ad2antlr 763	. . . . . . . . . 10 |- ( ( ( ( u e. { -u 1 , 0 , 1 } /\ v e. { -u 1 , 0 , 1 } /\ w e. { -u 1 , 0 , 1 } ) /\ w = 0 ) /\ -. v = 0 ) -> if ( w = 0 , ( u .+^ v ) , w ) = ( u .+^ v ) )
34	10	ad2antrr 762	. . . . . . . . . . 11 |- ( ( ( ( u e. { -u 1 , 0 , 1 } /\ v e. { -u 1 , 0 , 1 } /\ w e. { -u 1 , 0 , 1 } ) /\ w = 0 ) /\ -. v = 0 ) -> ( u .+^ v ) = if ( v = 0 , u , v ) )
35		iffalse 4095	. . . . . . . . . . . 12 |- ( -. v = 0 -> if ( v = 0 , u , v ) = v )
36	35	adantl 482	. . . . . . . . . . 11 |- ( ( ( ( u e. { -u 1 , 0 , 1 } /\ v e. { -u 1 , 0 , 1 } /\ w e. { -u 1 , 0 , 1 } ) /\ w = 0 ) /\ -. v = 0 ) -> if ( v = 0 , u , v ) = v )
37	34, 36	eqtrd 2656	. . . . . . . . . 10 |- ( ( ( ( u e. { -u 1 , 0 , 1 } /\ v e. { -u 1 , 0 , 1 } /\ w e. { -u 1 , 0 , 1 } ) /\ w = 0 ) /\ -. v = 0 ) -> ( u .+^ v ) = v )
38	32, 33, 37	3eqtrd 2660	. . . . . . . . 9 |- ( ( ( ( u e. { -u 1 , 0 , 1 } /\ v e. { -u 1 , 0 , 1 } /\ w e. { -u 1 , 0 , 1 } ) /\ w = 0 ) /\ -. v = 0 ) -> ( ( u .+^ v ) .+^ w ) = v )
39	23	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( ( u e. { -u 1 , 0 , 1 } /\ v e. { -u 1 , 0 , 1 } /\ w e. { -u 1 , 0 , 1 } ) /\ w = 0 ) /\ -. v = 0 ) -> ( u .+^ ( v .+^ w ) ) = if ( ( v .+^ w ) = 0 , u , ( v .+^ w ) ) )
40		simpr 477	. . . . . . . . . . . 12 |- ( ( ( ( u e. { -u 1 , 0 , 1 } /\ v e. { -u 1 , 0 , 1 } /\ w e. { -u 1 , 0 , 1 } ) /\ w = 0 ) /\ -. v = 0 ) -> -. v = 0 )
41	17	ad2antrr 762	. . . . . . . . . . . . . 14 |- ( ( ( ( u e. { -u 1 , 0 , 1 } /\ v e. { -u 1 , 0 , 1 } /\ w e. { -u 1 , 0 , 1 } ) /\ w = 0 ) /\ -. v = 0 ) -> ( v .+^ w ) = if ( w = 0 , v , w ) )
42	25	ad2antlr 763	. . . . . . . . . . . . . 14 |- ( ( ( ( u e. { -u 1 , 0 , 1 } /\ v e. { -u 1 , 0 , 1 } /\ w e. { -u 1 , 0 , 1 } ) /\ w = 0 ) /\ -. v = 0 ) -> if ( w = 0 , v , w ) = v )
43	41, 42	eqtrd 2656	. . . . . . . . . . . . 13 |- ( ( ( ( u e. { -u 1 , 0 , 1 } /\ v e. { -u 1 , 0 , 1 } /\ w e. { -u 1 , 0 , 1 } ) /\ w = 0 ) /\ -. v = 0 ) -> ( v .+^ w ) = v )
44	43	eqeq1d 2624	. . . . . . . . . . . 12 |- ( ( ( ( u e. { -u 1 , 0 , 1 } /\ v e. { -u 1 , 0 , 1 } /\ w e. { -u 1 , 0 , 1 } ) /\ w = 0 ) /\ -. v = 0 ) -> ( ( v .+^ w ) = 0 <-> v = 0 ) )
45	40, 44	mtbird 315	. . . . . . . . . . 11 |- ( ( ( ( u e. { -u 1 , 0 , 1 } /\ v e. { -u 1 , 0 , 1 } /\ w e. { -u 1 , 0 , 1 } ) /\ w = 0 ) /\ -. v = 0 ) -> -. ( v .+^ w ) = 0 )
46	45	iffalsed 4097	. . . . . . . . . 10 |- ( ( ( ( u e. { -u 1 , 0 , 1 } /\ v e. { -u 1 , 0 , 1 } /\ w e. { -u 1 , 0 , 1 } ) /\ w = 0 ) /\ -. v = 0 ) -> if ( ( v .+^ w ) = 0 , u , ( v .+^ w ) ) = ( v .+^ w ) )
47	39, 46, 43	3eqtrd 2660	. . . . . . . . 9 |- ( ( ( ( u e. { -u 1 , 0 , 1 } /\ v e. { -u 1 , 0 , 1 } /\ w e. { -u 1 , 0 , 1 } ) /\ w = 0 ) /\ -. v = 0 ) -> ( u .+^ ( v .+^ w ) ) = v )
48	38, 47	eqtr4d 2659	. . . . . . . 8 |- ( ( ( ( u e. { -u 1 , 0 , 1 } /\ v e. { -u 1 , 0 , 1 } /\ w e. { -u 1 , 0 , 1 } ) /\ w = 0 ) /\ -. v = 0 ) -> ( ( u .+^ v ) .+^ w ) = ( u .+^ ( v .+^ w ) ) )
49	31, 48	pm2.61dan 832	. . . . . . 7 |- ( ( ( u e. { -u 1 , 0 , 1 } /\ v e. { -u 1 , 0 , 1 } /\ w e. { -u 1 , 0 , 1 } ) /\ w = 0 ) -> ( ( u .+^ v ) .+^ w ) = ( u .+^ ( v .+^ w ) ) )
50		iffalse 4095	. . . . . . . . 9 |- ( -. w = 0 -> if ( w = 0 , ( u .+^ v ) , w ) = w )
51	6, 50	sylan9eq 2676	. . . . . . . 8 |- ( ( ( u e. { -u 1 , 0 , 1 } /\ v e. { -u 1 , 0 , 1 } /\ w e. { -u 1 , 0 , 1 } ) /\ -. w = 0 ) -> ( ( u .+^ v ) .+^ w ) = w )
52	23	adantr 481	. . . . . . . . 9 |- ( ( ( u e. { -u 1 , 0 , 1 } /\ v e. { -u 1 , 0 , 1 } /\ w e. { -u 1 , 0 , 1 } ) /\ -. w = 0 ) -> ( u .+^ ( v .+^ w ) ) = if ( ( v .+^ w ) = 0 , u , ( v .+^ w ) ) )
53		simpr 477	. . . . . . . . . . 11 |- ( ( ( u e. { -u 1 , 0 , 1 } /\ v e. { -u 1 , 0 , 1 } /\ w e. { -u 1 , 0 , 1 } ) /\ -. w = 0 ) -> -. w = 0 )
54		iffalse 4095	. . . . . . . . . . . . 13 |- ( -. w = 0 -> if ( w = 0 , v , w ) = w )
55	17, 54	sylan9eq 2676	. . . . . . . . . . . 12 |- ( ( ( u e. { -u 1 , 0 , 1 } /\ v e. { -u 1 , 0 , 1 } /\ w e. { -u 1 , 0 , 1 } ) /\ -. w = 0 ) -> ( v .+^ w ) = w )
56	55	eqeq1d 2624	. . . . . . . . . . 11 |- ( ( ( u e. { -u 1 , 0 , 1 } /\ v e. { -u 1 , 0 , 1 } /\ w e. { -u 1 , 0 , 1 } ) /\ -. w = 0 ) -> ( ( v .+^ w ) = 0 <-> w = 0 ) )
57	53, 56	mtbird 315	. . . . . . . . . 10 |- ( ( ( u e. { -u 1 , 0 , 1 } /\ v e. { -u 1 , 0 , 1 } /\ w e. { -u 1 , 0 , 1 } ) /\ -. w = 0 ) -> -. ( v .+^ w ) = 0 )
58	57	iffalsed 4097	. . . . . . . . 9 |- ( ( ( u e. { -u 1 , 0 , 1 } /\ v e. { -u 1 , 0 , 1 } /\ w e. { -u 1 , 0 , 1 } ) /\ -. w = 0 ) -> if ( ( v .+^ w ) = 0 , u , ( v .+^ w ) ) = ( v .+^ w ) )
59	52, 58, 55	3eqtrd 2660	. . . . . . . 8 |- ( ( ( u e. { -u 1 , 0 , 1 } /\ v e. { -u 1 , 0 , 1 } /\ w e. { -u 1 , 0 , 1 } ) /\ -. w = 0 ) -> ( u .+^ ( v .+^ w ) ) = w )
60	51, 59	eqtr4d 2659	. . . . . . 7 |- ( ( ( u e. { -u 1 , 0 , 1 } /\ v e. { -u 1 , 0 , 1 } /\ w e. { -u 1 , 0 , 1 } ) /\ -. w = 0 ) -> ( ( u .+^ v ) .+^ w ) = ( u .+^ ( v .+^ w ) ) )
61	49, 60	pm2.61dan 832	. . . . . 6 |- ( ( u e. { -u 1 , 0 , 1 } /\ v e. { -u 1 , 0 , 1 } /\ w e. { -u 1 , 0 , 1 } ) -> ( ( u .+^ v ) .+^ w ) = ( u .+^ ( v .+^ w ) ) )
62	61	3expa 1265	. . . . 5 |- ( ( ( u e. { -u 1 , 0 , 1 } /\ v e. { -u 1 , 0 , 1 } ) /\ w e. { -u 1 , 0 , 1 } ) -> ( ( u .+^ v ) .+^ w ) = ( u .+^ ( v .+^ w ) ) )
63	62	ralrimiva 2966	. . . 4 |- ( ( u e. { -u 1 , 0 , 1 } /\ v e. { -u 1 , 0 , 1 } ) -> A. w e. { -u 1 , 0 , 1 } ( ( u .+^ v ) .+^ w ) = ( u .+^ ( v .+^ w ) ) )
64	4, 63	jca 554	. . 3 |- ( ( u e. { -u 1 , 0 , 1 } /\ v e. { -u 1 , 0 , 1 } ) -> ( ( u .+^ v ) e. { -u 1 , 0 , 1 } /\ A. w e. { -u 1 , 0 , 1 } ( ( u .+^ v ) .+^ w ) = ( u .+^ ( v .+^ w ) ) ) )
65	64	rgen2a 2977	. 2 |- A. u e. { -u 1 , 0 , 1 } A. v e. { -u 1 , 0 , 1 } ( ( u .+^ v ) e. { -u 1 , 0 , 1 } /\ A. w e. { -u 1 , 0 , 1 } ( ( u .+^ v ) .+^ w ) = ( u .+^ ( v .+^ w ) ) )
66		c0ex 10034	. . . 4 |- 0 e. _V
67	66	tpid2 4304	. . 3 |- 0 e. { -u 1 , 0 , 1 }
68	1	signsw0glem 30630	. . 3 |- A. u e. { -u 1 , 0 , 1 } ( ( 0 .+^ u ) = u /\ ( u .+^ 0 ) = u )
69		oveq1 6657	. . . . . . 7 |- ( e = 0 -> ( e .+^ u ) = ( 0 .+^ u ) )
70	69	eqeq1d 2624	. . . . . 6 |- ( e = 0 -> ( ( e .+^ u ) = u <-> ( 0 .+^ u ) = u ) )
71		oveq2 6658	. . . . . . 7 |- ( e = 0 -> ( u .+^ e ) = ( u .+^ 0 ) )
72	71	eqeq1d 2624	. . . . . 6 |- ( e = 0 -> ( ( u .+^ e ) = u <-> ( u .+^ 0 ) = u ) )
73	70, 72	anbi12d 747	. . . . 5 |- ( e = 0 -> ( ( ( e .+^ u ) = u /\ ( u .+^ e ) = u ) <-> ( ( 0 .+^ u ) = u /\ ( u .+^ 0 ) = u ) ) )
74	73	ralbidv 2986	. . . 4 |- ( e = 0 -> ( A. u e. { -u 1 , 0 , 1 } ( ( e .+^ u ) = u /\ ( u .+^ e ) = u ) <-> A. u e. { -u 1 , 0 , 1 } ( ( 0 .+^ u ) = u /\ ( u .+^ 0 ) = u ) ) )
75	74	rspcev 3309	. . 3 |- ( ( 0 e. { -u 1 , 0 , 1 } /\ A. u e. { -u 1 , 0 , 1 } ( ( 0 .+^ u ) = u /\ ( u .+^ 0 ) = u ) ) -> E. e e. { -u 1 , 0 , 1 } A. u e. { -u 1 , 0 , 1 } ( ( e .+^ u ) = u /\ ( u .+^ e ) = u ) )
76	67, 68, 75	mp2an 708	. 2 |- E. e e. { -u 1 , 0 , 1 } A. u e. { -u 1 , 0 , 1 } ( ( e .+^ u ) = u /\ ( u .+^ e ) = u )
77		signsw.w	. . . 4 |- W = { <. ( Base ` ndx ) , { -u 1 , 0 , 1 } >. , <. ( +g ` ndx ) , .+^ >. }
78	1, 77	signswbase 30631	. . 3 |- { -u 1 , 0 , 1 } = ( Base ` W )
79	1, 77	signswplusg 30632	. . 3 |- .+^ = ( +g ` W )
80	78, 79	ismnd 17297	. 2 |- ( W e. Mnd <-> ( A. u e. { -u 1 , 0 , 1 } A. v e. { -u 1 , 0 , 1 } ( ( u .+^ v ) e. { -u 1 , 0 , 1 } /\ A. w e. { -u 1 , 0 , 1 } ( ( u .+^ v ) .+^ w ) = ( u .+^ ( v .+^ w ) ) ) /\ E. e e. { -u 1 , 0 , 1 } A. u e. { -u 1 , 0 , 1 } ( ( e .+^ u ) = u /\ ( u .+^ e ) = u ) ) )
81	65, 76, 80	mpbir2an 955	1 |- W e. Mnd

Syntax hints:   -. wn 3   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   ifcif 4086   {cpr 4179   {ctp 4181   <.cop 4183   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   0cc0 9936   1c1 9937   -ucneg 10267   ndxcnx 15854   Basecbs 15857   +g cplusg 15941   Mndcmnd 17294

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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  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-pss 3590  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-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-plusg 15954  df-mgm 17242  df-sgrp 17284  df-mnd 17295

This theorem is referenced by:  signstcl  30642  signstf  30643  signstf0  30645  signstfvn  30646