Theorem ltsrprg 7942

Description: Ordering of signed reals in terms of positive reals. (Contributed by Jim Kingdon, 2-Jan-2019.)

Assertion

Ref	Expression
ltsrprg	|- ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) -> ( [ <. A , B >. ] ~R <R [ <. C , D >. ] ~R <-> ( A +P. D ) <P ( B +P. C ) ) )

Proof of Theorem ltsrprg

Dummy variables x y z w v u f are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		enrex 7932	. 2 |- ~R e. _V
2		enrer 7930	. 2 |- ~R Er ( P. X. P. )
3		df-nr 7922	. 2 |- R. = ( ( P. X. P. ) /. ~R )
4		df-ltr 7925	. 2 |- <R = { <. x , y >. | ( ( x e. R. /\ y e. R. ) /\ E. z E. w E. v E. u ( ( x = [ <. z , w >. ] ~R /\ y = [ <. v , u >. ] ~R ) /\ ( z +P. u ) <P ( w +P. v ) ) ) }
5		enreceq 7931	. . . . 5 |- ( ( ( z e. P. /\ w e. P. ) /\ ( A e. P. /\ B e. P. ) ) -> ( [ <. z , w >. ] ~R = [ <. A , B >. ] ~R <-> ( z +P. B ) = ( w +P. A ) ) )
6		enreceq 7931	. . . . . 6 |- ( ( ( v e. P. /\ u e. P. ) /\ ( C e. P. /\ D e. P. ) ) -> ( [ <. v , u >. ] ~R = [ <. C , D >. ] ~R <-> ( v +P. D ) = ( u +P. C ) ) )
7		eqcom 2231	. . . . . 6 |- ( ( v +P. D ) = ( u +P. C ) <-> ( u +P. C ) = ( v +P. D ) )
8	6, 7	bitrdi 196	. . . . 5 |- ( ( ( v e. P. /\ u e. P. ) /\ ( C e. P. /\ D e. P. ) ) -> ( [ <. v , u >. ] ~R = [ <. C , D >. ] ~R <-> ( u +P. C ) = ( v +P. D ) ) )
9	5, 8	bi2anan9 608	. . . 4 |- ( ( ( ( z e. P. /\ w e. P. ) /\ ( A e. P. /\ B e. P. ) ) /\ ( ( v e. P. /\ u e. P. ) /\ ( C e. P. /\ D e. P. ) ) ) -> ( ( [ <. z , w >. ] ~R = [ <. A , B >. ] ~R /\ [ <. v , u >. ] ~R = [ <. C , D >. ] ~R ) <-> ( ( z +P. B ) = ( w +P. A ) /\ ( u +P. C ) = ( v +P. D ) ) ) )
10		oveq12 6016	. . . . . . 7 |- ( ( ( z +P. B ) = ( w +P. A ) /\ ( u +P. C ) = ( v +P. D ) ) -> ( ( z +P. B ) +P. ( u +P. C ) ) = ( ( w +P. A ) +P. ( v +P. D ) ) )
11	10	adantl 277	. . . . . 6 |- ( ( ( ( ( z e. P. /\ w e. P. ) /\ ( A e. P. /\ B e. P. ) ) /\ ( ( v e. P. /\ u e. P. ) /\ ( C e. P. /\ D e. P. ) ) ) /\ ( ( z +P. B ) = ( w +P. A ) /\ ( u +P. C ) = ( v +P. D ) ) ) -> ( ( z +P. B ) +P. ( u +P. C ) ) = ( ( w +P. A ) +P. ( v +P. D ) ) )
12		simprlr 538	. . . . . . . . . . 11 |- ( ( ( ( z e. P. /\ w e. P. ) /\ ( A e. P. /\ B e. P. ) ) /\ ( ( v e. P. /\ u e. P. ) /\ ( C e. P. /\ D e. P. ) ) ) -> u e. P. )
13		simplrr 536	. . . . . . . . . . 11 |- ( ( ( ( z e. P. /\ w e. P. ) /\ ( A e. P. /\ B e. P. ) ) /\ ( ( v e. P. /\ u e. P. ) /\ ( C e. P. /\ D e. P. ) ) ) -> B e. P. )
14		addcomprg 7773	. . . . . . . . . . . 12 |- ( ( u e. P. /\ B e. P. ) -> ( u +P. B ) = ( B +P. u ) )
15	14	oveq1d 6022	. . . . . . . . . . 11 |- ( ( u e. P. /\ B e. P. ) -> ( ( u +P. B ) +P. C ) = ( ( B +P. u ) +P. C ) )
16	12, 13, 15	syl2anc 411	. . . . . . . . . 10 |- ( ( ( ( z e. P. /\ w e. P. ) /\ ( A e. P. /\ B e. P. ) ) /\ ( ( v e. P. /\ u e. P. ) /\ ( C e. P. /\ D e. P. ) ) ) -> ( ( u +P. B ) +P. C ) = ( ( B +P. u ) +P. C ) )
17		simprrl 539	. . . . . . . . . . 11 |- ( ( ( ( z e. P. /\ w e. P. ) /\ ( A e. P. /\ B e. P. ) ) /\ ( ( v e. P. /\ u e. P. ) /\ ( C e. P. /\ D e. P. ) ) ) -> C e. P. )
18		addassprg 7774	. . . . . . . . . . 11 |- ( ( u e. P. /\ B e. P. /\ C e. P. ) -> ( ( u +P. B ) +P. C ) = ( u +P. ( B +P. C ) ) )
19	12, 13, 17, 18	syl3anc 1271	. . . . . . . . . 10 |- ( ( ( ( z e. P. /\ w e. P. ) /\ ( A e. P. /\ B e. P. ) ) /\ ( ( v e. P. /\ u e. P. ) /\ ( C e. P. /\ D e. P. ) ) ) -> ( ( u +P. B ) +P. C ) = ( u +P. ( B +P. C ) ) )
20		addassprg 7774	. . . . . . . . . . 11 |- ( ( B e. P. /\ u e. P. /\ C e. P. ) -> ( ( B +P. u ) +P. C ) = ( B +P. ( u +P. C ) ) )
21	13, 12, 17, 20	syl3anc 1271	. . . . . . . . . 10 |- ( ( ( ( z e. P. /\ w e. P. ) /\ ( A e. P. /\ B e. P. ) ) /\ ( ( v e. P. /\ u e. P. ) /\ ( C e. P. /\ D e. P. ) ) ) -> ( ( B +P. u ) +P. C ) = ( B +P. ( u +P. C ) ) )
22	16, 19, 21	3eqtr3d 2270	. . . . . . . . 9 |- ( ( ( ( z e. P. /\ w e. P. ) /\ ( A e. P. /\ B e. P. ) ) /\ ( ( v e. P. /\ u e. P. ) /\ ( C e. P. /\ D e. P. ) ) ) -> ( u +P. ( B +P. C ) ) = ( B +P. ( u +P. C ) ) )
23	22	oveq2d 6023	. . . . . . . 8 |- ( ( ( ( z e. P. /\ w e. P. ) /\ ( A e. P. /\ B e. P. ) ) /\ ( ( v e. P. /\ u e. P. ) /\ ( C e. P. /\ D e. P. ) ) ) -> ( z +P. ( u +P. ( B +P. C ) ) ) = ( z +P. ( B +P. ( u +P. C ) ) ) )
24		simplll 533	. . . . . . . . 9 |- ( ( ( ( z e. P. /\ w e. P. ) /\ ( A e. P. /\ B e. P. ) ) /\ ( ( v e. P. /\ u e. P. ) /\ ( C e. P. /\ D e. P. ) ) ) -> z e. P. )
25		addclpr 7732	. . . . . . . . . . . . 13 |- ( ( w e. P. /\ v e. P. ) -> ( w +P. v ) e. P. )
26	25	ad2ant2lr 510	. . . . . . . . . . . 12 |- ( ( ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( w +P. v ) e. P. )
27		addclpr 7732	. . . . . . . . . . . . 13 |- ( ( B e. P. /\ C e. P. ) -> ( B +P. C ) e. P. )
28	27	ad2ant2lr 510	. . . . . . . . . . . 12 |- ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) -> ( B +P. C ) e. P. )
29	26, 28	anim12ci 339	. . . . . . . . . . 11 |- ( ( ( ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) /\ ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) ) -> ( ( B +P. C ) e. P. /\ ( w +P. v ) e. P. ) )
30	29	an4s 590	. . . . . . . . . 10 |- ( ( ( ( z e. P. /\ w e. P. ) /\ ( A e. P. /\ B e. P. ) ) /\ ( ( v e. P. /\ u e. P. ) /\ ( C e. P. /\ D e. P. ) ) ) -> ( ( B +P. C ) e. P. /\ ( w +P. v ) e. P. ) )
31	30	simpld 112	. . . . . . . . 9 |- ( ( ( ( z e. P. /\ w e. P. ) /\ ( A e. P. /\ B e. P. ) ) /\ ( ( v e. P. /\ u e. P. ) /\ ( C e. P. /\ D e. P. ) ) ) -> ( B +P. C ) e. P. )
32		addassprg 7774	. . . . . . . . 9 |- ( ( z e. P. /\ u e. P. /\ ( B +P. C ) e. P. ) -> ( ( z +P. u ) +P. ( B +P. C ) ) = ( z +P. ( u +P. ( B +P. C ) ) ) )
33	24, 12, 31, 32	syl3anc 1271	. . . . . . . 8 |- ( ( ( ( z e. P. /\ w e. P. ) /\ ( A e. P. /\ B e. P. ) ) /\ ( ( v e. P. /\ u e. P. ) /\ ( C e. P. /\ D e. P. ) ) ) -> ( ( z +P. u ) +P. ( B +P. C ) ) = ( z +P. ( u +P. ( B +P. C ) ) ) )
34		addclpr 7732	. . . . . . . . . 10 |- ( ( u e. P. /\ C e. P. ) -> ( u +P. C ) e. P. )
35	12, 17, 34	syl2anc 411	. . . . . . . . 9 |- ( ( ( ( z e. P. /\ w e. P. ) /\ ( A e. P. /\ B e. P. ) ) /\ ( ( v e. P. /\ u e. P. ) /\ ( C e. P. /\ D e. P. ) ) ) -> ( u +P. C ) e. P. )
36		addassprg 7774	. . . . . . . . 9 |- ( ( z e. P. /\ B e. P. /\ ( u +P. C ) e. P. ) -> ( ( z +P. B ) +P. ( u +P. C ) ) = ( z +P. ( B +P. ( u +P. C ) ) ) )
37	24, 13, 35, 36	syl3anc 1271	. . . . . . . 8 |- ( ( ( ( z e. P. /\ w e. P. ) /\ ( A e. P. /\ B e. P. ) ) /\ ( ( v e. P. /\ u e. P. ) /\ ( C e. P. /\ D e. P. ) ) ) -> ( ( z +P. B ) +P. ( u +P. C ) ) = ( z +P. ( B +P. ( u +P. C ) ) ) )
38	23, 33, 37	3eqtr4d 2272	. . . . . . 7 |- ( ( ( ( z e. P. /\ w e. P. ) /\ ( A e. P. /\ B e. P. ) ) /\ ( ( v e. P. /\ u e. P. ) /\ ( C e. P. /\ D e. P. ) ) ) -> ( ( z +P. u ) +P. ( B +P. C ) ) = ( ( z +P. B ) +P. ( u +P. C ) ) )
39	38	adantr 276	. . . . . 6 |- ( ( ( ( ( z e. P. /\ w e. P. ) /\ ( A e. P. /\ B e. P. ) ) /\ ( ( v e. P. /\ u e. P. ) /\ ( C e. P. /\ D e. P. ) ) ) /\ ( ( z +P. B ) = ( w +P. A ) /\ ( u +P. C ) = ( v +P. D ) ) ) -> ( ( z +P. u ) +P. ( B +P. C ) ) = ( ( z +P. B ) +P. ( u +P. C ) ) )
40		simprll 537	. . . . . . . . . . . 12 |- ( ( ( ( z e. P. /\ w e. P. ) /\ ( A e. P. /\ B e. P. ) ) /\ ( ( v e. P. /\ u e. P. ) /\ ( C e. P. /\ D e. P. ) ) ) -> v e. P. )
41		simplrl 535	. . . . . . . . . . . 12 |- ( ( ( ( z e. P. /\ w e. P. ) /\ ( A e. P. /\ B e. P. ) ) /\ ( ( v e. P. /\ u e. P. ) /\ ( C e. P. /\ D e. P. ) ) ) -> A e. P. )
42		addcomprg 7773	. . . . . . . . . . . 12 |- ( ( v e. P. /\ A e. P. ) -> ( v +P. A ) = ( A +P. v ) )
43	40, 41, 42	syl2anc 411	. . . . . . . . . . 11 |- ( ( ( ( z e. P. /\ w e. P. ) /\ ( A e. P. /\ B e. P. ) ) /\ ( ( v e. P. /\ u e. P. ) /\ ( C e. P. /\ D e. P. ) ) ) -> ( v +P. A ) = ( A +P. v ) )
44	43	oveq1d 6022	. . . . . . . . . 10 |- ( ( ( ( z e. P. /\ w e. P. ) /\ ( A e. P. /\ B e. P. ) ) /\ ( ( v e. P. /\ u e. P. ) /\ ( C e. P. /\ D e. P. ) ) ) -> ( ( v +P. A ) +P. D ) = ( ( A +P. v ) +P. D ) )
45		simprrr 540	. . . . . . . . . . 11 |- ( ( ( ( z e. P. /\ w e. P. ) /\ ( A e. P. /\ B e. P. ) ) /\ ( ( v e. P. /\ u e. P. ) /\ ( C e. P. /\ D e. P. ) ) ) -> D e. P. )
46		addassprg 7774	. . . . . . . . . . 11 |- ( ( v e. P. /\ A e. P. /\ D e. P. ) -> ( ( v +P. A ) +P. D ) = ( v +P. ( A +P. D ) ) )
47	40, 41, 45, 46	syl3anc 1271	. . . . . . . . . 10 |- ( ( ( ( z e. P. /\ w e. P. ) /\ ( A e. P. /\ B e. P. ) ) /\ ( ( v e. P. /\ u e. P. ) /\ ( C e. P. /\ D e. P. ) ) ) -> ( ( v +P. A ) +P. D ) = ( v +P. ( A +P. D ) ) )
48		addassprg 7774	. . . . . . . . . . 11 |- ( ( A e. P. /\ v e. P. /\ D e. P. ) -> ( ( A +P. v ) +P. D ) = ( A +P. ( v +P. D ) ) )
49	41, 40, 45, 48	syl3anc 1271	. . . . . . . . . 10 |- ( ( ( ( z e. P. /\ w e. P. ) /\ ( A e. P. /\ B e. P. ) ) /\ ( ( v e. P. /\ u e. P. ) /\ ( C e. P. /\ D e. P. ) ) ) -> ( ( A +P. v ) +P. D ) = ( A +P. ( v +P. D ) ) )
50	44, 47, 49	3eqtr3d 2270	. . . . . . . . 9 |- ( ( ( ( z e. P. /\ w e. P. ) /\ ( A e. P. /\ B e. P. ) ) /\ ( ( v e. P. /\ u e. P. ) /\ ( C e. P. /\ D e. P. ) ) ) -> ( v +P. ( A +P. D ) ) = ( A +P. ( v +P. D ) ) )
51	50	oveq2d 6023	. . . . . . . 8 |- ( ( ( ( z e. P. /\ w e. P. ) /\ ( A e. P. /\ B e. P. ) ) /\ ( ( v e. P. /\ u e. P. ) /\ ( C e. P. /\ D e. P. ) ) ) -> ( w +P. ( v +P. ( A +P. D ) ) ) = ( w +P. ( A +P. ( v +P. D ) ) ) )
52		simpllr 534	. . . . . . . . 9 |- ( ( ( ( z e. P. /\ w e. P. ) /\ ( A e. P. /\ B e. P. ) ) /\ ( ( v e. P. /\ u e. P. ) /\ ( C e. P. /\ D e. P. ) ) ) -> w e. P. )
53		addclpr 7732	. . . . . . . . . 10 |- ( ( A e. P. /\ D e. P. ) -> ( A +P. D ) e. P. )
54	41, 45, 53	syl2anc 411	. . . . . . . . 9 |- ( ( ( ( z e. P. /\ w e. P. ) /\ ( A e. P. /\ B e. P. ) ) /\ ( ( v e. P. /\ u e. P. ) /\ ( C e. P. /\ D e. P. ) ) ) -> ( A +P. D ) e. P. )
55		addassprg 7774	. . . . . . . . 9 |- ( ( w e. P. /\ v e. P. /\ ( A +P. D ) e. P. ) -> ( ( w +P. v ) +P. ( A +P. D ) ) = ( w +P. ( v +P. ( A +P. D ) ) ) )
56	52, 40, 54, 55	syl3anc 1271	. . . . . . . 8 |- ( ( ( ( z e. P. /\ w e. P. ) /\ ( A e. P. /\ B e. P. ) ) /\ ( ( v e. P. /\ u e. P. ) /\ ( C e. P. /\ D e. P. ) ) ) -> ( ( w +P. v ) +P. ( A +P. D ) ) = ( w +P. ( v +P. ( A +P. D ) ) ) )
57		addclpr 7732	. . . . . . . . . 10 |- ( ( v e. P. /\ D e. P. ) -> ( v +P. D ) e. P. )
58	40, 45, 57	syl2anc 411	. . . . . . . . 9 |- ( ( ( ( z e. P. /\ w e. P. ) /\ ( A e. P. /\ B e. P. ) ) /\ ( ( v e. P. /\ u e. P. ) /\ ( C e. P. /\ D e. P. ) ) ) -> ( v +P. D ) e. P. )
59		addassprg 7774	. . . . . . . . 9 |- ( ( w e. P. /\ A e. P. /\ ( v +P. D ) e. P. ) -> ( ( w +P. A ) +P. ( v +P. D ) ) = ( w +P. ( A +P. ( v +P. D ) ) ) )
60	52, 41, 58, 59	syl3anc 1271	. . . . . . . 8 |- ( ( ( ( z e. P. /\ w e. P. ) /\ ( A e. P. /\ B e. P. ) ) /\ ( ( v e. P. /\ u e. P. ) /\ ( C e. P. /\ D e. P. ) ) ) -> ( ( w +P. A ) +P. ( v +P. D ) ) = ( w +P. ( A +P. ( v +P. D ) ) ) )
61	51, 56, 60	3eqtr4d 2272	. . . . . . 7 |- ( ( ( ( z e. P. /\ w e. P. ) /\ ( A e. P. /\ B e. P. ) ) /\ ( ( v e. P. /\ u e. P. ) /\ ( C e. P. /\ D e. P. ) ) ) -> ( ( w +P. v ) +P. ( A +P. D ) ) = ( ( w +P. A ) +P. ( v +P. D ) ) )
62	61	adantr 276	. . . . . 6 |- ( ( ( ( ( z e. P. /\ w e. P. ) /\ ( A e. P. /\ B e. P. ) ) /\ ( ( v e. P. /\ u e. P. ) /\ ( C e. P. /\ D e. P. ) ) ) /\ ( ( z +P. B ) = ( w +P. A ) /\ ( u +P. C ) = ( v +P. D ) ) ) -> ( ( w +P. v ) +P. ( A +P. D ) ) = ( ( w +P. A ) +P. ( v +P. D ) ) )
63	11, 39, 62	3eqtr4d 2272	. . . . 5 |- ( ( ( ( ( z e. P. /\ w e. P. ) /\ ( A e. P. /\ B e. P. ) ) /\ ( ( v e. P. /\ u e. P. ) /\ ( C e. P. /\ D e. P. ) ) ) /\ ( ( z +P. B ) = ( w +P. A ) /\ ( u +P. C ) = ( v +P. D ) ) ) -> ( ( z +P. u ) +P. ( B +P. C ) ) = ( ( w +P. v ) +P. ( A +P. D ) ) )
64	63	ex 115	. . . 4 |- ( ( ( ( z e. P. /\ w e. P. ) /\ ( A e. P. /\ B e. P. ) ) /\ ( ( v e. P. /\ u e. P. ) /\ ( C e. P. /\ D e. P. ) ) ) -> ( ( ( z +P. B ) = ( w +P. A ) /\ ( u +P. C ) = ( v +P. D ) ) -> ( ( z +P. u ) +P. ( B +P. C ) ) = ( ( w +P. v ) +P. ( A +P. D ) ) ) )
65	9, 64	sylbid 150	. . 3 |- ( ( ( ( z e. P. /\ w e. P. ) /\ ( A e. P. /\ B e. P. ) ) /\ ( ( v e. P. /\ u e. P. ) /\ ( C e. P. /\ D e. P. ) ) ) -> ( ( [ <. z , w >. ] ~R = [ <. A , B >. ] ~R /\ [ <. v , u >. ] ~R = [ <. C , D >. ] ~R ) -> ( ( z +P. u ) +P. ( B +P. C ) ) = ( ( w +P. v ) +P. ( A +P. D ) ) ) )
66		ltaprg 7814	. . . . 5 |- ( ( x e. P. /\ y e. P. /\ f e. P. ) -> ( x <P y <-> ( f +P. x ) <P ( f +P. y ) ) )
67	66	adantl 277	. . . 4 |- ( ( ( ( ( z e. P. /\ w e. P. ) /\ ( A e. P. /\ B e. P. ) ) /\ ( ( v e. P. /\ u e. P. ) /\ ( C e. P. /\ D e. P. ) ) ) /\ ( x e. P. /\ y e. P. /\ f e. P. ) ) -> ( x <P y <-> ( f +P. x ) <P ( f +P. y ) ) )
68		addclpr 7732	. . . . 5 |- ( ( z e. P. /\ u e. P. ) -> ( z +P. u ) e. P. )
69	24, 12, 68	syl2anc 411	. . . 4 |- ( ( ( ( z e. P. /\ w e. P. ) /\ ( A e. P. /\ B e. P. ) ) /\ ( ( v e. P. /\ u e. P. ) /\ ( C e. P. /\ D e. P. ) ) ) -> ( z +P. u ) e. P. )
70	30	simprd 114	. . . 4 |- ( ( ( ( z e. P. /\ w e. P. ) /\ ( A e. P. /\ B e. P. ) ) /\ ( ( v e. P. /\ u e. P. ) /\ ( C e. P. /\ D e. P. ) ) ) -> ( w +P. v ) e. P. )
71		addcomprg 7773	. . . . 5 |- ( ( x e. P. /\ y e. P. ) -> ( x +P. y ) = ( y +P. x ) )
72	71	adantl 277	. . . 4 |- ( ( ( ( ( z e. P. /\ w e. P. ) /\ ( A e. P. /\ B e. P. ) ) /\ ( ( v e. P. /\ u e. P. ) /\ ( C e. P. /\ D e. P. ) ) ) /\ ( x e. P. /\ y e. P. ) ) -> ( x +P. y ) = ( y +P. x ) )
73	67, 69, 31, 70, 72, 54	caovord3d 6182	. . 3 |- ( ( ( ( z e. P. /\ w e. P. ) /\ ( A e. P. /\ B e. P. ) ) /\ ( ( v e. P. /\ u e. P. ) /\ ( C e. P. /\ D e. P. ) ) ) -> ( ( ( z +P. u ) +P. ( B +P. C ) ) = ( ( w +P. v ) +P. ( A +P. D ) ) -> ( ( z +P. u ) <P ( w +P. v ) <-> ( A +P. D ) <P ( B +P. C ) ) ) )
74	65, 73	syld 45	. 2 |- ( ( ( ( z e. P. /\ w e. P. ) /\ ( A e. P. /\ B e. P. ) ) /\ ( ( v e. P. /\ u e. P. ) /\ ( C e. P. /\ D e. P. ) ) ) -> ( ( [ <. z , w >. ] ~R = [ <. A , B >. ] ~R /\ [ <. v , u >. ] ~R = [ <. C , D >. ] ~R ) -> ( ( z +P. u ) <P ( w +P. v ) <-> ( A +P. D ) <P ( B +P. C ) ) ) )
75	1, 2, 3, 4, 74	brecop 6780	1 |- ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) -> ( [ <. A , B >. ] ~R <R [ <. C , D >. ] ~R <-> ( A +P. D ) <P ( B +P. C ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1002   = wceq 1395   e. wcel 2200   <.cop 3669   class class class wbr 4083  (class class class)co 6007   [cec 6686   P.cnp 7486   +P. cpp 7488   <P cltp 7490   ~R cer 7491   R.cnr 7492   <R cltr 7498

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-eprel 4380  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-recs 6457  df-irdg 6522  df-1o 6568  df-2o 6569  df-oadd 6572  df-omul 6573  df-er 6688  df-ec 6690  df-qs 6694  df-ni 7499  df-pli 7500  df-mi 7501  df-lti 7502  df-plpq 7539  df-mpq 7540  df-enq 7542  df-nqqs 7543  df-plqqs 7544  df-mqqs 7545  df-1nqqs 7546  df-rq 7547  df-ltnqqs 7548  df-enq0 7619  df-nq0 7620  df-0nq0 7621  df-plq0 7622  df-mq0 7623  df-inp 7661  df-iplp 7663  df-iltp 7665  df-enr 7921  df-nr 7922  df-ltr 7925

This theorem is referenced by:  gt0srpr  7943  lttrsr  7957  ltposr  7958  ltsosr  7959  0lt1sr  7960  ltasrg  7965  aptisr  7974  mulextsr1  7976  archsr  7977  prsrlt  7982  ltpsrprg  7998  mappsrprg  7999  map2psrprg  8000  pitoregt0  8044