Theorem mulgt0sr 9926

Description: The product of two positive signed reals is positive. (Contributed by NM, 13-May-1996.) (New usage is discouraged.)

Assertion

Ref	Expression
mulgt0sr	|- ( ( 0R <R A /\ 0R <R B ) -> 0R <R ( A .R B ) )

Proof of Theorem mulgt0sr

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

Step	Hyp	Ref	Expression
1		ltrelsr 9889	. . . . 5 |- <R C_ ( R. X. R. )
2	1	brel 5168	. . . 4 |- ( 0R <R A -> ( 0R e. R. /\ A e. R. ) )
3	2	simprd 479	. . 3 |- ( 0R <R A -> A e. R. )
4	1	brel 5168	. . . 4 |- ( 0R <R B -> ( 0R e. R. /\ B e. R. ) )
5	4	simprd 479	. . 3 |- ( 0R <R B -> B e. R. )
6	3, 5	anim12i 590	. 2 |- ( ( 0R <R A /\ 0R <R B ) -> ( A e. R. /\ B e. R. ) )
7		df-nr 9878	. . 3 |- R. = ( ( P. X. P. ) /. ~R )
8		breq2 4657	. . . . 5 |- ( [ <. x , y >. ] ~R = A -> ( 0R <R [ <. x , y >. ] ~R <-> 0R <R A ) )
9	8	anbi1d 741	. . . 4 |- ( [ <. x , y >. ] ~R = A -> ( ( 0R <R [ <. x , y >. ] ~R /\ 0R <R [ <. z , w >. ] ~R ) <-> ( 0R <R A /\ 0R <R [ <. z , w >. ] ~R ) ) )
10		oveq1 6657	. . . . 5 |- ( [ <. x , y >. ] ~R = A -> ( [ <. x , y >. ] ~R .R [ <. z , w >. ] ~R ) = ( A .R [ <. z , w >. ] ~R ) )
11	10	breq2d 4665	. . . 4 |- ( [ <. x , y >. ] ~R = A -> ( 0R <R ( [ <. x , y >. ] ~R .R [ <. z , w >. ] ~R ) <-> 0R <R ( A .R [ <. z , w >. ] ~R ) ) )
12	9, 11	imbi12d 334	. . 3 |- ( [ <. x , y >. ] ~R = A -> ( ( ( 0R <R [ <. x , y >. ] ~R /\ 0R <R [ <. z , w >. ] ~R ) -> 0R <R ( [ <. x , y >. ] ~R .R [ <. z , w >. ] ~R ) ) <-> ( ( 0R <R A /\ 0R <R [ <. z , w >. ] ~R ) -> 0R <R ( A .R [ <. z , w >. ] ~R ) ) ) )
13		breq2 4657	. . . . 5 |- ( [ <. z , w >. ] ~R = B -> ( 0R <R [ <. z , w >. ] ~R <-> 0R <R B ) )
14	13	anbi2d 740	. . . 4 |- ( [ <. z , w >. ] ~R = B -> ( ( 0R <R A /\ 0R <R [ <. z , w >. ] ~R ) <-> ( 0R <R A /\ 0R <R B ) ) )
15		oveq2 6658	. . . . 5 |- ( [ <. z , w >. ] ~R = B -> ( A .R [ <. z , w >. ] ~R ) = ( A .R B ) )
16	15	breq2d 4665	. . . 4 |- ( [ <. z , w >. ] ~R = B -> ( 0R <R ( A .R [ <. z , w >. ] ~R ) <-> 0R <R ( A .R B ) ) )
17	14, 16	imbi12d 334	. . 3 |- ( [ <. z , w >. ] ~R = B -> ( ( ( 0R <R A /\ 0R <R [ <. z , w >. ] ~R ) -> 0R <R ( A .R [ <. z , w >. ] ~R ) ) <-> ( ( 0R <R A /\ 0R <R B ) -> 0R <R ( A .R B ) ) ) )
18		gt0srpr 9899	. . . . 5 |- ( 0R <R [ <. x , y >. ] ~R <-> y <P x )
19		gt0srpr 9899	. . . . 5 |- ( 0R <R [ <. z , w >. ] ~R <-> w <P z )
20	18, 19	anbi12i 733	. . . 4 |- ( ( 0R <R [ <. x , y >. ] ~R /\ 0R <R [ <. z , w >. ] ~R ) <-> ( y <P x /\ w <P z ) )
21		simprr 796	. . . . . 6 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> w e. P. )
22		mulclpr 9842	. . . . . . . 8 |- ( ( x e. P. /\ z e. P. ) -> ( x .P. z ) e. P. )
23		mulclpr 9842	. . . . . . . 8 |- ( ( y e. P. /\ w e. P. ) -> ( y .P. w ) e. P. )
24		addclpr 9840	. . . . . . . 8 |- ( ( ( x .P. z ) e. P. /\ ( y .P. w ) e. P. ) -> ( ( x .P. z ) +P. ( y .P. w ) ) e. P. )
25	22, 23, 24	syl2an 494	. . . . . . 7 |- ( ( ( x e. P. /\ z e. P. ) /\ ( y e. P. /\ w e. P. ) ) -> ( ( x .P. z ) +P. ( y .P. w ) ) e. P. )
26	25	an4s 869	. . . . . 6 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( ( x .P. z ) +P. ( y .P. w ) ) e. P. )
27		ltexpri 9865	. . . . . . . . 9 |- ( y <P x -> E. v e. P. ( y +P. v ) = x )
28		ltexpri 9865	. . . . . . . . 9 |- ( w <P z -> E. u e. P. ( w +P. u ) = z )
29		mulclpr 9842	. . . . . . . . . . . . . . . . 17 |- ( ( v e. P. /\ w e. P. ) -> ( v .P. w ) e. P. )
30		oveq12 6659	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( y +P. v ) = x /\ ( w +P. u ) = z ) -> ( ( y +P. v ) .P. ( w +P. u ) ) = ( x .P. z ) )
31	30	oveq1d 6665	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( y +P. v ) = x /\ ( w +P. u ) = z ) -> ( ( ( y +P. v ) .P. ( w +P. u ) ) +P. ( ( y .P. w ) +P. ( v .P. w ) ) ) = ( ( x .P. z ) +P. ( ( y .P. w ) +P. ( v .P. w ) ) ) )
32		distrpr 9850	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( y .P. ( w +P. u ) ) = ( ( y .P. w ) +P. ( y .P. u ) )
33		oveq2 6658	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( w +P. u ) = z -> ( y .P. ( w +P. u ) ) = ( y .P. z ) )
34	32, 33	syl5eqr 2670	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( w +P. u ) = z -> ( ( y .P. w ) +P. ( y .P. u ) ) = ( y .P. z ) )
35	34	oveq1d 6665	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( w +P. u ) = z -> ( ( ( y .P. w ) +P. ( y .P. u ) ) +P. ( ( v .P. w ) +P. ( v .P. u ) ) ) = ( ( y .P. z ) +P. ( ( v .P. w ) +P. ( v .P. u ) ) ) )
36		vex 3203	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- y e. _V
37		vex 3203	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- v e. _V
38		vex 3203	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- w e. _V
39		mulcompr 9845	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( f .P. g ) = ( g .P. f )
40		distrpr 9850	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( f .P. ( g +P. h ) ) = ( ( f .P. g ) +P. ( f .P. h ) )
41	36, 37, 38, 39, 40	caovdir 6868	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( y +P. v ) .P. w ) = ( ( y .P. w ) +P. ( v .P. w ) )
42		vex 3203	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- u e. _V
43	36, 37, 42, 39, 40	caovdir 6868	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( y +P. v ) .P. u ) = ( ( y .P. u ) +P. ( v .P. u ) )
44	41, 43	oveq12i 6662	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( ( y +P. v ) .P. w ) +P. ( ( y +P. v ) .P. u ) ) = ( ( ( y .P. w ) +P. ( v .P. w ) ) +P. ( ( y .P. u ) +P. ( v .P. u ) ) )
45		distrpr 9850	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( y +P. v ) .P. ( w +P. u ) ) = ( ( ( y +P. v ) .P. w ) +P. ( ( y +P. v ) .P. u ) )
46		ovex 6678	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( y .P. w ) e. _V
47		ovex 6678	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( y .P. u ) e. _V
48		ovex 6678	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( v .P. w ) e. _V
49		addcompr 9843	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( f +P. g ) = ( g +P. f )
50		addasspr 9844	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( f +P. g ) +P. h ) = ( f +P. ( g +P. h ) )
51		ovex 6678	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( v .P. u ) e. _V
52	46, 47, 48, 49, 50, 51	caov4 6865	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( ( y .P. w ) +P. ( y .P. u ) ) +P. ( ( v .P. w ) +P. ( v .P. u ) ) ) = ( ( ( y .P. w ) +P. ( v .P. w ) ) +P. ( ( y .P. u ) +P. ( v .P. u ) ) )
53	44, 45, 52	3eqtr4i 2654	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( y +P. v ) .P. ( w +P. u ) ) = ( ( ( y .P. w ) +P. ( y .P. u ) ) +P. ( ( v .P. w ) +P. ( v .P. u ) ) )
54		ovex 6678	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( y .P. z ) e. _V
55	48, 54, 51, 49, 50	caov12 6862	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( v .P. w ) +P. ( ( y .P. z ) +P. ( v .P. u ) ) ) = ( ( y .P. z ) +P. ( ( v .P. w ) +P. ( v .P. u ) ) )
56	35, 53, 55	3eqtr4g 2681	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( w +P. u ) = z -> ( ( y +P. v ) .P. ( w +P. u ) ) = ( ( v .P. w ) +P. ( ( y .P. z ) +P. ( v .P. u ) ) ) )
57		oveq1 6657	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( y +P. v ) = x -> ( ( y +P. v ) .P. w ) = ( x .P. w ) )
58	41, 57	syl5eqr 2670	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( y +P. v ) = x -> ( ( y .P. w ) +P. ( v .P. w ) ) = ( x .P. w ) )
59	56, 58	oveqan12rd 6670	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( y +P. v ) = x /\ ( w +P. u ) = z ) -> ( ( ( y +P. v ) .P. ( w +P. u ) ) +P. ( ( y .P. w ) +P. ( v .P. w ) ) ) = ( ( ( v .P. w ) +P. ( ( y .P. z ) +P. ( v .P. u ) ) ) +P. ( x .P. w ) ) )
60	31, 59	eqtr3d 2658	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( y +P. v ) = x /\ ( w +P. u ) = z ) -> ( ( x .P. z ) +P. ( ( y .P. w ) +P. ( v .P. w ) ) ) = ( ( ( v .P. w ) +P. ( ( y .P. z ) +P. ( v .P. u ) ) ) +P. ( x .P. w ) ) )
61		addasspr 9844	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( x .P. z ) +P. ( y .P. w ) ) +P. ( v .P. w ) ) = ( ( x .P. z ) +P. ( ( y .P. w ) +P. ( v .P. w ) ) )
62		addcompr 9843	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( x .P. z ) +P. ( y .P. w ) ) +P. ( v .P. w ) ) = ( ( v .P. w ) +P. ( ( x .P. z ) +P. ( y .P. w ) ) )
63	61, 62	eqtr3i 2646	. . . . . . . . . . . . . . . . . . . 20 |- ( ( x .P. z ) +P. ( ( y .P. w ) +P. ( v .P. w ) ) ) = ( ( v .P. w ) +P. ( ( x .P. z ) +P. ( y .P. w ) ) )
64		addasspr 9844	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( v .P. w ) +P. ( x .P. w ) ) +P. ( ( y .P. z ) +P. ( v .P. u ) ) ) = ( ( v .P. w ) +P. ( ( x .P. w ) +P. ( ( y .P. z ) +P. ( v .P. u ) ) ) )
65		ovex 6678	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( y .P. z ) +P. ( v .P. u ) ) e. _V
66		ovex 6678	. . . . . . . . . . . . . . . . . . . . . 22 |- ( x .P. w ) e. _V
67	48, 65, 66, 49, 50	caov32 6861	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( v .P. w ) +P. ( ( y .P. z ) +P. ( v .P. u ) ) ) +P. ( x .P. w ) ) = ( ( ( v .P. w ) +P. ( x .P. w ) ) +P. ( ( y .P. z ) +P. ( v .P. u ) ) )
68		addasspr 9844	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( x .P. w ) +P. ( y .P. z ) ) +P. ( v .P. u ) ) = ( ( x .P. w ) +P. ( ( y .P. z ) +P. ( v .P. u ) ) )
69	68	oveq2i 6661	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( v .P. w ) +P. ( ( ( x .P. w ) +P. ( y .P. z ) ) +P. ( v .P. u ) ) ) = ( ( v .P. w ) +P. ( ( x .P. w ) +P. ( ( y .P. z ) +P. ( v .P. u ) ) ) )
70	64, 67, 69	3eqtr4i 2654	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( v .P. w ) +P. ( ( y .P. z ) +P. ( v .P. u ) ) ) +P. ( x .P. w ) ) = ( ( v .P. w ) +P. ( ( ( x .P. w ) +P. ( y .P. z ) ) +P. ( v .P. u ) ) )
71	60, 63, 70	3eqtr3g 2679	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( y +P. v ) = x /\ ( w +P. u ) = z ) -> ( ( v .P. w ) +P. ( ( x .P. z ) +P. ( y .P. w ) ) ) = ( ( v .P. w ) +P. ( ( ( x .P. w ) +P. ( y .P. z ) ) +P. ( v .P. u ) ) ) )
72		addcanpr 9868	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( v .P. w ) e. P. /\ ( ( x .P. z ) +P. ( y .P. w ) ) e. P. ) -> ( ( ( v .P. w ) +P. ( ( x .P. z ) +P. ( y .P. w ) ) ) = ( ( v .P. w ) +P. ( ( ( x .P. w ) +P. ( y .P. z ) ) +P. ( v .P. u ) ) ) -> ( ( x .P. z ) +P. ( y .P. w ) ) = ( ( ( x .P. w ) +P. ( y .P. z ) ) +P. ( v .P. u ) ) ) )
73	71, 72	syl5 34	. . . . . . . . . . . . . . . . . 18 |- ( ( ( v .P. w ) e. P. /\ ( ( x .P. z ) +P. ( y .P. w ) ) e. P. ) -> ( ( ( y +P. v ) = x /\ ( w +P. u ) = z ) -> ( ( x .P. z ) +P. ( y .P. w ) ) = ( ( ( x .P. w ) +P. ( y .P. z ) ) +P. ( v .P. u ) ) ) )
74		eqcom 2629	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( x .P. z ) +P. ( y .P. w ) ) = ( ( ( x .P. w ) +P. ( y .P. z ) ) +P. ( v .P. u ) ) <-> ( ( ( x .P. w ) +P. ( y .P. z ) ) +P. ( v .P. u ) ) = ( ( x .P. z ) +P. ( y .P. w ) ) )
75		ltaddpr2 9857	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( x .P. z ) +P. ( y .P. w ) ) e. P. -> ( ( ( ( x .P. w ) +P. ( y .P. z ) ) +P. ( v .P. u ) ) = ( ( x .P. z ) +P. ( y .P. w ) ) -> ( ( x .P. w ) +P. ( y .P. z ) ) <P ( ( x .P. z ) +P. ( y .P. w ) ) ) )
76	74, 75	syl5bi 232	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( x .P. z ) +P. ( y .P. w ) ) e. P. -> ( ( ( x .P. z ) +P. ( y .P. w ) ) = ( ( ( x .P. w ) +P. ( y .P. z ) ) +P. ( v .P. u ) ) -> ( ( x .P. w ) +P. ( y .P. z ) ) <P ( ( x .P. z ) +P. ( y .P. w ) ) ) )
77	76	adantl 482	. . . . . . . . . . . . . . . . . 18 |- ( ( ( v .P. w ) e. P. /\ ( ( x .P. z ) +P. ( y .P. w ) ) e. P. ) -> ( ( ( x .P. z ) +P. ( y .P. w ) ) = ( ( ( x .P. w ) +P. ( y .P. z ) ) +P. ( v .P. u ) ) -> ( ( x .P. w ) +P. ( y .P. z ) ) <P ( ( x .P. z ) +P. ( y .P. w ) ) ) )
78	73, 77	syld 47	. . . . . . . . . . . . . . . . 17 |- ( ( ( v .P. w ) e. P. /\ ( ( x .P. z ) +P. ( y .P. w ) ) e. P. ) -> ( ( ( y +P. v ) = x /\ ( w +P. u ) = z ) -> ( ( x .P. w ) +P. ( y .P. z ) ) <P ( ( x .P. z ) +P. ( y .P. w ) ) ) )
79	29, 78	sylan 488	. . . . . . . . . . . . . . . 16 |- ( ( ( v e. P. /\ w e. P. ) /\ ( ( x .P. z ) +P. ( y .P. w ) ) e. P. ) -> ( ( ( y +P. v ) = x /\ ( w +P. u ) = z ) -> ( ( x .P. w ) +P. ( y .P. z ) ) <P ( ( x .P. z ) +P. ( y .P. w ) ) ) )
80	79	a1d 25	. . . . . . . . . . . . . . 15 |- ( ( ( v e. P. /\ w e. P. ) /\ ( ( x .P. z ) +P. ( y .P. w ) ) e. P. ) -> ( u e. P. -> ( ( ( y +P. v ) = x /\ ( w +P. u ) = z ) -> ( ( x .P. w ) +P. ( y .P. z ) ) <P ( ( x .P. z ) +P. ( y .P. w ) ) ) ) )
81	80	exp4a 633	. . . . . . . . . . . . . 14 |- ( ( ( v e. P. /\ w e. P. ) /\ ( ( x .P. z ) +P. ( y .P. w ) ) e. P. ) -> ( u e. P. -> ( ( y +P. v ) = x -> ( ( w +P. u ) = z -> ( ( x .P. w ) +P. ( y .P. z ) ) <P ( ( x .P. z ) +P. ( y .P. w ) ) ) ) ) )
82	81	com34 91	. . . . . . . . . . . . 13 |- ( ( ( v e. P. /\ w e. P. ) /\ ( ( x .P. z ) +P. ( y .P. w ) ) e. P. ) -> ( u e. P. -> ( ( w +P. u ) = z -> ( ( y +P. v ) = x -> ( ( x .P. w ) +P. ( y .P. z ) ) <P ( ( x .P. z ) +P. ( y .P. w ) ) ) ) ) )
83	82	rexlimdv 3030	. . . . . . . . . . . 12 |- ( ( ( v e. P. /\ w e. P. ) /\ ( ( x .P. z ) +P. ( y .P. w ) ) e. P. ) -> ( E. u e. P. ( w +P. u ) = z -> ( ( y +P. v ) = x -> ( ( x .P. w ) +P. ( y .P. z ) ) <P ( ( x .P. z ) +P. ( y .P. w ) ) ) ) )
84	83	expl 648	. . . . . . . . . . 11 |- ( v e. P. -> ( ( w e. P. /\ ( ( x .P. z ) +P. ( y .P. w ) ) e. P. ) -> ( E. u e. P. ( w +P. u ) = z -> ( ( y +P. v ) = x -> ( ( x .P. w ) +P. ( y .P. z ) ) <P ( ( x .P. z ) +P. ( y .P. w ) ) ) ) ) )
85	84	com24 95	. . . . . . . . . 10 |- ( v e. P. -> ( ( y +P. v ) = x -> ( E. u e. P. ( w +P. u ) = z -> ( ( w e. P. /\ ( ( x .P. z ) +P. ( y .P. w ) ) e. P. ) -> ( ( x .P. w ) +P. ( y .P. z ) ) <P ( ( x .P. z ) +P. ( y .P. w ) ) ) ) ) )
86	85	rexlimiv 3027	. . . . . . . . 9 |- ( E. v e. P. ( y +P. v ) = x -> ( E. u e. P. ( w +P. u ) = z -> ( ( w e. P. /\ ( ( x .P. z ) +P. ( y .P. w ) ) e. P. ) -> ( ( x .P. w ) +P. ( y .P. z ) ) <P ( ( x .P. z ) +P. ( y .P. w ) ) ) ) )
87	27, 28, 86	syl2im 40	. . . . . . . 8 |- ( y <P x -> ( w <P z -> ( ( w e. P. /\ ( ( x .P. z ) +P. ( y .P. w ) ) e. P. ) -> ( ( x .P. w ) +P. ( y .P. z ) ) <P ( ( x .P. z ) +P. ( y .P. w ) ) ) ) )
88	87	imp 445	. . . . . . 7 |- ( ( y <P x /\ w <P z ) -> ( ( w e. P. /\ ( ( x .P. z ) +P. ( y .P. w ) ) e. P. ) -> ( ( x .P. w ) +P. ( y .P. z ) ) <P ( ( x .P. z ) +P. ( y .P. w ) ) ) )
89	88	com12 32	. . . . . 6 |- ( ( w e. P. /\ ( ( x .P. z ) +P. ( y .P. w ) ) e. P. ) -> ( ( y <P x /\ w <P z ) -> ( ( x .P. w ) +P. ( y .P. z ) ) <P ( ( x .P. z ) +P. ( y .P. w ) ) ) )
90	21, 26, 89	syl2anc 693	. . . . 5 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( ( y <P x /\ w <P z ) -> ( ( x .P. w ) +P. ( y .P. z ) ) <P ( ( x .P. z ) +P. ( y .P. w ) ) ) )
91		mulsrpr 9897	. . . . . . 7 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( [ <. x , y >. ] ~R .R [ <. z , w >. ] ~R ) = [ <. ( ( x .P. z ) +P. ( y .P. w ) ) , ( ( x .P. w ) +P. ( y .P. z ) ) >. ] ~R )
92	91	breq2d 4665	. . . . . 6 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( 0R <R ( [ <. x , y >. ] ~R .R [ <. z , w >. ] ~R ) <-> 0R <R [ <. ( ( x .P. z ) +P. ( y .P. w ) ) , ( ( x .P. w ) +P. ( y .P. z ) ) >. ] ~R ) )
93		gt0srpr 9899	. . . . . 6 |- ( 0R <R [ <. ( ( x .P. z ) +P. ( y .P. w ) ) , ( ( x .P. w ) +P. ( y .P. z ) ) >. ] ~R <-> ( ( x .P. w ) +P. ( y .P. z ) ) <P ( ( x .P. z ) +P. ( y .P. w ) ) )
94	92, 93	syl6bb 276	. . . . 5 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( 0R <R ( [ <. x , y >. ] ~R .R [ <. z , w >. ] ~R ) <-> ( ( x .P. w ) +P. ( y .P. z ) ) <P ( ( x .P. z ) +P. ( y .P. w ) ) ) )
95	90, 94	sylibrd 249	. . . 4 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( ( y <P x /\ w <P z ) -> 0R <R ( [ <. x , y >. ] ~R .R [ <. z , w >. ] ~R ) ) )
96	20, 95	syl5bi 232	. . 3 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( ( 0R <R [ <. x , y >. ] ~R /\ 0R <R [ <. z , w >. ] ~R ) -> 0R <R ( [ <. x , y >. ] ~R .R [ <. z , w >. ] ~R ) ) )
97	7, 12, 17, 96	2ecoptocl 7838	. 2 |- ( ( A e. R. /\ B e. R. ) -> ( ( 0R <R A /\ 0R <R B ) -> 0R <R ( A .R B ) ) )
98	6, 97	mpcom 38	1 |- ( ( 0R <R A /\ 0R <R B ) -> 0R <R ( A .R B ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   <.cop 4183   class class class wbr 4653  (class class class)co 6650   [cec 7740   P.cnp 9681   +P. cpp 9683   .P. cmp 9684   <P cltp 9685   ~R cer 9686   R.cnr 9687   0Rc0r 9688   .R cmr 9692   <R cltr 9693

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-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538

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-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  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-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-omul 7565  df-er 7742  df-ec 7744  df-qs 7748  df-ni 9694  df-pli 9695  df-mi 9696  df-lti 9697  df-plpq 9730  df-mpq 9731  df-ltpq 9732  df-enq 9733  df-nq 9734  df-erq 9735  df-plq 9736  df-mq 9737  df-1nq 9738  df-rq 9739  df-ltnq 9740  df-np 9803  df-1p 9804  df-plp 9805  df-mp 9806  df-ltp 9807  df-enr 9877  df-nr 9878  df-mr 9880  df-ltr 9881  df-0r 9882

This theorem is referenced by:  sqgt0sr  9927  axpre-mulgt0  9989