Theorem mulgt0sr 7768

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

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 7728	. . . . 5 |- <R C_ ( R. X. R. )
2	1	brel 4675	. . . 4 |- ( 0R <R A -> ( 0R e. R. /\ A e. R. ) )
3	2	simprd 114	. . 3 |- ( 0R <R A -> A e. R. )
4	1	brel 4675	. . . 4 |- ( 0R <R B -> ( 0R e. R. /\ B e. R. ) )
5	4	simprd 114	. . 3 |- ( 0R <R B -> B e. R. )
6	3, 5	anim12i 338	. 2 |- ( ( 0R <R A /\ 0R <R B ) -> ( A e. R. /\ B e. R. ) )
7		df-nr 7717	. . 3 |- R. = ( ( P. X. P. ) /. ~R )
8		breq2 4004	. . . . 5 |- ( [ <. x , y >. ] ~R = A -> ( 0R <R [ <. x , y >. ] ~R <-> 0R <R A ) )
9	8	anbi1d 465	. . . 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 5876	. . . . 5 |- ( [ <. x , y >. ] ~R = A -> ( [ <. x , y >. ] ~R .R [ <. z , w >. ] ~R ) = ( A .R [ <. z , w >. ] ~R ) )
11	10	breq2d 4012	. . . 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 234	. . 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 4004	. . . . 5 |- ( [ <. z , w >. ] ~R = B -> ( 0R <R [ <. z , w >. ] ~R <-> 0R <R B ) )
14	13	anbi2d 464	. . . 4 |- ( [ <. z , w >. ] ~R = B -> ( ( 0R <R A /\ 0R <R [ <. z , w >. ] ~R ) <-> ( 0R <R A /\ 0R <R B ) ) )
15		oveq2 5877	. . . . 5 |- ( [ <. z , w >. ] ~R = B -> ( A .R [ <. z , w >. ] ~R ) = ( A .R B ) )
16	15	breq2d 4012	. . . 4 |- ( [ <. z , w >. ] ~R = B -> ( 0R <R ( A .R [ <. z , w >. ] ~R ) <-> 0R <R ( A .R B ) ) )
17	14, 16	imbi12d 234	. . 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 7738	. . . . 5 |- ( 0R <R [ <. x , y >. ] ~R <-> y <P x )
19		gt0srpr 7738	. . . . 5 |- ( 0R <R [ <. z , w >. ] ~R <-> w <P z )
20	18, 19	anbi12i 460	. . . 4 |- ( ( 0R <R [ <. x , y >. ] ~R /\ 0R <R [ <. z , w >. ] ~R ) <-> ( y <P x /\ w <P z ) )
21		ltexpri 7603	. . . . . . 7 |- ( y <P x -> E. v e. P. ( y +P. v ) = x )
22		ltexpri 7603	. . . . . . . . 9 |- ( w <P z -> E. u e. P. ( w +P. u ) = z )
23		addclpr 7527	. . . . . . . . . . . . . 14 |- ( ( f e. P. /\ g e. P. ) -> ( f +P. g ) e. P. )
24	23	adantl 277	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) /\ ( v e. P. /\ ( y +P. v ) = x ) ) /\ ( u e. P. /\ ( w +P. u ) = z ) ) /\ ( f e. P. /\ g e. P. ) ) -> ( f +P. g ) e. P. )
25		simplrr 536	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) /\ ( v e. P. /\ ( y +P. v ) = x ) ) /\ ( u e. P. /\ ( w +P. u ) = z ) ) -> ( y +P. v ) = x )
26		simplr 528	. . . . . . . . . . . . . . . . 17 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> y e. P. )
27	26	ad2antrr 488	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) /\ ( v e. P. /\ ( y +P. v ) = x ) ) /\ ( u e. P. /\ ( w +P. u ) = z ) ) -> y e. P. )
28		simplrl 535	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) /\ ( v e. P. /\ ( y +P. v ) = x ) ) /\ ( u e. P. /\ ( w +P. u ) = z ) ) -> v e. P. )
29	24, 27, 28	caovcld 6022	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) /\ ( v e. P. /\ ( y +P. v ) = x ) ) /\ ( u e. P. /\ ( w +P. u ) = z ) ) -> ( y +P. v ) e. P. )
30	25, 29	eqeltrrd 2255	. . . . . . . . . . . . . 14 |- ( ( ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) /\ ( v e. P. /\ ( y +P. v ) = x ) ) /\ ( u e. P. /\ ( w +P. u ) = z ) ) -> x e. P. )
31		simplrr 536	. . . . . . . . . . . . . . 15 |- ( ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) /\ ( v e. P. /\ ( y +P. v ) = x ) ) -> w e. P. )
32	31	adantr 276	. . . . . . . . . . . . . 14 |- ( ( ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) /\ ( v e. P. /\ ( y +P. v ) = x ) ) /\ ( u e. P. /\ ( w +P. u ) = z ) ) -> w e. P. )
33		mulclpr 7562	. . . . . . . . . . . . . 14 |- ( ( x e. P. /\ w e. P. ) -> ( x .P. w ) e. P. )
34	30, 32, 33	syl2anc 411	. . . . . . . . . . . . 13 |- ( ( ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) /\ ( v e. P. /\ ( y +P. v ) = x ) ) /\ ( u e. P. /\ ( w +P. u ) = z ) ) -> ( x .P. w ) e. P. )
35		simplrl 535	. . . . . . . . . . . . . . 15 |- ( ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) /\ ( v e. P. /\ ( y +P. v ) = x ) ) -> z e. P. )
36	35	adantr 276	. . . . . . . . . . . . . 14 |- ( ( ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) /\ ( v e. P. /\ ( y +P. v ) = x ) ) /\ ( u e. P. /\ ( w +P. u ) = z ) ) -> z e. P. )
37		mulclpr 7562	. . . . . . . . . . . . . 14 |- ( ( y e. P. /\ z e. P. ) -> ( y .P. z ) e. P. )
38	27, 36, 37	syl2anc 411	. . . . . . . . . . . . 13 |- ( ( ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) /\ ( v e. P. /\ ( y +P. v ) = x ) ) /\ ( u e. P. /\ ( w +P. u ) = z ) ) -> ( y .P. z ) e. P. )
39	24, 34, 38	caovcld 6022	. . . . . . . . . . . 12 |- ( ( ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) /\ ( v e. P. /\ ( y +P. v ) = x ) ) /\ ( u e. P. /\ ( w +P. u ) = z ) ) -> ( ( x .P. w ) +P. ( y .P. z ) ) e. P. )
40		simprl 529	. . . . . . . . . . . . 13 |- ( ( ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) /\ ( v e. P. /\ ( y +P. v ) = x ) ) /\ ( u e. P. /\ ( w +P. u ) = z ) ) -> u e. P. )
41		mulclpr 7562	. . . . . . . . . . . . 13 |- ( ( v e. P. /\ u e. P. ) -> ( v .P. u ) e. P. )
42	28, 40, 41	syl2anc 411	. . . . . . . . . . . 12 |- ( ( ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) /\ ( v e. P. /\ ( y +P. v ) = x ) ) /\ ( u e. P. /\ ( w +P. u ) = z ) ) -> ( v .P. u ) e. P. )
43		ltaddpr 7587	. . . . . . . . . . . 12 |- ( ( ( ( x .P. w ) +P. ( y .P. z ) ) e. P. /\ ( v .P. u ) e. P. ) -> ( ( x .P. w ) +P. ( y .P. z ) ) <P ( ( ( x .P. w ) +P. ( y .P. z ) ) +P. ( v .P. u ) ) )
44	39, 42, 43	syl2anc 411	. . . . . . . . . . 11 |- ( ( ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) /\ ( v e. P. /\ ( y +P. v ) = x ) ) /\ ( u e. P. /\ ( w +P. u ) = z ) ) -> ( ( x .P. w ) +P. ( y .P. z ) ) <P ( ( ( x .P. w ) +P. ( y .P. z ) ) +P. ( v .P. u ) ) )
45		simprr 531	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) /\ ( v e. P. /\ ( y +P. v ) = x ) ) /\ ( u e. P. /\ ( w +P. u ) = z ) ) -> ( w +P. u ) = z )
46		oveq12 5878	. . . . . . . . . . . . . . . 16 |- ( ( ( y +P. v ) = x /\ ( w +P. u ) = z ) -> ( ( y +P. v ) .P. ( w +P. u ) ) = ( x .P. z ) )
47	46	oveq1d 5884	. . . . . . . . . . . . . . 15 |- ( ( ( 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 ) ) ) )
48	25, 45, 47	syl2anc 411	. . . . . . . . . . . . . 14 |- ( ( ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) /\ ( v e. P. /\ ( y +P. v ) = x ) ) /\ ( u e. P. /\ ( 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 ) ) ) )
49		distrprg 7578	. . . . . . . . . . . . . . . . . . 19 |- ( ( y e. P. /\ w e. P. /\ u e. P. ) -> ( y .P. ( w +P. u ) ) = ( ( y .P. w ) +P. ( y .P. u ) ) )
50	27, 32, 40, 49	syl3anc 1238	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) /\ ( v e. P. /\ ( y +P. v ) = x ) ) /\ ( u e. P. /\ ( w +P. u ) = z ) ) -> ( y .P. ( w +P. u ) ) = ( ( y .P. w ) +P. ( y .P. u ) ) )
51		oveq2 5877	. . . . . . . . . . . . . . . . . . . 20 |- ( ( w +P. u ) = z -> ( y .P. ( w +P. u ) ) = ( y .P. z ) )
52	51	adantl 277	. . . . . . . . . . . . . . . . . . 19 |- ( ( u e. P. /\ ( w +P. u ) = z ) -> ( y .P. ( w +P. u ) ) = ( y .P. z ) )
53	52	adantl 277	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) /\ ( v e. P. /\ ( y +P. v ) = x ) ) /\ ( u e. P. /\ ( w +P. u ) = z ) ) -> ( y .P. ( w +P. u ) ) = ( y .P. z ) )
54	50, 53	eqtr3d 2212	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) /\ ( v e. P. /\ ( y +P. v ) = x ) ) /\ ( u e. P. /\ ( w +P. u ) = z ) ) -> ( ( y .P. w ) +P. ( y .P. u ) ) = ( y .P. z ) )
55	54	oveq1d 5884	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) /\ ( v e. P. /\ ( y +P. v ) = x ) ) /\ ( u e. P. /\ ( 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 ) ) ) )
56		distrprg 7578	. . . . . . . . . . . . . . . . . . . 20 |- ( ( f e. P. /\ g e. P. /\ h e. P. ) -> ( f .P. ( g +P. h ) ) = ( ( f .P. g ) +P. ( f .P. h ) ) )
57	56	adantl 277	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) /\ ( v e. P. /\ ( y +P. v ) = x ) ) /\ ( u e. P. /\ ( w +P. u ) = z ) ) /\ ( f e. P. /\ g e. P. /\ h e. P. ) ) -> ( f .P. ( g +P. h ) ) = ( ( f .P. g ) +P. ( f .P. h ) ) )
58		mulcomprg 7570	. . . . . . . . . . . . . . . . . . . 20 |- ( ( f e. P. /\ g e. P. ) -> ( f .P. g ) = ( g .P. f ) )
59	58	adantl 277	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) /\ ( v e. P. /\ ( y +P. v ) = x ) ) /\ ( u e. P. /\ ( w +P. u ) = z ) ) /\ ( f e. P. /\ g e. P. ) ) -> ( f .P. g ) = ( g .P. f ) )
60	57, 27, 28, 32, 24, 59	caovdir2d 6045	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) /\ ( v e. P. /\ ( y +P. v ) = x ) ) /\ ( u e. P. /\ ( w +P. u ) = z ) ) -> ( ( y +P. v ) .P. w ) = ( ( y .P. w ) +P. ( v .P. w ) ) )
61	57, 27, 28, 40, 24, 59	caovdir2d 6045	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) /\ ( v e. P. /\ ( y +P. v ) = x ) ) /\ ( u e. P. /\ ( w +P. u ) = z ) ) -> ( ( y +P. v ) .P. u ) = ( ( y .P. u ) +P. ( v .P. u ) ) )
62	60, 61	oveq12d 5887	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) /\ ( v e. P. /\ ( y +P. v ) = x ) ) /\ ( u e. P. /\ ( w +P. u ) = z ) ) -> ( ( ( 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 ) ) ) )
63		distrprg 7578	. . . . . . . . . . . . . . . . . 18 |- ( ( ( y +P. v ) e. P. /\ w e. P. /\ u e. P. ) -> ( ( y +P. v ) .P. ( w +P. u ) ) = ( ( ( y +P. v ) .P. w ) +P. ( ( y +P. v ) .P. u ) ) )
64	29, 32, 40, 63	syl3anc 1238	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) /\ ( v e. P. /\ ( y +P. v ) = x ) ) /\ ( u e. P. /\ ( w +P. u ) = z ) ) -> ( ( y +P. v ) .P. ( w +P. u ) ) = ( ( ( y +P. v ) .P. w ) +P. ( ( y +P. v ) .P. u ) ) )
65		mulclpr 7562	. . . . . . . . . . . . . . . . . . 19 |- ( ( y e. P. /\ w e. P. ) -> ( y .P. w ) e. P. )
66	27, 32, 65	syl2anc 411	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) /\ ( v e. P. /\ ( y +P. v ) = x ) ) /\ ( u e. P. /\ ( w +P. u ) = z ) ) -> ( y .P. w ) e. P. )
67		mulclpr 7562	. . . . . . . . . . . . . . . . . . 19 |- ( ( y e. P. /\ u e. P. ) -> ( y .P. u ) e. P. )
68	27, 40, 67	syl2anc 411	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) /\ ( v e. P. /\ ( y +P. v ) = x ) ) /\ ( u e. P. /\ ( w +P. u ) = z ) ) -> ( y .P. u ) e. P. )
69		mulclpr 7562	. . . . . . . . . . . . . . . . . . 19 |- ( ( v e. P. /\ w e. P. ) -> ( v .P. w ) e. P. )
70	28, 32, 69	syl2anc 411	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) /\ ( v e. P. /\ ( y +P. v ) = x ) ) /\ ( u e. P. /\ ( w +P. u ) = z ) ) -> ( v .P. w ) e. P. )
71		addcomprg 7568	. . . . . . . . . . . . . . . . . . 19 |- ( ( f e. P. /\ g e. P. ) -> ( f +P. g ) = ( g +P. f ) )
72	71	adantl 277	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) /\ ( v e. P. /\ ( y +P. v ) = x ) ) /\ ( u e. P. /\ ( w +P. u ) = z ) ) /\ ( f e. P. /\ g e. P. ) ) -> ( f +P. g ) = ( g +P. f ) )
73		addassprg 7569	. . . . . . . . . . . . . . . . . . 19 |- ( ( f e. P. /\ g e. P. /\ h e. P. ) -> ( ( f +P. g ) +P. h ) = ( f +P. ( g +P. h ) ) )
74	73	adantl 277	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) /\ ( v e. P. /\ ( y +P. v ) = x ) ) /\ ( u e. P. /\ ( w +P. u ) = z ) ) /\ ( f e. P. /\ g e. P. /\ h e. P. ) ) -> ( ( f +P. g ) +P. h ) = ( f +P. ( g +P. h ) ) )
75	66, 68, 70, 72, 74, 42, 24	caov4d 6053	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) /\ ( v e. P. /\ ( y +P. v ) = x ) ) /\ ( u e. P. /\ ( w +P. u ) = z ) ) -> ( ( ( 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 ) ) ) )
76	62, 64, 75	3eqtr4d 2220	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) /\ ( v e. P. /\ ( y +P. v ) = x ) ) /\ ( u e. P. /\ ( w +P. u ) = z ) ) -> ( ( y +P. v ) .P. ( w +P. u ) ) = ( ( ( y .P. w ) +P. ( y .P. u ) ) +P. ( ( v .P. w ) +P. ( v .P. u ) ) ) )
77	70, 38, 42, 72, 74	caov12d 6050	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) /\ ( v e. P. /\ ( y +P. v ) = x ) ) /\ ( u e. P. /\ ( w +P. u ) = z ) ) -> ( ( v .P. w ) +P. ( ( y .P. z ) +P. ( v .P. u ) ) ) = ( ( y .P. z ) +P. ( ( v .P. w ) +P. ( v .P. u ) ) ) )
78	55, 76, 77	3eqtr4d 2220	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) /\ ( v e. P. /\ ( y +P. v ) = x ) ) /\ ( u e. P. /\ ( w +P. u ) = z ) ) -> ( ( y +P. v ) .P. ( w +P. u ) ) = ( ( v .P. w ) +P. ( ( y .P. z ) +P. ( v .P. u ) ) ) )
79		oveq1 5876	. . . . . . . . . . . . . . . . . 18 |- ( ( y +P. v ) = x -> ( ( y +P. v ) .P. w ) = ( x .P. w ) )
80	79	adantl 277	. . . . . . . . . . . . . . . . 17 |- ( ( v e. P. /\ ( y +P. v ) = x ) -> ( ( y +P. v ) .P. w ) = ( x .P. w ) )
81	80	ad2antlr 489	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) /\ ( v e. P. /\ ( y +P. v ) = x ) ) /\ ( u e. P. /\ ( w +P. u ) = z ) ) -> ( ( y +P. v ) .P. w ) = ( x .P. w ) )
82	60, 81	eqtr3d 2212	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) /\ ( v e. P. /\ ( y +P. v ) = x ) ) /\ ( u e. P. /\ ( w +P. u ) = z ) ) -> ( ( y .P. w ) +P. ( v .P. w ) ) = ( x .P. w ) )
83	78, 82	oveq12d 5887	. . . . . . . . . . . . . 14 |- ( ( ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) /\ ( v e. P. /\ ( y +P. v ) = x ) ) /\ ( u e. P. /\ ( 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 ) ) )
84	48, 83	eqtr3d 2212	. . . . . . . . . . . . 13 |- ( ( ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) /\ ( v e. P. /\ ( y +P. v ) = x ) ) /\ ( u e. P. /\ ( 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 ) ) )
85		mulclpr 7562	. . . . . . . . . . . . . . . 16 |- ( ( x e. P. /\ z e. P. ) -> ( x .P. z ) e. P. )
86	30, 36, 85	syl2anc 411	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) /\ ( v e. P. /\ ( y +P. v ) = x ) ) /\ ( u e. P. /\ ( w +P. u ) = z ) ) -> ( x .P. z ) e. P. )
87		addassprg 7569	. . . . . . . . . . . . . . 15 |- ( ( ( x .P. z ) e. P. /\ ( y .P. w ) e. P. /\ ( v .P. w ) e. P. ) -> ( ( ( x .P. z ) +P. ( y .P. w ) ) +P. ( v .P. w ) ) = ( ( x .P. z ) +P. ( ( y .P. w ) +P. ( v .P. w ) ) ) )
88	86, 66, 70, 87	syl3anc 1238	. . . . . . . . . . . . . 14 |- ( ( ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) /\ ( v e. P. /\ ( y +P. v ) = x ) ) /\ ( u e. P. /\ ( w +P. u ) = z ) ) -> ( ( ( x .P. z ) +P. ( y .P. w ) ) +P. ( v .P. w ) ) = ( ( x .P. z ) +P. ( ( y .P. w ) +P. ( v .P. w ) ) ) )
89		addclpr 7527	. . . . . . . . . . . . . . . 16 |- ( ( ( x .P. z ) e. P. /\ ( y .P. w ) e. P. ) -> ( ( x .P. z ) +P. ( y .P. w ) ) e. P. )
90	86, 66, 89	syl2anc 411	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) /\ ( v e. P. /\ ( y +P. v ) = x ) ) /\ ( u e. P. /\ ( w +P. u ) = z ) ) -> ( ( x .P. z ) +P. ( y .P. w ) ) e. P. )
91		addcomprg 7568	. . . . . . . . . . . . . . 15 |- ( ( ( ( x .P. z ) +P. ( y .P. w ) ) e. P. /\ ( v .P. w ) e. P. ) -> ( ( ( x .P. z ) +P. ( y .P. w ) ) +P. ( v .P. w ) ) = ( ( v .P. w ) +P. ( ( x .P. z ) +P. ( y .P. w ) ) ) )
92	90, 70, 91	syl2anc 411	. . . . . . . . . . . . . 14 |- ( ( ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) /\ ( v e. P. /\ ( y +P. v ) = x ) ) /\ ( u e. P. /\ ( w +P. u ) = z ) ) -> ( ( ( x .P. z ) +P. ( y .P. w ) ) +P. ( v .P. w ) ) = ( ( v .P. w ) +P. ( ( x .P. z ) +P. ( y .P. w ) ) ) )
93	88, 92	eqtr3d 2212	. . . . . . . . . . . . 13 |- ( ( ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) /\ ( v e. P. /\ ( y +P. v ) = x ) ) /\ ( u e. P. /\ ( w +P. u ) = z ) ) -> ( ( x .P. z ) +P. ( ( y .P. w ) +P. ( v .P. w ) ) ) = ( ( v .P. w ) +P. ( ( x .P. z ) +P. ( y .P. w ) ) ) )
94	24, 38, 42	caovcld 6022	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) /\ ( v e. P. /\ ( y +P. v ) = x ) ) /\ ( u e. P. /\ ( w +P. u ) = z ) ) -> ( ( y .P. z ) +P. ( v .P. u ) ) e. P. )
95		addassprg 7569	. . . . . . . . . . . . . . 15 |- ( ( ( v .P. w ) e. P. /\ ( x .P. w ) e. P. /\ ( ( y .P. z ) +P. ( v .P. u ) ) e. P. ) -> ( ( ( 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 ) ) ) ) )
96	70, 34, 94, 95	syl3anc 1238	. . . . . . . . . . . . . 14 |- ( ( ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) /\ ( v e. P. /\ ( y +P. v ) = x ) ) /\ ( u e. P. /\ ( w +P. u ) = z ) ) -> ( ( ( 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 ) ) ) ) )
97	70, 94, 34, 72, 74	caov32d 6049	. . . . . . . . . . . . . 14 |- ( ( ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) /\ ( v e. P. /\ ( y +P. v ) = x ) ) /\ ( u e. P. /\ ( w +P. u ) = z ) ) -> ( ( ( 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 ) ) ) )
98		addassprg 7569	. . . . . . . . . . . . . . . 16 |- ( ( ( x .P. w ) e. P. /\ ( y .P. z ) e. P. /\ ( v .P. u ) e. P. ) -> ( ( ( x .P. w ) +P. ( y .P. z ) ) +P. ( v .P. u ) ) = ( ( x .P. w ) +P. ( ( y .P. z ) +P. ( v .P. u ) ) ) )
99	34, 38, 42, 98	syl3anc 1238	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) /\ ( v e. P. /\ ( y +P. v ) = x ) ) /\ ( u e. P. /\ ( w +P. u ) = z ) ) -> ( ( ( x .P. w ) +P. ( y .P. z ) ) +P. ( v .P. u ) ) = ( ( x .P. w ) +P. ( ( y .P. z ) +P. ( v .P. u ) ) ) )
100	99	oveq2d 5885	. . . . . . . . . . . . . 14 |- ( ( ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) /\ ( v e. P. /\ ( y +P. v ) = x ) ) /\ ( u e. P. /\ ( w +P. u ) = z ) ) -> ( ( 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 ) ) ) ) )
101	96, 97, 100	3eqtr4d 2220	. . . . . . . . . . . . 13 |- ( ( ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) /\ ( v e. P. /\ ( y +P. v ) = x ) ) /\ ( u e. P. /\ ( w +P. u ) = z ) ) -> ( ( ( 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 ) ) ) )
102	84, 93, 101	3eqtr3d 2218	. . . . . . . . . . . 12 |- ( ( ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) /\ ( v e. P. /\ ( y +P. v ) = x ) ) /\ ( u e. P. /\ ( 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 ) ) ) )
103	24, 39, 42	caovcld 6022	. . . . . . . . . . . . 13 |- ( ( ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) /\ ( v e. P. /\ ( y +P. v ) = x ) ) /\ ( u e. P. /\ ( w +P. u ) = z ) ) -> ( ( ( x .P. w ) +P. ( y .P. z ) ) +P. ( v .P. u ) ) e. P. )
104		addcanprg 7606	. . . . . . . . . . . . 13 |- ( ( ( v .P. w ) e. P. /\ ( ( x .P. z ) +P. ( y .P. w ) ) e. P. /\ ( ( ( x .P. w ) +P. ( y .P. z ) ) +P. ( v .P. u ) ) 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 ) ) ) )
105	70, 90, 103, 104	syl3anc 1238	. . . . . . . . . . . 12 |- ( ( ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) /\ ( v e. P. /\ ( y +P. v ) = x ) ) /\ ( u e. P. /\ ( 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 ) ) ) -> ( ( x .P. z ) +P. ( y .P. w ) ) = ( ( ( x .P. w ) +P. ( y .P. z ) ) +P. ( v .P. u ) ) ) )
106	102, 105	mpd 13	. . . . . . . . . . 11 |- ( ( ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) /\ ( v e. P. /\ ( y +P. v ) = x ) ) /\ ( u e. P. /\ ( w +P. u ) = z ) ) -> ( ( x .P. z ) +P. ( y .P. w ) ) = ( ( ( x .P. w ) +P. ( y .P. z ) ) +P. ( v .P. u ) ) )
107	44, 106	breqtrrd 4028	. . . . . . . . . 10 |- ( ( ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) /\ ( v e. P. /\ ( y +P. v ) = x ) ) /\ ( u e. P. /\ ( w +P. u ) = z ) ) -> ( ( x .P. w ) +P. ( y .P. z ) ) <P ( ( x .P. z ) +P. ( y .P. w ) ) )
108	107	rexlimdvaa 2595	. . . . . . . . 9 |- ( ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) /\ ( v e. P. /\ ( y +P. v ) = x ) ) -> ( E. u e. P. ( w +P. u ) = z -> ( ( x .P. w ) +P. ( y .P. z ) ) <P ( ( x .P. z ) +P. ( y .P. w ) ) ) )
109	22, 108	syl5 32	. . . . . . . 8 |- ( ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) /\ ( v e. P. /\ ( y +P. v ) = x ) ) -> ( w <P z -> ( ( x .P. w ) +P. ( y .P. z ) ) <P ( ( x .P. z ) +P. ( y .P. w ) ) ) )
110	109	rexlimdvaa 2595	. . . . . . 7 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( E. v e. P. ( y +P. v ) = x -> ( w <P z -> ( ( x .P. w ) +P. ( y .P. z ) ) <P ( ( x .P. z ) +P. ( y .P. w ) ) ) ) )
111	21, 110	syl5 32	. . . . . 6 |- ( ( ( 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 ) ) ) ) )
112	111	impd 254	. . . . 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 ) ) ) )
113		mulsrpr 7736	. . . . . . 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 )
114	113	breq2d 4012	. . . . . 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 ) )
115		gt0srpr 7738	. . . . . 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 ) ) )
116	114, 115	bitrdi 196	. . . . 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 ) ) ) )
117	112, 116	sylibrd 169	. . . 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 ) ) )
118	20, 117	biimtrid 152	. . 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 ) ) )
119	7, 12, 17, 118	2ecoptocl 6617	. 2 |- ( ( A e. R. /\ B e. R. ) -> ( ( 0R <R A /\ 0R <R B ) -> 0R <R ( A .R B ) ) )
120	6, 119	mpcom 36	1 |- ( ( 0R <R A /\ 0R <R B ) -> 0R <R ( A .R B ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 978   = wceq 1353   e. wcel 2148   E.wrex 2456   <.cop 3594   class class class wbr 4000  (class class class)co 5869   [cec 6527   P.cnp 7281   +P. cpp 7283   .P. cmp 7284   <P cltp 7285   ~R cer 7286   R.cnr 7287   0Rc0r 7288   .R cmr 7292   <R cltr 7293

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-eprel 4286  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-irdg 6365  df-1o 6411  df-2o 6412  df-oadd 6415  df-omul 6416  df-er 6529  df-ec 6531  df-qs 6535  df-ni 7294  df-pli 7295  df-mi 7296  df-lti 7297  df-plpq 7334  df-mpq 7335  df-enq 7337  df-nqqs 7338  df-plqqs 7339  df-mqqs 7340  df-1nqqs 7341  df-rq 7342  df-ltnqqs 7343  df-enq0 7414  df-nq0 7415  df-0nq0 7416  df-plq0 7417  df-mq0 7418  df-inp 7456  df-i1p 7457  df-iplp 7458  df-imp 7459  df-iltp 7460  df-enr 7716  df-nr 7717  df-mr 7719  df-ltr 7720  df-0r 7721

This theorem is referenced by:  axpre-mulgt0  7877