Theorem mulgt0sr 8093

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 8053	. . . . 5 |- <R C_ ( R. X. R. )
2	1	brel 4802	. . . 4 |- ( 0R <R A -> ( 0R e. R. /\ A e. R. ) )
3	2	simprd 114	. . 3 |- ( 0R <R A -> A e. R. )
4	1	brel 4802	. . . 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 8042	. . 3 |- R. = ( ( P. X. P. ) /. ~R )
8		breq2 4113	. . . . 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 6057	. . . . 5 |- ( [ <. x , y >. ] ~R = A -> ( [ <. x , y >. ] ~R .R [ <. z , w >. ] ~R ) = ( A .R [ <. z , w >. ] ~R ) )
11	10	breq2d 4121	. . . 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 4113	. . . . 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 6058	. . . . 5 |- ( [ <. z , w >. ] ~R = B -> ( A .R [ <. z , w >. ] ~R ) = ( A .R B ) )
16	15	breq2d 4121	. . . 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 8063	. . . . 5 |- ( 0R <R [ <. x , y >. ] ~R <-> y <P x )
19		gt0srpr 8063	. . . . 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 7928	. . . . . . 7 |- ( y <P x -> E. v e. P. ( y +P. v ) = x )
22		ltexpri 7928	. . . . . . . . 9 |- ( w <P z -> E. u e. P. ( w +P. u ) = z )
23		addclpr 7852	. . . . . . . . . . . . . 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 538	. . . . . . . . . . . . . . 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 529	. . . . . . . . . . . . . . . . 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 537	. . . . . . . . . . . . . . . 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 6208	. . . . . . . . . . . . . . 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 2310	. . . . . . . . . . . . . 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 538	. . . . . . . . . . . . . . 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 7887	. . . . . . . . . . . . . 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 537	. . . . . . . . . . . . . . 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 7887	. . . . . . . . . . . . . 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 6208	. . . . . . . . . . . 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 531	. . . . . . . . . . . . 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 7887	. . . . . . . . . . . . 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 7912	. . . . . . . . . . . 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 533	. . . . . . . . . . . . . . 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 6059	. . . . . . . . . . . . . . . 16 |- ( ( ( y +P. v ) = x /\ ( w +P. u ) = z ) -> ( ( y +P. v ) .P. ( w +P. u ) ) = ( x .P. z ) )
47	46	oveq1d 6065	. . . . . . . . . . . . . . 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 7903	. . . . . . . . . . . . . . . . . . 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 1274	. . . . . . . . . . . . . . . . . 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 6058	. . . . . . . . . . . . . . . . . . . 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 2267	. . . . . . . . . . . . . . . . 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 6065	. . . . . . . . . . . . . . . 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 7903	. . . . . . . . . . . . . . . . . . . 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 7895	. . . . . . . . . . . . . . . . . . . 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 6231	. . . . . . . . . . . . . . . . . 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 6231	. . . . . . . . . . . . . . . . . 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 6068	. . . . . . . . . . . . . . . . 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 7903	. . . . . . . . . . . . . . . . . 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 1274	. . . . . . . . . . . . . . . . 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 7887	. . . . . . . . . . . . . . . . . . 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 7887	. . . . . . . . . . . . . . . . . . 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 7887	. . . . . . . . . . . . . . . . . . 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 7893	. . . . . . . . . . . . . . . . . . 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 7894	. . . . . . . . . . . . . . . . . . 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 6239	. . . . . . . . . . . . . . . . 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 2275	. . . . . . . . . . . . . . . 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 6236	. . . . . . . . . . . . . . . 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 2275	. . . . . . . . . . . . . . 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 6057	. . . . . . . . . . . . . . . . . 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 2267	. . . . . . . . . . . . . . 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 6068	. . . . . . . . . . . . . 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 2267	. . . . . . . . . . . . 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 7887	. . . . . . . . . . . . . . . 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 7894	. . . . . . . . . . . . . . 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 1274	. . . . . . . . . . . . . 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 7852	. . . . . . . . . . . . . . . 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 7893	. . . . . . . . . . . . . . 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 2267	. . . . . . . . . . . . 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 6208	. . . . . . . . . . . . . . 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 7894	. . . . . . . . . . . . . . 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 1274	. . . . . . . . . . . . . 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 6235	. . . . . . . . . . . . . 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 7894	. . . . . . . . . . . . . . . 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 1274	. . . . . . . . . . . . . . 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 6066	. . . . . . . . . . . . . 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 2275	. . . . . . . . . . . . 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 2273	. . . . . . . . . . . 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 6208	. . . . . . . . . . . . 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 7931	. . . . . . . . . . . . 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 1274	. . . . . . . . . . . 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 4137	. . . . . . . . . 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 2661	. . . . . . . . 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 2661	. . . . . . 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 8061	. . . . . . 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 4121	. . . . . 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 8063	. . . . . 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 6857	. 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 1005   = wceq 1398   e. wcel 2203   E.wrex 2521   <.cop 3692   class class class wbr 4109  (class class class)co 6050   [cec 6765   P.cnp 7606   +P. cpp 7608   .P. cmp 7609   <P cltp 7610   ~R cer 7611   R.cnr 7612   0Rc0r 7613   .R cmr 7617   <R cltr 7618

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-eprel 4410  df-id 4414  df-po 4417  df-iso 4418  df-iord 4487  df-on 4489  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-irdg 6601  df-1o 6647  df-2o 6648  df-oadd 6651  df-omul 6652  df-er 6767  df-ec 6769  df-qs 6773  df-ni 7619  df-pli 7620  df-mi 7621  df-lti 7622  df-plpq 7659  df-mpq 7660  df-enq 7662  df-nqqs 7663  df-plqqs 7664  df-mqqs 7665  df-1nqqs 7666  df-rq 7667  df-ltnqqs 7668  df-enq0 7739  df-nq0 7740  df-0nq0 7741  df-plq0 7742  df-mq0 7743  df-inp 7781  df-i1p 7782  df-iplp 7783  df-imp 7784  df-iltp 7785  df-enr 8041  df-nr 8042  df-mr 8044  df-ltr 8045  df-0r 8046

This theorem is referenced by:  axpre-mulgt0  8202