Theorem mulgt0sr 8058

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 8018	. . . . 5 |- <R C_ ( R. X. R. )
2	1	brel 4784	. . . 4 |- ( 0R <R A -> ( 0R e. R. /\ A e. R. ) )
3	2	simprd 114	. . 3 |- ( 0R <R A -> A e. R. )
4	1	brel 4784	. . . 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 8007	. . 3 |- R. = ( ( P. X. P. ) /. ~R )
8		breq2 4097	. . . . 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 6035	. . . . 5 |- ( [ <. x , y >. ] ~R = A -> ( [ <. x , y >. ] ~R .R [ <. z , w >. ] ~R ) = ( A .R [ <. z , w >. ] ~R ) )
11	10	breq2d 4105	. . . 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 4097	. . . . 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 6036	. . . . 5 |- ( [ <. z , w >. ] ~R = B -> ( A .R [ <. z , w >. ] ~R ) = ( A .R B ) )
16	15	breq2d 4105	. . . 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 8028	. . . . 5 |- ( 0R <R [ <. x , y >. ] ~R <-> y <P x )
19		gt0srpr 8028	. . . . 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 7893	. . . . . . 7 |- ( y <P x -> E. v e. P. ( y +P. v ) = x )
22		ltexpri 7893	. . . . . . . . 9 |- ( w <P z -> E. u e. P. ( w +P. u ) = z )
23		addclpr 7817	. . . . . . . . . . . . . 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 6186	. . . . . . . . . . . . . . 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 2309	. . . . . . . . . . . . . 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 7852	. . . . . . . . . . . . . 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 7852	. . . . . . . . . . . . . 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 6186	. . . . . . . . . . . 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 7852	. . . . . . . . . . . . 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 7877	. . . . . . . . . . . 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 6037	. . . . . . . . . . . . . . . 16 |- ( ( ( y +P. v ) = x /\ ( w +P. u ) = z ) -> ( ( y +P. v ) .P. ( w +P. u ) ) = ( x .P. z ) )
47	46	oveq1d 6043	. . . . . . . . . . . . . . 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 7868	. . . . . . . . . . . . . . . . . . 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 6036	. . . . . . . . . . . . . . . . . . . 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 2266	. . . . . . . . . . . . . . . . 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 6043	. . . . . . . . . . . . . . . 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 7868	. . . . . . . . . . . . . . . . . . . 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 7860	. . . . . . . . . . . . . . . . . . . 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 6209	. . . . . . . . . . . . . . . . . 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 6209	. . . . . . . . . . . . . . . . . 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 6046	. . . . . . . . . . . . . . . . 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 7868	. . . . . . . . . . . . . . . . . 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 7852	. . . . . . . . . . . . . . . . . . 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 7852	. . . . . . . . . . . . . . . . . . 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 7852	. . . . . . . . . . . . . . . . . . 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 7858	. . . . . . . . . . . . . . . . . . 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 7859	. . . . . . . . . . . . . . . . . . 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 6217	. . . . . . . . . . . . . . . . 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 2274	. . . . . . . . . . . . . . . 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 6214	. . . . . . . . . . . . . . . 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 2274	. . . . . . . . . . . . . . 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 6035	. . . . . . . . . . . . . . . . . 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 2266	. . . . . . . . . . . . . . 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 6046	. . . . . . . . . . . . . 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 2266	. . . . . . . . . . . . 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 7852	. . . . . . . . . . . . . . . 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 7859	. . . . . . . . . . . . . . 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 7817	. . . . . . . . . . . . . . . 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 7858	. . . . . . . . . . . . . . 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 2266	. . . . . . . . . . . . 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 6186	. . . . . . . . . . . . . . 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 7859	. . . . . . . . . . . . . . 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 6213	. . . . . . . . . . . . . 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 7859	. . . . . . . . . . . . . . . 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 6044	. . . . . . . . . . . . . 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 2274	. . . . . . . . . . . . 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 2272	. . . . . . . . . . . 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 6186	. . . . . . . . . . . . 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 7896	. . . . . . . . . . . . 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 4121	. . . . . . . . . 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 2652	. . . . . . . . 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 2652	. . . . . . 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 8026	. . . . . . 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 4105	. . . . . 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 8028	. . . . . 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 6835	. 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 2202   E.wrex 2512   <.cop 3676   class class class wbr 4093  (class class class)co 6028   [cec 6743   P.cnp 7571   +P. cpp 7573   .P. cmp 7574   <P cltp 7575   ~R cer 7576   R.cnr 7577   0Rc0r 7578   .R cmr 7582   <R cltr 7583

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 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692

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 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-eprel 4392  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-irdg 6579  df-1o 6625  df-2o 6626  df-oadd 6629  df-omul 6630  df-er 6745  df-ec 6747  df-qs 6751  df-ni 7584  df-pli 7585  df-mi 7586  df-lti 7587  df-plpq 7624  df-mpq 7625  df-enq 7627  df-nqqs 7628  df-plqqs 7629  df-mqqs 7630  df-1nqqs 7631  df-rq 7632  df-ltnqqs 7633  df-enq0 7704  df-nq0 7705  df-0nq0 7706  df-plq0 7707  df-mq0 7708  df-inp 7746  df-i1p 7747  df-iplp 7748  df-imp 7749  df-iltp 7750  df-enr 8006  df-nr 8007  df-mr 8009  df-ltr 8010  df-0r 8011

This theorem is referenced by:  axpre-mulgt0  8167