Theorem mulgt0sr 7845

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 7805	. . . . 5 |- <R C_ ( R. X. R. )
2	1	brel 4715	. . . 4 |- ( 0R <R A -> ( 0R e. R. /\ A e. R. ) )
3	2	simprd 114	. . 3 |- ( 0R <R A -> A e. R. )
4	1	brel 4715	. . . 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 7794	. . 3 |- R. = ( ( P. X. P. ) /. ~R )
8		breq2 4037	. . . . 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 5929	. . . . 5 |- ( [ <. x , y >. ] ~R = A -> ( [ <. x , y >. ] ~R .R [ <. z , w >. ] ~R ) = ( A .R [ <. z , w >. ] ~R ) )
11	10	breq2d 4045	. . . 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 4037	. . . . 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 5930	. . . . 5 |- ( [ <. z , w >. ] ~R = B -> ( A .R [ <. z , w >. ] ~R ) = ( A .R B ) )
16	15	breq2d 4045	. . . 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 7815	. . . . 5 |- ( 0R <R [ <. x , y >. ] ~R <-> y <P x )
19		gt0srpr 7815	. . . . 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 7680	. . . . . . 7 |- ( y <P x -> E. v e. P. ( y +P. v ) = x )
22		ltexpri 7680	. . . . . . . . 9 |- ( w <P z -> E. u e. P. ( w +P. u ) = z )
23		addclpr 7604	. . . . . . . . . . . . . 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 6077	. . . . . . . . . . . . . . 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 2274	. . . . . . . . . . . . . 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 7639	. . . . . . . . . . . . . 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 7639	. . . . . . . . . . . . . 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 6077	. . . . . . . . . . . 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 7639	. . . . . . . . . . . . 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 7664	. . . . . . . . . . . 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 5931	. . . . . . . . . . . . . . . 16 |- ( ( ( y +P. v ) = x /\ ( w +P. u ) = z ) -> ( ( y +P. v ) .P. ( w +P. u ) ) = ( x .P. z ) )
47	46	oveq1d 5937	. . . . . . . . . . . . . . 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 7655	. . . . . . . . . . . . . . . . . . 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 1249	. . . . . . . . . . . . . . . . . 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 5930	. . . . . . . . . . . . . . . . . . . 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 2231	. . . . . . . . . . . . . . . . 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 5937	. . . . . . . . . . . . . . . 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 7655	. . . . . . . . . . . . . . . . . . . 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 7647	. . . . . . . . . . . . . . . . . . . 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 6100	. . . . . . . . . . . . . . . . . 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 6100	. . . . . . . . . . . . . . . . . 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 5940	. . . . . . . . . . . . . . . . 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 7655	. . . . . . . . . . . . . . . . . 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 1249	. . . . . . . . . . . . . . . . 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 7639	. . . . . . . . . . . . . . . . . . 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 7639	. . . . . . . . . . . . . . . . . . 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 7639	. . . . . . . . . . . . . . . . . . 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 7645	. . . . . . . . . . . . . . . . . . 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 7646	. . . . . . . . . . . . . . . . . . 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 6108	. . . . . . . . . . . . . . . . 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 2239	. . . . . . . . . . . . . . . 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 6105	. . . . . . . . . . . . . . . 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 2239	. . . . . . . . . . . . . . 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 5929	. . . . . . . . . . . . . . . . . 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 2231	. . . . . . . . . . . . . . 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 5940	. . . . . . . . . . . . . 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 2231	. . . . . . . . . . . . 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 7639	. . . . . . . . . . . . . . . 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 7646	. . . . . . . . . . . . . . 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 1249	. . . . . . . . . . . . . 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 7604	. . . . . . . . . . . . . . . 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 7645	. . . . . . . . . . . . . . 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 2231	. . . . . . . . . . . . 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 6077	. . . . . . . . . . . . . . 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 7646	. . . . . . . . . . . . . . 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 1249	. . . . . . . . . . . . . 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 6104	. . . . . . . . . . . . . 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 7646	. . . . . . . . . . . . . . . 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 1249	. . . . . . . . . . . . . . 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 5938	. . . . . . . . . . . . . 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 2239	. . . . . . . . . . . . 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 2237	. . . . . . . . . . . 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 6077	. . . . . . . . . . . . 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 7683	. . . . . . . . . . . . 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 1249	. . . . . . . . . . . 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 4061	. . . . . . . . . 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 2615	. . . . . . . . 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 2615	. . . . . . 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 7813	. . . . . . 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 4045	. . . . . 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 7815	. . . . . 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 6682	. 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 980   = wceq 1364   e. wcel 2167   E.wrex 2476   <.cop 3625   class class class wbr 4033  (class class class)co 5922   [cec 6590   P.cnp 7358   +P. cpp 7360   .P. cmp 7361   <P cltp 7362   ~R cer 7363   R.cnr 7364   0Rc0r 7365   .R cmr 7369   <R cltr 7370

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-eprel 4324  df-id 4328  df-po 4331  df-iso 4332  df-iord 4401  df-on 4403  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-irdg 6428  df-1o 6474  df-2o 6475  df-oadd 6478  df-omul 6479  df-er 6592  df-ec 6594  df-qs 6598  df-ni 7371  df-pli 7372  df-mi 7373  df-lti 7374  df-plpq 7411  df-mpq 7412  df-enq 7414  df-nqqs 7415  df-plqqs 7416  df-mqqs 7417  df-1nqqs 7418  df-rq 7419  df-ltnqqs 7420  df-enq0 7491  df-nq0 7492  df-0nq0 7493  df-plq0 7494  df-mq0 7495  df-inp 7533  df-i1p 7534  df-iplp 7535  df-imp 7536  df-iltp 7537  df-enr 7793  df-nr 7794  df-mr 7796  df-ltr 7797  df-0r 7798

This theorem is referenced by:  axpre-mulgt0  7954