Theorem distrlem4prl 7122

Description: Lemma for distributive law for positive reals. (Contributed by Jim Kingdon, 12-Dec-2019.)

Assertion

Ref	Expression
distrlem4prl	|- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 1st ` A ) /\ y e. ( 1st ` B ) ) /\ ( f e. ( 1st ` A ) /\ z e. ( 1st ` C ) ) ) ) -> ( ( x .Q y ) +Q ( f .Q z ) ) e. ( 1st ` ( A .P. ( B +P. C ) ) ) )

Distinct variable groups:   x, y, z, f, A   x, B, y, z, f   x, C, y, z, f

Proof of Theorem distrlem4prl

Dummy variables w v u g h are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ltmnqg 6939	. . . . . . 7 |- ( ( w e. Q. /\ v e. Q. /\ u e. Q. ) -> ( w <Q v <-> ( u .Q w ) <Q ( u .Q v ) ) )
2	1	adantl 271	. . . . . 6 |- ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 1st ` A ) /\ y e. ( 1st ` B ) ) /\ ( f e. ( 1st ` A ) /\ z e. ( 1st ` C ) ) ) ) /\ ( w e. Q. /\ v e. Q. /\ u e. Q. ) ) -> ( w <Q v <-> ( u .Q w ) <Q ( u .Q v ) ) )
3		simp1 943	. . . . . . 7 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> A e. P. )
4		simpll 496	. . . . . . 7 |- ( ( ( x e. ( 1st ` A ) /\ y e. ( 1st ` B ) ) /\ ( f e. ( 1st ` A ) /\ z e. ( 1st ` C ) ) ) -> x e. ( 1st ` A ) )
5		prop 7013	. . . . . . . 8 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
6		elprnql 7019	. . . . . . . 8 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ x e. ( 1st ` A ) ) -> x e. Q. )
7	5, 6	sylan 277	. . . . . . 7 |- ( ( A e. P. /\ x e. ( 1st ` A ) ) -> x e. Q. )
8	3, 4, 7	syl2an 283	. . . . . 6 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 1st ` A ) /\ y e. ( 1st ` B ) ) /\ ( f e. ( 1st ` A ) /\ z e. ( 1st ` C ) ) ) ) -> x e. Q. )
9		simprl 498	. . . . . . 7 |- ( ( ( x e. ( 1st ` A ) /\ y e. ( 1st ` B ) ) /\ ( f e. ( 1st ` A ) /\ z e. ( 1st ` C ) ) ) -> f e. ( 1st ` A ) )
10		elprnql 7019	. . . . . . . 8 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ f e. ( 1st ` A ) ) -> f e. Q. )
11	5, 10	sylan 277	. . . . . . 7 |- ( ( A e. P. /\ f e. ( 1st ` A ) ) -> f e. Q. )
12	3, 9, 11	syl2an 283	. . . . . 6 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 1st ` A ) /\ y e. ( 1st ` B ) ) /\ ( f e. ( 1st ` A ) /\ z e. ( 1st ` C ) ) ) ) -> f e. Q. )
13		simpl2 947	. . . . . . 7 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 1st ` A ) /\ y e. ( 1st ` B ) ) /\ ( f e. ( 1st ` A ) /\ z e. ( 1st ` C ) ) ) ) -> B e. P. )
14		simprlr 505	. . . . . . 7 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 1st ` A ) /\ y e. ( 1st ` B ) ) /\ ( f e. ( 1st ` A ) /\ z e. ( 1st ` C ) ) ) ) -> y e. ( 1st ` B ) )
15		prop 7013	. . . . . . . 8 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
16		elprnql 7019	. . . . . . . 8 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ y e. ( 1st ` B ) ) -> y e. Q. )
17	15, 16	sylan 277	. . . . . . 7 |- ( ( B e. P. /\ y e. ( 1st ` B ) ) -> y e. Q. )
18	13, 14, 17	syl2anc 403	. . . . . 6 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 1st ` A ) /\ y e. ( 1st ` B ) ) /\ ( f e. ( 1st ` A ) /\ z e. ( 1st ` C ) ) ) ) -> y e. Q. )
19		mulcomnqg 6921	. . . . . . 7 |- ( ( w e. Q. /\ v e. Q. ) -> ( w .Q v ) = ( v .Q w ) )
20	19	adantl 271	. . . . . 6 |- ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 1st ` A ) /\ y e. ( 1st ` B ) ) /\ ( f e. ( 1st ` A ) /\ z e. ( 1st ` C ) ) ) ) /\ ( w e. Q. /\ v e. Q. ) ) -> ( w .Q v ) = ( v .Q w ) )
21	2, 8, 12, 18, 20	caovord2d 5796	. . . . 5 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 1st ` A ) /\ y e. ( 1st ` B ) ) /\ ( f e. ( 1st ` A ) /\ z e. ( 1st ` C ) ) ) ) -> ( x <Q f <-> ( x .Q y ) <Q ( f .Q y ) ) )
22		ltanqg 6938	. . . . . . 7 |- ( ( w e. Q. /\ v e. Q. /\ u e. Q. ) -> ( w <Q v <-> ( u +Q w ) <Q ( u +Q v ) ) )
23	22	adantl 271	. . . . . 6 |- ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 1st ` A ) /\ y e. ( 1st ` B ) ) /\ ( f e. ( 1st ` A ) /\ z e. ( 1st ` C ) ) ) ) /\ ( w e. Q. /\ v e. Q. /\ u e. Q. ) ) -> ( w <Q v <-> ( u +Q w ) <Q ( u +Q v ) ) )
24		mulclnq 6914	. . . . . . 7 |- ( ( x e. Q. /\ y e. Q. ) -> ( x .Q y ) e. Q. )
25	8, 18, 24	syl2anc 403	. . . . . 6 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 1st ` A ) /\ y e. ( 1st ` B ) ) /\ ( f e. ( 1st ` A ) /\ z e. ( 1st ` C ) ) ) ) -> ( x .Q y ) e. Q. )
26		mulclnq 6914	. . . . . . 7 |- ( ( f e. Q. /\ y e. Q. ) -> ( f .Q y ) e. Q. )
27	12, 18, 26	syl2anc 403	. . . . . 6 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 1st ` A ) /\ y e. ( 1st ` B ) ) /\ ( f e. ( 1st ` A ) /\ z e. ( 1st ` C ) ) ) ) -> ( f .Q y ) e. Q. )
28		simpl3 948	. . . . . . . 8 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 1st ` A ) /\ y e. ( 1st ` B ) ) /\ ( f e. ( 1st ` A ) /\ z e. ( 1st ` C ) ) ) ) -> C e. P. )
29		simprrr 507	. . . . . . . 8 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 1st ` A ) /\ y e. ( 1st ` B ) ) /\ ( f e. ( 1st ` A ) /\ z e. ( 1st ` C ) ) ) ) -> z e. ( 1st ` C ) )
30		prop 7013	. . . . . . . . 9 |- ( C e. P. -> <. ( 1st ` C ) , ( 2nd ` C ) >. e. P. )
31		elprnql 7019	. . . . . . . . 9 |- ( ( <. ( 1st ` C ) , ( 2nd ` C ) >. e. P. /\ z e. ( 1st ` C ) ) -> z e. Q. )
32	30, 31	sylan 277	. . . . . . . 8 |- ( ( C e. P. /\ z e. ( 1st ` C ) ) -> z e. Q. )
33	28, 29, 32	syl2anc 403	. . . . . . 7 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 1st ` A ) /\ y e. ( 1st ` B ) ) /\ ( f e. ( 1st ` A ) /\ z e. ( 1st ` C ) ) ) ) -> z e. Q. )
34		mulclnq 6914	. . . . . . 7 |- ( ( f e. Q. /\ z e. Q. ) -> ( f .Q z ) e. Q. )
35	12, 33, 34	syl2anc 403	. . . . . 6 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 1st ` A ) /\ y e. ( 1st ` B ) ) /\ ( f e. ( 1st ` A ) /\ z e. ( 1st ` C ) ) ) ) -> ( f .Q z ) e. Q. )
36		addcomnqg 6919	. . . . . . 7 |- ( ( w e. Q. /\ v e. Q. ) -> ( w +Q v ) = ( v +Q w ) )
37	36	adantl 271	. . . . . 6 |- ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 1st ` A ) /\ y e. ( 1st ` B ) ) /\ ( f e. ( 1st ` A ) /\ z e. ( 1st ` C ) ) ) ) /\ ( w e. Q. /\ v e. Q. ) ) -> ( w +Q v ) = ( v +Q w ) )
38	23, 25, 27, 35, 37	caovord2d 5796	. . . . 5 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 1st ` A ) /\ y e. ( 1st ` B ) ) /\ ( f e. ( 1st ` A ) /\ z e. ( 1st ` C ) ) ) ) -> ( ( x .Q y ) <Q ( f .Q y ) <-> ( ( x .Q y ) +Q ( f .Q z ) ) <Q ( ( f .Q y ) +Q ( f .Q z ) ) ) )
39	21, 38	bitrd 186	. . . 4 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 1st ` A ) /\ y e. ( 1st ` B ) ) /\ ( f e. ( 1st ` A ) /\ z e. ( 1st ` C ) ) ) ) -> ( x <Q f <-> ( ( x .Q y ) +Q ( f .Q z ) ) <Q ( ( f .Q y ) +Q ( f .Q z ) ) ) )
40		simpl1 946	. . . . . 6 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 1st ` A ) /\ y e. ( 1st ` B ) ) /\ ( f e. ( 1st ` A ) /\ z e. ( 1st ` C ) ) ) ) -> A e. P. )
41		addclpr 7075	. . . . . . . 8 |- ( ( B e. P. /\ C e. P. ) -> ( B +P. C ) e. P. )
42	41	3adant1 961	. . . . . . 7 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( B +P. C ) e. P. )
43	42	adantr 270	. . . . . 6 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 1st ` A ) /\ y e. ( 1st ` B ) ) /\ ( f e. ( 1st ` A ) /\ z e. ( 1st ` C ) ) ) ) -> ( B +P. C ) e. P. )
44		mulclpr 7110	. . . . . 6 |- ( ( A e. P. /\ ( B +P. C ) e. P. ) -> ( A .P. ( B +P. C ) ) e. P. )
45	40, 43, 44	syl2anc 403	. . . . 5 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 1st ` A ) /\ y e. ( 1st ` B ) ) /\ ( f e. ( 1st ` A ) /\ z e. ( 1st ` C ) ) ) ) -> ( A .P. ( B +P. C ) ) e. P. )
46		distrnqg 6925	. . . . . . 7 |- ( ( f e. Q. /\ y e. Q. /\ z e. Q. ) -> ( f .Q ( y +Q z ) ) = ( ( f .Q y ) +Q ( f .Q z ) ) )
47	12, 18, 33, 46	syl3anc 1174	. . . . . 6 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 1st ` A ) /\ y e. ( 1st ` B ) ) /\ ( f e. ( 1st ` A ) /\ z e. ( 1st ` C ) ) ) ) -> ( f .Q ( y +Q z ) ) = ( ( f .Q y ) +Q ( f .Q z ) ) )
48		simprrl 506	. . . . . . 7 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 1st ` A ) /\ y e. ( 1st ` B ) ) /\ ( f e. ( 1st ` A ) /\ z e. ( 1st ` C ) ) ) ) -> f e. ( 1st ` A ) )
49		df-iplp 7006	. . . . . . . . . 10 |- +P. = ( u e. P. , v e. P. |-> <. { w e. Q. | E. g e. Q. E. h e. Q. ( g e. ( 1st ` u ) /\ h e. ( 1st ` v ) /\ w = ( g +Q h ) ) } , { w e. Q. | E. g e. Q. E. h e. Q. ( g e. ( 2nd ` u ) /\ h e. ( 2nd ` v ) /\ w = ( g +Q h ) ) } >. )
50		addclnq 6913	. . . . . . . . . 10 |- ( ( g e. Q. /\ h e. Q. ) -> ( g +Q h ) e. Q. )
51	49, 50	genpprecll 7052	. . . . . . . . 9 |- ( ( B e. P. /\ C e. P. ) -> ( ( y e. ( 1st ` B ) /\ z e. ( 1st ` C ) ) -> ( y +Q z ) e. ( 1st ` ( B +P. C ) ) ) )
52	51	imp 122	. . . . . . . 8 |- ( ( ( B e. P. /\ C e. P. ) /\ ( y e. ( 1st ` B ) /\ z e. ( 1st ` C ) ) ) -> ( y +Q z ) e. ( 1st ` ( B +P. C ) ) )
53	13, 28, 14, 29, 52	syl22anc 1175	. . . . . . 7 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 1st ` A ) /\ y e. ( 1st ` B ) ) /\ ( f e. ( 1st ` A ) /\ z e. ( 1st ` C ) ) ) ) -> ( y +Q z ) e. ( 1st ` ( B +P. C ) ) )
54		df-imp 7007	. . . . . . . . 9 |- .P. = ( u e. P. , v e. P. |-> <. { w e. Q. | E. g e. Q. E. h e. Q. ( g e. ( 1st ` u ) /\ h e. ( 1st ` v ) /\ w = ( g .Q h ) ) } , { w e. Q. | E. g e. Q. E. h e. Q. ( g e. ( 2nd ` u ) /\ h e. ( 2nd ` v ) /\ w = ( g .Q h ) ) } >. )
55		mulclnq 6914	. . . . . . . . 9 |- ( ( g e. Q. /\ h e. Q. ) -> ( g .Q h ) e. Q. )
56	54, 55	genpprecll 7052	. . . . . . . 8 |- ( ( A e. P. /\ ( B +P. C ) e. P. ) -> ( ( f e. ( 1st ` A ) /\ ( y +Q z ) e. ( 1st ` ( B +P. C ) ) ) -> ( f .Q ( y +Q z ) ) e. ( 1st ` ( A .P. ( B +P. C ) ) ) ) )
57	56	imp 122	. . . . . . 7 |- ( ( ( A e. P. /\ ( B +P. C ) e. P. ) /\ ( f e. ( 1st ` A ) /\ ( y +Q z ) e. ( 1st ` ( B +P. C ) ) ) ) -> ( f .Q ( y +Q z ) ) e. ( 1st ` ( A .P. ( B +P. C ) ) ) )
58	40, 43, 48, 53, 57	syl22anc 1175	. . . . . 6 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 1st ` A ) /\ y e. ( 1st ` B ) ) /\ ( f e. ( 1st ` A ) /\ z e. ( 1st ` C ) ) ) ) -> ( f .Q ( y +Q z ) ) e. ( 1st ` ( A .P. ( B +P. C ) ) ) )
59	47, 58	eqeltrrd 2165	. . . . 5 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 1st ` A ) /\ y e. ( 1st ` B ) ) /\ ( f e. ( 1st ` A ) /\ z e. ( 1st ` C ) ) ) ) -> ( ( f .Q y ) +Q ( f .Q z ) ) e. ( 1st ` ( A .P. ( B +P. C ) ) ) )
60		prop 7013	. . . . . 6 |- ( ( A .P. ( B +P. C ) ) e. P. -> <. ( 1st ` ( A .P. ( B +P. C ) ) ) , ( 2nd ` ( A .P. ( B +P. C ) ) ) >. e. P. )
61		prcdnql 7022	. . . . . 6 |- ( ( <. ( 1st ` ( A .P. ( B +P. C ) ) ) , ( 2nd ` ( A .P. ( B +P. C ) ) ) >. e. P. /\ ( ( f .Q y ) +Q ( f .Q z ) ) e. ( 1st ` ( A .P. ( B +P. C ) ) ) ) -> ( ( ( x .Q y ) +Q ( f .Q z ) ) <Q ( ( f .Q y ) +Q ( f .Q z ) ) -> ( ( x .Q y ) +Q ( f .Q z ) ) e. ( 1st ` ( A .P. ( B +P. C ) ) ) ) )
62	60, 61	sylan 277	. . . . 5 |- ( ( ( A .P. ( B +P. C ) ) e. P. /\ ( ( f .Q y ) +Q ( f .Q z ) ) e. ( 1st ` ( A .P. ( B +P. C ) ) ) ) -> ( ( ( x .Q y ) +Q ( f .Q z ) ) <Q ( ( f .Q y ) +Q ( f .Q z ) ) -> ( ( x .Q y ) +Q ( f .Q z ) ) e. ( 1st ` ( A .P. ( B +P. C ) ) ) ) )
63	45, 59, 62	syl2anc 403	. . . 4 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 1st ` A ) /\ y e. ( 1st ` B ) ) /\ ( f e. ( 1st ` A ) /\ z e. ( 1st ` C ) ) ) ) -> ( ( ( x .Q y ) +Q ( f .Q z ) ) <Q ( ( f .Q y ) +Q ( f .Q z ) ) -> ( ( x .Q y ) +Q ( f .Q z ) ) e. ( 1st ` ( A .P. ( B +P. C ) ) ) ) )
64	39, 63	sylbid 148	. . 3 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 1st ` A ) /\ y e. ( 1st ` B ) ) /\ ( f e. ( 1st ` A ) /\ z e. ( 1st ` C ) ) ) ) -> ( x <Q f -> ( ( x .Q y ) +Q ( f .Q z ) ) e. ( 1st ` ( A .P. ( B +P. C ) ) ) ) )
65	2, 12, 8, 33, 20	caovord2d 5796	. . . . 5 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 1st ` A ) /\ y e. ( 1st ` B ) ) /\ ( f e. ( 1st ` A ) /\ z e. ( 1st ` C ) ) ) ) -> ( f <Q x <-> ( f .Q z ) <Q ( x .Q z ) ) )
66		mulclnq 6914	. . . . . . 7 |- ( ( x e. Q. /\ z e. Q. ) -> ( x .Q z ) e. Q. )
67	8, 33, 66	syl2anc 403	. . . . . 6 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 1st ` A ) /\ y e. ( 1st ` B ) ) /\ ( f e. ( 1st ` A ) /\ z e. ( 1st ` C ) ) ) ) -> ( x .Q z ) e. Q. )
68		ltanqg 6938	. . . . . 6 |- ( ( ( f .Q z ) e. Q. /\ ( x .Q z ) e. Q. /\ ( x .Q y ) e. Q. ) -> ( ( f .Q z ) <Q ( x .Q z ) <-> ( ( x .Q y ) +Q ( f .Q z ) ) <Q ( ( x .Q y ) +Q ( x .Q z ) ) ) )
69	35, 67, 25, 68	syl3anc 1174	. . . . 5 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 1st ` A ) /\ y e. ( 1st ` B ) ) /\ ( f e. ( 1st ` A ) /\ z e. ( 1st ` C ) ) ) ) -> ( ( f .Q z ) <Q ( x .Q z ) <-> ( ( x .Q y ) +Q ( f .Q z ) ) <Q ( ( x .Q y ) +Q ( x .Q z ) ) ) )
70	65, 69	bitrd 186	. . . 4 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 1st ` A ) /\ y e. ( 1st ` B ) ) /\ ( f e. ( 1st ` A ) /\ z e. ( 1st ` C ) ) ) ) -> ( f <Q x <-> ( ( x .Q y ) +Q ( f .Q z ) ) <Q ( ( x .Q y ) +Q ( x .Q z ) ) ) )
71		distrnqg 6925	. . . . . . 7 |- ( ( x e. Q. /\ y e. Q. /\ z e. Q. ) -> ( x .Q ( y +Q z ) ) = ( ( x .Q y ) +Q ( x .Q z ) ) )
72	8, 18, 33, 71	syl3anc 1174	. . . . . 6 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 1st ` A ) /\ y e. ( 1st ` B ) ) /\ ( f e. ( 1st ` A ) /\ z e. ( 1st ` C ) ) ) ) -> ( x .Q ( y +Q z ) ) = ( ( x .Q y ) +Q ( x .Q z ) ) )
73		simprll 504	. . . . . . 7 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 1st ` A ) /\ y e. ( 1st ` B ) ) /\ ( f e. ( 1st ` A ) /\ z e. ( 1st ` C ) ) ) ) -> x e. ( 1st ` A ) )
74	54, 55	genpprecll 7052	. . . . . . . 8 |- ( ( A e. P. /\ ( B +P. C ) e. P. ) -> ( ( x e. ( 1st ` A ) /\ ( y +Q z ) e. ( 1st ` ( B +P. C ) ) ) -> ( x .Q ( y +Q z ) ) e. ( 1st ` ( A .P. ( B +P. C ) ) ) ) )
75	74	imp 122	. . . . . . 7 |- ( ( ( A e. P. /\ ( B +P. C ) e. P. ) /\ ( x e. ( 1st ` A ) /\ ( y +Q z ) e. ( 1st ` ( B +P. C ) ) ) ) -> ( x .Q ( y +Q z ) ) e. ( 1st ` ( A .P. ( B +P. C ) ) ) )
76	40, 43, 73, 53, 75	syl22anc 1175	. . . . . 6 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 1st ` A ) /\ y e. ( 1st ` B ) ) /\ ( f e. ( 1st ` A ) /\ z e. ( 1st ` C ) ) ) ) -> ( x .Q ( y +Q z ) ) e. ( 1st ` ( A .P. ( B +P. C ) ) ) )
77	72, 76	eqeltrrd 2165	. . . . 5 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 1st ` A ) /\ y e. ( 1st ` B ) ) /\ ( f e. ( 1st ` A ) /\ z e. ( 1st ` C ) ) ) ) -> ( ( x .Q y ) +Q ( x .Q z ) ) e. ( 1st ` ( A .P. ( B +P. C ) ) ) )
78		prcdnql 7022	. . . . . 6 |- ( ( <. ( 1st ` ( A .P. ( B +P. C ) ) ) , ( 2nd ` ( A .P. ( B +P. C ) ) ) >. e. P. /\ ( ( x .Q y ) +Q ( x .Q z ) ) e. ( 1st ` ( A .P. ( B +P. C ) ) ) ) -> ( ( ( x .Q y ) +Q ( f .Q z ) ) <Q ( ( x .Q y ) +Q ( x .Q z ) ) -> ( ( x .Q y ) +Q ( f .Q z ) ) e. ( 1st ` ( A .P. ( B +P. C ) ) ) ) )
79	60, 78	sylan 277	. . . . 5 |- ( ( ( A .P. ( B +P. C ) ) e. P. /\ ( ( x .Q y ) +Q ( x .Q z ) ) e. ( 1st ` ( A .P. ( B +P. C ) ) ) ) -> ( ( ( x .Q y ) +Q ( f .Q z ) ) <Q ( ( x .Q y ) +Q ( x .Q z ) ) -> ( ( x .Q y ) +Q ( f .Q z ) ) e. ( 1st ` ( A .P. ( B +P. C ) ) ) ) )
80	45, 77, 79	syl2anc 403	. . . 4 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 1st ` A ) /\ y e. ( 1st ` B ) ) /\ ( f e. ( 1st ` A ) /\ z e. ( 1st ` C ) ) ) ) -> ( ( ( x .Q y ) +Q ( f .Q z ) ) <Q ( ( x .Q y ) +Q ( x .Q z ) ) -> ( ( x .Q y ) +Q ( f .Q z ) ) e. ( 1st ` ( A .P. ( B +P. C ) ) ) ) )
81	70, 80	sylbid 148	. . 3 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 1st ` A ) /\ y e. ( 1st ` B ) ) /\ ( f e. ( 1st ` A ) /\ z e. ( 1st ` C ) ) ) ) -> ( f <Q x -> ( ( x .Q y ) +Q ( f .Q z ) ) e. ( 1st ` ( A .P. ( B +P. C ) ) ) ) )
82	64, 81	jaod 672	. 2 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 1st ` A ) /\ y e. ( 1st ` B ) ) /\ ( f e. ( 1st ` A ) /\ z e. ( 1st ` C ) ) ) ) -> ( ( x <Q f \/ f <Q x ) -> ( ( x .Q y ) +Q ( f .Q z ) ) e. ( 1st ` ( A .P. ( B +P. C ) ) ) ) )
83		ltsonq 6936	. . . . 5 |- <Q Or Q.
84		nqtri3or 6934	. . . . 5 |- ( ( x e. Q. /\ f e. Q. ) -> ( x <Q f \/ x = f \/ f <Q x ) )
85	83, 84	sotritrieq 4143	. . . 4 |- ( ( x e. Q. /\ f e. Q. ) -> ( x = f <-> -. ( x <Q f \/ f <Q x ) ) )
86	8, 12, 85	syl2anc 403	. . 3 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 1st ` A ) /\ y e. ( 1st ` B ) ) /\ ( f e. ( 1st ` A ) /\ z e. ( 1st ` C ) ) ) ) -> ( x = f <-> -. ( x <Q f \/ f <Q x ) ) )
87		oveq1 5641	. . . . . . 7 |- ( x = f -> ( x .Q z ) = ( f .Q z ) )
88	87	oveq2d 5650	. . . . . 6 |- ( x = f -> ( ( x .Q y ) +Q ( x .Q z ) ) = ( ( x .Q y ) +Q ( f .Q z ) ) )
89	72, 88	sylan9eq 2140	. . . . 5 |- ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 1st ` A ) /\ y e. ( 1st ` B ) ) /\ ( f e. ( 1st ` A ) /\ z e. ( 1st ` C ) ) ) ) /\ x = f ) -> ( x .Q ( y +Q z ) ) = ( ( x .Q y ) +Q ( f .Q z ) ) )
90	76	adantr 270	. . . . 5 |- ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 1st ` A ) /\ y e. ( 1st ` B ) ) /\ ( f e. ( 1st ` A ) /\ z e. ( 1st ` C ) ) ) ) /\ x = f ) -> ( x .Q ( y +Q z ) ) e. ( 1st ` ( A .P. ( B +P. C ) ) ) )
91	89, 90	eqeltrrd 2165	. . . 4 |- ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 1st ` A ) /\ y e. ( 1st ` B ) ) /\ ( f e. ( 1st ` A ) /\ z e. ( 1st ` C ) ) ) ) /\ x = f ) -> ( ( x .Q y ) +Q ( f .Q z ) ) e. ( 1st ` ( A .P. ( B +P. C ) ) ) )
92	91	ex 113	. . 3 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 1st ` A ) /\ y e. ( 1st ` B ) ) /\ ( f e. ( 1st ` A ) /\ z e. ( 1st ` C ) ) ) ) -> ( x = f -> ( ( x .Q y ) +Q ( f .Q z ) ) e. ( 1st ` ( A .P. ( B +P. C ) ) ) ) )
93	86, 92	sylbird 168	. 2 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 1st ` A ) /\ y e. ( 1st ` B ) ) /\ ( f e. ( 1st ` A ) /\ z e. ( 1st ` C ) ) ) ) -> ( -. ( x <Q f \/ f <Q x ) -> ( ( x .Q y ) +Q ( f .Q z ) ) e. ( 1st ` ( A .P. ( B +P. C ) ) ) ) )
94		ltdcnq 6935	. . . . 5 |- ( ( x e. Q. /\ f e. Q. ) -> DECID x <Q f )
95		ltdcnq 6935	. . . . . 6 |- ( ( f e. Q. /\ x e. Q. ) -> DECID f <Q x )
96	95	ancoms 264	. . . . 5 |- ( ( x e. Q. /\ f e. Q. ) -> DECID f <Q x )
97		dcor 881	. . . . 5 |- (DECID x <Q f -> (DECID f <Q x -> DECID ( x <Q f \/ f <Q x ) ) )
98	94, 96, 97	sylc 61	. . . 4 |- ( ( x e. Q. /\ f e. Q. ) -> DECID ( x <Q f \/ f <Q x ) )
99	8, 12, 98	syl2anc 403	. . 3 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 1st ` A ) /\ y e. ( 1st ` B ) ) /\ ( f e. ( 1st ` A ) /\ z e. ( 1st ` C ) ) ) ) -> DECID ( x <Q f \/ f <Q x ) )
100		df-dc 781	. . 3 |- (DECID ( x <Q f \/ f <Q x ) <-> ( ( x <Q f \/ f <Q x ) \/ -. ( x <Q f \/ f <Q x ) ) )
101	99, 100	sylib 120	. 2 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 1st ` A ) /\ y e. ( 1st ` B ) ) /\ ( f e. ( 1st ` A ) /\ z e. ( 1st ` C ) ) ) ) -> ( ( x <Q f \/ f <Q x ) \/ -. ( x <Q f \/ f <Q x ) ) )
102	82, 93, 101	mpjaod 673	1 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 1st ` A ) /\ y e. ( 1st ` B ) ) /\ ( f e. ( 1st ` A ) /\ z e. ( 1st ` C ) ) ) ) -> ( ( x .Q y ) +Q ( f .Q z ) ) e. ( 1st ` ( A .P. ( B +P. C ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 102   <-> wb 103   \/ wo 664  DECID wdc 780   /\ w3a 924   = wceq 1289   e. wcel 1438   <.cop 3444   class class class wbr 3837   ` cfv 5002  (class class class)co 5634   1stc1st 5891   2ndc2nd 5892   Q.cnq 6818   +Q cplq 6820   .Q cmq 6821   <Q cltq 6823   P.cnp 6829   +P. cpp 6831   .P. cmp 6832

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-iinf 4393

This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-eprel 4107  df-id 4111  df-po 4114  df-iso 4115  df-iord 4184  df-on 4186  df-suc 4189  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894  df-recs 6052  df-irdg 6117  df-1o 6163  df-2o 6164  df-oadd 6167  df-omul 6168  df-er 6272  df-ec 6274  df-qs 6278  df-ni 6842  df-pli 6843  df-mi 6844  df-lti 6845  df-plpq 6882  df-mpq 6883  df-enq 6885  df-nqqs 6886  df-plqqs 6887  df-mqqs 6888  df-1nqqs 6889  df-rq 6890  df-ltnqqs 6891  df-enq0 6962  df-nq0 6963  df-0nq0 6964  df-plq0 6965  df-mq0 6966  df-inp 7004  df-iplp 7006  df-imp 7007

This theorem is referenced by:  distrlem5prl  7124