Theorem distrlem4pru 7341

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

Assertion

Ref	Expression
distrlem4pru	|- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) -> ( ( x .Q y ) +Q ( f .Q z ) ) e. ( 2nd ` ( 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 distrlem4pru

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

Step	Hyp	Ref	Expression
1		ltmnqg 7157	. . . . . . 7 |- ( ( w e. Q. /\ v e. Q. /\ u e. Q. ) -> ( w <Q v <-> ( u .Q w ) <Q ( u .Q v ) ) )
2	1	adantl 273	. . . . . 6 |- ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) /\ ( w e. Q. /\ v e. Q. /\ u e. Q. ) ) -> ( w <Q v <-> ( u .Q w ) <Q ( u .Q v ) ) )
3		simp1 964	. . . . . . 7 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> A e. P. )
4		simpll 501	. . . . . . 7 |- ( ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) -> x e. ( 2nd ` A ) )
5		prop 7231	. . . . . . . 8 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
6		elprnqu 7238	. . . . . . . 8 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ x e. ( 2nd ` A ) ) -> x e. Q. )
7	5, 6	sylan 279	. . . . . . 7 |- ( ( A e. P. /\ x e. ( 2nd ` A ) ) -> x e. Q. )
8	3, 4, 7	syl2an 285	. . . . . 6 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) -> x e. Q. )
9		simprl 503	. . . . . . 7 |- ( ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) -> f e. ( 2nd ` A ) )
10		elprnqu 7238	. . . . . . . 8 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ f e. ( 2nd ` A ) ) -> f e. Q. )
11	5, 10	sylan 279	. . . . . . 7 |- ( ( A e. P. /\ f e. ( 2nd ` A ) ) -> f e. Q. )
12	3, 9, 11	syl2an 285	. . . . . 6 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) -> f e. Q. )
13		simpl3 969	. . . . . . 7 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) -> C e. P. )
14		simprrr 512	. . . . . . 7 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) -> z e. ( 2nd ` C ) )
15		prop 7231	. . . . . . . 8 |- ( C e. P. -> <. ( 1st ` C ) , ( 2nd ` C ) >. e. P. )
16		elprnqu 7238	. . . . . . . 8 |- ( ( <. ( 1st ` C ) , ( 2nd ` C ) >. e. P. /\ z e. ( 2nd ` C ) ) -> z e. Q. )
17	15, 16	sylan 279	. . . . . . 7 |- ( ( C e. P. /\ z e. ( 2nd ` C ) ) -> z e. Q. )
18	13, 14, 17	syl2anc 406	. . . . . 6 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) -> z e. Q. )
19		mulcomnqg 7139	. . . . . . 7 |- ( ( w e. Q. /\ v e. Q. ) -> ( w .Q v ) = ( v .Q w ) )
20	19	adantl 273	. . . . . 6 |- ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) /\ ( w e. Q. /\ v e. Q. ) ) -> ( w .Q v ) = ( v .Q w ) )
21	2, 8, 12, 18, 20	caovord2d 5894	. . . . 5 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) -> ( x <Q f <-> ( x .Q z ) <Q ( f .Q z ) ) )
22		mulclnq 7132	. . . . . . 7 |- ( ( x e. Q. /\ z e. Q. ) -> ( x .Q z ) e. Q. )
23	8, 18, 22	syl2anc 406	. . . . . 6 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) -> ( x .Q z ) e. Q. )
24		mulclnq 7132	. . . . . . 7 |- ( ( f e. Q. /\ z e. Q. ) -> ( f .Q z ) e. Q. )
25	12, 18, 24	syl2anc 406	. . . . . 6 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) -> ( f .Q z ) e. Q. )
26		simpl2 968	. . . . . . . 8 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) -> B e. P. )
27		simprlr 510	. . . . . . . 8 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) -> y e. ( 2nd ` B ) )
28		prop 7231	. . . . . . . . 9 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
29		elprnqu 7238	. . . . . . . . 9 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ y e. ( 2nd ` B ) ) -> y e. Q. )
30	28, 29	sylan 279	. . . . . . . 8 |- ( ( B e. P. /\ y e. ( 2nd ` B ) ) -> y e. Q. )
31	26, 27, 30	syl2anc 406	. . . . . . 7 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) -> y e. Q. )
32		mulclnq 7132	. . . . . . 7 |- ( ( x e. Q. /\ y e. Q. ) -> ( x .Q y ) e. Q. )
33	8, 31, 32	syl2anc 406	. . . . . 6 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) -> ( x .Q y ) e. Q. )
34		ltanqg 7156	. . . . . 6 |- ( ( ( x .Q z ) e. Q. /\ ( f .Q z ) e. Q. /\ ( x .Q y ) e. Q. ) -> ( ( x .Q z ) <Q ( f .Q z ) <-> ( ( x .Q y ) +Q ( x .Q z ) ) <Q ( ( x .Q y ) +Q ( f .Q z ) ) ) )
35	23, 25, 33, 34	syl3anc 1199	. . . . 5 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) -> ( ( x .Q z ) <Q ( f .Q z ) <-> ( ( x .Q y ) +Q ( x .Q z ) ) <Q ( ( x .Q y ) +Q ( f .Q z ) ) ) )
36	21, 35	bitrd 187	. . . 4 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) -> ( x <Q f <-> ( ( x .Q y ) +Q ( x .Q z ) ) <Q ( ( x .Q y ) +Q ( f .Q z ) ) ) )
37		simpl1 967	. . . . . 6 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) -> A e. P. )
38		addclpr 7293	. . . . . . . 8 |- ( ( B e. P. /\ C e. P. ) -> ( B +P. C ) e. P. )
39	38	3adant1 982	. . . . . . 7 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( B +P. C ) e. P. )
40	39	adantr 272	. . . . . 6 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) -> ( B +P. C ) e. P. )
41		mulclpr 7328	. . . . . 6 |- ( ( A e. P. /\ ( B +P. C ) e. P. ) -> ( A .P. ( B +P. C ) ) e. P. )
42	37, 40, 41	syl2anc 406	. . . . 5 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) -> ( A .P. ( B +P. C ) ) e. P. )
43		distrnqg 7143	. . . . . . 7 |- ( ( x e. Q. /\ y e. Q. /\ z e. Q. ) -> ( x .Q ( y +Q z ) ) = ( ( x .Q y ) +Q ( x .Q z ) ) )
44	8, 31, 18, 43	syl3anc 1199	. . . . . 6 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) -> ( x .Q ( y +Q z ) ) = ( ( x .Q y ) +Q ( x .Q z ) ) )
45		simprll 509	. . . . . . 7 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) -> x e. ( 2nd ` A ) )
46		df-iplp 7224	. . . . . . . . . 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 ) ) } >. )
47		addclnq 7131	. . . . . . . . . 10 |- ( ( g e. Q. /\ h e. Q. ) -> ( g +Q h ) e. Q. )
48	46, 47	genppreclu 7271	. . . . . . . . 9 |- ( ( B e. P. /\ C e. P. ) -> ( ( y e. ( 2nd ` B ) /\ z e. ( 2nd ` C ) ) -> ( y +Q z ) e. ( 2nd ` ( B +P. C ) ) ) )
49	48	imp 123	. . . . . . . 8 |- ( ( ( B e. P. /\ C e. P. ) /\ ( y e. ( 2nd ` B ) /\ z e. ( 2nd ` C ) ) ) -> ( y +Q z ) e. ( 2nd ` ( B +P. C ) ) )
50	26, 13, 27, 14, 49	syl22anc 1200	. . . . . . 7 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) -> ( y +Q z ) e. ( 2nd ` ( B +P. C ) ) )
51		df-imp 7225	. . . . . . . . 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 ) ) } >. )
52		mulclnq 7132	. . . . . . . . 9 |- ( ( g e. Q. /\ h e. Q. ) -> ( g .Q h ) e. Q. )
53	51, 52	genppreclu 7271	. . . . . . . 8 |- ( ( A e. P. /\ ( B +P. C ) e. P. ) -> ( ( x e. ( 2nd ` A ) /\ ( y +Q z ) e. ( 2nd ` ( B +P. C ) ) ) -> ( x .Q ( y +Q z ) ) e. ( 2nd ` ( A .P. ( B +P. C ) ) ) ) )
54	53	imp 123	. . . . . . 7 |- ( ( ( A e. P. /\ ( B +P. C ) e. P. ) /\ ( x e. ( 2nd ` A ) /\ ( y +Q z ) e. ( 2nd ` ( B +P. C ) ) ) ) -> ( x .Q ( y +Q z ) ) e. ( 2nd ` ( A .P. ( B +P. C ) ) ) )
55	37, 40, 45, 50, 54	syl22anc 1200	. . . . . 6 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) -> ( x .Q ( y +Q z ) ) e. ( 2nd ` ( A .P. ( B +P. C ) ) ) )
56	44, 55	eqeltrrd 2192	. . . . 5 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) -> ( ( x .Q y ) +Q ( x .Q z ) ) e. ( 2nd ` ( A .P. ( B +P. C ) ) ) )
57		prop 7231	. . . . . 6 |- ( ( A .P. ( B +P. C ) ) e. P. -> <. ( 1st ` ( A .P. ( B +P. C ) ) ) , ( 2nd ` ( A .P. ( B +P. C ) ) ) >. e. P. )
58		prcunqu 7241	. . . . . 6 |- ( ( <. ( 1st ` ( A .P. ( B +P. C ) ) ) , ( 2nd ` ( A .P. ( B +P. C ) ) ) >. e. P. /\ ( ( x .Q y ) +Q ( x .Q z ) ) e. ( 2nd ` ( A .P. ( B +P. C ) ) ) ) -> ( ( ( x .Q y ) +Q ( x .Q z ) ) <Q ( ( x .Q y ) +Q ( f .Q z ) ) -> ( ( x .Q y ) +Q ( f .Q z ) ) e. ( 2nd ` ( A .P. ( B +P. C ) ) ) ) )
59	57, 58	sylan 279	. . . . 5 |- ( ( ( A .P. ( B +P. C ) ) e. P. /\ ( ( x .Q y ) +Q ( x .Q z ) ) e. ( 2nd ` ( A .P. ( B +P. C ) ) ) ) -> ( ( ( x .Q y ) +Q ( x .Q z ) ) <Q ( ( x .Q y ) +Q ( f .Q z ) ) -> ( ( x .Q y ) +Q ( f .Q z ) ) e. ( 2nd ` ( A .P. ( B +P. C ) ) ) ) )
60	42, 56, 59	syl2anc 406	. . . 4 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) -> ( ( ( x .Q y ) +Q ( x .Q z ) ) <Q ( ( x .Q y ) +Q ( f .Q z ) ) -> ( ( x .Q y ) +Q ( f .Q z ) ) e. ( 2nd ` ( A .P. ( B +P. C ) ) ) ) )
61	36, 60	sylbid 149	. . 3 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) -> ( x <Q f -> ( ( x .Q y ) +Q ( f .Q z ) ) e. ( 2nd ` ( A .P. ( B +P. C ) ) ) ) )
62	2, 12, 8, 31, 20	caovord2d 5894	. . . . 5 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) -> ( f <Q x <-> ( f .Q y ) <Q ( x .Q y ) ) )
63		ltanqg 7156	. . . . . . 7 |- ( ( w e. Q. /\ v e. Q. /\ u e. Q. ) -> ( w <Q v <-> ( u +Q w ) <Q ( u +Q v ) ) )
64	63	adantl 273	. . . . . 6 |- ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) /\ ( w e. Q. /\ v e. Q. /\ u e. Q. ) ) -> ( w <Q v <-> ( u +Q w ) <Q ( u +Q v ) ) )
65		mulclnq 7132	. . . . . . 7 |- ( ( f e. Q. /\ y e. Q. ) -> ( f .Q y ) e. Q. )
66	12, 31, 65	syl2anc 406	. . . . . 6 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) -> ( f .Q y ) e. Q. )
67		addcomnqg 7137	. . . . . . 7 |- ( ( w e. Q. /\ v e. Q. ) -> ( w +Q v ) = ( v +Q w ) )
68	67	adantl 273	. . . . . 6 |- ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) /\ ( w e. Q. /\ v e. Q. ) ) -> ( w +Q v ) = ( v +Q w ) )
69	64, 66, 33, 25, 68	caovord2d 5894	. . . . 5 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) -> ( ( f .Q y ) <Q ( x .Q y ) <-> ( ( f .Q y ) +Q ( f .Q z ) ) <Q ( ( x .Q y ) +Q ( f .Q z ) ) ) )
70	62, 69	bitrd 187	. . . 4 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) -> ( f <Q x <-> ( ( f .Q y ) +Q ( f .Q z ) ) <Q ( ( x .Q y ) +Q ( f .Q z ) ) ) )
71		distrnqg 7143	. . . . . . 7 |- ( ( f e. Q. /\ y e. Q. /\ z e. Q. ) -> ( f .Q ( y +Q z ) ) = ( ( f .Q y ) +Q ( f .Q z ) ) )
72	12, 31, 18, 71	syl3anc 1199	. . . . . 6 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) -> ( f .Q ( y +Q z ) ) = ( ( f .Q y ) +Q ( f .Q z ) ) )
73		simprrl 511	. . . . . . 7 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) -> f e. ( 2nd ` A ) )
74	51, 52	genppreclu 7271	. . . . . . . 8 |- ( ( A e. P. /\ ( B +P. C ) e. P. ) -> ( ( f e. ( 2nd ` A ) /\ ( y +Q z ) e. ( 2nd ` ( B +P. C ) ) ) -> ( f .Q ( y +Q z ) ) e. ( 2nd ` ( A .P. ( B +P. C ) ) ) ) )
75	74	imp 123	. . . . . . 7 |- ( ( ( A e. P. /\ ( B +P. C ) e. P. ) /\ ( f e. ( 2nd ` A ) /\ ( y +Q z ) e. ( 2nd ` ( B +P. C ) ) ) ) -> ( f .Q ( y +Q z ) ) e. ( 2nd ` ( A .P. ( B +P. C ) ) ) )
76	37, 40, 73, 50, 75	syl22anc 1200	. . . . . 6 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) -> ( f .Q ( y +Q z ) ) e. ( 2nd ` ( A .P. ( B +P. C ) ) ) )
77	72, 76	eqeltrrd 2192	. . . . 5 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) -> ( ( f .Q y ) +Q ( f .Q z ) ) e. ( 2nd ` ( A .P. ( B +P. C ) ) ) )
78		prcunqu 7241	. . . . . 6 |- ( ( <. ( 1st ` ( A .P. ( B +P. C ) ) ) , ( 2nd ` ( A .P. ( B +P. C ) ) ) >. e. P. /\ ( ( f .Q y ) +Q ( f .Q z ) ) e. ( 2nd ` ( A .P. ( B +P. C ) ) ) ) -> ( ( ( f .Q y ) +Q ( f .Q z ) ) <Q ( ( x .Q y ) +Q ( f .Q z ) ) -> ( ( x .Q y ) +Q ( f .Q z ) ) e. ( 2nd ` ( A .P. ( B +P. C ) ) ) ) )
79	57, 78	sylan 279	. . . . 5 |- ( ( ( A .P. ( B +P. C ) ) e. P. /\ ( ( f .Q y ) +Q ( f .Q z ) ) e. ( 2nd ` ( A .P. ( B +P. C ) ) ) ) -> ( ( ( f .Q y ) +Q ( f .Q z ) ) <Q ( ( x .Q y ) +Q ( f .Q z ) ) -> ( ( x .Q y ) +Q ( f .Q z ) ) e. ( 2nd ` ( A .P. ( B +P. C ) ) ) ) )
80	42, 77, 79	syl2anc 406	. . . 4 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) -> ( ( ( f .Q y ) +Q ( f .Q z ) ) <Q ( ( x .Q y ) +Q ( f .Q z ) ) -> ( ( x .Q y ) +Q ( f .Q z ) ) e. ( 2nd ` ( A .P. ( B +P. C ) ) ) ) )
81	70, 80	sylbid 149	. . 3 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) -> ( f <Q x -> ( ( x .Q y ) +Q ( f .Q z ) ) e. ( 2nd ` ( A .P. ( B +P. C ) ) ) ) )
82	61, 81	jaod 689	. 2 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) -> ( ( x <Q f \/ f <Q x ) -> ( ( x .Q y ) +Q ( f .Q z ) ) e. ( 2nd ` ( A .P. ( B +P. C ) ) ) ) )
83		ltsonq 7154	. . . . 5 |- <Q Or Q.
84		nqtri3or 7152	. . . . 5 |- ( ( x e. Q. /\ f e. Q. ) -> ( x <Q f \/ x = f \/ f <Q x ) )
85	83, 84	sotritrieq 4207	. . . 4 |- ( ( x e. Q. /\ f e. Q. ) -> ( x = f <-> -. ( x <Q f \/ f <Q x ) ) )
86	8, 12, 85	syl2anc 406	. . 3 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) -> ( x = f <-> -. ( x <Q f \/ f <Q x ) ) )
87		oveq1 5735	. . . . . . 7 |- ( x = f -> ( x .Q z ) = ( f .Q z ) )
88	87	oveq2d 5744	. . . . . 6 |- ( x = f -> ( ( x .Q y ) +Q ( x .Q z ) ) = ( ( x .Q y ) +Q ( f .Q z ) ) )
89	44, 88	sylan9eq 2167	. . . . 5 |- ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) /\ x = f ) -> ( x .Q ( y +Q z ) ) = ( ( x .Q y ) +Q ( f .Q z ) ) )
90	55	adantr 272	. . . . 5 |- ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) /\ x = f ) -> ( x .Q ( y +Q z ) ) e. ( 2nd ` ( A .P. ( B +P. C ) ) ) )
91	89, 90	eqeltrrd 2192	. . . 4 |- ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) /\ x = f ) -> ( ( x .Q y ) +Q ( f .Q z ) ) e. ( 2nd ` ( A .P. ( B +P. C ) ) ) )
92	91	ex 114	. . 3 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) -> ( x = f -> ( ( x .Q y ) +Q ( f .Q z ) ) e. ( 2nd ` ( A .P. ( B +P. C ) ) ) ) )
93	86, 92	sylbird 169	. 2 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) -> ( -. ( x <Q f \/ f <Q x ) -> ( ( x .Q y ) +Q ( f .Q z ) ) e. ( 2nd ` ( A .P. ( B +P. C ) ) ) ) )
94		ltdcnq 7153	. . . . 5 |- ( ( x e. Q. /\ f e. Q. ) -> DECID x <Q f )
95		ltdcnq 7153	. . . . . 6 |- ( ( f e. Q. /\ x e. Q. ) -> DECID f <Q x )
96	95	ancoms 266	. . . . 5 |- ( ( x e. Q. /\ f e. Q. ) -> DECID f <Q x )
97		dcor 902	. . . . 5 |- (DECID x <Q f -> (DECID f <Q x -> DECID ( x <Q f \/ f <Q x ) ) )
98	94, 96, 97	sylc 62	. . . 4 |- ( ( x e. Q. /\ f e. Q. ) -> DECID ( x <Q f \/ f <Q x ) )
99	8, 12, 98	syl2anc 406	. . 3 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) -> DECID ( x <Q f \/ f <Q x ) )
100		df-dc 803	. . 3 |- (DECID ( x <Q f \/ f <Q x ) <-> ( ( x <Q f \/ f <Q x ) \/ -. ( x <Q f \/ f <Q x ) ) )
101	99, 100	sylib 121	. 2 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) -> ( ( x <Q f \/ f <Q x ) \/ -. ( x <Q f \/ f <Q x ) ) )
102	82, 93, 101	mpjaod 690	1 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) -> ( ( x .Q y ) +Q ( f .Q z ) ) e. ( 2nd ` ( A .P. ( B +P. C ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 680  DECID wdc 802   /\ w3a 945   = wceq 1314   e. wcel 1463   <.cop 3496   class class class wbr 3895   ` cfv 5081  (class class class)co 5728   1stc1st 5990   2ndc2nd 5991   Q.cnq 7036   +Q cplq 7038   .Q cmq 7039   <Q cltq 7041   P.cnp 7047   +P. cpp 7049   .P. cmp 7050

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4003  ax-sep 4006  ax-nul 4014  ax-pow 4058  ax-pr 4091  ax-un 4315  ax-setind 4412  ax-iinf 4462

This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ne 2283  df-ral 2395  df-rex 2396  df-reu 2397  df-rab 2399  df-v 2659  df-sbc 2879  df-csb 2972  df-dif 3039  df-un 3041  df-in 3043  df-ss 3050  df-nul 3330  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-int 3738  df-iun 3781  df-br 3896  df-opab 3950  df-mpt 3951  df-tr 3987  df-eprel 4171  df-id 4175  df-po 4178  df-iso 4179  df-iord 4248  df-on 4250  df-suc 4253  df-iom 4465  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-rn 4510  df-res 4511  df-ima 4512  df-iota 5046  df-fun 5083  df-fn 5084  df-f 5085  df-f1 5086  df-fo 5087  df-f1o 5088  df-fv 5089  df-ov 5731  df-oprab 5732  df-mpo 5733  df-1st 5992  df-2nd 5993  df-recs 6156  df-irdg 6221  df-1o 6267  df-2o 6268  df-oadd 6271  df-omul 6272  df-er 6383  df-ec 6385  df-qs 6389  df-ni 7060  df-pli 7061  df-mi 7062  df-lti 7063  df-plpq 7100  df-mpq 7101  df-enq 7103  df-nqqs 7104  df-plqqs 7105  df-mqqs 7106  df-1nqqs 7107  df-rq 7108  df-ltnqqs 7109  df-enq0 7180  df-nq0 7181  df-0nq0 7182  df-plq0 7183  df-mq0 7184  df-inp 7222  df-iplp 7224  df-imp 7225

This theorem is referenced by:  distrlem5pru  7343