Theorem distrlem4pru 7652

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 7468	. . . . . . 7 |- ( ( w e. Q. /\ v e. Q. /\ u e. Q. ) -> ( w <Q v <-> ( u .Q w ) <Q ( u .Q v ) ) )
2	1	adantl 277	. . . . . 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 999	. . . . . . 7 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> A e. P. )
4		simpll 527	. . . . . . 7 |- ( ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) -> x e. ( 2nd ` A ) )
5		prop 7542	. . . . . . . 8 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
6		elprnqu 7549	. . . . . . . 8 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ x e. ( 2nd ` A ) ) -> x e. Q. )
7	5, 6	sylan 283	. . . . . . 7 |- ( ( A e. P. /\ x e. ( 2nd ` A ) ) -> x e. Q. )
8	3, 4, 7	syl2an 289	. . . . . 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 529	. . . . . . 7 |- ( ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) -> f e. ( 2nd ` A ) )
10		elprnqu 7549	. . . . . . . 8 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ f e. ( 2nd ` A ) ) -> f e. Q. )
11	5, 10	sylan 283	. . . . . . 7 |- ( ( A e. P. /\ f e. ( 2nd ` A ) ) -> f e. Q. )
12	3, 9, 11	syl2an 289	. . . . . 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 1004	. . . . . . 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 540	. . . . . . 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 7542	. . . . . . . 8 |- ( C e. P. -> <. ( 1st ` C ) , ( 2nd ` C ) >. e. P. )
16		elprnqu 7549	. . . . . . . 8 |- ( ( <. ( 1st ` C ) , ( 2nd ` C ) >. e. P. /\ z e. ( 2nd ` C ) ) -> z e. Q. )
17	15, 16	sylan 283	. . . . . . 7 |- ( ( C e. P. /\ z e. ( 2nd ` C ) ) -> z e. Q. )
18	13, 14, 17	syl2anc 411	. . . . . 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 7450	. . . . . . 7 |- ( ( w e. Q. /\ v e. Q. ) -> ( w .Q v ) = ( v .Q w ) )
20	19	adantl 277	. . . . . 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 6093	. . . . 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 7443	. . . . . . 7 |- ( ( x e. Q. /\ z e. Q. ) -> ( x .Q z ) e. Q. )
23	8, 18, 22	syl2anc 411	. . . . . 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 7443	. . . . . . 7 |- ( ( f e. Q. /\ z e. Q. ) -> ( f .Q z ) e. Q. )
25	12, 18, 24	syl2anc 411	. . . . . 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 1003	. . . . . . . 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 538	. . . . . . . 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 7542	. . . . . . . . 9 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
29		elprnqu 7549	. . . . . . . . 9 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ y e. ( 2nd ` B ) ) -> y e. Q. )
30	28, 29	sylan 283	. . . . . . . 8 |- ( ( B e. P. /\ y e. ( 2nd ` B ) ) -> y e. Q. )
31	26, 27, 30	syl2anc 411	. . . . . . 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 7443	. . . . . . 7 |- ( ( x e. Q. /\ y e. Q. ) -> ( x .Q y ) e. Q. )
33	8, 31, 32	syl2anc 411	. . . . . 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 7467	. . . . . 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 1249	. . . . 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 188	. . . 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 1002	. . . . . 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 7604	. . . . . . . 8 |- ( ( B e. P. /\ C e. P. ) -> ( B +P. C ) e. P. )
39	38	3adant1 1017	. . . . . . 7 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( B +P. C ) e. P. )
40	39	adantr 276	. . . . . 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 7639	. . . . . 6 |- ( ( A e. P. /\ ( B +P. C ) e. P. ) -> ( A .P. ( B +P. C ) ) e. P. )
42	37, 40, 41	syl2anc 411	. . . . 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 7454	. . . . . . 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 1249	. . . . . 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 537	. . . . . . 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 7535	. . . . . . . . . 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 7442	. . . . . . . . . 10 |- ( ( g e. Q. /\ h e. Q. ) -> ( g +Q h ) e. Q. )
48	46, 47	genppreclu 7582	. . . . . . . . 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 124	. . . . . . . 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 1250	. . . . . . 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 7536	. . . . . . . . 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 7443	. . . . . . . . 9 |- ( ( g e. Q. /\ h e. Q. ) -> ( g .Q h ) e. Q. )
53	51, 52	genppreclu 7582	. . . . . . . 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 124	. . . . . . 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 1250	. . . . . 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 2274	. . . . 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 7542	. . . . . 6 |- ( ( A .P. ( B +P. C ) ) e. P. -> <. ( 1st ` ( A .P. ( B +P. C ) ) ) , ( 2nd ` ( A .P. ( B +P. C ) ) ) >. e. P. )
58		prcunqu 7552	. . . . . 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 283	. . . . 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 411	. . . 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 150	. . 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 6093	. . . . 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 7467	. . . . . . 7 |- ( ( w e. Q. /\ v e. Q. /\ u e. Q. ) -> ( w <Q v <-> ( u +Q w ) <Q ( u +Q v ) ) )
64	63	adantl 277	. . . . . 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 7443	. . . . . . 7 |- ( ( f e. Q. /\ y e. Q. ) -> ( f .Q y ) e. Q. )
66	12, 31, 65	syl2anc 411	. . . . . 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 7448	. . . . . . 7 |- ( ( w e. Q. /\ v e. Q. ) -> ( w +Q v ) = ( v +Q w ) )
68	67	adantl 277	. . . . . 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 6093	. . . . 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 188	. . . 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 7454	. . . . . . 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 1249	. . . . . 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 539	. . . . . . 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 7582	. . . . . . . 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 124	. . . . . . 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 1250	. . . . . 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 2274	. . . . 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 7552	. . . . . 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 283	. . . . 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 411	. . . 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 150	. . 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 718	. 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 7465	. . . . 5 |- <Q Or Q.
84		nqtri3or 7463	. . . . 5 |- ( ( x e. Q. /\ f e. Q. ) -> ( x <Q f \/ x = f \/ f <Q x ) )
85	83, 84	sotritrieq 4360	. . . 4 |- ( ( x e. Q. /\ f e. Q. ) -> ( x = f <-> -. ( x <Q f \/ f <Q x ) ) )
86	8, 12, 85	syl2anc 411	. . 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 5929	. . . . . . 7 |- ( x = f -> ( x .Q z ) = ( f .Q z ) )
88	87	oveq2d 5938	. . . . . 6 |- ( x = f -> ( ( x .Q y ) +Q ( x .Q z ) ) = ( ( x .Q y ) +Q ( f .Q z ) ) )
89	44, 88	sylan9eq 2249	. . . . 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 276	. . . . 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 2274	. . . 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 115	. . 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 170	. 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 7464	. . . . 5 |- ( ( x e. Q. /\ f e. Q. ) -> DECID x <Q f )
95		ltdcnq 7464	. . . . . 6 |- ( ( f e. Q. /\ x e. Q. ) -> DECID f <Q x )
96	95	ancoms 268	. . . . 5 |- ( ( x e. Q. /\ f e. Q. ) -> DECID f <Q x )
97		dcor 937	. . . . 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 411	. . 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 836	. . 3 |- (DECID ( x <Q f \/ f <Q x ) <-> ( ( x <Q f \/ f <Q x ) \/ -. ( x <Q f \/ f <Q x ) ) )
101	99, 100	sylib 122	. 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 719	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 104   <-> wb 105   \/ wo 709  DECID wdc 835   /\ w3a 980   = wceq 1364   e. wcel 2167   <.cop 3625   class class class wbr 4033   ` cfv 5258  (class class class)co 5922   1stc1st 6196   2ndc2nd 6197   Q.cnq 7347   +Q cplq 7349   .Q cmq 7350   <Q cltq 7352   P.cnp 7358   +P. cpp 7360   .P. cmp 7361

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-iplp 7535  df-imp 7536

This theorem is referenced by:  distrlem5pru  7654