Theorem distrlem4prl 7546

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 7363	. . . . . . 7 |- ( ( w e. Q. /\ v e. Q. /\ u e. Q. ) -> ( w <Q v <-> ( u .Q w ) <Q ( u .Q v ) ) )
2	1	adantl 275	. . . . . 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 992	. . . . . . 7 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> A e. P. )
4		simpll 524	. . . . . . 7 |- ( ( ( x e. ( 1st ` A ) /\ y e. ( 1st ` B ) ) /\ ( f e. ( 1st ` A ) /\ z e. ( 1st ` C ) ) ) -> x e. ( 1st ` A ) )
5		prop 7437	. . . . . . . 8 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
6		elprnql 7443	. . . . . . . 8 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ x e. ( 1st ` A ) ) -> x e. Q. )
7	5, 6	sylan 281	. . . . . . 7 |- ( ( A e. P. /\ x e. ( 1st ` A ) ) -> x e. Q. )
8	3, 4, 7	syl2an 287	. . . . . 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 526	. . . . . . 7 |- ( ( ( x e. ( 1st ` A ) /\ y e. ( 1st ` B ) ) /\ ( f e. ( 1st ` A ) /\ z e. ( 1st ` C ) ) ) -> f e. ( 1st ` A ) )
10		elprnql 7443	. . . . . . . 8 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ f e. ( 1st ` A ) ) -> f e. Q. )
11	5, 10	sylan 281	. . . . . . 7 |- ( ( A e. P. /\ f e. ( 1st ` A ) ) -> f e. Q. )
12	3, 9, 11	syl2an 287	. . . . . 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 996	. . . . . . 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 533	. . . . . . 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 7437	. . . . . . . 8 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
16		elprnql 7443	. . . . . . . 8 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ y e. ( 1st ` B ) ) -> y e. Q. )
17	15, 16	sylan 281	. . . . . . 7 |- ( ( B e. P. /\ y e. ( 1st ` B ) ) -> y e. Q. )
18	13, 14, 17	syl2anc 409	. . . . . 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 7345	. . . . . . 7 |- ( ( w e. Q. /\ v e. Q. ) -> ( w .Q v ) = ( v .Q w ) )
20	19	adantl 275	. . . . . 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 6022	. . . . 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 7362	. . . . . . 7 |- ( ( w e. Q. /\ v e. Q. /\ u e. Q. ) -> ( w <Q v <-> ( u +Q w ) <Q ( u +Q v ) ) )
23	22	adantl 275	. . . . . 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 7338	. . . . . . 7 |- ( ( x e. Q. /\ y e. Q. ) -> ( x .Q y ) e. Q. )
25	8, 18, 24	syl2anc 409	. . . . . 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 7338	. . . . . . 7 |- ( ( f e. Q. /\ y e. Q. ) -> ( f .Q y ) e. Q. )
27	12, 18, 26	syl2anc 409	. . . . . 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 997	. . . . . . . 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 535	. . . . . . . 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 7437	. . . . . . . . 9 |- ( C e. P. -> <. ( 1st ` C ) , ( 2nd ` C ) >. e. P. )
31		elprnql 7443	. . . . . . . . 9 |- ( ( <. ( 1st ` C ) , ( 2nd ` C ) >. e. P. /\ z e. ( 1st ` C ) ) -> z e. Q. )
32	30, 31	sylan 281	. . . . . . . 8 |- ( ( C e. P. /\ z e. ( 1st ` C ) ) -> z e. Q. )
33	28, 29, 32	syl2anc 409	. . . . . . 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 7338	. . . . . . 7 |- ( ( f e. Q. /\ z e. Q. ) -> ( f .Q z ) e. Q. )
35	12, 33, 34	syl2anc 409	. . . . . 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 7343	. . . . . . 7 |- ( ( w e. Q. /\ v e. Q. ) -> ( w +Q v ) = ( v +Q w ) )
37	36	adantl 275	. . . . . 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 6022	. . . . 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 187	. . . 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 995	. . . . . 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 7499	. . . . . . . 8 |- ( ( B e. P. /\ C e. P. ) -> ( B +P. C ) e. P. )
42	41	3adant1 1010	. . . . . . 7 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( B +P. C ) e. P. )
43	42	adantr 274	. . . . . 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 7534	. . . . . 6 |- ( ( A e. P. /\ ( B +P. C ) e. P. ) -> ( A .P. ( B +P. C ) ) e. P. )
45	40, 43, 44	syl2anc 409	. . . . 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 7349	. . . . . . 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 1233	. . . . . 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 534	. . . . . . 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 7430	. . . . . . . . . 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 7337	. . . . . . . . . 10 |- ( ( g e. Q. /\ h e. Q. ) -> ( g +Q h ) e. Q. )
51	49, 50	genpprecll 7476	. . . . . . . . 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 123	. . . . . . . 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 1234	. . . . . . 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 7431	. . . . . . . . 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 7338	. . . . . . . . 9 |- ( ( g e. Q. /\ h e. Q. ) -> ( g .Q h ) e. Q. )
56	54, 55	genpprecll 7476	. . . . . . . 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 123	. . . . . . 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 1234	. . . . . 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 2248	. . . . 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 7437	. . . . . 6 |- ( ( A .P. ( B +P. C ) ) e. P. -> <. ( 1st ` ( A .P. ( B +P. C ) ) ) , ( 2nd ` ( A .P. ( B +P. C ) ) ) >. e. P. )
61		prcdnql 7446	. . . . . 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 281	. . . . 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 409	. . . 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 149	. . 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 6022	. . . . 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 7338	. . . . . . 7 |- ( ( x e. Q. /\ z e. Q. ) -> ( x .Q z ) e. Q. )
67	8, 33, 66	syl2anc 409	. . . . . 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 7362	. . . . . 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 1233	. . . . 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 187	. . . 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 7349	. . . . . . 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 1233	. . . . . 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 532	. . . . . . 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 7476	. . . . . . . 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 123	. . . . . . 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 1234	. . . . . 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 2248	. . . . 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 7446	. . . . . 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 281	. . . . 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 409	. . . 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 149	. . 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 712	. 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 7360	. . . . 5 |- <Q Or Q.
84		nqtri3or 7358	. . . . 5 |- ( ( x e. Q. /\ f e. Q. ) -> ( x <Q f \/ x = f \/ f <Q x ) )
85	83, 84	sotritrieq 4310	. . . 4 |- ( ( x e. Q. /\ f e. Q. ) -> ( x = f <-> -. ( x <Q f \/ f <Q x ) ) )
86	8, 12, 85	syl2anc 409	. . 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 5860	. . . . . . 7 |- ( x = f -> ( x .Q z ) = ( f .Q z ) )
88	87	oveq2d 5869	. . . . . 6 |- ( x = f -> ( ( x .Q y ) +Q ( x .Q z ) ) = ( ( x .Q y ) +Q ( f .Q z ) ) )
89	72, 88	sylan9eq 2223	. . . . 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 274	. . . . 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 2248	. . . 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 114	. . 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 169	. 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 7359	. . . . 5 |- ( ( x e. Q. /\ f e. Q. ) -> DECID x <Q f )
95		ltdcnq 7359	. . . . . 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 930	. . . . 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 409	. . 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 830	. . 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. ( 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 713	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 103   <-> wb 104   \/ wo 703  DECID wdc 829   /\ w3a 973   = wceq 1348   e. wcel 2141   <.cop 3586   class class class wbr 3989   ` cfv 5198  (class class class)co 5853   1stc1st 6117   2ndc2nd 6118   Q.cnq 7242   +Q cplq 7244   .Q cmq 7245   <Q cltq 7247   P.cnp 7253   +P. cpp 7255   .P. cmp 7256

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-eprel 4274  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-1o 6395  df-2o 6396  df-oadd 6399  df-omul 6400  df-er 6513  df-ec 6515  df-qs 6519  df-ni 7266  df-pli 7267  df-mi 7268  df-lti 7269  df-plpq 7306  df-mpq 7307  df-enq 7309  df-nqqs 7310  df-plqqs 7311  df-mqqs 7312  df-1nqqs 7313  df-rq 7314  df-ltnqqs 7315  df-enq0 7386  df-nq0 7387  df-0nq0 7388  df-plq0 7389  df-mq0 7390  df-inp 7428  df-iplp 7430  df-imp 7431

This theorem is referenced by:  distrlem5prl  7548