Theorem distrnq 9126

Description: Multiplication of positive fractions is distributive. (Contributed by NM, 2-Sep-1995.) (Revised by Mario Carneiro, 28-Apr-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
distrnq	|- ( A .Q ( B +Q C ) ) = ( ( A .Q B ) +Q ( A .Q C ) )

Proof of Theorem distrnq

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mulcompi 9061	. . . . . . . . . . . . 13 |- ( ( 1st ` A ) .N ( 1st ` B ) ) = ( ( 1st ` B ) .N ( 1st ` A ) )
2	1	oveq1i 6100	. . . . . . . . . . . 12 |- ( ( ( 1st ` A ) .N ( 1st ` B ) ) .N ( ( 2nd ` A ) .N ( 2nd ` C ) ) ) = ( ( ( 1st ` B ) .N ( 1st ` A ) ) .N ( ( 2nd ` A ) .N ( 2nd ` C ) ) )
3		fvex 5698	. . . . . . . . . . . . 13 |- ( 1st ` B ) e. _V
4		fvex 5698	. . . . . . . . . . . . 13 |- ( 1st ` A ) e. _V
5		fvex 5698	. . . . . . . . . . . . 13 |- ( 2nd ` A ) e. _V
6		mulcompi 9061	. . . . . . . . . . . . 13 |- ( x .N y ) = ( y .N x )
7		mulasspi 9062	. . . . . . . . . . . . 13 |- ( ( x .N y ) .N z ) = ( x .N ( y .N z ) )
8		fvex 5698	. . . . . . . . . . . . 13 |- ( 2nd ` C ) e. _V
9	3, 4, 5, 6, 7, 8	caov411 6294	. . . . . . . . . . . 12 |- ( ( ( 1st ` B ) .N ( 1st ` A ) ) .N ( ( 2nd ` A ) .N ( 2nd ` C ) ) ) = ( ( ( 2nd ` A ) .N ( 1st ` A ) ) .N ( ( 1st ` B ) .N ( 2nd ` C ) ) )
10	2, 9	eqtri 2461	. . . . . . . . . . 11 |- ( ( ( 1st ` A ) .N ( 1st ` B ) ) .N ( ( 2nd ` A ) .N ( 2nd ` C ) ) ) = ( ( ( 2nd ` A ) .N ( 1st ` A ) ) .N ( ( 1st ` B ) .N ( 2nd ` C ) ) )
11		mulcompi 9061	. . . . . . . . . . . . 13 |- ( ( 1st ` A ) .N ( 1st ` C ) ) = ( ( 1st ` C ) .N ( 1st ` A ) )
12	11	oveq1i 6100	. . . . . . . . . . . 12 |- ( ( ( 1st ` A ) .N ( 1st ` C ) ) .N ( ( 2nd ` A ) .N ( 2nd ` B ) ) ) = ( ( ( 1st ` C ) .N ( 1st ` A ) ) .N ( ( 2nd ` A ) .N ( 2nd ` B ) ) )
13		fvex 5698	. . . . . . . . . . . . 13 |- ( 1st ` C ) e. _V
14		fvex 5698	. . . . . . . . . . . . 13 |- ( 2nd ` B ) e. _V
15	13, 4, 5, 6, 7, 14	caov411 6294	. . . . . . . . . . . 12 |- ( ( ( 1st ` C ) .N ( 1st ` A ) ) .N ( ( 2nd ` A ) .N ( 2nd ` B ) ) ) = ( ( ( 2nd ` A ) .N ( 1st ` A ) ) .N ( ( 1st ` C ) .N ( 2nd ` B ) ) )
16	12, 15	eqtri 2461	. . . . . . . . . . 11 |- ( ( ( 1st ` A ) .N ( 1st ` C ) ) .N ( ( 2nd ` A ) .N ( 2nd ` B ) ) ) = ( ( ( 2nd ` A ) .N ( 1st ` A ) ) .N ( ( 1st ` C ) .N ( 2nd ` B ) ) )
17	10, 16	oveq12i 6102	. . . . . . . . . 10 |- ( ( ( ( 1st ` A ) .N ( 1st ` B ) ) .N ( ( 2nd ` A ) .N ( 2nd ` C ) ) ) +N ( ( ( 1st ` A ) .N ( 1st ` C ) ) .N ( ( 2nd ` A ) .N ( 2nd ` B ) ) ) ) = ( ( ( ( 2nd ` A ) .N ( 1st ` A ) ) .N ( ( 1st ` B ) .N ( 2nd ` C ) ) ) +N ( ( ( 2nd ` A ) .N ( 1st ` A ) ) .N ( ( 1st ` C ) .N ( 2nd ` B ) ) ) )
18		distrpi 9063	. . . . . . . . . 10 |- ( ( ( 2nd ` A ) .N ( 1st ` A ) ) .N ( ( ( 1st ` B ) .N ( 2nd ` C ) ) +N ( ( 1st ` C ) .N ( 2nd ` B ) ) ) ) = ( ( ( ( 2nd ` A ) .N ( 1st ` A ) ) .N ( ( 1st ` B ) .N ( 2nd ` C ) ) ) +N ( ( ( 2nd ` A ) .N ( 1st ` A ) ) .N ( ( 1st ` C ) .N ( 2nd ` B ) ) ) )
19		mulasspi 9062	. . . . . . . . . 10 |- ( ( ( 2nd ` A ) .N ( 1st ` A ) ) .N ( ( ( 1st ` B ) .N ( 2nd ` C ) ) +N ( ( 1st ` C ) .N ( 2nd ` B ) ) ) ) = ( ( 2nd ` A ) .N ( ( 1st ` A ) .N ( ( ( 1st ` B ) .N ( 2nd ` C ) ) +N ( ( 1st ` C ) .N ( 2nd ` B ) ) ) ) )
20	17, 18, 19	3eqtr2i 2467	. . . . . . . . 9 |- ( ( ( ( 1st ` A ) .N ( 1st ` B ) ) .N ( ( 2nd ` A ) .N ( 2nd ` C ) ) ) +N ( ( ( 1st ` A ) .N ( 1st ` C ) ) .N ( ( 2nd ` A ) .N ( 2nd ` B ) ) ) ) = ( ( 2nd ` A ) .N ( ( 1st ` A ) .N ( ( ( 1st ` B ) .N ( 2nd ` C ) ) +N ( ( 1st ` C ) .N ( 2nd ` B ) ) ) ) )
21		mulasspi 9062	. . . . . . . . . 10 |- ( ( ( 2nd ` A ) .N ( 2nd ` B ) ) .N ( ( 2nd ` A ) .N ( 2nd ` C ) ) ) = ( ( 2nd ` A ) .N ( ( 2nd ` B ) .N ( ( 2nd ` A ) .N ( 2nd ` C ) ) ) )
22	14, 5, 8, 6, 7	caov12 6290	. . . . . . . . . . 11 |- ( ( 2nd ` B ) .N ( ( 2nd ` A ) .N ( 2nd ` C ) ) ) = ( ( 2nd ` A ) .N ( ( 2nd ` B ) .N ( 2nd ` C ) ) )
23	22	oveq2i 6101	. . . . . . . . . 10 |- ( ( 2nd ` A ) .N ( ( 2nd ` B ) .N ( ( 2nd ` A ) .N ( 2nd ` C ) ) ) ) = ( ( 2nd ` A ) .N ( ( 2nd ` A ) .N ( ( 2nd ` B ) .N ( 2nd ` C ) ) ) )
24	21, 23	eqtri 2461	. . . . . . . . 9 |- ( ( ( 2nd ` A ) .N ( 2nd ` B ) ) .N ( ( 2nd ` A ) .N ( 2nd ` C ) ) ) = ( ( 2nd ` A ) .N ( ( 2nd ` A ) .N ( ( 2nd ` B ) .N ( 2nd ` C ) ) ) )
25	20, 24	opeq12i 4061	. . . . . . . 8 |- <. ( ( ( ( 1st ` A ) .N ( 1st ` B ) ) .N ( ( 2nd ` A ) .N ( 2nd ` C ) ) ) +N ( ( ( 1st ` A ) .N ( 1st ` C ) ) .N ( ( 2nd ` A ) .N ( 2nd ` B ) ) ) ) , ( ( ( 2nd ` A ) .N ( 2nd ` B ) ) .N ( ( 2nd ` A ) .N ( 2nd ` C ) ) ) >. = <. ( ( 2nd ` A ) .N ( ( 1st ` A ) .N ( ( ( 1st ` B ) .N ( 2nd ` C ) ) +N ( ( 1st ` C ) .N ( 2nd ` B ) ) ) ) ) , ( ( 2nd ` A ) .N ( ( 2nd ` A ) .N ( ( 2nd ` B ) .N ( 2nd ` C ) ) ) ) >.
26		elpqn 9090	. . . . . . . . . . 11 |- ( A e. Q. -> A e. ( N. X. N. ) )
27	26	3ad2ant1 1004	. . . . . . . . . 10 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> A e. ( N. X. N. ) )
28		xp2nd 6606	. . . . . . . . . 10 |- ( A e. ( N. X. N. ) -> ( 2nd ` A ) e. N. )
29	27, 28	syl 16	. . . . . . . . 9 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( 2nd ` A ) e. N. )
30		xp1st 6605	. . . . . . . . . . 11 |- ( A e. ( N. X. N. ) -> ( 1st ` A ) e. N. )
31	27, 30	syl 16	. . . . . . . . . 10 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( 1st ` A ) e. N. )
32		elpqn 9090	. . . . . . . . . . . . . 14 |- ( B e. Q. -> B e. ( N. X. N. ) )
33	32	3ad2ant2 1005	. . . . . . . . . . . . 13 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> B e. ( N. X. N. ) )
34		xp1st 6605	. . . . . . . . . . . . 13 |- ( B e. ( N. X. N. ) -> ( 1st ` B ) e. N. )
35	33, 34	syl 16	. . . . . . . . . . . 12 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( 1st ` B ) e. N. )
36		elpqn 9090	. . . . . . . . . . . . . 14 |- ( C e. Q. -> C e. ( N. X. N. ) )
37	36	3ad2ant3 1006	. . . . . . . . . . . . 13 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> C e. ( N. X. N. ) )
38		xp2nd 6606	. . . . . . . . . . . . 13 |- ( C e. ( N. X. N. ) -> ( 2nd ` C ) e. N. )
39	37, 38	syl 16	. . . . . . . . . . . 12 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( 2nd ` C ) e. N. )
40		mulclpi 9058	. . . . . . . . . . . 12 |- ( ( ( 1st ` B ) e. N. /\ ( 2nd ` C ) e. N. ) -> ( ( 1st ` B ) .N ( 2nd ` C ) ) e. N. )
41	35, 39, 40	syl2anc 656	. . . . . . . . . . 11 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( ( 1st ` B ) .N ( 2nd ` C ) ) e. N. )
42		xp1st 6605	. . . . . . . . . . . . 13 |- ( C e. ( N. X. N. ) -> ( 1st ` C ) e. N. )
43	37, 42	syl 16	. . . . . . . . . . . 12 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( 1st ` C ) e. N. )
44		xp2nd 6606	. . . . . . . . . . . . 13 |- ( B e. ( N. X. N. ) -> ( 2nd ` B ) e. N. )
45	33, 44	syl 16	. . . . . . . . . . . 12 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( 2nd ` B ) e. N. )
46		mulclpi 9058	. . . . . . . . . . . 12 |- ( ( ( 1st ` C ) e. N. /\ ( 2nd ` B ) e. N. ) -> ( ( 1st ` C ) .N ( 2nd ` B ) ) e. N. )
47	43, 45, 46	syl2anc 656	. . . . . . . . . . 11 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( ( 1st ` C ) .N ( 2nd ` B ) ) e. N. )
48		addclpi 9057	. . . . . . . . . . 11 |- ( ( ( ( 1st ` B ) .N ( 2nd ` C ) ) e. N. /\ ( ( 1st ` C ) .N ( 2nd ` B ) ) e. N. ) -> ( ( ( 1st ` B ) .N ( 2nd ` C ) ) +N ( ( 1st ` C ) .N ( 2nd ` B ) ) ) e. N. )
49	41, 47, 48	syl2anc 656	. . . . . . . . . 10 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( ( ( 1st ` B ) .N ( 2nd ` C ) ) +N ( ( 1st ` C ) .N ( 2nd ` B ) ) ) e. N. )
50		mulclpi 9058	. . . . . . . . . 10 |- ( ( ( 1st ` A ) e. N. /\ ( ( ( 1st ` B ) .N ( 2nd ` C ) ) +N ( ( 1st ` C ) .N ( 2nd ` B ) ) ) e. N. ) -> ( ( 1st ` A ) .N ( ( ( 1st ` B ) .N ( 2nd ` C ) ) +N ( ( 1st ` C ) .N ( 2nd ` B ) ) ) ) e. N. )
51	31, 49, 50	syl2anc 656	. . . . . . . . 9 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( ( 1st ` A ) .N ( ( ( 1st ` B ) .N ( 2nd ` C ) ) +N ( ( 1st ` C ) .N ( 2nd ` B ) ) ) ) e. N. )
52		mulclpi 9058	. . . . . . . . . . 11 |- ( ( ( 2nd ` B ) e. N. /\ ( 2nd ` C ) e. N. ) -> ( ( 2nd ` B ) .N ( 2nd ` C ) ) e. N. )
53	45, 39, 52	syl2anc 656	. . . . . . . . . 10 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( ( 2nd ` B ) .N ( 2nd ` C ) ) e. N. )
54		mulclpi 9058	. . . . . . . . . 10 |- ( ( ( 2nd ` A ) e. N. /\ ( ( 2nd ` B ) .N ( 2nd ` C ) ) e. N. ) -> ( ( 2nd ` A ) .N ( ( 2nd ` B ) .N ( 2nd ` C ) ) ) e. N. )
55	29, 53, 54	syl2anc 656	. . . . . . . . 9 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( ( 2nd ` A ) .N ( ( 2nd ` B ) .N ( 2nd ` C ) ) ) e. N. )
56		mulcanenq 9125	. . . . . . . . 9 |- ( ( ( 2nd ` A ) e. N. /\ ( ( 1st ` A ) .N ( ( ( 1st ` B ) .N ( 2nd ` C ) ) +N ( ( 1st ` C ) .N ( 2nd ` B ) ) ) ) e. N. /\ ( ( 2nd ` A ) .N ( ( 2nd ` B ) .N ( 2nd ` C ) ) ) e. N. ) -> <. ( ( 2nd ` A ) .N ( ( 1st ` A ) .N ( ( ( 1st ` B ) .N ( 2nd ` C ) ) +N ( ( 1st ` C ) .N ( 2nd ` B ) ) ) ) ) , ( ( 2nd ` A ) .N ( ( 2nd ` A ) .N ( ( 2nd ` B ) .N ( 2nd ` C ) ) ) ) >. ~Q <. ( ( 1st ` A ) .N ( ( ( 1st ` B ) .N ( 2nd ` C ) ) +N ( ( 1st ` C ) .N ( 2nd ` B ) ) ) ) , ( ( 2nd ` A ) .N ( ( 2nd ` B ) .N ( 2nd ` C ) ) ) >. )
57	29, 51, 55, 56	syl3anc 1213	. . . . . . . 8 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> <. ( ( 2nd ` A ) .N ( ( 1st ` A ) .N ( ( ( 1st ` B ) .N ( 2nd ` C ) ) +N ( ( 1st ` C ) .N ( 2nd ` B ) ) ) ) ) , ( ( 2nd ` A ) .N ( ( 2nd ` A ) .N ( ( 2nd ` B ) .N ( 2nd ` C ) ) ) ) >. ~Q <. ( ( 1st ` A ) .N ( ( ( 1st ` B ) .N ( 2nd ` C ) ) +N ( ( 1st ` C ) .N ( 2nd ` B ) ) ) ) , ( ( 2nd ` A ) .N ( ( 2nd ` B ) .N ( 2nd ` C ) ) ) >. )
58	25, 57	syl5eqbr 4322	. . . . . . 7 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> <. ( ( ( ( 1st ` A ) .N ( 1st ` B ) ) .N ( ( 2nd ` A ) .N ( 2nd ` C ) ) ) +N ( ( ( 1st ` A ) .N ( 1st ` C ) ) .N ( ( 2nd ` A ) .N ( 2nd ` B ) ) ) ) , ( ( ( 2nd ` A ) .N ( 2nd ` B ) ) .N ( ( 2nd ` A ) .N ( 2nd ` C ) ) ) >. ~Q <. ( ( 1st ` A ) .N ( ( ( 1st ` B ) .N ( 2nd ` C ) ) +N ( ( 1st ` C ) .N ( 2nd ` B ) ) ) ) , ( ( 2nd ` A ) .N ( ( 2nd ` B ) .N ( 2nd ` C ) ) ) >. )
59		mulpipq2 9104	. . . . . . . . . 10 |- ( ( A e. ( N. X. N. ) /\ B e. ( N. X. N. ) ) -> ( A .pQ B ) = <. ( ( 1st ` A ) .N ( 1st ` B ) ) , ( ( 2nd ` A ) .N ( 2nd ` B ) ) >. )
60	27, 33, 59	syl2anc 656	. . . . . . . . 9 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( A .pQ B ) = <. ( ( 1st ` A ) .N ( 1st ` B ) ) , ( ( 2nd ` A ) .N ( 2nd ` B ) ) >. )
61		mulpipq2 9104	. . . . . . . . . 10 |- ( ( A e. ( N. X. N. ) /\ C e. ( N. X. N. ) ) -> ( A .pQ C ) = <. ( ( 1st ` A ) .N ( 1st ` C ) ) , ( ( 2nd ` A ) .N ( 2nd ` C ) ) >. )
62	27, 37, 61	syl2anc 656	. . . . . . . . 9 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( A .pQ C ) = <. ( ( 1st ` A ) .N ( 1st ` C ) ) , ( ( 2nd ` A ) .N ( 2nd ` C ) ) >. )
63	60, 62	oveq12d 6108	. . . . . . . 8 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( ( A .pQ B ) +pQ ( A .pQ C ) ) = ( <. ( ( 1st ` A ) .N ( 1st ` B ) ) , ( ( 2nd ` A ) .N ( 2nd ` B ) ) >. +pQ <. ( ( 1st ` A ) .N ( 1st ` C ) ) , ( ( 2nd ` A ) .N ( 2nd ` C ) ) >. ) )
64		mulclpi 9058	. . . . . . . . . 10 |- ( ( ( 1st ` A ) e. N. /\ ( 1st ` B ) e. N. ) -> ( ( 1st ` A ) .N ( 1st ` B ) ) e. N. )
65	31, 35, 64	syl2anc 656	. . . . . . . . 9 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( ( 1st ` A ) .N ( 1st ` B ) ) e. N. )
66		mulclpi 9058	. . . . . . . . . 10 |- ( ( ( 2nd ` A ) e. N. /\ ( 2nd ` B ) e. N. ) -> ( ( 2nd ` A ) .N ( 2nd ` B ) ) e. N. )
67	29, 45, 66	syl2anc 656	. . . . . . . . 9 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( ( 2nd ` A ) .N ( 2nd ` B ) ) e. N. )
68		mulclpi 9058	. . . . . . . . . 10 |- ( ( ( 1st ` A ) e. N. /\ ( 1st ` C ) e. N. ) -> ( ( 1st ` A ) .N ( 1st ` C ) ) e. N. )
69	31, 43, 68	syl2anc 656	. . . . . . . . 9 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( ( 1st ` A ) .N ( 1st ` C ) ) e. N. )
70		mulclpi 9058	. . . . . . . . . 10 |- ( ( ( 2nd ` A ) e. N. /\ ( 2nd ` C ) e. N. ) -> ( ( 2nd ` A ) .N ( 2nd ` C ) ) e. N. )
71	29, 39, 70	syl2anc 656	. . . . . . . . 9 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( ( 2nd ` A ) .N ( 2nd ` C ) ) e. N. )
72		addpipq 9102	. . . . . . . . 9 |- ( ( ( ( ( 1st ` A ) .N ( 1st ` B ) ) e. N. /\ ( ( 2nd ` A ) .N ( 2nd ` B ) ) e. N. ) /\ ( ( ( 1st ` A ) .N ( 1st ` C ) ) e. N. /\ ( ( 2nd ` A ) .N ( 2nd ` C ) ) e. N. ) ) -> ( <. ( ( 1st ` A ) .N ( 1st ` B ) ) , ( ( 2nd ` A ) .N ( 2nd ` B ) ) >. +pQ <. ( ( 1st ` A ) .N ( 1st ` C ) ) , ( ( 2nd ` A ) .N ( 2nd ` C ) ) >. ) = <. ( ( ( ( 1st ` A ) .N ( 1st ` B ) ) .N ( ( 2nd ` A ) .N ( 2nd ` C ) ) ) +N ( ( ( 1st ` A ) .N ( 1st ` C ) ) .N ( ( 2nd ` A ) .N ( 2nd ` B ) ) ) ) , ( ( ( 2nd ` A ) .N ( 2nd ` B ) ) .N ( ( 2nd ` A ) .N ( 2nd ` C ) ) ) >. )
73	65, 67, 69, 71, 72	syl22anc 1214	. . . . . . . 8 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( <. ( ( 1st ` A ) .N ( 1st ` B ) ) , ( ( 2nd ` A ) .N ( 2nd ` B ) ) >. +pQ <. ( ( 1st ` A ) .N ( 1st ` C ) ) , ( ( 2nd ` A ) .N ( 2nd ` C ) ) >. ) = <. ( ( ( ( 1st ` A ) .N ( 1st ` B ) ) .N ( ( 2nd ` A ) .N ( 2nd ` C ) ) ) +N ( ( ( 1st ` A ) .N ( 1st ` C ) ) .N ( ( 2nd ` A ) .N ( 2nd ` B ) ) ) ) , ( ( ( 2nd ` A ) .N ( 2nd ` B ) ) .N ( ( 2nd ` A ) .N ( 2nd ` C ) ) ) >. )
74	63, 73	eqtrd 2473	. . . . . . 7 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( ( A .pQ B ) +pQ ( A .pQ C ) ) = <. ( ( ( ( 1st ` A ) .N ( 1st ` B ) ) .N ( ( 2nd ` A ) .N ( 2nd ` C ) ) ) +N ( ( ( 1st ` A ) .N ( 1st ` C ) ) .N ( ( 2nd ` A ) .N ( 2nd ` B ) ) ) ) , ( ( ( 2nd ` A ) .N ( 2nd ` B ) ) .N ( ( 2nd ` A ) .N ( 2nd ` C ) ) ) >. )
75		relxp 4943	. . . . . . . . . 10 |- Rel ( N. X. N. )
76		1st2nd 6619	. . . . . . . . . 10 |- ( ( Rel ( N. X. N. ) /\ A e. ( N. X. N. ) ) -> A = <. ( 1st ` A ) , ( 2nd ` A ) >. )
77	75, 27, 76	sylancr 658	. . . . . . . . 9 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> A = <. ( 1st ` A ) , ( 2nd ` A ) >. )
78		addpipq2 9101	. . . . . . . . . 10 |- ( ( B e. ( N. X. N. ) /\ C e. ( N. X. N. ) ) -> ( B +pQ C ) = <. ( ( ( 1st ` B ) .N ( 2nd ` C ) ) +N ( ( 1st ` C ) .N ( 2nd ` B ) ) ) , ( ( 2nd ` B ) .N ( 2nd ` C ) ) >. )
79	33, 37, 78	syl2anc 656	. . . . . . . . 9 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( B +pQ C ) = <. ( ( ( 1st ` B ) .N ( 2nd ` C ) ) +N ( ( 1st ` C ) .N ( 2nd ` B ) ) ) , ( ( 2nd ` B ) .N ( 2nd ` C ) ) >. )
80	77, 79	oveq12d 6108	. . . . . . . 8 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( A .pQ ( B +pQ C ) ) = ( <. ( 1st ` A ) , ( 2nd ` A ) >. .pQ <. ( ( ( 1st ` B ) .N ( 2nd ` C ) ) +N ( ( 1st ` C ) .N ( 2nd ` B ) ) ) , ( ( 2nd ` B ) .N ( 2nd ` C ) ) >. ) )
81		mulpipq 9105	. . . . . . . . 9 |- ( ( ( ( 1st ` A ) e. N. /\ ( 2nd ` A ) e. N. ) /\ ( ( ( ( 1st ` B ) .N ( 2nd ` C ) ) +N ( ( 1st ` C ) .N ( 2nd ` B ) ) ) e. N. /\ ( ( 2nd ` B ) .N ( 2nd ` C ) ) e. N. ) ) -> ( <. ( 1st ` A ) , ( 2nd ` A ) >. .pQ <. ( ( ( 1st ` B ) .N ( 2nd ` C ) ) +N ( ( 1st ` C ) .N ( 2nd ` B ) ) ) , ( ( 2nd ` B ) .N ( 2nd ` C ) ) >. ) = <. ( ( 1st ` A ) .N ( ( ( 1st ` B ) .N ( 2nd ` C ) ) +N ( ( 1st ` C ) .N ( 2nd ` B ) ) ) ) , ( ( 2nd ` A ) .N ( ( 2nd ` B ) .N ( 2nd ` C ) ) ) >. )
82	31, 29, 49, 53, 81	syl22anc 1214	. . . . . . . 8 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( <. ( 1st ` A ) , ( 2nd ` A ) >. .pQ <. ( ( ( 1st ` B ) .N ( 2nd ` C ) ) +N ( ( 1st ` C ) .N ( 2nd ` B ) ) ) , ( ( 2nd ` B ) .N ( 2nd ` C ) ) >. ) = <. ( ( 1st ` A ) .N ( ( ( 1st ` B ) .N ( 2nd ` C ) ) +N ( ( 1st ` C ) .N ( 2nd ` B ) ) ) ) , ( ( 2nd ` A ) .N ( ( 2nd ` B ) .N ( 2nd ` C ) ) ) >. )
83	80, 82	eqtrd 2473	. . . . . . 7 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( A .pQ ( B +pQ C ) ) = <. ( ( 1st ` A ) .N ( ( ( 1st ` B ) .N ( 2nd ` C ) ) +N ( ( 1st ` C ) .N ( 2nd ` B ) ) ) ) , ( ( 2nd ` A ) .N ( ( 2nd ` B ) .N ( 2nd ` C ) ) ) >. )
84	58, 74, 83	3brtr4d 4319	. . . . . 6 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( ( A .pQ B ) +pQ ( A .pQ C ) ) ~Q ( A .pQ ( B +pQ C ) ) )
85		mulpqf 9111	. . . . . . . . . 10 |- .pQ : ( ( N. X. N. ) X. ( N. X. N. ) ) --> ( N. X. N. )
86	85	fovcl 6194	. . . . . . . . 9 |- ( ( A e. ( N. X. N. ) /\ B e. ( N. X. N. ) ) -> ( A .pQ B ) e. ( N. X. N. ) )
87	27, 33, 86	syl2anc 656	. . . . . . . 8 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( A .pQ B ) e. ( N. X. N. ) )
88	85	fovcl 6194	. . . . . . . . 9 |- ( ( A e. ( N. X. N. ) /\ C e. ( N. X. N. ) ) -> ( A .pQ C ) e. ( N. X. N. ) )
89	27, 37, 88	syl2anc 656	. . . . . . . 8 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( A .pQ C ) e. ( N. X. N. ) )
90		addpqf 9109	. . . . . . . . 9 |- +pQ : ( ( N. X. N. ) X. ( N. X. N. ) ) --> ( N. X. N. )
91	90	fovcl 6194	. . . . . . . 8 |- ( ( ( A .pQ B ) e. ( N. X. N. ) /\ ( A .pQ C ) e. ( N. X. N. ) ) -> ( ( A .pQ B ) +pQ ( A .pQ C ) ) e. ( N. X. N. ) )
92	87, 89, 91	syl2anc 656	. . . . . . 7 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( ( A .pQ B ) +pQ ( A .pQ C ) ) e. ( N. X. N. ) )
93	90	fovcl 6194	. . . . . . . . 9 |- ( ( B e. ( N. X. N. ) /\ C e. ( N. X. N. ) ) -> ( B +pQ C ) e. ( N. X. N. ) )
94	33, 37, 93	syl2anc 656	. . . . . . . 8 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( B +pQ C ) e. ( N. X. N. ) )
95	85	fovcl 6194	. . . . . . . 8 |- ( ( A e. ( N. X. N. ) /\ ( B +pQ C ) e. ( N. X. N. ) ) -> ( A .pQ ( B +pQ C ) ) e. ( N. X. N. ) )
96	27, 94, 95	syl2anc 656	. . . . . . 7 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( A .pQ ( B +pQ C ) ) e. ( N. X. N. ) )
97		nqereq 9100	. . . . . . 7 |- ( ( ( ( A .pQ B ) +pQ ( A .pQ C ) ) e. ( N. X. N. ) /\ ( A .pQ ( B +pQ C ) ) e. ( N. X. N. ) ) -> ( ( ( A .pQ B ) +pQ ( A .pQ C ) ) ~Q ( A .pQ ( B +pQ C ) ) <-> ( /Q ` ( ( A .pQ B ) +pQ ( A .pQ C ) ) ) = ( /Q ` ( A .pQ ( B +pQ C ) ) ) ) )
98	92, 96, 97	syl2anc 656	. . . . . 6 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( ( ( A .pQ B ) +pQ ( A .pQ C ) ) ~Q ( A .pQ ( B +pQ C ) ) <-> ( /Q ` ( ( A .pQ B ) +pQ ( A .pQ C ) ) ) = ( /Q ` ( A .pQ ( B +pQ C ) ) ) ) )
99	84, 98	mpbid 210	. . . . 5 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( /Q ` ( ( A .pQ B ) +pQ ( A .pQ C ) ) ) = ( /Q ` ( A .pQ ( B +pQ C ) ) ) )
100	99	eqcomd 2446	. . . 4 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( /Q ` ( A .pQ ( B +pQ C ) ) ) = ( /Q ` ( ( A .pQ B ) +pQ ( A .pQ C ) ) ) )
101		mulerpq 9122	. . . 4 |- ( ( /Q ` A ) .Q ( /Q ` ( B +pQ C ) ) ) = ( /Q ` ( A .pQ ( B +pQ C ) ) )
102		adderpq 9121	. . . 4 |- ( ( /Q ` ( A .pQ B ) ) +Q ( /Q ` ( A .pQ C ) ) ) = ( /Q ` ( ( A .pQ B ) +pQ ( A .pQ C ) ) )
103	100, 101, 102	3eqtr4g 2498	. . 3 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( ( /Q ` A ) .Q ( /Q ` ( B +pQ C ) ) ) = ( ( /Q ` ( A .pQ B ) ) +Q ( /Q ` ( A .pQ C ) ) ) )
104		nqerid 9098	. . . . . 6 |- ( A e. Q. -> ( /Q ` A ) = A )
105	104	eqcomd 2446	. . . . 5 |- ( A e. Q. -> A = ( /Q ` A ) )
106	105	3ad2ant1 1004	. . . 4 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> A = ( /Q ` A ) )
107		addpqnq 9103	. . . . 5 |- ( ( B e. Q. /\ C e. Q. ) -> ( B +Q C ) = ( /Q ` ( B +pQ C ) ) )
108	107	3adant1 1001	. . . 4 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( B +Q C ) = ( /Q ` ( B +pQ C ) ) )
109	106, 108	oveq12d 6108	. . 3 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( A .Q ( B +Q C ) ) = ( ( /Q ` A ) .Q ( /Q ` ( B +pQ C ) ) ) )
110		mulpqnq 9106	. . . . 5 |- ( ( A e. Q. /\ B e. Q. ) -> ( A .Q B ) = ( /Q ` ( A .pQ B ) ) )
111	110	3adant3 1003	. . . 4 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( A .Q B ) = ( /Q ` ( A .pQ B ) ) )
112		mulpqnq 9106	. . . . 5 |- ( ( A e. Q. /\ C e. Q. ) -> ( A .Q C ) = ( /Q ` ( A .pQ C ) ) )
113	112	3adant2 1002	. . . 4 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( A .Q C ) = ( /Q ` ( A .pQ C ) ) )
114	111, 113	oveq12d 6108	. . 3 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( ( A .Q B ) +Q ( A .Q C ) ) = ( ( /Q ` ( A .pQ B ) ) +Q ( /Q ` ( A .pQ C ) ) ) )
115	103, 109, 114	3eqtr4d 2483	. 2 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( A .Q ( B +Q C ) ) = ( ( A .Q B ) +Q ( A .Q C ) ) )
116		addnqf 9113	. . . 4 |- +Q : ( Q. X. Q. ) --> Q.
117	116	fdmi 5561	. . 3 |- dom +Q = ( Q. X. Q. )
118		0nnq 9089	. . 3 |- -. (/) e. Q.
119		mulnqf 9114	. . . 4 |- .Q : ( Q. X. Q. ) --> Q.
120	119	fdmi 5561	. . 3 |- dom .Q = ( Q. X. Q. )
121	117, 118, 120	ndmovdistr 6251	. 2 |- ( -. ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( A .Q ( B +Q C ) ) = ( ( A .Q B ) +Q ( A .Q C ) ) )
122	115, 121	pm2.61i 164	1 |- ( A .Q ( B +Q C ) ) = ( ( A .Q B ) +Q ( A .Q C ) )

Syntax hints:   <-> wb 184   /\ w3a 960   = wceq 1364   e. wcel 1761   <.cop 3880   class class class wbr 4289   X. cxp 4834   Rel wrel 4841   ` cfv 5415  (class class class)co 6090   1stc1st 6574   2ndc2nd 6575   N.cnpi 9007   +N cpli 9008   .N cmi 9009   +pQ cplpq 9011   .pQ cmpq 9012   ~Q ceq 9014   Q.cnq 9015   /Qcerq 9017   +Q cplq 9018   .Q cmq 9019

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-omul 6921  df-er 7097  df-ni 9037  df-pli 9038  df-mi 9039  df-lti 9040  df-plpq 9073  df-mpq 9074  df-enq 9076  df-nq 9077  df-erq 9078  df-plq 9079  df-mq 9080  df-1nq 9081

This theorem is referenced by:  ltaddnq  9139  halfnq  9141  addclprlem2  9182  distrlem1pr  9190  distrlem4pr  9191  prlem934  9198  prlem936  9212