Theorem distrnq 8794

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 8729	. . . . . . . . . . . . 13 |- ( ( 1st ` A ) .N ( 1st ` B ) ) = ( ( 1st ` B ) .N ( 1st ` A ) )
2	1	oveq1i 6050	. . . . . . . . . . . 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 5701	. . . . . . . . . . . . 13 |- ( 1st ` B ) e. _V
4		fvex 5701	. . . . . . . . . . . . 13 |- ( 1st ` A ) e. _V
5		fvex 5701	. . . . . . . . . . . . 13 |- ( 2nd ` A ) e. _V
6		mulcompi 8729	. . . . . . . . . . . . 13 |- ( x .N y ) = ( y .N x )
7		mulasspi 8730	. . . . . . . . . . . . 13 |- ( ( x .N y ) .N z ) = ( x .N ( y .N z ) )
8		fvex 5701	. . . . . . . . . . . . 13 |- ( 2nd ` C ) e. _V
9	3, 4, 5, 6, 7, 8	caov411 6238	. . . . . . . . . . . 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 2424	. . . . . . . . . . 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 8729	. . . . . . . . . . . . 13 |- ( ( 1st ` A ) .N ( 1st ` C ) ) = ( ( 1st ` C ) .N ( 1st ` A ) )
12	11	oveq1i 6050	. . . . . . . . . . . 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 5701	. . . . . . . . . . . . 13 |- ( 1st ` C ) e. _V
14		fvex 5701	. . . . . . . . . . . . 13 |- ( 2nd ` B ) e. _V
15	13, 4, 5, 6, 7, 14	caov411 6238	. . . . . . . . . . . 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 2424	. . . . . . . . . . 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 6052	. . . . . . . . . 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 8731	. . . . . . . . . 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 8730	. . . . . . . . . 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 2430	. . . . . . . . 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 8730	. . . . . . . . . 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 6234	. . . . . . . . . . 11 |- ( ( 2nd ` B ) .N ( ( 2nd ` A ) .N ( 2nd ` C ) ) ) = ( ( 2nd ` A ) .N ( ( 2nd ` B ) .N ( 2nd ` C ) ) )
23	22	oveq2i 6051	. . . . . . . . . 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 2424	. . . . . . . . 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 3949	. . . . . . . 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 8758	. . . . . . . . . . 11 |- ( A e. Q. -> A e. ( N. X. N. ) )
27	26	3ad2ant1 978	. . . . . . . . . 10 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> A e. ( N. X. N. ) )
28		xp2nd 6336	. . . . . . . . . 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 6335	. . . . . . . . . . 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 8758	. . . . . . . . . . . . . 14 |- ( B e. Q. -> B e. ( N. X. N. ) )
33	32	3ad2ant2 979	. . . . . . . . . . . . 13 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> B e. ( N. X. N. ) )
34		xp1st 6335	. . . . . . . . . . . . 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 8758	. . . . . . . . . . . . . 14 |- ( C e. Q. -> C e. ( N. X. N. ) )
37	36	3ad2ant3 980	. . . . . . . . . . . . 13 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> C e. ( N. X. N. ) )
38		xp2nd 6336	. . . . . . . . . . . . 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 8726	. . . . . . . . . . . 12 |- ( ( ( 1st ` B ) e. N. /\ ( 2nd ` C ) e. N. ) -> ( ( 1st ` B ) .N ( 2nd ` C ) ) e. N. )
41	35, 39, 40	syl2anc 643	. . . . . . . . . . 11 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( ( 1st ` B ) .N ( 2nd ` C ) ) e. N. )
42		xp1st 6335	. . . . . . . . . . . . 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 6336	. . . . . . . . . . . . 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 8726	. . . . . . . . . . . 12 |- ( ( ( 1st ` C ) e. N. /\ ( 2nd ` B ) e. N. ) -> ( ( 1st ` C ) .N ( 2nd ` B ) ) e. N. )
47	43, 45, 46	syl2anc 643	. . . . . . . . . . 11 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( ( 1st ` C ) .N ( 2nd ` B ) ) e. N. )
48		addclpi 8725	. . . . . . . . . . 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 643	. . . . . . . . . 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 8726	. . . . . . . . . 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 643	. . . . . . . . 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 8726	. . . . . . . . . . 11 |- ( ( ( 2nd ` B ) e. N. /\ ( 2nd ` C ) e. N. ) -> ( ( 2nd ` B ) .N ( 2nd ` C ) ) e. N. )
53	45, 39, 52	syl2anc 643	. . . . . . . . . 10 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( ( 2nd ` B ) .N ( 2nd ` C ) ) e. N. )
54		mulclpi 8726	. . . . . . . . . 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 643	. . . . . . . . 9 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( ( 2nd ` A ) .N ( ( 2nd ` B ) .N ( 2nd ` C ) ) ) e. N. )
56		mulcanenq 8793	. . . . . . . . 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 1184	. . . . . . . 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 4205	. . . . . . 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 8772	. . . . . . . . . 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 643	. . . . . . . . 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 8772	. . . . . . . . . 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 643	. . . . . . . . 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 6058	. . . . . . . 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 8726	. . . . . . . . . 10 |- ( ( ( 1st ` A ) e. N. /\ ( 1st ` B ) e. N. ) -> ( ( 1st ` A ) .N ( 1st ` B ) ) e. N. )
65	31, 35, 64	syl2anc 643	. . . . . . . . 9 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( ( 1st ` A ) .N ( 1st ` B ) ) e. N. )
66		mulclpi 8726	. . . . . . . . . 10 |- ( ( ( 2nd ` A ) e. N. /\ ( 2nd ` B ) e. N. ) -> ( ( 2nd ` A ) .N ( 2nd ` B ) ) e. N. )
67	29, 45, 66	syl2anc 643	. . . . . . . . 9 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( ( 2nd ` A ) .N ( 2nd ` B ) ) e. N. )
68		mulclpi 8726	. . . . . . . . . 10 |- ( ( ( 1st ` A ) e. N. /\ ( 1st ` C ) e. N. ) -> ( ( 1st ` A ) .N ( 1st ` C ) ) e. N. )
69	31, 43, 68	syl2anc 643	. . . . . . . . 9 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( ( 1st ` A ) .N ( 1st ` C ) ) e. N. )
70		mulclpi 8726	. . . . . . . . . 10 |- ( ( ( 2nd ` A ) e. N. /\ ( 2nd ` C ) e. N. ) -> ( ( 2nd ` A ) .N ( 2nd ` C ) ) e. N. )
71	29, 39, 70	syl2anc 643	. . . . . . . . 9 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( ( 2nd ` A ) .N ( 2nd ` C ) ) e. N. )
72		addpipq 8770	. . . . . . . . 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 1185	. . . . . . . 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 2436	. . . . . . 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 4942	. . . . . . . . . 10 |- Rel ( N. X. N. )
76		1st2nd 6352	. . . . . . . . . 10 |- ( ( Rel ( N. X. N. ) /\ A e. ( N. X. N. ) ) -> A = <. ( 1st ` A ) , ( 2nd ` A ) >. )
77	75, 27, 76	sylancr 645	. . . . . . . . 9 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> A = <. ( 1st ` A ) , ( 2nd ` A ) >. )
78		addpipq2 8769	. . . . . . . . . 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 643	. . . . . . . . 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 6058	. . . . . . . 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 8773	. . . . . . . . 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 1185	. . . . . . . 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 2436	. . . . . . 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 4202	. . . . . 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 8779	. . . . . . . . . 10 |- .pQ : ( ( N. X. N. ) X. ( N. X. N. ) ) --> ( N. X. N. )
86	85	fovcl 6134	. . . . . . . . 9 |- ( ( A e. ( N. X. N. ) /\ B e. ( N. X. N. ) ) -> ( A .pQ B ) e. ( N. X. N. ) )
87	27, 33, 86	syl2anc 643	. . . . . . . 8 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( A .pQ B ) e. ( N. X. N. ) )
88	85	fovcl 6134	. . . . . . . . 9 |- ( ( A e. ( N. X. N. ) /\ C e. ( N. X. N. ) ) -> ( A .pQ C ) e. ( N. X. N. ) )
89	27, 37, 88	syl2anc 643	. . . . . . . 8 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( A .pQ C ) e. ( N. X. N. ) )
90		addpqf 8777	. . . . . . . . 9 |- +pQ : ( ( N. X. N. ) X. ( N. X. N. ) ) --> ( N. X. N. )
91	90	fovcl 6134	. . . . . . . 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 643	. . . . . . 7 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( ( A .pQ B ) +pQ ( A .pQ C ) ) e. ( N. X. N. ) )
93	90	fovcl 6134	. . . . . . . . 9 |- ( ( B e. ( N. X. N. ) /\ C e. ( N. X. N. ) ) -> ( B +pQ C ) e. ( N. X. N. ) )
94	33, 37, 93	syl2anc 643	. . . . . . . 8 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( B +pQ C ) e. ( N. X. N. ) )
95	85	fovcl 6134	. . . . . . . 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 643	. . . . . . 7 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( A .pQ ( B +pQ C ) ) e. ( N. X. N. ) )
97		nqereq 8768	. . . . . . 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 643	. . . . . 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 202	. . . . 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 2409	. . . 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 8790	. . . 4 |- ( ( /Q ` A ) .Q ( /Q ` ( B +pQ C ) ) ) = ( /Q ` ( A .pQ ( B +pQ C ) ) )
102		adderpq 8789	. . . 4 |- ( ( /Q ` ( A .pQ B ) ) +Q ( /Q ` ( A .pQ C ) ) ) = ( /Q ` ( ( A .pQ B ) +pQ ( A .pQ C ) ) )
103	100, 101, 102	3eqtr4g 2461	. . 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 8766	. . . . . 6 |- ( A e. Q. -> ( /Q ` A ) = A )
105	104	eqcomd 2409	. . . . 5 |- ( A e. Q. -> A = ( /Q ` A ) )
106	105	3ad2ant1 978	. . . 4 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> A = ( /Q ` A ) )
107		addpqnq 8771	. . . . 5 |- ( ( B e. Q. /\ C e. Q. ) -> ( B +Q C ) = ( /Q ` ( B +pQ C ) ) )
108	107	3adant1 975	. . . 4 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( B +Q C ) = ( /Q ` ( B +pQ C ) ) )
109	106, 108	oveq12d 6058	. . 3 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( A .Q ( B +Q C ) ) = ( ( /Q ` A ) .Q ( /Q ` ( B +pQ C ) ) ) )
110		mulpqnq 8774	. . . . 5 |- ( ( A e. Q. /\ B e. Q. ) -> ( A .Q B ) = ( /Q ` ( A .pQ B ) ) )
111	110	3adant3 977	. . . 4 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( A .Q B ) = ( /Q ` ( A .pQ B ) ) )
112		mulpqnq 8774	. . . . 5 |- ( ( A e. Q. /\ C e. Q. ) -> ( A .Q C ) = ( /Q ` ( A .pQ C ) ) )
113	112	3adant2 976	. . . 4 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( A .Q C ) = ( /Q ` ( A .pQ C ) ) )
114	111, 113	oveq12d 6058	. . 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 2446	. 2 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( A .Q ( B +Q C ) ) = ( ( A .Q B ) +Q ( A .Q C ) ) )
116		addnqf 8781	. . . 4 |- +Q : ( Q. X. Q. ) --> Q.
117	116	fdmi 5555	. . 3 |- dom +Q = ( Q. X. Q. )
118		0nnq 8757	. . 3 |- -. (/) e. Q.
119		mulnqf 8782	. . . 4 |- .Q : ( Q. X. Q. ) --> Q.
120	119	fdmi 5555	. . 3 |- dom .Q = ( Q. X. Q. )
121	117, 118, 120	ndmovdistr 6195	. 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 158	1 |- ( A .Q ( B +Q C ) ) = ( ( A .Q B ) +Q ( A .Q C ) )

Syntax hints:   <-> wb 177   /\ w3a 936   = wceq 1649   e. wcel 1721   <.cop 3777   class class class wbr 4172   X. cxp 4835   Rel wrel 4842   ` cfv 5413  (class class class)co 6040   1stc1st 6306   2ndc2nd 6307   N.cnpi 8675   +N cpli 8676   .N cmi 8677   +pQ cplpq 8679   .pQ cmpq 8680   ~Q ceq 8682   Q.cnq 8683   /Qcerq 8685   +Q cplq 8686   .Q cmq 8687

This theorem is referenced by:  ltaddnq  8807  halfnq  8809  addclprlem2  8850  distrlem1pr  8858  distrlem4pr  8859  prlem934  8866  prlem936  8880

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-omul 6688  df-er 6864  df-ni 8705  df-pli 8706  df-mi 8707  df-lti 8708  df-plpq 8741  df-mpq 8742  df-enq 8744  df-nq 8745  df-erq 8746  df-plq 8747  df-mq 8748  df-1nq 8749