Theorem distrnq 9130

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 9065	. . . . . . . . . . . . 13 |- ( ( 1st ` A ) .N ( 1st ` B ) ) = ( ( 1st ` B ) .N ( 1st ` A ) )
2	1	oveq1i 6101	. . . . . . . . . . . 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 9065	. . . . . . . . . . . . 13 |- ( x .N y ) = ( y .N x )
7		mulasspi 9066	. . . . . . . . . . . . 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 6295	. . . . . . . . . . . 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 2463	. . . . . . . . . . 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 9065	. . . . . . . . . . . . 13 |- ( ( 1st ` A ) .N ( 1st ` C ) ) = ( ( 1st ` C ) .N ( 1st ` A ) )
12	11	oveq1i 6101	. . . . . . . . . . . 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 6295	. . . . . . . . . . . 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 2463	. . . . . . . . . . 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 6103	. . . . . . . . . 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 9067	. . . . . . . . . 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 9066	. . . . . . . . . 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 2469	. . . . . . . . 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 9066	. . . . . . . . . 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 6291	. . . . . . . . . . 11 |- ( ( 2nd ` B ) .N ( ( 2nd ` A ) .N ( 2nd ` C ) ) ) = ( ( 2nd ` A ) .N ( ( 2nd ` B ) .N ( 2nd ` C ) ) )
23	22	oveq2i 6102	. . . . . . . . . 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 2463	. . . . . . . . 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 4064	. . . . . . . 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 9094	. . . . . . . . . . 11 |- ( A e. Q. -> A e. ( N. X. N. ) )
27	26	3ad2ant1 1009	. . . . . . . . . 10 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> A e. ( N. X. N. ) )
28		xp2nd 6607	. . . . . . . . . 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 6606	. . . . . . . . . . 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 9094	. . . . . . . . . . . . . 14 |- ( B e. Q. -> B e. ( N. X. N. ) )
33	32	3ad2ant2 1010	. . . . . . . . . . . . 13 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> B e. ( N. X. N. ) )
34		xp1st 6606	. . . . . . . . . . . . 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 9094	. . . . . . . . . . . . . 14 |- ( C e. Q. -> C e. ( N. X. N. ) )
37	36	3ad2ant3 1011	. . . . . . . . . . . . 13 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> C e. ( N. X. N. ) )
38		xp2nd 6607	. . . . . . . . . . . . 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 9062	. . . . . . . . . . . 12 |- ( ( ( 1st ` B ) e. N. /\ ( 2nd ` C ) e. N. ) -> ( ( 1st ` B ) .N ( 2nd ` C ) ) e. N. )
41	35, 39, 40	syl2anc 661	. . . . . . . . . . 11 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( ( 1st ` B ) .N ( 2nd ` C ) ) e. N. )
42		xp1st 6606	. . . . . . . . . . . . 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 6607	. . . . . . . . . . . . 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 9062	. . . . . . . . . . . 12 |- ( ( ( 1st ` C ) e. N. /\ ( 2nd ` B ) e. N. ) -> ( ( 1st ` C ) .N ( 2nd ` B ) ) e. N. )
47	43, 45, 46	syl2anc 661	. . . . . . . . . . 11 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( ( 1st ` C ) .N ( 2nd ` B ) ) e. N. )
48		addclpi 9061	. . . . . . . . . . 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 661	. . . . . . . . . 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 9062	. . . . . . . . . 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 661	. . . . . . . . 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 9062	. . . . . . . . . . 11 |- ( ( ( 2nd ` B ) e. N. /\ ( 2nd ` C ) e. N. ) -> ( ( 2nd ` B ) .N ( 2nd ` C ) ) e. N. )
53	45, 39, 52	syl2anc 661	. . . . . . . . . 10 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( ( 2nd ` B ) .N ( 2nd ` C ) ) e. N. )
54		mulclpi 9062	. . . . . . . . . 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 661	. . . . . . . . 9 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( ( 2nd ` A ) .N ( ( 2nd ` B ) .N ( 2nd ` C ) ) ) e. N. )
56		mulcanenq 9129	. . . . . . . . 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 1218	. . . . . . . 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 4325	. . . . . . 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 9108	. . . . . . . . . 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 661	. . . . . . . . 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 9108	. . . . . . . . . 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 661	. . . . . . . . 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 6109	. . . . . . . 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 9062	. . . . . . . . . 10 |- ( ( ( 1st ` A ) e. N. /\ ( 1st ` B ) e. N. ) -> ( ( 1st ` A ) .N ( 1st ` B ) ) e. N. )
65	31, 35, 64	syl2anc 661	. . . . . . . . 9 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( ( 1st ` A ) .N ( 1st ` B ) ) e. N. )
66		mulclpi 9062	. . . . . . . . . 10 |- ( ( ( 2nd ` A ) e. N. /\ ( 2nd ` B ) e. N. ) -> ( ( 2nd ` A ) .N ( 2nd ` B ) ) e. N. )
67	29, 45, 66	syl2anc 661	. . . . . . . . 9 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( ( 2nd ` A ) .N ( 2nd ` B ) ) e. N. )
68		mulclpi 9062	. . . . . . . . . 10 |- ( ( ( 1st ` A ) e. N. /\ ( 1st ` C ) e. N. ) -> ( ( 1st ` A ) .N ( 1st ` C ) ) e. N. )
69	31, 43, 68	syl2anc 661	. . . . . . . . 9 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( ( 1st ` A ) .N ( 1st ` C ) ) e. N. )
70		mulclpi 9062	. . . . . . . . . 10 |- ( ( ( 2nd ` A ) e. N. /\ ( 2nd ` C ) e. N. ) -> ( ( 2nd ` A ) .N ( 2nd ` C ) ) e. N. )
71	29, 39, 70	syl2anc 661	. . . . . . . . 9 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( ( 2nd ` A ) .N ( 2nd ` C ) ) e. N. )
72		addpipq 9106	. . . . . . . . 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 1219	. . . . . . . 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 2475	. . . . . . 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 4947	. . . . . . . . . 10 |- Rel ( N. X. N. )
76		1st2nd 6620	. . . . . . . . . 10 |- ( ( Rel ( N. X. N. ) /\ A e. ( N. X. N. ) ) -> A = <. ( 1st ` A ) , ( 2nd ` A ) >. )
77	75, 27, 76	sylancr 663	. . . . . . . . 9 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> A = <. ( 1st ` A ) , ( 2nd ` A ) >. )
78		addpipq2 9105	. . . . . . . . . 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 661	. . . . . . . . 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 6109	. . . . . . . 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 9109	. . . . . . . . 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 1219	. . . . . . . 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 2475	. . . . . . 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 4322	. . . . . 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 9115	. . . . . . . . . 10 |- .pQ : ( ( N. X. N. ) X. ( N. X. N. ) ) --> ( N. X. N. )
86	85	fovcl 6195	. . . . . . . . 9 |- ( ( A e. ( N. X. N. ) /\ B e. ( N. X. N. ) ) -> ( A .pQ B ) e. ( N. X. N. ) )
87	27, 33, 86	syl2anc 661	. . . . . . . 8 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( A .pQ B ) e. ( N. X. N. ) )
88	85	fovcl 6195	. . . . . . . . 9 |- ( ( A e. ( N. X. N. ) /\ C e. ( N. X. N. ) ) -> ( A .pQ C ) e. ( N. X. N. ) )
89	27, 37, 88	syl2anc 661	. . . . . . . 8 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( A .pQ C ) e. ( N. X. N. ) )
90		addpqf 9113	. . . . . . . . 9 |- +pQ : ( ( N. X. N. ) X. ( N. X. N. ) ) --> ( N. X. N. )
91	90	fovcl 6195	. . . . . . . 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 661	. . . . . . 7 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( ( A .pQ B ) +pQ ( A .pQ C ) ) e. ( N. X. N. ) )
93	90	fovcl 6195	. . . . . . . . 9 |- ( ( B e. ( N. X. N. ) /\ C e. ( N. X. N. ) ) -> ( B +pQ C ) e. ( N. X. N. ) )
94	33, 37, 93	syl2anc 661	. . . . . . . 8 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( B +pQ C ) e. ( N. X. N. ) )
95	85	fovcl 6195	. . . . . . . 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 661	. . . . . . 7 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( A .pQ ( B +pQ C ) ) e. ( N. X. N. ) )
97		nqereq 9104	. . . . . . 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 661	. . . . . 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 2448	. . . 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 9126	. . . 4 |- ( ( /Q ` A ) .Q ( /Q ` ( B +pQ C ) ) ) = ( /Q ` ( A .pQ ( B +pQ C ) ) )
102		adderpq 9125	. . . 4 |- ( ( /Q ` ( A .pQ B ) ) +Q ( /Q ` ( A .pQ C ) ) ) = ( /Q ` ( ( A .pQ B ) +pQ ( A .pQ C ) ) )
103	100, 101, 102	3eqtr4g 2500	. . 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 9102	. . . . . 6 |- ( A e. Q. -> ( /Q ` A ) = A )
105	104	eqcomd 2448	. . . . 5 |- ( A e. Q. -> A = ( /Q ` A ) )
106	105	3ad2ant1 1009	. . . 4 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> A = ( /Q ` A ) )
107		addpqnq 9107	. . . . 5 |- ( ( B e. Q. /\ C e. Q. ) -> ( B +Q C ) = ( /Q ` ( B +pQ C ) ) )
108	107	3adant1 1006	. . . 4 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( B +Q C ) = ( /Q ` ( B +pQ C ) ) )
109	106, 108	oveq12d 6109	. . 3 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( A .Q ( B +Q C ) ) = ( ( /Q ` A ) .Q ( /Q ` ( B +pQ C ) ) ) )
110		mulpqnq 9110	. . . . 5 |- ( ( A e. Q. /\ B e. Q. ) -> ( A .Q B ) = ( /Q ` ( A .pQ B ) ) )
111	110	3adant3 1008	. . . 4 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( A .Q B ) = ( /Q ` ( A .pQ B ) ) )
112		mulpqnq 9110	. . . . 5 |- ( ( A e. Q. /\ C e. Q. ) -> ( A .Q C ) = ( /Q ` ( A .pQ C ) ) )
113	112	3adant2 1007	. . . 4 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( A .Q C ) = ( /Q ` ( A .pQ C ) ) )
114	111, 113	oveq12d 6109	. . 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 2485	. 2 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( A .Q ( B +Q C ) ) = ( ( A .Q B ) +Q ( A .Q C ) ) )
116		addnqf 9117	. . . 4 |- +Q : ( Q. X. Q. ) --> Q.
117	116	fdmi 5564	. . 3 |- dom +Q = ( Q. X. Q. )
118		0nnq 9093	. . 3 |- -. (/) e. Q.
119		mulnqf 9118	. . . 4 |- .Q : ( Q. X. Q. ) --> Q.
120	119	fdmi 5564	. . 3 |- dom .Q = ( Q. X. Q. )
121	117, 118, 120	ndmovdistr 6252	. 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 965   = wceq 1369   e. wcel 1756   <.cop 3883   class class class wbr 4292   X. cxp 4838   Rel wrel 4845   ` cfv 5418  (class class class)co 6091   1stc1st 6575   2ndc2nd 6576   N.cnpi 9011   +N cpli 9012   .N cmi 9013   +pQ cplpq 9015   .pQ cmpq 9016   ~Q ceq 9018   Q.cnq 9019   /Qcerq 9021   +Q cplq 9022   .Q cmq 9023

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-1o 6920  df-oadd 6924  df-omul 6925  df-er 7101  df-ni 9041  df-pli 9042  df-mi 9043  df-lti 9044  df-plpq 9077  df-mpq 9078  df-enq 9080  df-nq 9081  df-erq 9082  df-plq 9083  df-mq 9084  df-1nq 9085

This theorem is referenced by:  ltaddnq  9143  halfnq  9145  addclprlem2  9186  distrlem1pr  9194  distrlem4pr  9195  prlem934  9202  prlem936  9216