Theorem distrnq 9783

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 9718	. . . . . . . . . . . . 13 |- ( ( 1st ` A ) .N ( 1st ` B ) ) = ( ( 1st ` B ) .N ( 1st ` A ) )
2	1	oveq1i 6660	. . . . . . . . . . . 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 6201	. . . . . . . . . . . . 13 |- ( 1st ` B ) e. _V
4		fvex 6201	. . . . . . . . . . . . 13 |- ( 1st ` A ) e. _V
5		fvex 6201	. . . . . . . . . . . . 13 |- ( 2nd ` A ) e. _V
6		mulcompi 9718	. . . . . . . . . . . . 13 |- ( x .N y ) = ( y .N x )
7		mulasspi 9719	. . . . . . . . . . . . 13 |- ( ( x .N y ) .N z ) = ( x .N ( y .N z ) )
8		fvex 6201	. . . . . . . . . . . . 13 |- ( 2nd ` C ) e. _V
9	3, 4, 5, 6, 7, 8	caov411 6866	. . . . . . . . . . . 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 2644	. . . . . . . . . . 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 9718	. . . . . . . . . . . . 13 |- ( ( 1st ` A ) .N ( 1st ` C ) ) = ( ( 1st ` C ) .N ( 1st ` A ) )
12	11	oveq1i 6660	. . . . . . . . . . . 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 6201	. . . . . . . . . . . . 13 |- ( 1st ` C ) e. _V
14		fvex 6201	. . . . . . . . . . . . 13 |- ( 2nd ` B ) e. _V
15	13, 4, 5, 6, 7, 14	caov411 6866	. . . . . . . . . . . 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 2644	. . . . . . . . . . 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 6662	. . . . . . . . . 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 9720	. . . . . . . . . 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 9719	. . . . . . . . . 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 2650	. . . . . . . . 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 9719	. . . . . . . . . 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 6862	. . . . . . . . . . 11 |- ( ( 2nd ` B ) .N ( ( 2nd ` A ) .N ( 2nd ` C ) ) ) = ( ( 2nd ` A ) .N ( ( 2nd ` B ) .N ( 2nd ` C ) ) )
23	22	oveq2i 6661	. . . . . . . . . 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 2644	. . . . . . . . 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 4407	. . . . . . . 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 9747	. . . . . . . . . . 11 |- ( A e. Q. -> A e. ( N. X. N. ) )
27	26	3ad2ant1 1082	. . . . . . . . . 10 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> A e. ( N. X. N. ) )
28		xp2nd 7199	. . . . . . . . . 10 |- ( A e. ( N. X. N. ) -> ( 2nd ` A ) e. N. )
29	27, 28	syl 17	. . . . . . . . 9 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( 2nd ` A ) e. N. )
30		xp1st 7198	. . . . . . . . . . 11 |- ( A e. ( N. X. N. ) -> ( 1st ` A ) e. N. )
31	27, 30	syl 17	. . . . . . . . . 10 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( 1st ` A ) e. N. )
32		elpqn 9747	. . . . . . . . . . . . . 14 |- ( B e. Q. -> B e. ( N. X. N. ) )
33	32	3ad2ant2 1083	. . . . . . . . . . . . 13 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> B e. ( N. X. N. ) )
34		xp1st 7198	. . . . . . . . . . . . 13 |- ( B e. ( N. X. N. ) -> ( 1st ` B ) e. N. )
35	33, 34	syl 17	. . . . . . . . . . . 12 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( 1st ` B ) e. N. )
36		elpqn 9747	. . . . . . . . . . . . . 14 |- ( C e. Q. -> C e. ( N. X. N. ) )
37	36	3ad2ant3 1084	. . . . . . . . . . . . 13 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> C e. ( N. X. N. ) )
38		xp2nd 7199	. . . . . . . . . . . . 13 |- ( C e. ( N. X. N. ) -> ( 2nd ` C ) e. N. )
39	37, 38	syl 17	. . . . . . . . . . . 12 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( 2nd ` C ) e. N. )
40		mulclpi 9715	. . . . . . . . . . . 12 |- ( ( ( 1st ` B ) e. N. /\ ( 2nd ` C ) e. N. ) -> ( ( 1st ` B ) .N ( 2nd ` C ) ) e. N. )
41	35, 39, 40	syl2anc 693	. . . . . . . . . . 11 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( ( 1st ` B ) .N ( 2nd ` C ) ) e. N. )
42		xp1st 7198	. . . . . . . . . . . . 13 |- ( C e. ( N. X. N. ) -> ( 1st ` C ) e. N. )
43	37, 42	syl 17	. . . . . . . . . . . 12 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( 1st ` C ) e. N. )
44		xp2nd 7199	. . . . . . . . . . . . 13 |- ( B e. ( N. X. N. ) -> ( 2nd ` B ) e. N. )
45	33, 44	syl 17	. . . . . . . . . . . 12 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( 2nd ` B ) e. N. )
46		mulclpi 9715	. . . . . . . . . . . 12 |- ( ( ( 1st ` C ) e. N. /\ ( 2nd ` B ) e. N. ) -> ( ( 1st ` C ) .N ( 2nd ` B ) ) e. N. )
47	43, 45, 46	syl2anc 693	. . . . . . . . . . 11 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( ( 1st ` C ) .N ( 2nd ` B ) ) e. N. )
48		addclpi 9714	. . . . . . . . . . 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 693	. . . . . . . . . 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 9715	. . . . . . . . . 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 693	. . . . . . . . 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 9715	. . . . . . . . . . 11 |- ( ( ( 2nd ` B ) e. N. /\ ( 2nd ` C ) e. N. ) -> ( ( 2nd ` B ) .N ( 2nd ` C ) ) e. N. )
53	45, 39, 52	syl2anc 693	. . . . . . . . . 10 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( ( 2nd ` B ) .N ( 2nd ` C ) ) e. N. )
54		mulclpi 9715	. . . . . . . . . 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 693	. . . . . . . . 9 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( ( 2nd ` A ) .N ( ( 2nd ` B ) .N ( 2nd ` C ) ) ) e. N. )
56		mulcanenq 9782	. . . . . . . . 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 1326	. . . . . . . 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 4688	. . . . . . 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 9761	. . . . . . . . . 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 693	. . . . . . . . 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 9761	. . . . . . . . . 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 693	. . . . . . . . 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 6668	. . . . . . . 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 9715	. . . . . . . . . 10 |- ( ( ( 1st ` A ) e. N. /\ ( 1st ` B ) e. N. ) -> ( ( 1st ` A ) .N ( 1st ` B ) ) e. N. )
65	31, 35, 64	syl2anc 693	. . . . . . . . 9 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( ( 1st ` A ) .N ( 1st ` B ) ) e. N. )
66		mulclpi 9715	. . . . . . . . . 10 |- ( ( ( 2nd ` A ) e. N. /\ ( 2nd ` B ) e. N. ) -> ( ( 2nd ` A ) .N ( 2nd ` B ) ) e. N. )
67	29, 45, 66	syl2anc 693	. . . . . . . . 9 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( ( 2nd ` A ) .N ( 2nd ` B ) ) e. N. )
68		mulclpi 9715	. . . . . . . . . 10 |- ( ( ( 1st ` A ) e. N. /\ ( 1st ` C ) e. N. ) -> ( ( 1st ` A ) .N ( 1st ` C ) ) e. N. )
69	31, 43, 68	syl2anc 693	. . . . . . . . 9 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( ( 1st ` A ) .N ( 1st ` C ) ) e. N. )
70		mulclpi 9715	. . . . . . . . . 10 |- ( ( ( 2nd ` A ) e. N. /\ ( 2nd ` C ) e. N. ) -> ( ( 2nd ` A ) .N ( 2nd ` C ) ) e. N. )
71	29, 39, 70	syl2anc 693	. . . . . . . . 9 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( ( 2nd ` A ) .N ( 2nd ` C ) ) e. N. )
72		addpipq 9759	. . . . . . . . 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 1327	. . . . . . . 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 2656	. . . . . . 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 5227	. . . . . . . . . 10 |- Rel ( N. X. N. )
76		1st2nd 7214	. . . . . . . . . 10 |- ( ( Rel ( N. X. N. ) /\ A e. ( N. X. N. ) ) -> A = <. ( 1st ` A ) , ( 2nd ` A ) >. )
77	75, 27, 76	sylancr 695	. . . . . . . . 9 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> A = <. ( 1st ` A ) , ( 2nd ` A ) >. )
78		addpipq2 9758	. . . . . . . . . 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 693	. . . . . . . . 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 6668	. . . . . . . 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 9762	. . . . . . . . 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 1327	. . . . . . . 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 2656	. . . . . . 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 4685	. . . . . 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 9768	. . . . . . . . . 10 |- .pQ : ( ( N. X. N. ) X. ( N. X. N. ) ) --> ( N. X. N. )
86	85	fovcl 6765	. . . . . . . . 9 |- ( ( A e. ( N. X. N. ) /\ B e. ( N. X. N. ) ) -> ( A .pQ B ) e. ( N. X. N. ) )
87	27, 33, 86	syl2anc 693	. . . . . . . 8 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( A .pQ B ) e. ( N. X. N. ) )
88	85	fovcl 6765	. . . . . . . . 9 |- ( ( A e. ( N. X. N. ) /\ C e. ( N. X. N. ) ) -> ( A .pQ C ) e. ( N. X. N. ) )
89	27, 37, 88	syl2anc 693	. . . . . . . 8 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( A .pQ C ) e. ( N. X. N. ) )
90		addpqf 9766	. . . . . . . . 9 |- +pQ : ( ( N. X. N. ) X. ( N. X. N. ) ) --> ( N. X. N. )
91	90	fovcl 6765	. . . . . . . 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 693	. . . . . . 7 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( ( A .pQ B ) +pQ ( A .pQ C ) ) e. ( N. X. N. ) )
93	90	fovcl 6765	. . . . . . . . 9 |- ( ( B e. ( N. X. N. ) /\ C e. ( N. X. N. ) ) -> ( B +pQ C ) e. ( N. X. N. ) )
94	33, 37, 93	syl2anc 693	. . . . . . . 8 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( B +pQ C ) e. ( N. X. N. ) )
95	85	fovcl 6765	. . . . . . . 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 693	. . . . . . 7 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( A .pQ ( B +pQ C ) ) e. ( N. X. N. ) )
97		nqereq 9757	. . . . . . 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 693	. . . . . 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 222	. . . . 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 2628	. . . 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 9779	. . . 4 |- ( ( /Q ` A ) .Q ( /Q ` ( B +pQ C ) ) ) = ( /Q ` ( A .pQ ( B +pQ C ) ) )
102		adderpq 9778	. . . 4 |- ( ( /Q ` ( A .pQ B ) ) +Q ( /Q ` ( A .pQ C ) ) ) = ( /Q ` ( ( A .pQ B ) +pQ ( A .pQ C ) ) )
103	100, 101, 102	3eqtr4g 2681	. . 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 9755	. . . . . 6 |- ( A e. Q. -> ( /Q ` A ) = A )
105	104	eqcomd 2628	. . . . 5 |- ( A e. Q. -> A = ( /Q ` A ) )
106	105	3ad2ant1 1082	. . . 4 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> A = ( /Q ` A ) )
107		addpqnq 9760	. . . . 5 |- ( ( B e. Q. /\ C e. Q. ) -> ( B +Q C ) = ( /Q ` ( B +pQ C ) ) )
108	107	3adant1 1079	. . . 4 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( B +Q C ) = ( /Q ` ( B +pQ C ) ) )
109	106, 108	oveq12d 6668	. . 3 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( A .Q ( B +Q C ) ) = ( ( /Q ` A ) .Q ( /Q ` ( B +pQ C ) ) ) )
110		mulpqnq 9763	. . . . 5 |- ( ( A e. Q. /\ B e. Q. ) -> ( A .Q B ) = ( /Q ` ( A .pQ B ) ) )
111	110	3adant3 1081	. . . 4 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( A .Q B ) = ( /Q ` ( A .pQ B ) ) )
112		mulpqnq 9763	. . . . 5 |- ( ( A e. Q. /\ C e. Q. ) -> ( A .Q C ) = ( /Q ` ( A .pQ C ) ) )
113	112	3adant2 1080	. . . 4 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( A .Q C ) = ( /Q ` ( A .pQ C ) ) )
114	111, 113	oveq12d 6668	. . 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 2666	. 2 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( A .Q ( B +Q C ) ) = ( ( A .Q B ) +Q ( A .Q C ) ) )
116		addnqf 9770	. . . 4 |- +Q : ( Q. X. Q. ) --> Q.
117	116	fdmi 6052	. . 3 |- dom +Q = ( Q. X. Q. )
118		0nnq 9746	. . 3 |- -. (/) e. Q.
119		mulnqf 9771	. . . 4 |- .Q : ( Q. X. Q. ) --> Q.
120	119	fdmi 6052	. . 3 |- dom .Q = ( Q. X. Q. )
121	117, 118, 120	ndmovdistr 6823	. 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 176	1 |- ( A .Q ( B +Q C ) ) = ( ( A .Q B ) +Q ( A .Q C ) )

Syntax hints:   <-> wb 196   /\ w3a 1037   = wceq 1483   e. wcel 1990   <.cop 4183   class class class wbr 4653   X. cxp 5112   Rel wrel 5119   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   N.cnpi 9666   +N cpli 9667   .N cmi 9668   +pQ cplpq 9670   .pQ cmpq 9671   ~Q ceq 9673   Q.cnq 9674   /Qcerq 9676   +Q cplq 9677   .Q cmq 9678

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-omul 7565  df-er 7742  df-ni 9694  df-pli 9695  df-mi 9696  df-lti 9697  df-plpq 9730  df-mpq 9731  df-enq 9733  df-nq 9734  df-erq 9735  df-plq 9736  df-mq 9737  df-1nq 9738

This theorem is referenced by:  ltaddnq  9796  halfnq  9798  addclprlem2  9839  distrlem1pr  9847  distrlem4pr  9848  prlem934  9855  prlem936  9869