Theorem mulassnq 9781

Description: Multiplication of positive fractions is associative. (Contributed by NM, 1-Sep-1995.) (Revised by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)

Assertion

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

Proof of Theorem mulassnq

Step	Hyp	Ref	Expression
1		mulasspi 9719	. . . . . . 7 |- ( ( ( 1st ` A ) .N ( 1st ` B ) ) .N ( 1st ` C ) ) = ( ( 1st ` A ) .N ( ( 1st ` B ) .N ( 1st ` C ) ) )
2		mulasspi 9719	. . . . . . 7 |- ( ( ( 2nd ` A ) .N ( 2nd ` B ) ) .N ( 2nd ` C ) ) = ( ( 2nd ` A ) .N ( ( 2nd ` B ) .N ( 2nd ` C ) ) )
3	1, 2	opeq12i 4407	. . . . . 6 |- <. ( ( ( 1st ` A ) .N ( 1st ` B ) ) .N ( 1st ` C ) ) , ( ( ( 2nd ` A ) .N ( 2nd ` B ) ) .N ( 2nd ` C ) ) >. = <. ( ( 1st ` A ) .N ( ( 1st ` B ) .N ( 1st ` C ) ) ) , ( ( 2nd ` A ) .N ( ( 2nd ` B ) .N ( 2nd ` C ) ) ) >.
4		elpqn 9747	. . . . . . . . . 10 |- ( A e. Q. -> A e. ( N. X. N. ) )
5	4	3ad2ant1 1082	. . . . . . . . 9 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> A e. ( N. X. N. ) )
6		elpqn 9747	. . . . . . . . . 10 |- ( B e. Q. -> B e. ( N. X. N. ) )
7	6	3ad2ant2 1083	. . . . . . . . 9 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> B e. ( N. X. N. ) )
8		mulpipq2 9761	. . . . . . . . 9 |- ( ( A e. ( N. X. N. ) /\ B e. ( N. X. N. ) ) -> ( A .pQ B ) = <. ( ( 1st ` A ) .N ( 1st ` B ) ) , ( ( 2nd ` A ) .N ( 2nd ` B ) ) >. )
9	5, 7, 8	syl2anc 693	. . . . . . . 8 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( A .pQ B ) = <. ( ( 1st ` A ) .N ( 1st ` B ) ) , ( ( 2nd ` A ) .N ( 2nd ` B ) ) >. )
10		relxp 5227	. . . . . . . . 9 |- Rel ( N. X. N. )
11		elpqn 9747	. . . . . . . . . 10 |- ( C e. Q. -> C e. ( N. X. N. ) )
12	11	3ad2ant3 1084	. . . . . . . . 9 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> C e. ( N. X. N. ) )
13		1st2nd 7214	. . . . . . . . 9 |- ( ( Rel ( N. X. N. ) /\ C e. ( N. X. N. ) ) -> C = <. ( 1st ` C ) , ( 2nd ` C ) >. )
14	10, 12, 13	sylancr 695	. . . . . . . 8 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> C = <. ( 1st ` C ) , ( 2nd ` C ) >. )
15	9, 14	oveq12d 6668	. . . . . . 7 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( ( A .pQ B ) .pQ C ) = ( <. ( ( 1st ` A ) .N ( 1st ` B ) ) , ( ( 2nd ` A ) .N ( 2nd ` B ) ) >. .pQ <. ( 1st ` C ) , ( 2nd ` C ) >. ) )
16		xp1st 7198	. . . . . . . . . 10 |- ( A e. ( N. X. N. ) -> ( 1st ` A ) e. N. )
17	5, 16	syl 17	. . . . . . . . 9 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( 1st ` A ) e. N. )
18		xp1st 7198	. . . . . . . . . 10 |- ( B e. ( N. X. N. ) -> ( 1st ` B ) e. N. )
19	7, 18	syl 17	. . . . . . . . 9 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( 1st ` B ) e. N. )
20		mulclpi 9715	. . . . . . . . 9 |- ( ( ( 1st ` A ) e. N. /\ ( 1st ` B ) e. N. ) -> ( ( 1st ` A ) .N ( 1st ` B ) ) e. N. )
21	17, 19, 20	syl2anc 693	. . . . . . . 8 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( ( 1st ` A ) .N ( 1st ` B ) ) e. N. )
22		xp2nd 7199	. . . . . . . . . 10 |- ( A e. ( N. X. N. ) -> ( 2nd ` A ) e. N. )
23	5, 22	syl 17	. . . . . . . . 9 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( 2nd ` A ) e. N. )
24		xp2nd 7199	. . . . . . . . . 10 |- ( B e. ( N. X. N. ) -> ( 2nd ` B ) e. N. )
25	7, 24	syl 17	. . . . . . . . 9 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( 2nd ` B ) e. N. )
26		mulclpi 9715	. . . . . . . . 9 |- ( ( ( 2nd ` A ) e. N. /\ ( 2nd ` B ) e. N. ) -> ( ( 2nd ` A ) .N ( 2nd ` B ) ) e. N. )
27	23, 25, 26	syl2anc 693	. . . . . . . 8 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( ( 2nd ` A ) .N ( 2nd ` B ) ) e. N. )
28		xp1st 7198	. . . . . . . . 9 |- ( C e. ( N. X. N. ) -> ( 1st ` C ) e. N. )
29	12, 28	syl 17	. . . . . . . 8 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( 1st ` C ) e. N. )
30		xp2nd 7199	. . . . . . . . 9 |- ( C e. ( N. X. N. ) -> ( 2nd ` C ) e. N. )
31	12, 30	syl 17	. . . . . . . 8 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( 2nd ` C ) e. N. )
32		mulpipq 9762	. . . . . . . 8 |- ( ( ( ( ( 1st ` A ) .N ( 1st ` B ) ) e. N. /\ ( ( 2nd ` A ) .N ( 2nd ` B ) ) e. N. ) /\ ( ( 1st ` C ) e. N. /\ ( 2nd ` C ) e. N. ) ) -> ( <. ( ( 1st ` A ) .N ( 1st ` B ) ) , ( ( 2nd ` A ) .N ( 2nd ` B ) ) >. .pQ <. ( 1st ` C ) , ( 2nd ` C ) >. ) = <. ( ( ( 1st ` A ) .N ( 1st ` B ) ) .N ( 1st ` C ) ) , ( ( ( 2nd ` A ) .N ( 2nd ` B ) ) .N ( 2nd ` C ) ) >. )
33	21, 27, 29, 31, 32	syl22anc 1327	. . . . . . 7 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( <. ( ( 1st ` A ) .N ( 1st ` B ) ) , ( ( 2nd ` A ) .N ( 2nd ` B ) ) >. .pQ <. ( 1st ` C ) , ( 2nd ` C ) >. ) = <. ( ( ( 1st ` A ) .N ( 1st ` B ) ) .N ( 1st ` C ) ) , ( ( ( 2nd ` A ) .N ( 2nd ` B ) ) .N ( 2nd ` C ) ) >. )
34	15, 33	eqtrd 2656	. . . . . 6 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( ( A .pQ B ) .pQ C ) = <. ( ( ( 1st ` A ) .N ( 1st ` B ) ) .N ( 1st ` C ) ) , ( ( ( 2nd ` A ) .N ( 2nd ` B ) ) .N ( 2nd ` C ) ) >. )
35		1st2nd 7214	. . . . . . . . 9 |- ( ( Rel ( N. X. N. ) /\ A e. ( N. X. N. ) ) -> A = <. ( 1st ` A ) , ( 2nd ` A ) >. )
36	10, 5, 35	sylancr 695	. . . . . . . 8 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> A = <. ( 1st ` A ) , ( 2nd ` A ) >. )
37		mulpipq2 9761	. . . . . . . . 9 |- ( ( B e. ( N. X. N. ) /\ C e. ( N. X. N. ) ) -> ( B .pQ C ) = <. ( ( 1st ` B ) .N ( 1st ` C ) ) , ( ( 2nd ` B ) .N ( 2nd ` C ) ) >. )
38	7, 12, 37	syl2anc 693	. . . . . . . 8 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( B .pQ C ) = <. ( ( 1st ` B ) .N ( 1st ` C ) ) , ( ( 2nd ` B ) .N ( 2nd ` C ) ) >. )
39	36, 38	oveq12d 6668	. . . . . . 7 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( A .pQ ( B .pQ C ) ) = ( <. ( 1st ` A ) , ( 2nd ` A ) >. .pQ <. ( ( 1st ` B ) .N ( 1st ` C ) ) , ( ( 2nd ` B ) .N ( 2nd ` C ) ) >. ) )
40		mulclpi 9715	. . . . . . . . 9 |- ( ( ( 1st ` B ) e. N. /\ ( 1st ` C ) e. N. ) -> ( ( 1st ` B ) .N ( 1st ` C ) ) e. N. )
41	19, 29, 40	syl2anc 693	. . . . . . . 8 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( ( 1st ` B ) .N ( 1st ` C ) ) e. N. )
42		mulclpi 9715	. . . . . . . . 9 |- ( ( ( 2nd ` B ) e. N. /\ ( 2nd ` C ) e. N. ) -> ( ( 2nd ` B ) .N ( 2nd ` C ) ) e. N. )
43	25, 31, 42	syl2anc 693	. . . . . . . 8 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( ( 2nd ` B ) .N ( 2nd ` C ) ) e. N. )
44		mulpipq 9762	. . . . . . . 8 |- ( ( ( ( 1st ` A ) e. N. /\ ( 2nd ` A ) e. N. ) /\ ( ( ( 1st ` B ) .N ( 1st ` C ) ) e. N. /\ ( ( 2nd ` B ) .N ( 2nd ` C ) ) e. N. ) ) -> ( <. ( 1st ` A ) , ( 2nd ` A ) >. .pQ <. ( ( 1st ` B ) .N ( 1st ` C ) ) , ( ( 2nd ` B ) .N ( 2nd ` C ) ) >. ) = <. ( ( 1st ` A ) .N ( ( 1st ` B ) .N ( 1st ` C ) ) ) , ( ( 2nd ` A ) .N ( ( 2nd ` B ) .N ( 2nd ` C ) ) ) >. )
45	17, 23, 41, 43, 44	syl22anc 1327	. . . . . . 7 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( <. ( 1st ` A ) , ( 2nd ` A ) >. .pQ <. ( ( 1st ` B ) .N ( 1st ` C ) ) , ( ( 2nd ` B ) .N ( 2nd ` C ) ) >. ) = <. ( ( 1st ` A ) .N ( ( 1st ` B ) .N ( 1st ` C ) ) ) , ( ( 2nd ` A ) .N ( ( 2nd ` B ) .N ( 2nd ` C ) ) ) >. )
46	39, 45	eqtrd 2656	. . . . . 6 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( A .pQ ( B .pQ C ) ) = <. ( ( 1st ` A ) .N ( ( 1st ` B ) .N ( 1st ` C ) ) ) , ( ( 2nd ` A ) .N ( ( 2nd ` B ) .N ( 2nd ` C ) ) ) >. )
47	3, 34, 46	3eqtr4a 2682	. . . . 5 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( ( A .pQ B ) .pQ C ) = ( A .pQ ( B .pQ C ) ) )
48	47	fveq2d 6195	. . . 4 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( /Q ` ( ( A .pQ B ) .pQ C ) ) = ( /Q ` ( A .pQ ( B .pQ C ) ) ) )
49		mulerpq 9779	. . . 4 |- ( ( /Q ` ( A .pQ B ) ) .Q ( /Q ` C ) ) = ( /Q ` ( ( A .pQ B ) .pQ C ) )
50		mulerpq 9779	. . . 4 |- ( ( /Q ` A ) .Q ( /Q ` ( B .pQ C ) ) ) = ( /Q ` ( A .pQ ( B .pQ C ) ) )
51	48, 49, 50	3eqtr4g 2681	. . 3 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( ( /Q ` ( A .pQ B ) ) .Q ( /Q ` C ) ) = ( ( /Q ` A ) .Q ( /Q ` ( B .pQ C ) ) ) )
52		mulpqnq 9763	. . . . 5 |- ( ( A e. Q. /\ B e. Q. ) -> ( A .Q B ) = ( /Q ` ( A .pQ B ) ) )
53	52	3adant3 1081	. . . 4 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( A .Q B ) = ( /Q ` ( A .pQ B ) ) )
54		nqerid 9755	. . . . . 6 |- ( C e. Q. -> ( /Q ` C ) = C )
55	54	eqcomd 2628	. . . . 5 |- ( C e. Q. -> C = ( /Q ` C ) )
56	55	3ad2ant3 1084	. . . 4 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> C = ( /Q ` C ) )
57	53, 56	oveq12d 6668	. . 3 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( ( A .Q B ) .Q C ) = ( ( /Q ` ( A .pQ B ) ) .Q ( /Q ` C ) ) )
58		nqerid 9755	. . . . . 6 |- ( A e. Q. -> ( /Q ` A ) = A )
59	58	eqcomd 2628	. . . . 5 |- ( A e. Q. -> A = ( /Q ` A ) )
60	59	3ad2ant1 1082	. . . 4 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> A = ( /Q ` A ) )
61		mulpqnq 9763	. . . . 5 |- ( ( B e. Q. /\ C e. Q. ) -> ( B .Q C ) = ( /Q ` ( B .pQ C ) ) )
62	61	3adant1 1079	. . . 4 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( B .Q C ) = ( /Q ` ( B .pQ C ) ) )
63	60, 62	oveq12d 6668	. . 3 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( A .Q ( B .Q C ) ) = ( ( /Q ` A ) .Q ( /Q ` ( B .pQ C ) ) ) )
64	51, 57, 63	3eqtr4d 2666	. 2 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( ( A .Q B ) .Q C ) = ( A .Q ( B .Q C ) ) )
65		mulnqf 9771	. . . 4 |- .Q : ( Q. X. Q. ) --> Q.
66	65	fdmi 6052	. . 3 |- dom .Q = ( Q. X. Q. )
67		0nnq 9746	. . 3 |- -. (/) e. Q.
68	66, 67	ndmovass 6822	. 2 |- ( -. ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( ( A .Q B ) .Q C ) = ( A .Q ( B .Q C ) ) )
69	64, 68	pm2.61i 176	1 |- ( ( A .Q B ) .Q C ) = ( A .Q ( B .Q C ) )

Syntax hints:   /\ w3a 1037   = wceq 1483   e. wcel 1990   <.cop 4183   X. cxp 5112   Rel wrel 5119   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   N.cnpi 9666   .N cmi 9668   .pQ cmpq 9671   Q.cnq 9674   /Qcerq 9676   .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-mi 9696  df-lti 9697  df-mpq 9731  df-enq 9733  df-nq 9734  df-erq 9735  df-mq 9737  df-1nq 9738

This theorem is referenced by:  recmulnq  9786  halfnq  9798  ltrnq  9801  addclprlem2  9839  mulclprlem  9841  mulasspr  9846  1idpr  9851  prlem934  9855  prlem936  9869  reclem3pr  9871