Theorem recmulnq 9786

Description: Relationship between reciprocal and multiplication on positive fractions. (Contributed by NM, 6-Mar-1996.) (Revised by Mario Carneiro, 28-Apr-2015.) (New usage is discouraged.)

Assertion

Ref	Expression
recmulnq	|- ( A e. Q. -> ( ( *Q ` A ) = B <-> ( A .Q B ) = 1Q ) )

Proof of Theorem recmulnq

Dummy variables x y s r t are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvex 6201	. . . 4 |- ( *Q ` A ) e. _V
2	1	a1i 11	. . 3 |- ( A e. Q. -> ( *Q ` A ) e. _V )
3		eleq1 2689	. . 3 |- ( ( *Q ` A ) = B -> ( ( *Q ` A ) e. _V <-> B e. _V ) )
4	2, 3	syl5ibcom 235	. 2 |- ( A e. Q. -> ( ( *Q ` A ) = B -> B e. _V ) )
5		id 22	. . . . . . 7 |- ( ( A .Q B ) = 1Q -> ( A .Q B ) = 1Q )
6		1nq 9750	. . . . . . 7 |- 1Q e. Q.
7	5, 6	syl6eqel 2709	. . . . . 6 |- ( ( A .Q B ) = 1Q -> ( A .Q B ) e. Q. )
8		mulnqf 9771	. . . . . . . 8 |- .Q : ( Q. X. Q. ) --> Q.
9	8	fdmi 6052	. . . . . . 7 |- dom .Q = ( Q. X. Q. )
10		0nnq 9746	. . . . . . 7 |- -. (/) e. Q.
11	9, 10	ndmovrcl 6820	. . . . . 6 |- ( ( A .Q B ) e. Q. -> ( A e. Q. /\ B e. Q. ) )
12	7, 11	syl 17	. . . . 5 |- ( ( A .Q B ) = 1Q -> ( A e. Q. /\ B e. Q. ) )
13	12	simprd 479	. . . 4 |- ( ( A .Q B ) = 1Q -> B e. Q. )
14		elex 3212	. . . 4 |- ( B e. Q. -> B e. _V )
15	13, 14	syl 17	. . 3 |- ( ( A .Q B ) = 1Q -> B e. _V )
16	15	a1i 11	. 2 |- ( A e. Q. -> ( ( A .Q B ) = 1Q -> B e. _V ) )
17		oveq1 6657	. . . . 5 |- ( x = A -> ( x .Q y ) = ( A .Q y ) )
18	17	eqeq1d 2624	. . . 4 |- ( x = A -> ( ( x .Q y ) = 1Q <-> ( A .Q y ) = 1Q ) )
19		oveq2 6658	. . . . 5 |- ( y = B -> ( A .Q y ) = ( A .Q B ) )
20	19	eqeq1d 2624	. . . 4 |- ( y = B -> ( ( A .Q y ) = 1Q <-> ( A .Q B ) = 1Q ) )
21		nqerid 9755	. . . . . . . . . 10 |- ( x e. Q. -> ( /Q ` x ) = x )
22		relxp 5227	. . . . . . . . . . . 12 |- Rel ( N. X. N. )
23		elpqn 9747	. . . . . . . . . . . 12 |- ( x e. Q. -> x e. ( N. X. N. ) )
24		1st2nd 7214	. . . . . . . . . . . 12 |- ( ( Rel ( N. X. N. ) /\ x e. ( N. X. N. ) ) -> x = <. ( 1st ` x ) , ( 2nd ` x ) >. )
25	22, 23, 24	sylancr 695	. . . . . . . . . . 11 |- ( x e. Q. -> x = <. ( 1st ` x ) , ( 2nd ` x ) >. )
26	25	fveq2d 6195	. . . . . . . . . 10 |- ( x e. Q. -> ( /Q ` x ) = ( /Q ` <. ( 1st ` x ) , ( 2nd ` x ) >. ) )
27	21, 26	eqtr3d 2658	. . . . . . . . 9 |- ( x e. Q. -> x = ( /Q ` <. ( 1st ` x ) , ( 2nd ` x ) >. ) )
28	27	oveq1d 6665	. . . . . . . 8 |- ( x e. Q. -> ( x .Q ( /Q ` <. ( 2nd ` x ) , ( 1st ` x ) >. ) ) = ( ( /Q ` <. ( 1st ` x ) , ( 2nd ` x ) >. ) .Q ( /Q ` <. ( 2nd ` x ) , ( 1st ` x ) >. ) ) )
29		mulerpq 9779	. . . . . . . 8 |- ( ( /Q ` <. ( 1st ` x ) , ( 2nd ` x ) >. ) .Q ( /Q ` <. ( 2nd ` x ) , ( 1st ` x ) >. ) ) = ( /Q ` ( <. ( 1st ` x ) , ( 2nd ` x ) >. .pQ <. ( 2nd ` x ) , ( 1st ` x ) >. ) )
30	28, 29	syl6eq 2672	. . . . . . 7 |- ( x e. Q. -> ( x .Q ( /Q ` <. ( 2nd ` x ) , ( 1st ` x ) >. ) ) = ( /Q ` ( <. ( 1st ` x ) , ( 2nd ` x ) >. .pQ <. ( 2nd ` x ) , ( 1st ` x ) >. ) ) )
31		xp1st 7198	. . . . . . . . . . 11 |- ( x e. ( N. X. N. ) -> ( 1st ` x ) e. N. )
32	23, 31	syl 17	. . . . . . . . . 10 |- ( x e. Q. -> ( 1st ` x ) e. N. )
33		xp2nd 7199	. . . . . . . . . . 11 |- ( x e. ( N. X. N. ) -> ( 2nd ` x ) e. N. )
34	23, 33	syl 17	. . . . . . . . . 10 |- ( x e. Q. -> ( 2nd ` x ) e. N. )
35		mulpipq 9762	. . . . . . . . . 10 |- ( ( ( ( 1st ` x ) e. N. /\ ( 2nd ` x ) e. N. ) /\ ( ( 2nd ` x ) e. N. /\ ( 1st ` x ) e. N. ) ) -> ( <. ( 1st ` x ) , ( 2nd ` x ) >. .pQ <. ( 2nd ` x ) , ( 1st ` x ) >. ) = <. ( ( 1st ` x ) .N ( 2nd ` x ) ) , ( ( 2nd ` x ) .N ( 1st ` x ) ) >. )
36	32, 34, 34, 32, 35	syl22anc 1327	. . . . . . . . 9 |- ( x e. Q. -> ( <. ( 1st ` x ) , ( 2nd ` x ) >. .pQ <. ( 2nd ` x ) , ( 1st ` x ) >. ) = <. ( ( 1st ` x ) .N ( 2nd ` x ) ) , ( ( 2nd ` x ) .N ( 1st ` x ) ) >. )
37		mulcompi 9718	. . . . . . . . . 10 |- ( ( 2nd ` x ) .N ( 1st ` x ) ) = ( ( 1st ` x ) .N ( 2nd ` x ) )
38	37	opeq2i 4406	. . . . . . . . 9 |- <. ( ( 1st ` x ) .N ( 2nd ` x ) ) , ( ( 2nd ` x ) .N ( 1st ` x ) ) >. = <. ( ( 1st ` x ) .N ( 2nd ` x ) ) , ( ( 1st ` x ) .N ( 2nd ` x ) ) >.
39	36, 38	syl6eq 2672	. . . . . . . 8 |- ( x e. Q. -> ( <. ( 1st ` x ) , ( 2nd ` x ) >. .pQ <. ( 2nd ` x ) , ( 1st ` x ) >. ) = <. ( ( 1st ` x ) .N ( 2nd ` x ) ) , ( ( 1st ` x ) .N ( 2nd ` x ) ) >. )
40	39	fveq2d 6195	. . . . . . 7 |- ( x e. Q. -> ( /Q ` ( <. ( 1st ` x ) , ( 2nd ` x ) >. .pQ <. ( 2nd ` x ) , ( 1st ` x ) >. ) ) = ( /Q ` <. ( ( 1st ` x ) .N ( 2nd ` x ) ) , ( ( 1st ` x ) .N ( 2nd ` x ) ) >. ) )
41		nqerid 9755	. . . . . . . . 9 |- ( 1Q e. Q. -> ( /Q ` 1Q ) = 1Q )
42	6, 41	ax-mp 5	. . . . . . . 8 |- ( /Q ` 1Q ) = 1Q
43		mulclpi 9715	. . . . . . . . . . 11 |- ( ( ( 1st ` x ) e. N. /\ ( 2nd ` x ) e. N. ) -> ( ( 1st ` x ) .N ( 2nd ` x ) ) e. N. )
44	32, 34, 43	syl2anc 693	. . . . . . . . . 10 |- ( x e. Q. -> ( ( 1st ` x ) .N ( 2nd ` x ) ) e. N. )
45		1nqenq 9784	. . . . . . . . . 10 |- ( ( ( 1st ` x ) .N ( 2nd ` x ) ) e. N. -> 1Q ~Q <. ( ( 1st ` x ) .N ( 2nd ` x ) ) , ( ( 1st ` x ) .N ( 2nd ` x ) ) >. )
46	44, 45	syl 17	. . . . . . . . 9 |- ( x e. Q. -> 1Q ~Q <. ( ( 1st ` x ) .N ( 2nd ` x ) ) , ( ( 1st ` x ) .N ( 2nd ` x ) ) >. )
47		elpqn 9747	. . . . . . . . . . 11 |- ( 1Q e. Q. -> 1Q e. ( N. X. N. ) )
48	6, 47	ax-mp 5	. . . . . . . . . 10 |- 1Q e. ( N. X. N. )
49		opelxpi 5148	. . . . . . . . . . 11 |- ( ( ( ( 1st ` x ) .N ( 2nd ` x ) ) e. N. /\ ( ( 1st ` x ) .N ( 2nd ` x ) ) e. N. ) -> <. ( ( 1st ` x ) .N ( 2nd ` x ) ) , ( ( 1st ` x ) .N ( 2nd ` x ) ) >. e. ( N. X. N. ) )
50	44, 44, 49	syl2anc 693	. . . . . . . . . 10 |- ( x e. Q. -> <. ( ( 1st ` x ) .N ( 2nd ` x ) ) , ( ( 1st ` x ) .N ( 2nd ` x ) ) >. e. ( N. X. N. ) )
51		nqereq 9757	. . . . . . . . . 10 |- ( ( 1Q e. ( N. X. N. ) /\ <. ( ( 1st ` x ) .N ( 2nd ` x ) ) , ( ( 1st ` x ) .N ( 2nd ` x ) ) >. e. ( N. X. N. ) ) -> ( 1Q ~Q <. ( ( 1st ` x ) .N ( 2nd ` x ) ) , ( ( 1st ` x ) .N ( 2nd ` x ) ) >. <-> ( /Q ` 1Q ) = ( /Q ` <. ( ( 1st ` x ) .N ( 2nd ` x ) ) , ( ( 1st ` x ) .N ( 2nd ` x ) ) >. ) ) )
52	48, 50, 51	sylancr 695	. . . . . . . . 9 |- ( x e. Q. -> ( 1Q ~Q <. ( ( 1st ` x ) .N ( 2nd ` x ) ) , ( ( 1st ` x ) .N ( 2nd ` x ) ) >. <-> ( /Q ` 1Q ) = ( /Q ` <. ( ( 1st ` x ) .N ( 2nd ` x ) ) , ( ( 1st ` x ) .N ( 2nd ` x ) ) >. ) ) )
53	46, 52	mpbid 222	. . . . . . . 8 |- ( x e. Q. -> ( /Q ` 1Q ) = ( /Q ` <. ( ( 1st ` x ) .N ( 2nd ` x ) ) , ( ( 1st ` x ) .N ( 2nd ` x ) ) >. ) )
54	42, 53	syl5reqr 2671	. . . . . . 7 |- ( x e. Q. -> ( /Q ` <. ( ( 1st ` x ) .N ( 2nd ` x ) ) , ( ( 1st ` x ) .N ( 2nd ` x ) ) >. ) = 1Q )
55	30, 40, 54	3eqtrd 2660	. . . . . 6 |- ( x e. Q. -> ( x .Q ( /Q ` <. ( 2nd ` x ) , ( 1st ` x ) >. ) ) = 1Q )
56		fvex 6201	. . . . . . 7 |- ( /Q ` <. ( 2nd ` x ) , ( 1st ` x ) >. ) e. _V
57		oveq2 6658	. . . . . . . 8 |- ( y = ( /Q ` <. ( 2nd ` x ) , ( 1st ` x ) >. ) -> ( x .Q y ) = ( x .Q ( /Q ` <. ( 2nd ` x ) , ( 1st ` x ) >. ) ) )
58	57	eqeq1d 2624	. . . . . . 7 |- ( y = ( /Q ` <. ( 2nd ` x ) , ( 1st ` x ) >. ) -> ( ( x .Q y ) = 1Q <-> ( x .Q ( /Q ` <. ( 2nd ` x ) , ( 1st ` x ) >. ) ) = 1Q ) )
59	56, 58	spcev 3300	. . . . . 6 |- ( ( x .Q ( /Q ` <. ( 2nd ` x ) , ( 1st ` x ) >. ) ) = 1Q -> E. y ( x .Q y ) = 1Q )
60	55, 59	syl 17	. . . . 5 |- ( x e. Q. -> E. y ( x .Q y ) = 1Q )
61		mulcomnq 9775	. . . . . . 7 |- ( r .Q s ) = ( s .Q r )
62		mulassnq 9781	. . . . . . 7 |- ( ( r .Q s ) .Q t ) = ( r .Q ( s .Q t ) )
63		mulidnq 9785	. . . . . . 7 |- ( r e. Q. -> ( r .Q 1Q ) = r )
64	6, 9, 10, 61, 62, 63	caovmo 6871	. . . . . 6 |- E* y ( x .Q y ) = 1Q
65		eu5 2496	. . . . . 6 |- ( E! y ( x .Q y ) = 1Q <-> ( E. y ( x .Q y ) = 1Q /\ E* y ( x .Q y ) = 1Q ) )
66	64, 65	mpbiran2 954	. . . . 5 |- ( E! y ( x .Q y ) = 1Q <-> E. y ( x .Q y ) = 1Q )
67	60, 66	sylibr 224	. . . 4 |- ( x e. Q. -> E! y ( x .Q y ) = 1Q )
68		cnvimass 5485	. . . . . . . 8 |- ( `' .Q " { 1Q } ) C_ dom .Q
69		df-rq 9739	. . . . . . . 8 |- *Q = ( `' .Q " { 1Q } )
70	9	eqcomi 2631	. . . . . . . 8 |- ( Q. X. Q. ) = dom .Q
71	68, 69, 70	3sstr4i 3644	. . . . . . 7 |- *Q C_ ( Q. X. Q. )
72		relxp 5227	. . . . . . 7 |- Rel ( Q. X. Q. )
73		relss 5206	. . . . . . 7 |- ( *Q C_ ( Q. X. Q. ) -> ( Rel ( Q. X. Q. ) -> Rel *Q ) )
74	71, 72, 73	mp2 9	. . . . . 6 |- Rel *Q
75	69	eleq2i 2693	. . . . . . . 8 |- ( <. x , y >. e. *Q <-> <. x , y >. e. ( `' .Q " { 1Q } ) )
76		ffn 6045	. . . . . . . . 9 |- ( .Q : ( Q. X. Q. ) --> Q. -> .Q Fn ( Q. X. Q. ) )
77		fniniseg 6338	. . . . . . . . 9 |- ( .Q Fn ( Q. X. Q. ) -> ( <. x , y >. e. ( `' .Q " { 1Q } ) <-> ( <. x , y >. e. ( Q. X. Q. ) /\ ( .Q ` <. x , y >. ) = 1Q ) ) )
78	8, 76, 77	mp2b 10	. . . . . . . 8 |- ( <. x , y >. e. ( `' .Q " { 1Q } ) <-> ( <. x , y >. e. ( Q. X. Q. ) /\ ( .Q ` <. x , y >. ) = 1Q ) )
79		ancom 466	. . . . . . . . 9 |- ( ( <. x , y >. e. ( Q. X. Q. ) /\ ( .Q ` <. x , y >. ) = 1Q ) <-> ( ( .Q ` <. x , y >. ) = 1Q /\ <. x , y >. e. ( Q. X. Q. ) ) )
80		ancom 466	. . . . . . . . . 10 |- ( ( x e. Q. /\ ( x .Q y ) = 1Q ) <-> ( ( x .Q y ) = 1Q /\ x e. Q. ) )
81		eleq1 2689	. . . . . . . . . . . . . . 15 |- ( ( x .Q y ) = 1Q -> ( ( x .Q y ) e. Q. <-> 1Q e. Q. ) )
82	6, 81	mpbiri 248	. . . . . . . . . . . . . 14 |- ( ( x .Q y ) = 1Q -> ( x .Q y ) e. Q. )
83	9, 10	ndmovrcl 6820	. . . . . . . . . . . . . 14 |- ( ( x .Q y ) e. Q. -> ( x e. Q. /\ y e. Q. ) )
84	82, 83	syl 17	. . . . . . . . . . . . 13 |- ( ( x .Q y ) = 1Q -> ( x e. Q. /\ y e. Q. ) )
85		opelxpi 5148	. . . . . . . . . . . . 13 |- ( ( x e. Q. /\ y e. Q. ) -> <. x , y >. e. ( Q. X. Q. ) )
86	84, 85	syl 17	. . . . . . . . . . . 12 |- ( ( x .Q y ) = 1Q -> <. x , y >. e. ( Q. X. Q. ) )
87	84	simpld 475	. . . . . . . . . . . 12 |- ( ( x .Q y ) = 1Q -> x e. Q. )
88	86, 87	2thd 255	. . . . . . . . . . 11 |- ( ( x .Q y ) = 1Q -> ( <. x , y >. e. ( Q. X. Q. ) <-> x e. Q. ) )
89	88	pm5.32i 669	. . . . . . . . . 10 |- ( ( ( x .Q y ) = 1Q /\ <. x , y >. e. ( Q. X. Q. ) ) <-> ( ( x .Q y ) = 1Q /\ x e. Q. ) )
90		df-ov 6653	. . . . . . . . . . . 12 |- ( x .Q y ) = ( .Q ` <. x , y >. )
91	90	eqeq1i 2627	. . . . . . . . . . 11 |- ( ( x .Q y ) = 1Q <-> ( .Q ` <. x , y >. ) = 1Q )
92	91	anbi1i 731	. . . . . . . . . 10 |- ( ( ( x .Q y ) = 1Q /\ <. x , y >. e. ( Q. X. Q. ) ) <-> ( ( .Q ` <. x , y >. ) = 1Q /\ <. x , y >. e. ( Q. X. Q. ) ) )
93	80, 89, 92	3bitr2ri 289	. . . . . . . . 9 |- ( ( ( .Q ` <. x , y >. ) = 1Q /\ <. x , y >. e. ( Q. X. Q. ) ) <-> ( x e. Q. /\ ( x .Q y ) = 1Q ) )
94	79, 93	bitri 264	. . . . . . . 8 |- ( ( <. x , y >. e. ( Q. X. Q. ) /\ ( .Q ` <. x , y >. ) = 1Q ) <-> ( x e. Q. /\ ( x .Q y ) = 1Q ) )
95	75, 78, 94	3bitri 286	. . . . . . 7 |- ( <. x , y >. e. *Q <-> ( x e. Q. /\ ( x .Q y ) = 1Q ) )
96	95	a1i 11	. . . . . 6 |- ( T. -> ( <. x , y >. e. *Q <-> ( x e. Q. /\ ( x .Q y ) = 1Q ) ) )
97	74, 96	opabbi2dv 5271	. . . . 5 |- ( T. -> *Q = { <. x , y >. | ( x e. Q. /\ ( x .Q y ) = 1Q ) } )
98	97	trud 1493	. . . 4 |- *Q = { <. x , y >. | ( x e. Q. /\ ( x .Q y ) = 1Q ) }
99	18, 20, 67, 98	fvopab3g 6277	. . 3 |- ( ( A e. Q. /\ B e. _V ) -> ( ( *Q ` A ) = B <-> ( A .Q B ) = 1Q ) )
100	99	ex 450	. 2 |- ( A e. Q. -> ( B e. _V -> ( ( *Q ` A ) = B <-> ( A .Q B ) = 1Q ) ) )
101	4, 16, 100	pm5.21ndd 369	1 |- ( A e. Q. -> ( ( *Q ` A ) = B <-> ( A .Q B ) = 1Q ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   T. wtru 1484   E.wex 1704   e. wcel 1990   E!weu 2470   E*wmo 2471   _Vcvv 3200   C_ wss 3574   {csn 4177   <.cop 4183   class class class wbr 4653   {copab 4712   X. cxp 5112   `'ccnv 5113   dom cdm 5114   "cima 5117   Rel wrel 5119   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   N.cnpi 9666   .N cmi 9668   .pQ cmpq 9671   ~Q ceq 9673   Q.cnq 9674   1Qc1q 9675   /Qcerq 9676   .Q cmq 9678   *Qcrq 9679

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  df-rq 9739

This theorem is referenced by:  recidnq  9787  recrecnq  9789  reclem3pr  9871