Theorem recexnq 7721

Description: Existence of positive fraction reciprocal. (Contributed by Jim Kingdon, 20-Sep-2019.)

Assertion

Ref	Expression
recexnq	|- ( A e. Q. -> E. y ( y e. Q. /\ ( A .Q y ) = 1Q ) )

Distinct variable group:   y, A

Proof of Theorem recexnq

Dummy variables x z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-nqqs 7679	. 2 |- Q. = ( ( N. X. N. ) /. ~Q )
2		oveq1 6065	. . . . 5 |- ( [ <. x , z >. ] ~Q = A -> ( [ <. x , z >. ] ~Q .Q y ) = ( A .Q y ) )
3	2	eqeq1d 2243	. . . 4 |- ( [ <. x , z >. ] ~Q = A -> ( ( [ <. x , z >. ] ~Q .Q y ) = 1Q <-> ( A .Q y ) = 1Q ) )
4	3	anbi2d 464	. . 3 |- ( [ <. x , z >. ] ~Q = A -> ( ( y e. Q. /\ ( [ <. x , z >. ] ~Q .Q y ) = 1Q ) <-> ( y e. Q. /\ ( A .Q y ) = 1Q ) ) )
5	4	exbidv 1874	. 2 |- ( [ <. x , z >. ] ~Q = A -> ( E. y ( y e. Q. /\ ( [ <. x , z >. ] ~Q .Q y ) = 1Q ) <-> E. y ( y e. Q. /\ ( A .Q y ) = 1Q ) ) )
6		opelxpi 4786	. . . . . 6 |- ( ( z e. N. /\ x e. N. ) -> <. z , x >. e. ( N. X. N. ) )
7	6	ancoms 268	. . . . 5 |- ( ( x e. N. /\ z e. N. ) -> <. z , x >. e. ( N. X. N. ) )
8		enqex 7691	. . . . . 6 |- ~Q e. _V
9	8	ecelqsi 6836	. . . . 5 |- ( <. z , x >. e. ( N. X. N. ) -> [ <. z , x >. ] ~Q e. ( ( N. X. N. ) /. ~Q ) )
10	7, 9	syl 14	. . . 4 |- ( ( x e. N. /\ z e. N. ) -> [ <. z , x >. ] ~Q e. ( ( N. X. N. ) /. ~Q ) )
11	10, 1	eleqtrrdi 2328	. . 3 |- ( ( x e. N. /\ z e. N. ) -> [ <. z , x >. ] ~Q e. Q. )
12		mulcompig 7662	. . . . . . 7 |- ( ( x e. N. /\ z e. N. ) -> ( x .N z ) = ( z .N x ) )
13	12	opeq2d 3895	. . . . . 6 |- ( ( x e. N. /\ z e. N. ) -> <. ( x .N z ) , ( x .N z ) >. = <. ( x .N z ) , ( z .N x ) >. )
14	13	eceq1d 6816	. . . . 5 |- ( ( x e. N. /\ z e. N. ) -> [ <. ( x .N z ) , ( x .N z ) >. ] ~Q = [ <. ( x .N z ) , ( z .N x ) >. ] ~Q )
15		mulclpi 7659	. . . . . 6 |- ( ( x e. N. /\ z e. N. ) -> ( x .N z ) e. N. )
16		1qec 7719	. . . . . 6 |- ( ( x .N z ) e. N. -> 1Q = [ <. ( x .N z ) , ( x .N z ) >. ] ~Q )
17	15, 16	syl 14	. . . . 5 |- ( ( x e. N. /\ z e. N. ) -> 1Q = [ <. ( x .N z ) , ( x .N z ) >. ] ~Q )
18		mulpipqqs 7704	. . . . . . 7 |- ( ( ( x e. N. /\ z e. N. ) /\ ( z e. N. /\ x e. N. ) ) -> ( [ <. x , z >. ] ~Q .Q [ <. z , x >. ] ~Q ) = [ <. ( x .N z ) , ( z .N x ) >. ] ~Q )
19	18	an42s 593	. . . . . 6 |- ( ( ( x e. N. /\ z e. N. ) /\ ( x e. N. /\ z e. N. ) ) -> ( [ <. x , z >. ] ~Q .Q [ <. z , x >. ] ~Q ) = [ <. ( x .N z ) , ( z .N x ) >. ] ~Q )
20	19	anidms 397	. . . . 5 |- ( ( x e. N. /\ z e. N. ) -> ( [ <. x , z >. ] ~Q .Q [ <. z , x >. ] ~Q ) = [ <. ( x .N z ) , ( z .N x ) >. ] ~Q )
21	14, 17, 20	3eqtr4rd 2278	. . . 4 |- ( ( x e. N. /\ z e. N. ) -> ( [ <. x , z >. ] ~Q .Q [ <. z , x >. ] ~Q ) = 1Q )
22	11, 21	jca 306	. . 3 |- ( ( x e. N. /\ z e. N. ) -> ( [ <. z , x >. ] ~Q e. Q. /\ ( [ <. x , z >. ] ~Q .Q [ <. z , x >. ] ~Q ) = 1Q ) )
23		eleq1 2297	. . . . 5 |- ( y = [ <. z , x >. ] ~Q -> ( y e. Q. <-> [ <. z , x >. ] ~Q e. Q. ) )
24		oveq2 6066	. . . . . 6 |- ( y = [ <. z , x >. ] ~Q -> ( [ <. x , z >. ] ~Q .Q y ) = ( [ <. x , z >. ] ~Q .Q [ <. z , x >. ] ~Q ) )
25	24	eqeq1d 2243	. . . . 5 |- ( y = [ <. z , x >. ] ~Q -> ( ( [ <. x , z >. ] ~Q .Q y ) = 1Q <-> ( [ <. x , z >. ] ~Q .Q [ <. z , x >. ] ~Q ) = 1Q ) )
26	23, 25	anbi12d 473	. . . 4 |- ( y = [ <. z , x >. ] ~Q -> ( ( y e. Q. /\ ( [ <. x , z >. ] ~Q .Q y ) = 1Q ) <-> ( [ <. z , x >. ] ~Q e. Q. /\ ( [ <. x , z >. ] ~Q .Q [ <. z , x >. ] ~Q ) = 1Q ) ) )
27	26	spcegv 2907	. . 3 |- ( [ <. z , x >. ] ~Q e. Q. -> ( ( [ <. z , x >. ] ~Q e. Q. /\ ( [ <. x , z >. ] ~Q .Q [ <. z , x >. ] ~Q ) = 1Q ) -> E. y ( y e. Q. /\ ( [ <. x , z >. ] ~Q .Q y ) = 1Q ) ) )
28	11, 22, 27	sylc 62	. 2 |- ( ( x e. N. /\ z e. N. ) -> E. y ( y e. Q. /\ ( [ <. x , z >. ] ~Q .Q y ) = 1Q ) )
29	1, 5, 28	ecoptocl 6869	1 |- ( A e. Q. -> E. y ( y e. Q. /\ ( A .Q y ) = 1Q ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   E.wex 1541   e. wcel 2205   <.cop 3697   X. cxp 4752  (class class class)co 6058   [cec 6778   /.cqs 6779   N.cnpi 7603   .N cmi 7605   ~Q ceq 7610   Q.cnq 7611   1Qc1q 7612   .Q cmq 7614

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-irdg 6614  df-1o 6660  df-oadd 6664  df-omul 6665  df-er 6780  df-ec 6782  df-qs 6786  df-ni 7635  df-mi 7637  df-mpq 7676  df-enq 7678  df-nqqs 7679  df-mqqs 7681  df-1nqqs 7682

This theorem is referenced by:  recmulnqg  7722  recclnq  7723