MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  receu Unicode version

Theorem receu 9623
Description: Existential uniqueness of reciprocals. Theorem I.8 of [Apostol] p. 18. (Contributed by NM, 29-Jan-1995.) (Revised by Mario Carneiro, 17-Feb-2014.)
Assertion
Ref Expression
receu  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  E! x  e.  CC  ( B  x.  x )  =  A )
Distinct variable groups:    x, A    x, B

Proof of Theorem receu
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 recex 9610 . . . 4  |-  ( ( B  e.  CC  /\  B  =/=  0 )  ->  E. y  e.  CC  ( B  x.  y
)  =  1 )
213adant1 975 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  E. y  e.  CC  ( B  x.  y )  =  1 )
3 simprl 733 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( y  e.  CC  /\  ( B  x.  y )  =  1 ) )  -> 
y  e.  CC )
4 simpll 731 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( y  e.  CC  /\  ( B  x.  y )  =  1 ) )  ->  A  e.  CC )
53, 4mulcld 9064 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( y  e.  CC  /\  ( B  x.  y )  =  1 ) )  -> 
( y  x.  A
)  e.  CC )
6 oveq1 6047 . . . . . . . 8  |-  ( ( B  x.  y )  =  1  ->  (
( B  x.  y
)  x.  A )  =  ( 1  x.  A ) )
76ad2antll 710 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( y  e.  CC  /\  ( B  x.  y )  =  1 ) )  -> 
( ( B  x.  y )  x.  A
)  =  ( 1  x.  A ) )
8 simplr 732 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( y  e.  CC  /\  ( B  x.  y )  =  1 ) )  ->  B  e.  CC )
98, 3, 4mulassd 9067 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( y  e.  CC  /\  ( B  x.  y )  =  1 ) )  -> 
( ( B  x.  y )  x.  A
)  =  ( B  x.  ( y  x.  A ) ) )
104mulid2d 9062 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( y  e.  CC  /\  ( B  x.  y )  =  1 ) )  -> 
( 1  x.  A
)  =  A )
117, 9, 103eqtr3d 2444 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( y  e.  CC  /\  ( B  x.  y )  =  1 ) )  -> 
( B  x.  (
y  x.  A ) )  =  A )
12 oveq2 6048 . . . . . . . 8  |-  ( x  =  ( y  x.  A )  ->  ( B  x.  x )  =  ( B  x.  ( y  x.  A
) ) )
1312eqeq1d 2412 . . . . . . 7  |-  ( x  =  ( y  x.  A )  ->  (
( B  x.  x
)  =  A  <->  ( B  x.  ( y  x.  A
) )  =  A ) )
1413rspcev 3012 . . . . . 6  |-  ( ( ( y  x.  A
)  e.  CC  /\  ( B  x.  (
y  x.  A ) )  =  A )  ->  E. x  e.  CC  ( B  x.  x
)  =  A )
155, 11, 14syl2anc 643 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( y  e.  CC  /\  ( B  x.  y )  =  1 ) )  ->  E. x  e.  CC  ( B  x.  x
)  =  A )
1615rexlimdvaa 2791 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( E. y  e.  CC  ( B  x.  y )  =  1  ->  E. x  e.  CC  ( B  x.  x
)  =  A ) )
17163adant3 977 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( E. y  e.  CC  ( B  x.  y
)  =  1  ->  E. x  e.  CC  ( B  x.  x
)  =  A ) )
182, 17mpd 15 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  E. x  e.  CC  ( B  x.  x )  =  A )
19 eqtr3 2423 . . . . . . 7  |-  ( ( ( B  x.  x
)  =  A  /\  ( B  x.  y
)  =  A )  ->  ( B  x.  x )  =  ( B  x.  y ) )
20 mulcan 9615 . . . . . . 7  |-  ( ( x  e.  CC  /\  y  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 ) )  -> 
( ( B  x.  x )  =  ( B  x.  y )  <-> 
x  =  y ) )
2119, 20syl5ib 211 . . . . . 6  |-  ( ( x  e.  CC  /\  y  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 ) )  -> 
( ( ( B  x.  x )  =  A  /\  ( B  x.  y )  =  A )  ->  x  =  y ) )
22213expa 1153 . . . . 5  |-  ( ( ( x  e.  CC  /\  y  e.  CC )  /\  ( B  e.  CC  /\  B  =/=  0 ) )  -> 
( ( ( B  x.  x )  =  A  /\  ( B  x.  y )  =  A )  ->  x  =  y ) )
2322expcom 425 . . . 4  |-  ( ( B  e.  CC  /\  B  =/=  0 )  -> 
( ( x  e.  CC  /\  y  e.  CC )  ->  (
( ( B  x.  x )  =  A  /\  ( B  x.  y )  =  A )  ->  x  =  y ) ) )
24233adant1 975 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  (
( x  e.  CC  /\  y  e.  CC )  ->  ( ( ( B  x.  x )  =  A  /\  ( B  x.  y )  =  A )  ->  x  =  y ) ) )
2524ralrimivv 2757 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  A. x  e.  CC  A. y  e.  CC  ( ( ( B  x.  x )  =  A  /\  ( B  x.  y )  =  A )  ->  x  =  y ) )
26 oveq2 6048 . . . 4  |-  ( x  =  y  ->  ( B  x.  x )  =  ( B  x.  y ) )
2726eqeq1d 2412 . . 3  |-  ( x  =  y  ->  (
( B  x.  x
)  =  A  <->  ( B  x.  y )  =  A ) )
2827reu4 3088 . 2  |-  ( E! x  e.  CC  ( B  x.  x )  =  A  <->  ( E. x  e.  CC  ( B  x.  x )  =  A  /\  A. x  e.  CC  A. y  e.  CC  ( ( ( B  x.  x )  =  A  /\  ( B  x.  y )  =  A )  ->  x  =  y ) ) )
2918, 25, 28sylanbrc 646 1  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  E! x  e.  CC  ( B  x.  x )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567   A.wral 2666   E.wrex 2667   E!wreu 2668  (class class class)co 6040   CCcc 8944   0cc0 8946   1c1 8947    x. cmul 8951
This theorem is referenced by:  divmul  9637  divcl  9640  rexdiv  24125
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-po 4463  df-so 4464  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-riota 6508  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250
  Copyright terms: Public domain W3C validator