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Theorem receuap 8397
Description: Existential uniqueness of reciprocals. (Contributed by Jim Kingdon, 21-Feb-2020.)
Assertion
Ref Expression
receuap  |-  ( ( 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 receuap
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 recexap 8381 . . . 4  |-  ( ( B  e.  CC  /\  B #  0 )  ->  E. y  e.  CC  ( B  x.  y )  =  1 )
213adant1 984 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  E. y  e.  CC  ( B  x.  y )  =  1 )
3 simprl 505 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( y  e.  CC  /\  ( B  x.  y )  =  1 ) )  -> 
y  e.  CC )
4 simpll 503 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( y  e.  CC  /\  ( B  x.  y )  =  1 ) )  ->  A  e.  CC )
53, 4mulcld 7754 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( y  e.  CC  /\  ( B  x.  y )  =  1 ) )  -> 
( y  x.  A
)  e.  CC )
6 oveq1 5749 . . . . . . . 8  |-  ( ( B  x.  y )  =  1  ->  (
( B  x.  y
)  x.  A )  =  ( 1  x.  A ) )
76ad2antll 482 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( y  e.  CC  /\  ( B  x.  y )  =  1 ) )  -> 
( ( B  x.  y )  x.  A
)  =  ( 1  x.  A ) )
8 simplr 504 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( y  e.  CC  /\  ( B  x.  y )  =  1 ) )  ->  B  e.  CC )
98, 3, 4mulassd 7757 . . . . . . 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 7752 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( y  e.  CC  /\  ( B  x.  y )  =  1 ) )  -> 
( 1  x.  A
)  =  A )
117, 9, 103eqtr3d 2158 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( y  e.  CC  /\  ( B  x.  y )  =  1 ) )  -> 
( B  x.  (
y  x.  A ) )  =  A )
12 oveq2 5750 . . . . . . . 8  |-  ( x  =  ( y  x.  A )  ->  ( B  x.  x )  =  ( B  x.  ( y  x.  A
) ) )
1312eqeq1d 2126 . . . . . . 7  |-  ( x  =  ( y  x.  A )  ->  (
( B  x.  x
)  =  A  <->  ( B  x.  ( y  x.  A
) )  =  A ) )
1413rspcev 2763 . . . . . 6  |-  ( ( ( y  x.  A
)  e.  CC  /\  ( B  x.  (
y  x.  A ) )  =  A )  ->  E. x  e.  CC  ( B  x.  x
)  =  A )
155, 11, 14syl2anc 408 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( y  e.  CC  /\  ( B  x.  y )  =  1 ) )  ->  E. x  e.  CC  ( B  x.  x
)  =  A )
1615rexlimdvaa 2527 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( E. y  e.  CC  ( B  x.  y )  =  1  ->  E. x  e.  CC  ( B  x.  x
)  =  A ) )
17163adant3 986 . . 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 13 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  E. x  e.  CC  ( B  x.  x )  =  A )
19 eqtr3 2137 . . . . . . 7  |-  ( ( ( B  x.  x
)  =  A  /\  ( B  x.  y
)  =  A )  ->  ( B  x.  x )  =  ( B  x.  y ) )
20 mulcanap 8393 . . . . . . 7  |-  ( ( x  e.  CC  /\  y  e.  CC  /\  ( B  e.  CC  /\  B #  0 ) )  -> 
( ( B  x.  x )  =  ( B  x.  y )  <-> 
x  =  y ) )
2119, 20syl5ib 153 . . . . . 6  |-  ( ( x  e.  CC  /\  y  e.  CC  /\  ( B  e.  CC  /\  B #  0 ) )  -> 
( ( ( B  x.  x )  =  A  /\  ( B  x.  y )  =  A )  ->  x  =  y ) )
22213expa 1166 . . . . 5  |-  ( ( ( x  e.  CC  /\  y  e.  CC )  /\  ( B  e.  CC  /\  B #  0 ) )  ->  (
( ( B  x.  x )  =  A  /\  ( B  x.  y )  =  A )  ->  x  =  y ) )
2322expcom 115 . . . 4  |-  ( ( B  e.  CC  /\  B #  0 )  ->  (
( x  e.  CC  /\  y  e.  CC )  ->  ( ( ( B  x.  x )  =  A  /\  ( B  x.  y )  =  A )  ->  x  =  y ) ) )
24233adant1 984 . . 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 2490 . 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 5750 . . . 4  |-  ( x  =  y  ->  ( B  x.  x )  =  ( B  x.  y ) )
2726eqeq1d 2126 . . 3  |-  ( x  =  y  ->  (
( B  x.  x
)  =  A  <->  ( B  x.  y )  =  A ) )
2827reu4 2851 . 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 413 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 103    /\ w3a 947    = wceq 1316    e. wcel 1465   A.wral 2393   E.wrex 2394   E!wreu 2395   class class class wbr 3899  (class class class)co 5742   CCcc 7586   0cc0 7588   1c1 7589    x. cmul 7593   # cap 8310
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-cnex 7679  ax-resscn 7680  ax-1cn 7681  ax-1re 7682  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-mulrcl 7687  ax-addcom 7688  ax-mulcom 7689  ax-addass 7690  ax-mulass 7691  ax-distr 7692  ax-i2m1 7693  ax-0lt1 7694  ax-1rid 7695  ax-0id 7696  ax-rnegex 7697  ax-precex 7698  ax-cnre 7699  ax-pre-ltirr 7700  ax-pre-ltwlin 7701  ax-pre-lttrn 7702  ax-pre-apti 7703  ax-pre-ltadd 7704  ax-pre-mulgt0 7705  ax-pre-mulext 7706
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-reu 2400  df-rmo 2401  df-rab 2402  df-v 2662  df-sbc 2883  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-id 4185  df-po 4188  df-iso 4189  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-iota 5058  df-fun 5095  df-fv 5101  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-pnf 7770  df-mnf 7771  df-xr 7772  df-ltxr 7773  df-le 7774  df-sub 7903  df-neg 7904  df-reap 8304  df-ap 8311
This theorem is referenced by:  divvalap  8401  divmulap  8402  divclap  8405
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