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Theorem receu 10111
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 10098 . . . 4  |-  ( ( B  e.  CC  /\  B  =/=  0 )  ->  E. y  e.  CC  ( B  x.  y
)  =  1 )
213adant1 1012 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  E. y  e.  CC  ( B  x.  y )  =  1 )
3 simprl 754 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( y  e.  CC  /\  ( B  x.  y )  =  1 ) )  -> 
y  e.  CC )
4 simpll 751 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( y  e.  CC  /\  ( B  x.  y )  =  1 ) )  ->  A  e.  CC )
53, 4mulcld 9527 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( y  e.  CC  /\  ( B  x.  y )  =  1 ) )  -> 
( y  x.  A
)  e.  CC )
6 oveq1 6203 . . . . . . . 8  |-  ( ( B  x.  y )  =  1  ->  (
( B  x.  y
)  x.  A )  =  ( 1  x.  A ) )
76ad2antll 726 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( y  e.  CC  /\  ( B  x.  y )  =  1 ) )  -> 
( ( B  x.  y )  x.  A
)  =  ( 1  x.  A ) )
8 simplr 753 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( y  e.  CC  /\  ( B  x.  y )  =  1 ) )  ->  B  e.  CC )
98, 3, 4mulassd 9530 . . . . . . 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 9525 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( y  e.  CC  /\  ( B  x.  y )  =  1 ) )  -> 
( 1  x.  A
)  =  A )
117, 9, 103eqtr3d 2431 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( y  e.  CC  /\  ( B  x.  y )  =  1 ) )  -> 
( B  x.  (
y  x.  A ) )  =  A )
12 oveq2 6204 . . . . . . . 8  |-  ( x  =  ( y  x.  A )  ->  ( B  x.  x )  =  ( B  x.  ( y  x.  A
) ) )
1312eqeq1d 2384 . . . . . . 7  |-  ( x  =  ( y  x.  A )  ->  (
( B  x.  x
)  =  A  <->  ( B  x.  ( y  x.  A
) )  =  A ) )
1413rspcev 3135 . . . . . 6  |-  ( ( ( y  x.  A
)  e.  CC  /\  ( B  x.  (
y  x.  A ) )  =  A )  ->  E. x  e.  CC  ( B  x.  x
)  =  A )
155, 11, 14syl2anc 659 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( y  e.  CC  /\  ( B  x.  y )  =  1 ) )  ->  E. x  e.  CC  ( B  x.  x
)  =  A )
1615rexlimdvaa 2875 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( E. y  e.  CC  ( B  x.  y )  =  1  ->  E. x  e.  CC  ( B  x.  x
)  =  A ) )
17163adant3 1014 . . 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 2410 . . . . . . 7  |-  ( ( ( B  x.  x
)  =  A  /\  ( B  x.  y
)  =  A )  ->  ( B  x.  x )  =  ( B  x.  y ) )
20 mulcan 10103 . . . . . . 7  |-  ( ( x  e.  CC  /\  y  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 ) )  -> 
( ( B  x.  x )  =  ( B  x.  y )  <-> 
x  =  y ) )
2119, 20syl5ib 219 . . . . . 6  |-  ( ( x  e.  CC  /\  y  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 ) )  -> 
( ( ( B  x.  x )  =  A  /\  ( B  x.  y )  =  A )  ->  x  =  y ) )
22213expa 1194 . . . . 5  |-  ( ( ( x  e.  CC  /\  y  e.  CC )  /\  ( B  e.  CC  /\  B  =/=  0 ) )  -> 
( ( ( B  x.  x )  =  A  /\  ( B  x.  y )  =  A )  ->  x  =  y ) )
2322expcom 433 . . . 4  |-  ( ( B  e.  CC  /\  B  =/=  0 )  -> 
( ( x  e.  CC  /\  y  e.  CC )  ->  (
( ( B  x.  x )  =  A  /\  ( B  x.  y )  =  A )  ->  x  =  y ) ) )
24233adant1 1012 . . 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 2802 . 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 6204 . . . 4  |-  ( x  =  y  ->  ( B  x.  x )  =  ( B  x.  y ) )
2726eqeq1d 2384 . . 3  |-  ( x  =  y  ->  (
( B  x.  x
)  =  A  <->  ( B  x.  y )  =  A ) )
2827reu4 3218 . 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 662 1  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  E! x  e.  CC  ( B  x.  x )  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 971    = wceq 1399    e. wcel 1826    =/= wne 2577   A.wral 2732   E.wrex 2733   E!wreu 2734  (class class class)co 6196   CCcc 9401   0cc0 9403   1c1 9404    x. cmul 9408
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-po 4714  df-so 4715  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-er 7229  df-en 7436  df-dom 7437  df-sdom 7438  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721
This theorem is referenced by:  divmul  10127  divcl  10130  rexdiv  27775
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