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Theorem receu 9408
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
Dummy variable  y is distinct from all other variables.

Proof of Theorem receu
StepHypRef Expression
1 recex 9395 . . . 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 734 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( y  e.  CC  /\  ( B  x.  y )  =  1 ) )  -> 
y  e.  CC )
4 simpll 732 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( y  e.  CC  /\  ( B  x.  y )  =  1 ) )  ->  A  e.  CC )
53, 4mulcld 8850 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( y  e.  CC  /\  ( B  x.  y )  =  1 ) )  -> 
( y  x.  A
)  e.  CC )
6 oveq1 5826 . . . . . . . . 9  |-  ( ( B  x.  y )  =  1  ->  (
( B  x.  y
)  x.  A )  =  ( 1  x.  A ) )
76ad2antll 711 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( y  e.  CC  /\  ( B  x.  y )  =  1 ) )  -> 
( ( B  x.  y )  x.  A
)  =  ( 1  x.  A ) )
8 simplr 733 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( y  e.  CC  /\  ( B  x.  y )  =  1 ) )  ->  B  e.  CC )
98, 3, 4mulassd 8853 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( y  e.  CC  /\  ( B  x.  y )  =  1 ) )  -> 
( ( B  x.  y )  x.  A
)  =  ( B  x.  ( y  x.  A ) ) )
104mulid2d 8848 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( y  e.  CC  /\  ( B  x.  y )  =  1 ) )  -> 
( 1  x.  A
)  =  A )
117, 9, 103eqtr3d 2324 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( y  e.  CC  /\  ( B  x.  y )  =  1 ) )  -> 
( B  x.  (
y  x.  A ) )  =  A )
12 oveq2 5827 . . . . . . . . 9  |-  ( x  =  ( y  x.  A )  ->  ( B  x.  x )  =  ( B  x.  ( y  x.  A
) ) )
1312eqeq1d 2292 . . . . . . . 8  |-  ( x  =  ( y  x.  A )  ->  (
( B  x.  x
)  =  A  <->  ( B  x.  ( y  x.  A
) )  =  A ) )
1413rspcev 2885 . . . . . . 7  |-  ( ( ( y  x.  A
)  e.  CC  /\  ( B  x.  (
y  x.  A ) )  =  A )  ->  E. x  e.  CC  ( B  x.  x
)  =  A )
155, 11, 14syl2anc 644 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( y  e.  CC  /\  ( B  x.  y )  =  1 ) )  ->  E. x  e.  CC  ( B  x.  x
)  =  A )
1615exp32 590 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( y  e.  CC  ->  ( ( B  x.  y )  =  1  ->  E. x  e.  CC  ( B  x.  x
)  =  A ) ) )
1716rexlimdv 2667 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( E. y  e.  CC  ( B  x.  y )  =  1  ->  E. x  e.  CC  ( B  x.  x
)  =  A ) )
18173adant3 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 ) )
192, 18mpd 16 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  E. x  e.  CC  ( B  x.  x )  =  A )
20 eqtr3 2303 . . . . . . 7  |-  ( ( ( B  x.  x
)  =  A  /\  ( B  x.  y
)  =  A )  ->  ( B  x.  x )  =  ( B  x.  y ) )
21 mulcan 9400 . . . . . . 7  |-  ( ( x  e.  CC  /\  y  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 ) )  -> 
( ( B  x.  x )  =  ( B  x.  y )  <-> 
x  =  y ) )
2220, 21syl5ib 212 . . . . . 6  |-  ( ( x  e.  CC  /\  y  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 ) )  -> 
( ( ( B  x.  x )  =  A  /\  ( B  x.  y )  =  A )  ->  x  =  y ) )
23223expa 1153 . . . . 5  |-  ( ( ( x  e.  CC  /\  y  e.  CC )  /\  ( B  e.  CC  /\  B  =/=  0 ) )  -> 
( ( ( B  x.  x )  =  A  /\  ( B  x.  y )  =  A )  ->  x  =  y ) )
2423expcom 426 . . . 4  |-  ( ( B  e.  CC  /\  B  =/=  0 )  -> 
( ( x  e.  CC  /\  y  e.  CC )  ->  (
( ( B  x.  x )  =  A  /\  ( B  x.  y )  =  A )  ->  x  =  y ) ) )
25243adant1 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 ) ) )
2625ralrimivv 2635 . 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 ) )
27 oveq2 5827 . . . 4  |-  ( x  =  y  ->  ( B  x.  x )  =  ( B  x.  y ) )
2827eqeq1d 2292 . . 3  |-  ( x  =  y  ->  (
( B  x.  x
)  =  A  <->  ( B  x.  y )  =  A ) )
2928reu4 2960 . 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 ) ) )
3019, 26, 29sylanbrc 647 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 6    /\ wa 360    /\ w3a 936    = wceq 1624    e. wcel 1685    =/= wne 2447   A.wral 2544   E.wrex 2545   E!wreu 2546  (class class class)co 5819   CCcc 8730   0cc0 8732   1c1 8733    x. cmul 8737
This theorem is referenced by:  divmul  9422  divcl  9425
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-resscn 8789  ax-1cn 8790  ax-icn 8791  ax-addcl 8792  ax-addrcl 8793  ax-mulcl 8794  ax-mulrcl 8795  ax-mulcom 8796  ax-addass 8797  ax-mulass 8798  ax-distr 8799  ax-i2m1 8800  ax-1ne0 8801  ax-1rid 8802  ax-rnegex 8803  ax-rrecex 8804  ax-cnre 8805  ax-pre-lttri 8806  ax-pre-lttrn 8807  ax-pre-ltadd 8808  ax-pre-mulgt0 8809
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-po 4313  df-so 4314  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-iota 6252  df-riota 6299  df-er 6655  df-en 6859  df-dom 6860  df-sdom 6861  df-pnf 8864  df-mnf 8865  df-xr 8866  df-ltxr 8867  df-le 8868  df-sub 9034  df-neg 9035
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