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Theorem divvalap 8401
Description: Value of division: the (unique) element  x such that  ( B  x.  x )  =  A. This is meaningful only when  B is apart from zero. (Contributed by Jim Kingdon, 21-Feb-2020.)
Assertion
Ref Expression
divvalap  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  ( A  /  B )  =  ( iota_ x  e.  CC  ( B  x.  x
)  =  A ) )
Distinct variable groups:    x, A    x, B

Proof of Theorem divvalap
Dummy variables  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp1 966 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  A  e.  CC )
2 simp2 967 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  B  e.  CC )
3 0cn 7726 . . . . . 6  |-  0  e.  CC
4 apne 8352 . . . . . 6  |-  ( ( B  e.  CC  /\  0  e.  CC )  ->  ( B #  0  ->  B  =/=  0 ) )
53, 4mpan2 421 . . . . 5  |-  ( B  e.  CC  ->  ( B #  0  ->  B  =/=  0 ) )
65adantl 275 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( B #  0  ->  B  =/=  0 ) )
763impia 1163 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  B  =/=  0 )
8 eldifsn 3620 . . 3  |-  ( B  e.  ( CC  \  { 0 } )  <-> 
( B  e.  CC  /\  B  =/=  0 ) )
92, 7, 8sylanbrc 413 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  B  e.  ( CC  \  {
0 } ) )
10 receuap 8397 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  E! x  e.  CC  ( B  x.  x )  =  A )
11 riotacl 5712 . . 3  |-  ( E! x  e.  CC  ( B  x.  x )  =  A  ->  ( iota_ x  e.  CC  ( B  x.  x )  =  A )  e.  CC )
1210, 11syl 14 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  ( iota_ x  e.  CC  ( B  x.  x )  =  A )  e.  CC )
13 eqeq2 2127 . . . 4  |-  ( z  =  A  ->  (
( y  x.  x
)  =  z  <->  ( y  x.  x )  =  A ) )
1413riotabidv 5700 . . 3  |-  ( z  =  A  ->  ( iota_ x  e.  CC  (
y  x.  x )  =  z )  =  ( iota_ x  e.  CC  ( y  x.  x
)  =  A ) )
15 oveq1 5749 . . . . 5  |-  ( y  =  B  ->  (
y  x.  x )  =  ( B  x.  x ) )
1615eqeq1d 2126 . . . 4  |-  ( y  =  B  ->  (
( y  x.  x
)  =  A  <->  ( B  x.  x )  =  A ) )
1716riotabidv 5700 . . 3  |-  ( y  =  B  ->  ( iota_ x  e.  CC  (
y  x.  x )  =  A )  =  ( iota_ x  e.  CC  ( B  x.  x
)  =  A ) )
18 df-div 8400 . . 3  |-  /  =  ( z  e.  CC ,  y  e.  ( CC  \  { 0 } )  |->  ( iota_ x  e.  CC  ( y  x.  x )  =  z ) )
1914, 17, 18ovmpog 5873 . 2  |-  ( ( A  e.  CC  /\  B  e.  ( CC  \  { 0 } )  /\  ( iota_ x  e.  CC  ( B  x.  x )  =  A )  e.  CC )  ->  ( A  /  B )  =  (
iota_ x  e.  CC  ( B  x.  x
)  =  A ) )
201, 9, 12, 19syl3anc 1201 1  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  ( A  /  B )  =  ( iota_ x  e.  CC  ( B  x.  x
)  =  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 947    = wceq 1316    e. wcel 1465    =/= wne 2285   E!wreu 2395    \ cdif 3038   {csn 3497   class class class wbr 3899   iota_crio 5697  (class class class)co 5742   CCcc 7586   0cc0 7588    x. cmul 7593   # cap 8310    / cdiv 8399
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  df-div 8400
This theorem is referenced by:  divmulap  8402  divclap  8405
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