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Theorem divvalap 8441
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 981 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  A  e.  CC )
2 simp2 982 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  B  e.  CC )
3 0cn 7765 . . . . . 6  |-  0  e.  CC
4 apne 8392 . . . . . 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 1178 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  B  =/=  0 )
8 eldifsn 3650 . . 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 8437 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  E! x  e.  CC  ( B  x.  x )  =  A )
11 riotacl 5744 . . 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 2149 . . . 4  |-  ( z  =  A  ->  (
( y  x.  x
)  =  z  <->  ( y  x.  x )  =  A ) )
1413riotabidv 5732 . . 3  |-  ( z  =  A  ->  ( iota_ x  e.  CC  (
y  x.  x )  =  z )  =  ( iota_ x  e.  CC  ( y  x.  x
)  =  A ) )
15 oveq1 5781 . . . . 5  |-  ( y  =  B  ->  (
y  x.  x )  =  ( B  x.  x ) )
1615eqeq1d 2148 . . . 4  |-  ( y  =  B  ->  (
( y  x.  x
)  =  A  <->  ( B  x.  x )  =  A ) )
1716riotabidv 5732 . . 3  |-  ( y  =  B  ->  ( iota_ x  e.  CC  (
y  x.  x )  =  A )  =  ( iota_ x  e.  CC  ( B  x.  x
)  =  A ) )
18 df-div 8440 . . 3  |-  /  =  ( z  e.  CC ,  y  e.  ( CC  \  { 0 } )  |->  ( iota_ x  e.  CC  ( y  x.  x )  =  z ) )
1914, 17, 18ovmpog 5905 . 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 1216 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 962    = wceq 1331    e. wcel 1480    =/= wne 2308   E!wreu 2418    \ cdif 3068   {csn 3527   class class class wbr 3929   iota_crio 5729  (class class class)co 5774   CCcc 7625   0cc0 7627    x. cmul 7632   # cap 8350    / cdiv 8439
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7718  ax-resscn 7719  ax-1cn 7720  ax-1re 7721  ax-icn 7722  ax-addcl 7723  ax-addrcl 7724  ax-mulcl 7725  ax-mulrcl 7726  ax-addcom 7727  ax-mulcom 7728  ax-addass 7729  ax-mulass 7730  ax-distr 7731  ax-i2m1 7732  ax-0lt1 7733  ax-1rid 7734  ax-0id 7735  ax-rnegex 7736  ax-precex 7737  ax-cnre 7738  ax-pre-ltirr 7739  ax-pre-ltwlin 7740  ax-pre-lttrn 7741  ax-pre-apti 7742  ax-pre-ltadd 7743  ax-pre-mulgt0 7744  ax-pre-mulext 7745
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-id 4215  df-po 4218  df-iso 4219  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-pnf 7809  df-mnf 7810  df-xr 7811  df-ltxr 7812  df-le 7813  df-sub 7942  df-neg 7943  df-reap 8344  df-ap 8351  df-div 8440
This theorem is referenced by:  divmulap  8442  divclap  8445
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