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Theorem divvalap 8644
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 998 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  A  e.  CC )
2 simp2 999 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  B  e.  CC )
3 0cn 7962 . . . . . 6  |-  0  e.  CC
4 apne 8593 . . . . . 6  |-  ( ( B  e.  CC  /\  0  e.  CC )  ->  ( B #  0  ->  B  =/=  0 ) )
53, 4mpan2 425 . . . . 5  |-  ( B  e.  CC  ->  ( B #  0  ->  B  =/=  0 ) )
65adantl 277 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( B #  0  ->  B  =/=  0 ) )
763impia 1201 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  B  =/=  0 )
8 eldifsn 3731 . . 3  |-  ( B  e.  ( CC  \  { 0 } )  <-> 
( B  e.  CC  /\  B  =/=  0 ) )
92, 7, 8sylanbrc 417 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  B  e.  ( CC  \  {
0 } ) )
10 receuap 8639 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  E! x  e.  CC  ( B  x.  x )  =  A )
11 riotacl 5858 . . 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 2197 . . . 4  |-  ( z  =  A  ->  (
( y  x.  x
)  =  z  <->  ( y  x.  x )  =  A ) )
1413riotabidv 5846 . . 3  |-  ( z  =  A  ->  ( iota_ x  e.  CC  (
y  x.  x )  =  z )  =  ( iota_ x  e.  CC  ( y  x.  x
)  =  A ) )
15 oveq1 5895 . . . . 5  |-  ( y  =  B  ->  (
y  x.  x )  =  ( B  x.  x ) )
1615eqeq1d 2196 . . . 4  |-  ( y  =  B  ->  (
( y  x.  x
)  =  A  <->  ( B  x.  x )  =  A ) )
1716riotabidv 5846 . . 3  |-  ( y  =  B  ->  ( iota_ x  e.  CC  (
y  x.  x )  =  A )  =  ( iota_ x  e.  CC  ( B  x.  x
)  =  A ) )
18 df-div 8643 . . 3  |-  /  =  ( z  e.  CC ,  y  e.  ( CC  \  { 0 } )  |->  ( iota_ x  e.  CC  ( y  x.  x )  =  z ) )
1914, 17, 18ovmpog 6022 . 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 1248 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 979    = wceq 1363    e. wcel 2158    =/= wne 2357   E!wreu 2467    \ cdif 3138   {csn 3604   class class class wbr 4015   iota_crio 5843  (class class class)co 5888   CCcc 7822   0cc0 7824    x. cmul 7829   # cap 8551    / cdiv 8642
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-cnex 7915  ax-resscn 7916  ax-1cn 7917  ax-1re 7918  ax-icn 7919  ax-addcl 7920  ax-addrcl 7921  ax-mulcl 7922  ax-mulrcl 7923  ax-addcom 7924  ax-mulcom 7925  ax-addass 7926  ax-mulass 7927  ax-distr 7928  ax-i2m1 7929  ax-0lt1 7930  ax-1rid 7931  ax-0id 7932  ax-rnegex 7933  ax-precex 7934  ax-cnre 7935  ax-pre-ltirr 7936  ax-pre-ltwlin 7937  ax-pre-lttrn 7938  ax-pre-apti 7939  ax-pre-ltadd 7940  ax-pre-mulgt0 7941  ax-pre-mulext 7942
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473  df-rab 2474  df-v 2751  df-sbc 2975  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-opab 4077  df-id 4305  df-po 4308  df-iso 4309  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-iota 5190  df-fun 5230  df-fv 5236  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-pnf 8007  df-mnf 8008  df-xr 8009  df-ltxr 8010  df-le 8011  df-sub 8143  df-neg 8144  df-reap 8545  df-ap 8552  df-div 8643
This theorem is referenced by:  divmulap  8645  divclap  8648
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