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Theorem divvalap 8115
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 943 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  A  e.  CC )
2 simp2 944 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  B  e.  CC )
3 0cn 7459 . . . . . 6  |-  0  e.  CC
4 apne 8076 . . . . . 6  |-  ( ( B  e.  CC  /\  0  e.  CC )  ->  ( B #  0  ->  B  =/=  0 ) )
53, 4mpan2 416 . . . . 5  |-  ( B  e.  CC  ->  ( B #  0  ->  B  =/=  0 ) )
65adantl 271 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( B #  0  ->  B  =/=  0 ) )
763impia 1140 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  B  =/=  0 )
8 eldifsn 3562 . . 3  |-  ( B  e.  ( CC  \  { 0 } )  <-> 
( B  e.  CC  /\  B  =/=  0 ) )
92, 7, 8sylanbrc 408 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  B  e.  ( CC  \  {
0 } ) )
10 receuap 8112 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  E! x  e.  CC  ( B  x.  x )  =  A )
11 riotacl 5604 . . 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 2097 . . . 4  |-  ( z  =  A  ->  (
( y  x.  x
)  =  z  <->  ( y  x.  x )  =  A ) )
1413riotabidv 5592 . . 3  |-  ( z  =  A  ->  ( iota_ x  e.  CC  (
y  x.  x )  =  z )  =  ( iota_ x  e.  CC  ( y  x.  x
)  =  A ) )
15 oveq1 5641 . . . . 5  |-  ( y  =  B  ->  (
y  x.  x )  =  ( B  x.  x ) )
1615eqeq1d 2096 . . . 4  |-  ( y  =  B  ->  (
( y  x.  x
)  =  A  <->  ( B  x.  x )  =  A ) )
1716riotabidv 5592 . . 3  |-  ( y  =  B  ->  ( iota_ x  e.  CC  (
y  x.  x )  =  A )  =  ( iota_ x  e.  CC  ( B  x.  x
)  =  A ) )
18 df-div 8114 . . 3  |-  /  =  ( z  e.  CC ,  y  e.  ( CC  \  { 0 } )  |->  ( iota_ x  e.  CC  ( y  x.  x )  =  z ) )
1914, 17, 18ovmpt2g 5761 . 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 1174 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 924    = wceq 1289    e. wcel 1438    =/= wne 2255   E!wreu 2361    \ cdif 2994   {csn 3441   class class class wbr 3837   iota_crio 5589  (class class class)co 5634   CCcc 7327   0cc0 7329    x. cmul 7334   # cap 8034    / cdiv 8113
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-cnex 7415  ax-resscn 7416  ax-1cn 7417  ax-1re 7418  ax-icn 7419  ax-addcl 7420  ax-addrcl 7421  ax-mulcl 7422  ax-mulrcl 7423  ax-addcom 7424  ax-mulcom 7425  ax-addass 7426  ax-mulass 7427  ax-distr 7428  ax-i2m1 7429  ax-0lt1 7430  ax-1rid 7431  ax-0id 7432  ax-rnegex 7433  ax-precex 7434  ax-cnre 7435  ax-pre-ltirr 7436  ax-pre-ltwlin 7437  ax-pre-lttrn 7438  ax-pre-apti 7439  ax-pre-ltadd 7440  ax-pre-mulgt0 7441  ax-pre-mulext 7442
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367  df-rab 2368  df-v 2621  df-sbc 2839  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-id 4111  df-po 4114  df-iso 4115  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-iota 4967  df-fun 5004  df-fv 5010  df-riota 5590  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-pnf 7503  df-mnf 7504  df-xr 7505  df-ltxr 7506  df-le 7507  df-sub 7634  df-neg 7635  df-reap 8028  df-ap 8035  df-div 8114
This theorem is referenced by:  divmulap  8116  divclap  8119
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