MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  divmul Structured version   Unicode version

Theorem divmul 10280
Description: Relationship between division and multiplication. (Contributed by NM, 2-Aug-2004.) (Revised by Mario Carneiro, 17-Feb-2014.)
Assertion
Ref Expression
divmul  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( ( A  /  C )  =  B  <-> 
( C  x.  B
)  =  A ) )

Proof of Theorem divmul
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 divval 10279 . . . . 5  |-  ( ( A  e.  CC  /\  C  e.  CC  /\  C  =/=  0 )  ->  ( A  /  C )  =  ( iota_ x  e.  CC  ( C  x.  x
)  =  A ) )
213expb 1206 . . . 4  |-  ( ( A  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  ->  ( A  /  C )  =  (
iota_ x  e.  CC  ( C  x.  x
)  =  A ) )
323adant2 1024 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( A  /  C
)  =  ( iota_ x  e.  CC  ( C  x.  x )  =  A ) )
43eqeq1d 2424 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( ( A  /  C )  =  B  <-> 
( iota_ x  e.  CC  ( C  x.  x
)  =  A )  =  B ) )
5 simp2 1006 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  ->  B  e.  CC )
6 receu 10264 . . . . 5  |-  ( ( A  e.  CC  /\  C  e.  CC  /\  C  =/=  0 )  ->  E! x  e.  CC  ( C  x.  x )  =  A )
763expb 1206 . . . 4  |-  ( ( A  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  ->  E! x  e.  CC  ( C  x.  x )  =  A )
873adant2 1024 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  ->  E! x  e.  CC  ( C  x.  x
)  =  A )
9 oveq2 6313 . . . . 5  |-  ( x  =  B  ->  ( C  x.  x )  =  ( C  x.  B ) )
109eqeq1d 2424 . . . 4  |-  ( x  =  B  ->  (
( C  x.  x
)  =  A  <->  ( C  x.  B )  =  A ) )
1110riota2 6289 . . 3  |-  ( ( B  e.  CC  /\  E! x  e.  CC  ( C  x.  x
)  =  A )  ->  ( ( C  x.  B )  =  A  <->  ( iota_ x  e.  CC  ( C  x.  x )  =  A )  =  B ) )
125, 8, 11syl2anc 665 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( ( C  x.  B )  =  A  <-> 
( iota_ x  e.  CC  ( C  x.  x
)  =  A )  =  B ) )
134, 12bitr4d 259 1  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( ( A  /  C )  =  B  <-> 
( C  x.  B
)  =  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872    =/= wne 2614   E!wreu 2773   iota_crio 6266  (class class class)co 6305   CCcc 9544   0cc0 9546    x. cmul 9551    / cdiv 10276
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597  ax-resscn 9603  ax-1cn 9604  ax-icn 9605  ax-addcl 9606  ax-addrcl 9607  ax-mulcl 9608  ax-mulrcl 9609  ax-mulcom 9610  ax-addass 9611  ax-mulass 9612  ax-distr 9613  ax-i2m1 9614  ax-1ne0 9615  ax-1rid 9616  ax-rnegex 9617  ax-rrecex 9618  ax-cnre 9619  ax-pre-lttri 9620  ax-pre-lttrn 9621  ax-pre-ltadd 9622  ax-pre-mulgt0 9623
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rmo 2779  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-br 4424  df-opab 4483  df-mpt 4484  df-id 4768  df-po 4774  df-so 4775  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-er 7374  df-en 7581  df-dom 7582  df-sdom 7583  df-pnf 9684  df-mnf 9685  df-xr 9686  df-ltxr 9687  df-le 9688  df-sub 9869  df-neg 9870  df-div 10277
This theorem is referenced by:  divmul2  10281  divcan2  10285  divrec  10293  divcan3  10301  div0  10305  div1  10306  recrec  10311  rec11  10312  divdivdiv  10315  ddcan  10328  rereccl  10332  div2neg  10337  divmulzi  10365  divmuld  10412  crreczi  12403  odd2np1  14364  sqgcd  14525
  Copyright terms: Public domain W3C validator