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Theorem mulnzcnopr 9668
Description: Multiplication maps nonzero complex numbers to nonzero complex numbers. (Contributed by Steve Rodriguez, 23-Feb-2007.)
Assertion
Ref Expression
mulnzcnopr  |-  (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC  \  { 0 } ) ) ) : ( ( CC  \  {
0 } )  X.  ( CC  \  {
0 } ) ) --> ( CC  \  {
0 } )

Proof of Theorem mulnzcnopr
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ax-mulf 9070 . . . . 5  |-  x.  :
( CC  X.  CC )
--> CC
2 ffnov 6174 . . . . 5  |-  (  x.  : ( CC  X.  CC ) --> CC  <->  (  x.  Fn  ( CC  X.  CC )  /\  A. x  e.  CC  A. y  e.  CC  ( x  x.  y )  e.  CC ) )
31, 2mpbi 200 . . . 4  |-  (  x.  Fn  ( CC  X.  CC )  /\  A. x  e.  CC  A. y  e.  CC  ( x  x.  y )  e.  CC )
43simpli 445 . . 3  |-  x.  Fn  ( CC  X.  CC )
5 difss 3474 . . . 4  |-  ( CC 
\  { 0 } )  C_  CC
6 xpss12 4981 . . . 4  |-  ( ( ( CC  \  {
0 } )  C_  CC  /\  ( CC  \  { 0 } ) 
C_  CC )  -> 
( ( CC  \  { 0 } )  X.  ( CC  \  { 0 } ) )  C_  ( CC  X.  CC ) )
75, 5, 6mp2an 654 . . 3  |-  ( ( CC  \  { 0 } )  X.  ( CC  \  { 0 } ) )  C_  ( CC  X.  CC )
8 fnssres 5558 . . 3  |-  ( (  x.  Fn  ( CC 
X.  CC )  /\  ( ( CC  \  { 0 } )  X.  ( CC  \  { 0 } ) )  C_  ( CC  X.  CC ) )  -> 
(  x.  |`  (
( CC  \  {
0 } )  X.  ( CC  \  {
0 } ) ) )  Fn  ( ( CC  \  { 0 } )  X.  ( CC  \  { 0 } ) ) )
94, 7, 8mp2an 654 . 2  |-  (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC  \  { 0 } ) ) )  Fn  (
( CC  \  {
0 } )  X.  ( CC  \  {
0 } ) )
10 ovres 6213 . . . 4  |-  ( ( x  e.  ( CC 
\  { 0 } )  /\  y  e.  ( CC  \  {
0 } ) )  ->  ( x (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC 
\  { 0 } ) ) ) y )  =  ( x  x.  y ) )
11 eldifsn 3927 . . . . . 6  |-  ( x  e.  ( CC  \  { 0 } )  <-> 
( x  e.  CC  /\  x  =/=  0 ) )
12 eldifsn 3927 . . . . . 6  |-  ( y  e.  ( CC  \  { 0 } )  <-> 
( y  e.  CC  /\  y  =/=  0 ) )
13 mulcl 9074 . . . . . . . 8  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  x.  y
)  e.  CC )
1413ad2ant2r 728 . . . . . . 7  |-  ( ( ( x  e.  CC  /\  x  =/=  0 )  /\  ( y  e.  CC  /\  y  =/=  0 ) )  -> 
( x  x.  y
)  e.  CC )
15 mulne0 9664 . . . . . . 7  |-  ( ( ( x  e.  CC  /\  x  =/=  0 )  /\  ( y  e.  CC  /\  y  =/=  0 ) )  -> 
( x  x.  y
)  =/=  0 )
1614, 15jca 519 . . . . . 6  |-  ( ( ( x  e.  CC  /\  x  =/=  0 )  /\  ( y  e.  CC  /\  y  =/=  0 ) )  -> 
( ( x  x.  y )  e.  CC  /\  ( x  x.  y
)  =/=  0 ) )
1711, 12, 16syl2anb 466 . . . . 5  |-  ( ( x  e.  ( CC 
\  { 0 } )  /\  y  e.  ( CC  \  {
0 } ) )  ->  ( ( x  x.  y )  e.  CC  /\  ( x  x.  y )  =/=  0 ) )
18 eldifsn 3927 . . . . 5  |-  ( ( x  x.  y )  e.  ( CC  \  { 0 } )  <-> 
( ( x  x.  y )  e.  CC  /\  ( x  x.  y
)  =/=  0 ) )
1917, 18sylibr 204 . . . 4  |-  ( ( x  e.  ( CC 
\  { 0 } )  /\  y  e.  ( CC  \  {
0 } ) )  ->  ( x  x.  y )  e.  ( CC  \  { 0 } ) )
2010, 19eqeltrd 2510 . . 3  |-  ( ( x  e.  ( CC 
\  { 0 } )  /\  y  e.  ( CC  \  {
0 } ) )  ->  ( x (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC 
\  { 0 } ) ) ) y )  e.  ( CC 
\  { 0 } ) )
2120rgen2a 2772 . 2  |-  A. x  e.  ( CC  \  {
0 } ) A. y  e.  ( CC  \  { 0 } ) ( x (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC  \  { 0 } ) ) ) y )  e.  ( CC  \  { 0 } )
22 ffnov 6174 . 2  |-  ( (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC 
\  { 0 } ) ) ) : ( ( CC  \  { 0 } )  X.  ( CC  \  { 0 } ) ) --> ( CC  \  { 0 } )  <-> 
( (  x.  |`  (
( CC  \  {
0 } )  X.  ( CC  \  {
0 } ) ) )  Fn  ( ( CC  \  { 0 } )  X.  ( CC  \  { 0 } ) )  /\  A. x  e.  ( CC  \  { 0 } ) A. y  e.  ( CC  \  { 0 } ) ( x (  x.  |`  (
( CC  \  {
0 } )  X.  ( CC  \  {
0 } ) ) ) y )  e.  ( CC  \  {
0 } ) ) )
239, 21, 22mpbir2an 887 1  |-  (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC  \  { 0 } ) ) ) : ( ( CC  \  {
0 } )  X.  ( CC  \  {
0 } ) ) --> ( CC  \  {
0 } )
Colors of variables: wff set class
Syntax hints:    /\ wa 359    e. wcel 1725    =/= wne 2599   A.wral 2705    \ cdif 3317    C_ wss 3320   {csn 3814    X. cxp 4876    |` cres 4880    Fn wfn 5449   -->wf 5450  (class class class)co 6081   CCcc 8988   0cc0 8990    x. cmul 8995
This theorem is referenced by:  ablomul  21943  mulid  21944
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-mulf 9070
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-po 4503  df-so 4504  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-riota 6549  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294
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