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Theorem mulnzcnopr 9430
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 8833 . . . . 5  |-  x.  :
( CC  X.  CC )
--> CC
2 ffnov 5964 . . . . 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 199 . . . 4  |-  (  x.  Fn  ( CC  X.  CC )  /\  A. x  e.  CC  A. y  e.  CC  ( x  x.  y )  e.  CC )
43simpli 444 . . 3  |-  x.  Fn  ( CC  X.  CC )
5 difss 3316 . . . 4  |-  ( CC 
\  { 0 } )  C_  CC
6 xpss12 4808 . . . 4  |-  ( ( ( CC  \  {
0 } )  C_  CC  /\  ( CC  \  { 0 } ) 
C_  CC )  -> 
( ( CC  \  { 0 } )  X.  ( CC  \  { 0 } ) )  C_  ( CC  X.  CC ) )
75, 5, 6mp2an 653 . . 3  |-  ( ( CC  \  { 0 } )  X.  ( CC  \  { 0 } ) )  C_  ( CC  X.  CC )
8 fnssres 5373 . . 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 653 . 2  |-  (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC  \  { 0 } ) ) )  Fn  (
( CC  \  {
0 } )  X.  ( CC  \  {
0 } ) )
10 ovres 6003 . . . 4  |-  ( ( x  e.  ( CC 
\  { 0 } )  /\  y  e.  ( CC  \  {
0 } ) )  ->  ( x (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC 
\  { 0 } ) ) ) y )  =  ( x  x.  y ) )
11 eldifsn 3762 . . . . . 6  |-  ( x  e.  ( CC  \  { 0 } )  <-> 
( x  e.  CC  /\  x  =/=  0 ) )
12 eldifsn 3762 . . . . . 6  |-  ( y  e.  ( CC  \  { 0 } )  <-> 
( y  e.  CC  /\  y  =/=  0 ) )
13 mulcl 8837 . . . . . . . 8  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  x.  y
)  e.  CC )
1413ad2ant2r 727 . . . . . . 7  |-  ( ( ( x  e.  CC  /\  x  =/=  0 )  /\  ( y  e.  CC  /\  y  =/=  0 ) )  -> 
( x  x.  y
)  e.  CC )
15 mulne0 9426 . . . . . . 7  |-  ( ( ( x  e.  CC  /\  x  =/=  0 )  /\  ( y  e.  CC  /\  y  =/=  0 ) )  -> 
( x  x.  y
)  =/=  0 )
1614, 15jca 518 . . . . . 6  |-  ( ( ( x  e.  CC  /\  x  =/=  0 )  /\  ( y  e.  CC  /\  y  =/=  0 ) )  -> 
( ( x  x.  y )  e.  CC  /\  ( x  x.  y
)  =/=  0 ) )
1711, 12, 16syl2anb 465 . . . . 5  |-  ( ( x  e.  ( CC 
\  { 0 } )  /\  y  e.  ( CC  \  {
0 } ) )  ->  ( ( x  x.  y )  e.  CC  /\  ( x  x.  y )  =/=  0 ) )
18 eldifsn 3762 . . . . 5  |-  ( ( x  x.  y )  e.  ( CC  \  { 0 } )  <-> 
( ( x  x.  y )  e.  CC  /\  ( x  x.  y
)  =/=  0 ) )
1917, 18sylibr 203 . . . 4  |-  ( ( x  e.  ( CC 
\  { 0 } )  /\  y  e.  ( CC  \  {
0 } ) )  ->  ( x  x.  y )  e.  ( CC  \  { 0 } ) )
2010, 19eqeltrd 2370 . . 3  |-  ( ( x  e.  ( CC 
\  { 0 } )  /\  y  e.  ( CC  \  {
0 } ) )  ->  ( x (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC 
\  { 0 } ) ) ) y )  e.  ( CC 
\  { 0 } ) )
2120rgen2a 2622 . 2  |-  A. x  e.  ( CC  \  {
0 } ) A. y  e.  ( CC  \  { 0 } ) ( x (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC  \  { 0 } ) ) ) y )  e.  ( CC  \  { 0 } )
22 ffnov 5964 . 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 886 1  |-  (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC  \  { 0 } ) ) ) : ( ( CC  \  {
0 } )  X.  ( CC  \  {
0 } ) ) --> ( CC  \  {
0 } )
Colors of variables: wff set class
Syntax hints:    /\ wa 358    e. wcel 1696    =/= wne 2459   A.wral 2556    \ cdif 3162    C_ wss 3165   {csn 3653    X. cxp 4703    |` cres 4707    Fn wfn 5266   -->wf 5267  (class class class)co 5874   CCcc 8751   0cc0 8753    x. cmul 8758
This theorem is referenced by:  ablomul  21038  mulid  21039
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-mulf 8833
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056
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