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Theorem mulnzcnopr 10088
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 9468 . . . . 5  |-  x.  :
( CC  X.  CC )
--> CC
2 ffnov 6299 . . . . 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 208 . . . 4  |-  (  x.  Fn  ( CC  X.  CC )  /\  A. x  e.  CC  A. y  e.  CC  ( x  x.  y )  e.  CC )
43simpli 458 . . 3  |-  x.  Fn  ( CC  X.  CC )
5 difss 3586 . . . 4  |-  ( CC 
\  { 0 } )  C_  CC
6 xpss12 5048 . . . 4  |-  ( ( ( CC  \  {
0 } )  C_  CC  /\  ( CC  \  { 0 } ) 
C_  CC )  -> 
( ( CC  \  { 0 } )  X.  ( CC  \  { 0 } ) )  C_  ( CC  X.  CC ) )
75, 5, 6mp2an 672 . . 3  |-  ( ( CC  \  { 0 } )  X.  ( CC  \  { 0 } ) )  C_  ( CC  X.  CC )
8 fnssres 5627 . . 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 672 . 2  |-  (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC  \  { 0 } ) ) )  Fn  (
( CC  \  {
0 } )  X.  ( CC  \  {
0 } ) )
10 ovres 6335 . . . 4  |-  ( ( x  e.  ( CC 
\  { 0 } )  /\  y  e.  ( CC  \  {
0 } ) )  ->  ( x (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC 
\  { 0 } ) ) ) y )  =  ( x  x.  y ) )
11 eldifsn 4103 . . . . . 6  |-  ( x  e.  ( CC  \  { 0 } )  <-> 
( x  e.  CC  /\  x  =/=  0 ) )
12 eldifsn 4103 . . . . . 6  |-  ( y  e.  ( CC  \  { 0 } )  <-> 
( y  e.  CC  /\  y  =/=  0 ) )
13 mulcl 9472 . . . . . . . 8  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  x.  y
)  e.  CC )
1413ad2ant2r 746 . . . . . . 7  |-  ( ( ( x  e.  CC  /\  x  =/=  0 )  /\  ( y  e.  CC  /\  y  =/=  0 ) )  -> 
( x  x.  y
)  e.  CC )
15 mulne0 10084 . . . . . . 7  |-  ( ( ( x  e.  CC  /\  x  =/=  0 )  /\  ( y  e.  CC  /\  y  =/=  0 ) )  -> 
( x  x.  y
)  =/=  0 )
1614, 15jca 532 . . . . . 6  |-  ( ( ( x  e.  CC  /\  x  =/=  0 )  /\  ( y  e.  CC  /\  y  =/=  0 ) )  -> 
( ( x  x.  y )  e.  CC  /\  ( x  x.  y
)  =/=  0 ) )
1711, 12, 16syl2anb 479 . . . . 5  |-  ( ( x  e.  ( CC 
\  { 0 } )  /\  y  e.  ( CC  \  {
0 } ) )  ->  ( ( x  x.  y )  e.  CC  /\  ( x  x.  y )  =/=  0 ) )
18 eldifsn 4103 . . . . 5  |-  ( ( x  x.  y )  e.  ( CC  \  { 0 } )  <-> 
( ( x  x.  y )  e.  CC  /\  ( x  x.  y
)  =/=  0 ) )
1917, 18sylibr 212 . . . 4  |-  ( ( x  e.  ( CC 
\  { 0 } )  /\  y  e.  ( CC  \  {
0 } ) )  ->  ( x  x.  y )  e.  ( CC  \  { 0 } ) )
2010, 19eqeltrd 2540 . . 3  |-  ( ( x  e.  ( CC 
\  { 0 } )  /\  y  e.  ( CC  \  {
0 } ) )  ->  ( x (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC 
\  { 0 } ) ) ) y )  e.  ( CC 
\  { 0 } ) )
2120rgen2a 2894 . 2  |-  A. x  e.  ( CC  \  {
0 } ) A. y  e.  ( CC  \  { 0 } ) ( x (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC  \  { 0 } ) ) ) y )  e.  ( CC  \  { 0 } )
22 ffnov 6299 . 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 911 1  |-  (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC  \  { 0 } ) ) ) : ( ( CC  \  {
0 } )  X.  ( CC  \  {
0 } ) ) --> ( CC  \  {
0 } )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    e. wcel 1758    =/= wne 2645   A.wral 2796    \ cdif 3428    C_ wss 3431   {csn 3980    X. cxp 4941    |` cres 4945    Fn wfn 5516   -->wf 5517  (class class class)co 6195   CCcc 9386   0cc0 9388    x. cmul 9393
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-resscn 9445  ax-1cn 9446  ax-icn 9447  ax-addcl 9448  ax-addrcl 9449  ax-mulcl 9450  ax-mulrcl 9451  ax-mulcom 9452  ax-addass 9453  ax-mulass 9454  ax-distr 9455  ax-i2m1 9456  ax-1ne0 9457  ax-1rid 9458  ax-rnegex 9459  ax-rrecex 9460  ax-cnre 9461  ax-pre-lttri 9462  ax-pre-lttrn 9463  ax-pre-ltadd 9464  ax-pre-mulgt0 9465  ax-mulf 9468
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-reu 2803  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-po 4744  df-so 4745  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-riota 6156  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-er 7206  df-en 7416  df-dom 7417  df-sdom 7418  df-pnf 9526  df-mnf 9527  df-xr 9528  df-ltxr 9529  df-le 9530  df-sub 9703  df-neg 9704
This theorem is referenced by:  ablomul  23989  mulid  23990
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