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Theorem mulnzcnopr 10112
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 9483 . . . . 5  |-  x.  :
( CC  X.  CC )
--> CC
2 ffnov 6305 . . . . 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 456 . . 3  |-  x.  Fn  ( CC  X.  CC )
5 difss 3545 . . . 4  |-  ( CC 
\  { 0 } )  C_  CC
6 xpss12 5021 . . . 4  |-  ( ( ( CC  \  {
0 } )  C_  CC  /\  ( CC  \  { 0 } ) 
C_  CC )  -> 
( ( CC  \  { 0 } )  X.  ( CC  \  { 0 } ) )  C_  ( CC  X.  CC ) )
75, 5, 6mp2an 670 . . 3  |-  ( ( CC  \  { 0 } )  X.  ( CC  \  { 0 } ) )  C_  ( CC  X.  CC )
8 fnssres 5602 . . 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 670 . 2  |-  (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC  \  { 0 } ) ) )  Fn  (
( CC  \  {
0 } )  X.  ( CC  \  {
0 } ) )
10 ovres 6341 . . . 4  |-  ( ( x  e.  ( CC 
\  { 0 } )  /\  y  e.  ( CC  \  {
0 } ) )  ->  ( x (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC 
\  { 0 } ) ) ) y )  =  ( x  x.  y ) )
11 eldifsn 4069 . . . . . 6  |-  ( x  e.  ( CC  \  { 0 } )  <-> 
( x  e.  CC  /\  x  =/=  0 ) )
12 eldifsn 4069 . . . . . 6  |-  ( y  e.  ( CC  \  { 0 } )  <-> 
( y  e.  CC  /\  y  =/=  0 ) )
13 mulcl 9487 . . . . . . . 8  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  x.  y
)  e.  CC )
1413ad2ant2r 744 . . . . . . 7  |-  ( ( ( x  e.  CC  /\  x  =/=  0 )  /\  ( y  e.  CC  /\  y  =/=  0 ) )  -> 
( x  x.  y
)  e.  CC )
15 mulne0 10108 . . . . . . 7  |-  ( ( ( x  e.  CC  /\  x  =/=  0 )  /\  ( y  e.  CC  /\  y  =/=  0 ) )  -> 
( x  x.  y
)  =/=  0 )
1614, 15jca 530 . . . . . 6  |-  ( ( ( x  e.  CC  /\  x  =/=  0 )  /\  ( y  e.  CC  /\  y  =/=  0 ) )  -> 
( ( x  x.  y )  e.  CC  /\  ( x  x.  y
)  =/=  0 ) )
1711, 12, 16syl2anb 477 . . . . 5  |-  ( ( x  e.  ( CC 
\  { 0 } )  /\  y  e.  ( CC  \  {
0 } ) )  ->  ( ( x  x.  y )  e.  CC  /\  ( x  x.  y )  =/=  0 ) )
18 eldifsn 4069 . . . . 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 2470 . . 3  |-  ( ( x  e.  ( CC 
\  { 0 } )  /\  y  e.  ( CC  \  {
0 } ) )  ->  ( x (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC 
\  { 0 } ) ) ) y )  e.  ( CC 
\  { 0 } ) )
2120rgen2a 2809 . 2  |-  A. x  e.  ( CC  \  {
0 } ) A. y  e.  ( CC  \  { 0 } ) ( x (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC  \  { 0 } ) ) ) y )  e.  ( CC  \  { 0 } )
22 ffnov 6305 . 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 918 1  |-  (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC  \  { 0 } ) ) ) : ( ( CC  \  {
0 } )  X.  ( CC  \  {
0 } ) ) --> ( CC  \  {
0 } )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 367    e. wcel 1826    =/= wne 2577   A.wral 2732    \ cdif 3386    C_ wss 3389   {csn 3944    X. cxp 4911    |` cres 4915    Fn wfn 5491   -->wf 5492  (class class class)co 6196   CCcc 9401   0cc0 9403    x. cmul 9408
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480  ax-mulf 9483
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-po 4714  df-so 4715  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-er 7229  df-en 7436  df-dom 7437  df-sdom 7438  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721
This theorem is referenced by:  ablomul  25474  mulid  25475
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