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Theorem mulnzcnopr 10286
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 9645 . . . . 5  |-  x.  :
( CC  X.  CC )
--> CC
2 ffnov 6427 . . . . 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 213 . . . 4  |-  (  x.  Fn  ( CC  X.  CC )  /\  A. x  e.  CC  A. y  e.  CC  ( x  x.  y )  e.  CC )
43simpli 464 . . 3  |-  x.  Fn  ( CC  X.  CC )
5 difss 3572 . . . 4  |-  ( CC 
\  { 0 } )  C_  CC
6 xpss12 4959 . . . 4  |-  ( ( ( CC  \  {
0 } )  C_  CC  /\  ( CC  \  { 0 } ) 
C_  CC )  -> 
( ( CC  \  { 0 } )  X.  ( CC  \  { 0 } ) )  C_  ( CC  X.  CC ) )
75, 5, 6mp2an 683 . . 3  |-  ( ( CC  \  { 0 } )  X.  ( CC  \  { 0 } ) )  C_  ( CC  X.  CC )
8 fnssres 5711 . . 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 683 . 2  |-  (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC  \  { 0 } ) ) )  Fn  (
( CC  \  {
0 } )  X.  ( CC  \  {
0 } ) )
10 ovres 6463 . . . 4  |-  ( ( x  e.  ( CC 
\  { 0 } )  /\  y  e.  ( CC  \  {
0 } ) )  ->  ( x (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC 
\  { 0 } ) ) ) y )  =  ( x  x.  y ) )
11 eldifsn 4110 . . . . . 6  |-  ( x  e.  ( CC  \  { 0 } )  <-> 
( x  e.  CC  /\  x  =/=  0 ) )
12 eldifsn 4110 . . . . . 6  |-  ( y  e.  ( CC  \  { 0 } )  <-> 
( y  e.  CC  /\  y  =/=  0 ) )
13 mulcl 9649 . . . . . . . 8  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  x.  y
)  e.  CC )
1413ad2ant2r 758 . . . . . . 7  |-  ( ( ( x  e.  CC  /\  x  =/=  0 )  /\  ( y  e.  CC  /\  y  =/=  0 ) )  -> 
( x  x.  y
)  e.  CC )
15 mulne0 10282 . . . . . . 7  |-  ( ( ( x  e.  CC  /\  x  =/=  0 )  /\  ( y  e.  CC  /\  y  =/=  0 ) )  -> 
( x  x.  y
)  =/=  0 )
1614, 15jca 539 . . . . . 6  |-  ( ( ( x  e.  CC  /\  x  =/=  0 )  /\  ( y  e.  CC  /\  y  =/=  0 ) )  -> 
( ( x  x.  y )  e.  CC  /\  ( x  x.  y
)  =/=  0 ) )
1711, 12, 16syl2anb 486 . . . . 5  |-  ( ( x  e.  ( CC 
\  { 0 } )  /\  y  e.  ( CC  \  {
0 } ) )  ->  ( ( x  x.  y )  e.  CC  /\  ( x  x.  y )  =/=  0 ) )
18 eldifsn 4110 . . . . 5  |-  ( ( x  x.  y )  e.  ( CC  \  { 0 } )  <-> 
( ( x  x.  y )  e.  CC  /\  ( x  x.  y
)  =/=  0 ) )
1917, 18sylibr 217 . . . 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 2827 . 2  |-  A. x  e.  ( CC  \  {
0 } ) A. y  e.  ( CC  \  { 0 } ) ( x (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC  \  { 0 } ) ) ) y )  e.  ( CC  \  { 0 } )
22 ffnov 6427 . 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 936 1  |-  (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC  \  { 0 } ) ) ) : ( ( CC  \  {
0 } )  X.  ( CC  \  {
0 } ) ) --> ( CC  \  {
0 } )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 375    e. wcel 1898    =/= wne 2633   A.wral 2749    \ cdif 3413    C_ wss 3416   {csn 3980    X. cxp 4851    |` cres 4855    Fn wfn 5596   -->wf 5597  (class class class)co 6315   CCcc 9563   0cc0 9565    x. cmul 9570
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6610  ax-resscn 9622  ax-1cn 9623  ax-icn 9624  ax-addcl 9625  ax-addrcl 9626  ax-mulcl 9627  ax-mulrcl 9628  ax-mulcom 9629  ax-addass 9630  ax-mulass 9631  ax-distr 9632  ax-i2m1 9633  ax-1ne0 9634  ax-1rid 9635  ax-rnegex 9636  ax-rrecex 9637  ax-cnre 9638  ax-pre-lttri 9639  ax-pre-lttrn 9640  ax-pre-ltadd 9641  ax-pre-mulgt0 9642  ax-mulf 9645
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-nel 2636  df-ral 2754  df-rex 2755  df-reu 2756  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-iun 4294  df-br 4417  df-opab 4476  df-mpt 4477  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 6277  df-ov 6318  df-oprab 6319  df-mpt2 6320  df-er 7389  df-en 7596  df-dom 7597  df-sdom 7598  df-pnf 9703  df-mnf 9704  df-xr 9705  df-ltxr 9706  df-le 9707  df-sub 9888  df-neg 9889
This theorem is referenced by:  ablomul  26132  mulid  26133
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