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Theorem axmulf 7670
Description: Multiplication is an operation on the complex numbers. This theorem can be used as an alternate axiom for complex numbers in place of the less specific axmulcl 7667. This construction-dependent theorem should not be referenced directly; instead, use ax-mulf 7736. (Contributed by NM, 8-Feb-2005.) (New usage is discouraged.)
Assertion
Ref Expression
axmulf  |-  x.  :
( CC  X.  CC )
--> CC

Proof of Theorem axmulf
Dummy variables  a  b  x  y  z  w  v  u  f are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 moeq 2854 . . . . . . . . 9  |-  E* z 
z  =  <. (
( w  .R  u
)  +R  ( -1R 
.R  ( v  .R  f ) ) ) ,  ( ( v  .R  u )  +R  ( w  .R  f
) ) >.
21mosubop 4600 . . . . . . . 8  |-  E* z E. u E. f ( y  =  <. u ,  f >.  /\  z  =  <. ( ( w  .R  u )  +R  ( -1R  .R  (
v  .R  f )
) ) ,  ( ( v  .R  u
)  +R  ( w  .R  f ) )
>. )
32mosubop 4600 . . . . . . 7  |-  E* z E. w E. v ( x  =  <. w ,  v >.  /\  E. u E. f ( y  =  <. u ,  f
>.  /\  z  =  <. ( ( w  .R  u
)  +R  ( -1R 
.R  ( v  .R  f ) ) ) ,  ( ( v  .R  u )  +R  ( w  .R  f
) ) >. )
)
4 anass 398 . . . . . . . . . . 11  |-  ( ( ( x  =  <. w ,  v >.  /\  y  =  <. u ,  f
>. )  /\  z  =  <. ( ( w  .R  u )  +R  ( -1R  .R  (
v  .R  f )
) ) ,  ( ( v  .R  u
)  +R  ( w  .R  f ) )
>. )  <->  ( x  = 
<. w ,  v >.  /\  ( y  =  <. u ,  f >.  /\  z  =  <. ( ( w  .R  u )  +R  ( -1R  .R  (
v  .R  f )
) ) ,  ( ( v  .R  u
)  +R  ( w  .R  f ) )
>. ) ) )
542exbii 1585 . . . . . . . . . 10  |-  ( E. u E. f ( ( x  =  <. w ,  v >.  /\  y  =  <. u ,  f
>. )  /\  z  =  <. ( ( w  .R  u )  +R  ( -1R  .R  (
v  .R  f )
) ) ,  ( ( v  .R  u
)  +R  ( w  .R  f ) )
>. )  <->  E. u E. f
( x  =  <. w ,  v >.  /\  (
y  =  <. u ,  f >.  /\  z  =  <. ( ( w  .R  u )  +R  ( -1R  .R  (
v  .R  f )
) ) ,  ( ( v  .R  u
)  +R  ( w  .R  f ) )
>. ) ) )
6 19.42vv 1883 . . . . . . . . . 10  |-  ( E. u E. f ( x  =  <. w ,  v >.  /\  (
y  =  <. u ,  f >.  /\  z  =  <. ( ( w  .R  u )  +R  ( -1R  .R  (
v  .R  f )
) ) ,  ( ( v  .R  u
)  +R  ( w  .R  f ) )
>. ) )  <->  ( x  =  <. w ,  v
>.  /\  E. u E. f ( y  = 
<. u ,  f >.  /\  z  =  <. ( ( w  .R  u
)  +R  ( -1R 
.R  ( v  .R  f ) ) ) ,  ( ( v  .R  u )  +R  ( w  .R  f
) ) >. )
) )
75, 6bitri 183 . . . . . . . . 9  |-  ( E. u E. f ( ( x  =  <. w ,  v >.  /\  y  =  <. u ,  f
>. )  /\  z  =  <. ( ( w  .R  u )  +R  ( -1R  .R  (
v  .R  f )
) ) ,  ( ( v  .R  u
)  +R  ( w  .R  f ) )
>. )  <->  ( x  = 
<. w ,  v >.  /\  E. u E. f
( y  =  <. u ,  f >.  /\  z  =  <. ( ( w  .R  u )  +R  ( -1R  .R  (
v  .R  f )
) ) ,  ( ( v  .R  u
)  +R  ( w  .R  f ) )
>. ) ) )
872exbii 1585 . . . . . . . 8  |-  ( E. w E. v E. u E. f ( ( x  =  <. w ,  v >.  /\  y  =  <. u ,  f
>. )  /\  z  =  <. ( ( w  .R  u )  +R  ( -1R  .R  (
v  .R  f )
) ) ,  ( ( v  .R  u
)  +R  ( w  .R  f ) )
>. )  <->  E. w E. v
( x  =  <. w ,  v >.  /\  E. u E. f ( y  =  <. u ,  f
>.  /\  z  =  <. ( ( w  .R  u
)  +R  ( -1R 
.R  ( v  .R  f ) ) ) ,  ( ( v  .R  u )  +R  ( w  .R  f
) ) >. )
) )
98mobii 2034 . . . . . . 7  |-  ( E* z E. w E. v E. u E. f
( ( x  = 
<. w ,  v >.  /\  y  =  <. u ,  f >. )  /\  z  =  <. ( ( w  .R  u
)  +R  ( -1R 
.R  ( v  .R  f ) ) ) ,  ( ( v  .R  u )  +R  ( w  .R  f
) ) >. )  <->  E* z E. w E. v ( x  = 
<. w ,  v >.  /\  E. u E. f
( y  =  <. u ,  f >.  /\  z  =  <. ( ( w  .R  u )  +R  ( -1R  .R  (
v  .R  f )
) ) ,  ( ( v  .R  u
)  +R  ( w  .R  f ) )
>. ) ) )
103, 9mpbir 145 . . . . . 6  |-  E* z E. w E. v E. u E. f ( ( x  =  <. w ,  v >.  /\  y  =  <. u ,  f
>. )  /\  z  =  <. ( ( w  .R  u )  +R  ( -1R  .R  (
v  .R  f )
) ) ,  ( ( v  .R  u
)  +R  ( w  .R  f ) )
>. )
1110moani 2067 . . . . 5  |-  E* z
( ( x  e.  CC  /\  y  e.  CC )  /\  E. w E. v E. u E. f ( ( x  =  <. w ,  v
>.  /\  y  =  <. u ,  f >. )  /\  z  =  <. ( ( w  .R  u
)  +R  ( -1R 
.R  ( v  .R  f ) ) ) ,  ( ( v  .R  u )  +R  ( w  .R  f
) ) >. )
)
1211funoprab 5864 . . . 4  |-  Fun  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  CC  /\  y  e.  CC )  /\  E. w E. v E. u E. f ( ( x  =  <. w ,  v
>.  /\  y  =  <. u ,  f >. )  /\  z  =  <. ( ( w  .R  u
)  +R  ( -1R 
.R  ( v  .R  f ) ) ) ,  ( ( v  .R  u )  +R  ( w  .R  f
) ) >. )
) }
13 df-mul 7625 . . . . 5  |-  x.  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  CC  /\  y  e.  CC )  /\  E. w E. v E. u E. f ( ( x  =  <. w ,  v
>.  /\  y  =  <. u ,  f >. )  /\  z  =  <. ( ( w  .R  u
)  +R  ( -1R 
.R  ( v  .R  f ) ) ) ,  ( ( v  .R  u )  +R  ( w  .R  f
) ) >. )
) }
1413funeqi 5139 . . . 4  |-  ( Fun 
x. 
<->  Fun  { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  CC  /\  y  e.  CC )  /\  E. w E. v E. u E. f ( ( x  =  <. w ,  v >.  /\  y  =  <. u ,  f
>. )  /\  z  =  <. ( ( w  .R  u )  +R  ( -1R  .R  (
v  .R  f )
) ) ,  ( ( v  .R  u
)  +R  ( w  .R  f ) )
>. ) ) } )
1512, 14mpbir 145 . . 3  |-  Fun  x.
1613dmeqi 4735 . . . . 5  |-  dom  x.  =  dom  { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  CC  /\  y  e.  CC )  /\  E. w E. v E. u E. f ( ( x  =  <. w ,  v >.  /\  y  =  <. u ,  f
>. )  /\  z  =  <. ( ( w  .R  u )  +R  ( -1R  .R  (
v  .R  f )
) ) ,  ( ( v  .R  u
)  +R  ( w  .R  f ) )
>. ) ) }
17 dmoprabss 5846 . . . . 5  |-  dom  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  CC  /\  y  e.  CC )  /\  E. w E. v E. u E. f ( ( x  =  <. w ,  v
>.  /\  y  =  <. u ,  f >. )  /\  z  =  <. ( ( w  .R  u
)  +R  ( -1R 
.R  ( v  .R  f ) ) ) ,  ( ( v  .R  u )  +R  ( w  .R  f
) ) >. )
) }  C_  ( CC  X.  CC )
1816, 17eqsstri 3124 . . . 4  |-  dom  x.  C_  ( CC  X.  CC )
19 cnm 7633 . . . . . . 7  |-  ( a  e.  CC  ->  E. b 
b  e.  a )
2019adantl 275 . . . . . 6  |-  ( ( T.  /\  a  e.  CC )  ->  E. b 
b  e.  a )
21 axmulcl 7667 . . . . . . 7  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  x.  y
)  e.  CC )
2221adantl 275 . . . . . 6  |-  ( ( T.  /\  ( x  e.  CC  /\  y  e.  CC ) )  -> 
( x  x.  y
)  e.  CC )
23 funrel 5135 . . . . . . 7  |-  ( Fun 
x.  ->  Rel  x.  )
2415, 23mp1i 10 . . . . . 6  |-  ( T. 
->  Rel  x.  )
2520, 22, 24oprssdmm 6062 . . . . 5  |-  ( T. 
->  ( CC  X.  CC )  C_  dom  x.  )
2625mptru 1340 . . . 4  |-  ( CC 
X.  CC )  C_  dom  x.
2718, 26eqssi 3108 . . 3  |-  dom  x.  =  ( CC  X.  CC )
28 df-fn 5121 . . 3  |-  (  x.  Fn  ( CC  X.  CC )  <->  ( Fun  x.  /\  dom  x.  =  ( CC  X.  CC ) ) )
2915, 27, 28mpbir2an 926 . 2  |-  x.  Fn  ( CC  X.  CC )
3021rgen2a 2484 . 2  |-  A. x  e.  CC  A. y  e.  CC  ( x  x.  y )  e.  CC
31 ffnov 5868 . 2  |-  (  x.  : ( CC  X.  CC ) --> CC  <->  (  x.  Fn  ( CC  X.  CC )  /\  A. x  e.  CC  A. y  e.  CC  ( x  x.  y )  e.  CC ) )
3229, 30, 31mpbir2an 926 1  |-  x.  :
( CC  X.  CC )
--> CC
Colors of variables: wff set class
Syntax hints:    /\ wa 103    = wceq 1331   T. wtru 1332   E.wex 1468    e. wcel 1480   E*wmo 1998   A.wral 2414    C_ wss 3066   <.cop 3525    X. cxp 4532   dom cdm 4534   Rel wrel 4539   Fun wfun 5112    Fn wfn 5113   -->wf 5114  (class class class)co 5767   {coprab 5768   -1Rcm1r 7101    +R cplr 7102    .R cmr 7103   CCcc 7611    x. cmul 7618
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-eprel 4206  df-id 4210  df-po 4213  df-iso 4214  df-iord 4283  df-on 4285  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-recs 6195  df-irdg 6260  df-1o 6306  df-2o 6307  df-oadd 6310  df-omul 6311  df-er 6422  df-ec 6424  df-qs 6428  df-ni 7105  df-pli 7106  df-mi 7107  df-lti 7108  df-plpq 7145  df-mpq 7146  df-enq 7148  df-nqqs 7149  df-plqqs 7150  df-mqqs 7151  df-1nqqs 7152  df-rq 7153  df-ltnqqs 7154  df-enq0 7225  df-nq0 7226  df-0nq0 7227  df-plq0 7228  df-mq0 7229  df-inp 7267  df-i1p 7268  df-iplp 7269  df-imp 7270  df-enr 7527  df-nr 7528  df-plr 7529  df-mr 7530  df-m1r 7534  df-c 7619  df-mul 7625
This theorem is referenced by: (None)
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