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Theorem divclrvd 25798
Description: Closure law for vector division by a scalar. (Contributed by FL, 29-May-2014.)
Hypothesis
Ref Expression
divclcvd.1  |-  / t  =  ( / cv `  N )
Assertion
Ref Expression
divclrvd  |-  ( ( N  e.  NN  /\  U  e.  ( RR  ^m  ( 1 ... N
) )  /\  S  e.  ( RR  \  {
0 } ) )  ->  ( U / t S )  e.  ( RR  ^m  ( 1 ... N ) ) )

Proof of Theorem divclrvd
StepHypRef Expression
1 id 19 . . 3  |-  ( N  e.  NN  ->  N  e.  NN )
2 ax-resscn 8810 . . . . 5  |-  RR  C_  CC
3 fss 5413 . . . . 5  |-  ( ( U : ( 1 ... N ) --> RR 
/\  RR  C_  CC )  ->  U : ( 1 ... N ) --> CC )
42, 3mpan2 652 . . . 4  |-  ( U : ( 1 ... N ) --> RR  ->  U : ( 1 ... N ) --> CC )
5 reex 8844 . . . . 5  |-  RR  e.  _V
6 ovex 5899 . . . . 5  |-  ( 1 ... N )  e. 
_V
75, 6elmap 6812 . . . 4  |-  ( U  e.  ( RR  ^m  ( 1 ... N
) )  <->  U :
( 1 ... N
) --> RR )
8 cnex 8834 . . . . 5  |-  CC  e.  _V
98, 6elmap 6812 . . . 4  |-  ( U  e.  ( CC  ^m  ( 1 ... N
) )  <->  U :
( 1 ... N
) --> CC )
104, 7, 93imtr4i 257 . . 3  |-  ( U  e.  ( RR  ^m  ( 1 ... N
) )  ->  U  e.  ( CC  ^m  (
1 ... N ) ) )
11 recn 8843 . . . . 5  |-  ( S  e.  RR  ->  S  e.  CC )
1211anim1i 551 . . . 4  |-  ( ( S  e.  RR  /\  S  =/=  0 )  -> 
( S  e.  CC  /\  S  =/=  0 ) )
13 eldifsn 3762 . . . 4  |-  ( S  e.  ( RR  \  { 0 } )  <-> 
( S  e.  RR  /\  S  =/=  0 ) )
14 eldifsn 3762 . . . 4  |-  ( S  e.  ( CC  \  { 0 } )  <-> 
( S  e.  CC  /\  S  =/=  0 ) )
1512, 13, 143imtr4i 257 . . 3  |-  ( S  e.  ( RR  \  { 0 } )  ->  S  e.  ( CC  \  { 0 } ) )
16 divclcvd.1 . . . 4  |-  / t  =  ( / cv `  N )
17 eqid 2296 . . . 4  |-  ( . cv `  N )  =  ( . cv `  N )
1816, 17isdivcv2 25796 . . 3  |-  ( ( N  e.  NN  /\  U  e.  ( CC  ^m  ( 1 ... N
) )  /\  S  e.  ( CC  \  {
0 } ) )  ->  ( U / t S )  =  ( ( 1  /  S
) ( . cv `  N ) U ) )
191, 10, 15, 18syl3an 1224 . 2  |-  ( ( N  e.  NN  /\  U  e.  ( RR  ^m  ( 1 ... N
) )  /\  S  e.  ( RR  \  {
0 } ) )  ->  ( U / t S )  =  ( ( 1  /  S
) ( . cv `  N ) U ) )
20 simp1 955 . . 3  |-  ( ( N  e.  NN  /\  U  e.  ( RR  ^m  ( 1 ... N
) )  /\  S  e.  ( RR  \  {
0 } ) )  ->  N  e.  NN )
21 rereccl 9494 . . . . 5  |-  ( ( S  e.  RR  /\  S  =/=  0 )  -> 
( 1  /  S
)  e.  RR )
2213, 21sylbi 187 . . . 4  |-  ( S  e.  ( RR  \  { 0 } )  ->  ( 1  /  S )  e.  RR )
23223ad2ant3 978 . . 3  |-  ( ( N  e.  NN  /\  U  e.  ( RR  ^m  ( 1 ... N
) )  /\  S  e.  ( RR  \  {
0 } ) )  ->  ( 1  /  S )  e.  RR )
24 simp2 956 . . 3  |-  ( ( N  e.  NN  /\  U  e.  ( RR  ^m  ( 1 ... N
) )  /\  S  e.  ( RR  \  {
0 } ) )  ->  U  e.  ( RR  ^m  ( 1 ... N ) ) )
2517clsmulrv 25786 . . 3  |-  ( ( N  e.  NN  /\  ( 1  /  S
)  e.  RR  /\  U  e.  ( RR  ^m  ( 1 ... N
) ) )  -> 
( ( 1  /  S ) ( . cv `  N ) U )  e.  ( RR  ^m  ( 1 ... N ) ) )
2620, 23, 24, 25syl3anc 1182 . 2  |-  ( ( N  e.  NN  /\  U  e.  ( RR  ^m  ( 1 ... N
) )  /\  S  e.  ( RR  \  {
0 } ) )  ->  ( ( 1  /  S ) ( . cv `  N
) U )  e.  ( RR  ^m  (
1 ... N ) ) )
2719, 26eqeltrd 2370 1  |-  ( ( N  e.  NN  /\  U  e.  ( RR  ^m  ( 1 ... N
) )  /\  S  e.  ( RR  \  {
0 } ) )  ->  ( U / t S )  e.  ( RR  ^m  ( 1 ... N ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459    \ cdif 3162    C_ wss 3165   {csn 3653   -->wf 5267   ` cfv 5271  (class class class)co 5874    ^m cmap 6788   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754    / cdiv 9439   NNcn 9762   ...cfz 10798   . cvcsmcv 25782   / cvcdivcv 25794
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-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  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
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-rmo 2564  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-1st 6138  df-2nd 6139  df-riota 6320  df-er 6676  df-map 6790  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  df-div 9440  df-mulcv 25783  df-divcv 25795
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