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Theorem imasmulr 13749
Description: The ring multiplication in an image structure. (Contributed by Mario Carneiro, 23-Feb-2015.) (Revised by Mario Carneiro, 11-Jul-2015.)
Hypotheses
Ref Expression
imasbas.u  |-  ( ph  ->  U  =  ( F 
"s  R ) )
imasbas.v  |-  ( ph  ->  V  =  ( Base `  R ) )
imasbas.f  |-  ( ph  ->  F : V -onto-> B
)
imasbas.r  |-  ( ph  ->  R  e.  Z )
imasmulr.p  |-  .x.  =  ( .r `  R )
imasmulr.t  |-  .xb  =  ( .r `  U )
Assertion
Ref Expression
imasmulr  |-  ( ph  -> 
.xb  =  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( F `  ( p 
.x.  q ) )
>. } )
Distinct variable groups:    q, p, F    R, p, q    ph, p, q    V, p, q
Allowed substitution hints:    B( q, p)    .xb ( q, p)    .x. ( q, p)    U( q, p)    Z( q, p)

Proof of Theorem imasmulr
Dummy variables  g  h  i  n  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 imasbas.u . . 3  |-  ( ph  ->  U  =  ( F 
"s  R ) )
2 imasbas.v . . 3  |-  ( ph  ->  V  =  ( Base `  R ) )
3 eqid 2438 . . 3  |-  ( +g  `  R )  =  ( +g  `  R )
4 imasmulr.p . . 3  |-  .x.  =  ( .r `  R )
5 eqid 2438 . . 3  |-  (Scalar `  R )  =  (Scalar `  R )
6 eqid 2438 . . 3  |-  ( Base `  (Scalar `  R )
)  =  ( Base `  (Scalar `  R )
)
7 eqid 2438 . . 3  |-  ( .s
`  R )  =  ( .s `  R
)
8 eqid 2438 . . 3  |-  ( TopOpen `  R )  =  (
TopOpen `  R )
9 eqid 2438 . . 3  |-  ( dist `  R )  =  (
dist `  R )
10 eqid 2438 . . 3  |-  ( le
`  R )  =  ( le `  R
)
11 imasbas.f . . . 4  |-  ( ph  ->  F : V -onto-> B
)
12 imasbas.r . . . 4  |-  ( ph  ->  R  e.  Z )
13 eqid 2438 . . . 4  |-  ( +g  `  U )  =  ( +g  `  U )
141, 2, 11, 12, 3, 13imasplusg 13748 . . 3  |-  ( ph  ->  ( +g  `  U
)  =  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( F `  ( p ( +g  `  R
) q ) )
>. } )
15 eqidd 2439 . . 3  |-  ( ph  ->  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q )
>. ,  ( F `  ( p  .x.  q
) ) >. }  =  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q )
>. ,  ( F `  ( p  .x.  q
) ) >. } )
16 eqidd 2439 . . 3  |-  ( ph  ->  U_ q  e.  V  ( p  e.  ( Base `  (Scalar `  R
) ) ,  x  e.  { ( F `  q ) }  |->  ( F `  ( p ( .s `  R
) q ) ) )  =  U_ q  e.  V  ( p  e.  ( Base `  (Scalar `  R ) ) ,  x  e.  { ( F `  q ) }  |->  ( F `  ( p ( .s
`  R ) q ) ) ) )
17 eqidd 2439 . . 3  |-  ( ph  ->  ( ( TopOpen `  R
) qTop  F )  =  ( ( TopOpen `  R ) qTop  F ) )
18 eqid 2438 . . . 4  |-  ( dist `  U )  =  (
dist `  U )
191, 2, 11, 12, 9, 18imasds 13744 . . 3  |-  ( ph  ->  ( dist `  U
)  =  ( x  e.  B ,  y  e.  B  |->  sup ( U_ n  e.  NN  ran  ( g  e.  {
h  e.  ( ( V  X.  V )  ^m  ( 1 ... n ) )  |  ( ( F `  ( 1st `  ( h `
 1 ) ) )  =  x  /\  ( F `  ( 2nd `  ( h `  n
) ) )  =  y  /\  A. i  e.  ( 1 ... (
n  -  1 ) ) ( F `  ( 2nd `  ( h `
 i ) ) )  =  ( F `
 ( 1st `  (
h `  ( i  +  1 ) ) ) ) ) } 
|->  ( RR* s  gsumg  ( (
dist `  R )  o.  g ) ) ) ,  RR* ,  `'  <  ) ) )
20 eqidd 2439 . . 3  |-  ( ph  ->  ( ( F  o.  ( le `  R ) )  o.  `' F
)  =  ( ( F  o.  ( le
`  R ) )  o.  `' F ) )
211, 2, 3, 4, 5, 6, 7, 8, 9, 10, 14, 15, 16, 17, 19, 20, 11, 12imasval 13742 . 2  |-  ( ph  ->  U  =  ( ( { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  ( +g  `  U ) >. ,  <. ( .r `  ndx ) ,  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q )
>. ,  ( F `  ( p  .x.  q
) ) >. } >. }  u.  { <. (Scalar ` 
ndx ) ,  (Scalar `  R ) >. ,  <. ( .s `  ndx ) ,  U_ q  e.  V  ( p  e.  ( Base `  (Scalar `  R
) ) ,  x  e.  { ( F `  q ) }  |->  ( F `  ( p ( .s `  R
) q ) ) ) >. } )  u. 
{ <. (TopSet `  ndx ) ,  ( ( TopOpen
`  R ) qTop  F
) >. ,  <. ( le `  ndx ) ,  ( ( F  o.  ( le `  R ) )  o.  `' F
) >. ,  <. ( dist `  ndx ) ,  ( dist `  U
) >. } ) )
22 eqid 2438 . . 3  |-  ( ( { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  ( +g  `  U ) >. ,  <. ( .r `  ndx ) ,  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q )
>. ,  ( F `  ( p  .x.  q
) ) >. } >. }  u.  { <. (Scalar ` 
ndx ) ,  (Scalar `  R ) >. ,  <. ( .s `  ndx ) ,  U_ q  e.  V  ( p  e.  ( Base `  (Scalar `  R
) ) ,  x  e.  { ( F `  q ) }  |->  ( F `  ( p ( .s `  R
) q ) ) ) >. } )  u. 
{ <. (TopSet `  ndx ) ,  ( ( TopOpen
`  R ) qTop  F
) >. ,  <. ( le `  ndx ) ,  ( ( F  o.  ( le `  R ) )  o.  `' F
) >. ,  <. ( dist `  ndx ) ,  ( dist `  U
) >. } )  =  ( ( { <. (
Base `  ndx ) ,  B >. ,  <. ( +g  `  ndx ) ,  ( +g  `  U
) >. ,  <. ( .r `  ndx ) , 
U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q )
>. ,  ( F `  ( p  .x.  q
) ) >. } >. }  u.  { <. (Scalar ` 
ndx ) ,  (Scalar `  R ) >. ,  <. ( .s `  ndx ) ,  U_ q  e.  V  ( p  e.  ( Base `  (Scalar `  R
) ) ,  x  e.  { ( F `  q ) }  |->  ( F `  ( p ( .s `  R
) q ) ) ) >. } )  u. 
{ <. (TopSet `  ndx ) ,  ( ( TopOpen
`  R ) qTop  F
) >. ,  <. ( le `  ndx ) ,  ( ( F  o.  ( le `  R ) )  o.  `' F
) >. ,  <. ( dist `  ndx ) ,  ( dist `  U
) >. } )
2322imasvalstr 13680 . 2  |-  ( ( { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  ( +g  `  U ) >. ,  <. ( .r `  ndx ) ,  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q )
>. ,  ( F `  ( p  .x.  q
) ) >. } >. }  u.  { <. (Scalar ` 
ndx ) ,  (Scalar `  R ) >. ,  <. ( .s `  ndx ) ,  U_ q  e.  V  ( p  e.  ( Base `  (Scalar `  R
) ) ,  x  e.  { ( F `  q ) }  |->  ( F `  ( p ( .s `  R
) q ) ) ) >. } )  u. 
{ <. (TopSet `  ndx ) ,  ( ( TopOpen
`  R ) qTop  F
) >. ,  <. ( le `  ndx ) ,  ( ( F  o.  ( le `  R ) )  o.  `' F
) >. ,  <. ( dist `  ndx ) ,  ( dist `  U
) >. } ) Struct  <. 1 , ; 1 2 >.
24 mulrid 13580 . 2  |-  .r  = Slot  ( .r `  ndx )
25 snsstp3 3953 . . . 4  |-  { <. ( .r `  ndx ) ,  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q )
>. ,  ( F `  ( p  .x.  q
) ) >. } >. } 
C_  { <. ( Base `  ndx ) ,  B >. ,  <. ( +g  `  ndx ) ,  ( +g  `  U
) >. ,  <. ( .r `  ndx ) , 
U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q )
>. ,  ( F `  ( p  .x.  q
) ) >. } >. }
26 ssun1 3512 . . . 4  |-  { <. (
Base `  ndx ) ,  B >. ,  <. ( +g  `  ndx ) ,  ( +g  `  U
) >. ,  <. ( .r `  ndx ) , 
U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q )
>. ,  ( F `  ( p  .x.  q
) ) >. } >. } 
C_  ( { <. (
Base `  ndx ) ,  B >. ,  <. ( +g  `  ndx ) ,  ( +g  `  U
) >. ,  <. ( .r `  ndx ) , 
U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q )
>. ,  ( F `  ( p  .x.  q
) ) >. } >. }  u.  { <. (Scalar ` 
ndx ) ,  (Scalar `  R ) >. ,  <. ( .s `  ndx ) ,  U_ q  e.  V  ( p  e.  ( Base `  (Scalar `  R
) ) ,  x  e.  { ( F `  q ) }  |->  ( F `  ( p ( .s `  R
) q ) ) ) >. } )
2725, 26sstri 3359 . . 3  |-  { <. ( .r `  ndx ) ,  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q )
>. ,  ( F `  ( p  .x.  q
) ) >. } >. } 
C_  ( { <. (
Base `  ndx ) ,  B >. ,  <. ( +g  `  ndx ) ,  ( +g  `  U
) >. ,  <. ( .r `  ndx ) , 
U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q )
>. ,  ( F `  ( p  .x.  q
) ) >. } >. }  u.  { <. (Scalar ` 
ndx ) ,  (Scalar `  R ) >. ,  <. ( .s `  ndx ) ,  U_ q  e.  V  ( p  e.  ( Base `  (Scalar `  R
) ) ,  x  e.  { ( F `  q ) }  |->  ( F `  ( p ( .s `  R
) q ) ) ) >. } )
28 ssun1 3512 . . 3  |-  ( {
<. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  ( +g  `  U ) >. ,  <. ( .r `  ndx ) ,  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q )
>. ,  ( F `  ( p  .x.  q
) ) >. } >. }  u.  { <. (Scalar ` 
ndx ) ,  (Scalar `  R ) >. ,  <. ( .s `  ndx ) ,  U_ q  e.  V  ( p  e.  ( Base `  (Scalar `  R
) ) ,  x  e.  { ( F `  q ) }  |->  ( F `  ( p ( .s `  R
) q ) ) ) >. } )  C_  ( ( { <. (
Base `  ndx ) ,  B >. ,  <. ( +g  `  ndx ) ,  ( +g  `  U
) >. ,  <. ( .r `  ndx ) , 
U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q )
>. ,  ( F `  ( p  .x.  q
) ) >. } >. }  u.  { <. (Scalar ` 
ndx ) ,  (Scalar `  R ) >. ,  <. ( .s `  ndx ) ,  U_ q  e.  V  ( p  e.  ( Base `  (Scalar `  R
) ) ,  x  e.  { ( F `  q ) }  |->  ( F `  ( p ( .s `  R
) q ) ) ) >. } )  u. 
{ <. (TopSet `  ndx ) ,  ( ( TopOpen
`  R ) qTop  F
) >. ,  <. ( le `  ndx ) ,  ( ( F  o.  ( le `  R ) )  o.  `' F
) >. ,  <. ( dist `  ndx ) ,  ( dist `  U
) >. } )
2927, 28sstri 3359 . 2  |-  { <. ( .r `  ndx ) ,  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q )
>. ,  ( F `  ( p  .x.  q
) ) >. } >. } 
C_  ( ( {
<. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  ( +g  `  U ) >. ,  <. ( .r `  ndx ) ,  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q )
>. ,  ( F `  ( p  .x.  q
) ) >. } >. }  u.  { <. (Scalar ` 
ndx ) ,  (Scalar `  R ) >. ,  <. ( .s `  ndx ) ,  U_ q  e.  V  ( p  e.  ( Base `  (Scalar `  R
) ) ,  x  e.  { ( F `  q ) }  |->  ( F `  ( p ( .s `  R
) q ) ) ) >. } )  u. 
{ <. (TopSet `  ndx ) ,  ( ( TopOpen
`  R ) qTop  F
) >. ,  <. ( le `  ndx ) ,  ( ( F  o.  ( le `  R ) )  o.  `' F
) >. ,  <. ( dist `  ndx ) ,  ( dist `  U
) >. } )
30 fvex 5745 . . . 4  |-  ( Base `  R )  e.  _V
312, 30syl6eqel 2526 . . 3  |-  ( ph  ->  V  e.  _V )
32 snex 4408 . . . . . 6  |-  { <. <.
( F `  p
) ,  ( F `
 q ) >. ,  ( F `  ( p  .x.  q ) ) >. }  e.  _V
3332rgenw 2775 . . . . 5  |-  A. q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( F `  ( p 
.x.  q ) )
>. }  e.  _V
34 iunexg 5990 . . . . 5  |-  ( ( V  e.  _V  /\  A. q  e.  V  { <. <. ( F `  p ) ,  ( F `  q )
>. ,  ( F `  ( p  .x.  q
) ) >. }  e.  _V )  ->  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( F `  ( p 
.x.  q ) )
>. }  e.  _V )
3531, 33, 34sylancl 645 . . . 4  |-  ( ph  ->  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q )
>. ,  ( F `  ( p  .x.  q
) ) >. }  e.  _V )
3635ralrimivw 2792 . . 3  |-  ( ph  ->  A. p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q )
>. ,  ( F `  ( p  .x.  q
) ) >. }  e.  _V )
37 iunexg 5990 . . 3  |-  ( ( V  e.  _V  /\  A. p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q )
>. ,  ( F `  ( p  .x.  q
) ) >. }  e.  _V )  ->  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( F `  ( p 
.x.  q ) )
>. }  e.  _V )
3831, 36, 37syl2anc 644 . 2  |-  ( ph  ->  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q )
>. ,  ( F `  ( p  .x.  q
) ) >. }  e.  _V )
39 imasmulr.t . 2  |-  .xb  =  ( .r `  U )
4021, 23, 24, 29, 38, 39strfv3 13507 1  |-  ( ph  -> 
.xb  =  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( F `  ( p 
.x.  q ) )
>. } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726   A.wral 2707   _Vcvv 2958    u. cun 3320   {csn 3816   {cpr 3817   {ctp 3818   <.cop 3819   U_ciun 4095   `'ccnv 4880    o. ccom 4885   -onto->wfo 5455   ` cfv 5457  (class class class)co 6084    e. cmpt2 6086   1c1 8996   2c2 10054  ;cdc 10387   ndxcnx 13471   Basecbs 13474   +g cplusg 13534   .rcmulr 13535  Scalarcsca 13537   .scvsca 13538  TopSetcts 13540   lecple 13541   distcds 13543   TopOpenctopn 13654   qTop cqtop 13734    "s cimas 13735
This theorem is referenced by:  imassca  13750  imasvsca  13751  imastset  13752  imasle  13753  imasmulfn  13764  imasmulval  13765  imasmulf  13766  divsmulval  13785  divsmulf  13786
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-oadd 6731  df-er 6908  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-sup 7449  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-nn 10006  df-2 10063  df-3 10064  df-4 10065  df-5 10066  df-6 10067  df-7 10068  df-8 10069  df-9 10070  df-10 10071  df-n0 10227  df-z 10288  df-dec 10388  df-uz 10494  df-fz 11049  df-struct 13476  df-ndx 13477  df-slot 13478  df-base 13479  df-plusg 13547  df-mulr 13548  df-sca 13550  df-vsca 13551  df-tset 13553  df-ple 13554  df-ds 13556  df-imas 13739
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