MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  srngmul Unicode version

Theorem srngmul 15639
Description: The involution function in a star ring distributes over multiplication, with a change in the order of the factors. (Contributed by Mario Carneiro, 6-Oct-2015.)
Hypotheses
Ref Expression
srngcl.i  |-  .*  =  ( * r `  R )
srngcl.b  |-  B  =  ( Base `  R
)
srngmul.t  |-  .x.  =  ( .r `  R )
Assertion
Ref Expression
srngmul  |-  ( ( R  e.  *Ring  /\  X  e.  B  /\  Y  e.  B )  ->  (  .*  `  ( X  .x.  Y ) )  =  ( (  .*  `  Y )  .x.  (  .*  `  X ) ) )

Proof of Theorem srngmul
StepHypRef Expression
1 eqid 2296 . . . . 5  |-  (oppr `  R
)  =  (oppr `  R
)
2 eqid 2296 . . . . 5  |-  ( * r f `  R
)  =  ( * r f `  R
)
31, 2srngrhm 15632 . . . 4  |-  ( R  e.  *Ring  ->  ( * r f `  R )  e.  ( R RingHom  (oppr `  R
) ) )
4 srngcl.b . . . . 5  |-  B  =  ( Base `  R
)
5 srngmul.t . . . . 5  |-  .x.  =  ( .r `  R )
6 eqid 2296 . . . . 5  |-  ( .r
`  (oppr
`  R ) )  =  ( .r `  (oppr `  R ) )
74, 5, 6rhmmul 15521 . . . 4  |-  ( ( ( * r f `
 R )  e.  ( R RingHom  (oppr
`  R ) )  /\  X  e.  B  /\  Y  e.  B
)  ->  ( (
* r f `  R ) `  ( X  .x.  Y ) )  =  ( ( ( * r f `  R ) `  X
) ( .r `  (oppr `  R ) ) ( ( * r f `
 R ) `  Y ) ) )
83, 7syl3an1 1215 . . 3  |-  ( ( R  e.  *Ring  /\  X  e.  B  /\  Y  e.  B )  ->  (
( * r f `
 R ) `  ( X  .x.  Y ) )  =  ( ( ( * r f `
 R ) `  X ) ( .r
`  (oppr
`  R ) ) ( ( * r f `  R ) `
 Y ) ) )
94, 5, 1, 6opprmul 15424 . . 3  |-  ( ( ( * r f `
 R ) `  X ) ( .r
`  (oppr
`  R ) ) ( ( * r f `  R ) `
 Y ) )  =  ( ( ( * r f `  R ) `  Y
)  .x.  ( (
* r f `  R ) `  X
) )
108, 9syl6eq 2344 . 2  |-  ( ( R  e.  *Ring  /\  X  e.  B  /\  Y  e.  B )  ->  (
( * r f `
 R ) `  ( X  .x.  Y ) )  =  ( ( ( * r f `
 R ) `  Y )  .x.  (
( * r f `
 R ) `  X ) ) )
11 srngrng 15633 . . . 4  |-  ( R  e.  *Ring  ->  R  e.  Ring )
124, 5rngcl 15370 . . . 4  |-  ( ( R  e.  Ring  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .x.  Y )  e.  B )
1311, 12syl3an1 1215 . . 3  |-  ( ( R  e.  *Ring  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .x.  Y )  e.  B )
14 srngcl.i . . . 4  |-  .*  =  ( * r `  R )
154, 14, 2stafval 15629 . . 3  |-  ( ( X  .x.  Y )  e.  B  ->  (
( * r f `
 R ) `  ( X  .x.  Y ) )  =  (  .* 
`  ( X  .x.  Y ) ) )
1613, 15syl 15 . 2  |-  ( ( R  e.  *Ring  /\  X  e.  B  /\  Y  e.  B )  ->  (
( * r f `
 R ) `  ( X  .x.  Y ) )  =  (  .* 
`  ( X  .x.  Y ) ) )
174, 14, 2stafval 15629 . . . 4  |-  ( Y  e.  B  ->  (
( * r f `
 R ) `  Y )  =  (  .*  `  Y ) )
18173ad2ant3 978 . . 3  |-  ( ( R  e.  *Ring  /\  X  e.  B  /\  Y  e.  B )  ->  (
( * r f `
 R ) `  Y )  =  (  .*  `  Y ) )
194, 14, 2stafval 15629 . . . 4  |-  ( X  e.  B  ->  (
( * r f `
 R ) `  X )  =  (  .*  `  X ) )
20193ad2ant2 977 . . 3  |-  ( ( R  e.  *Ring  /\  X  e.  B  /\  Y  e.  B )  ->  (
( * r f `
 R ) `  X )  =  (  .*  `  X ) )
2118, 20oveq12d 5892 . 2  |-  ( ( R  e.  *Ring  /\  X  e.  B  /\  Y  e.  B )  ->  (
( ( * r f `  R ) `
 Y )  .x.  ( ( * r f `  R ) `
 X ) )  =  ( (  .* 
`  Y )  .x.  (  .*  `  X ) ) )
2210, 16, 213eqtr3d 2336 1  |-  ( ( R  e.  *Ring  /\  X  e.  B  /\  Y  e.  B )  ->  (  .*  `  ( X  .x.  Y ) )  =  ( (  .*  `  Y )  .x.  (  .*  `  X ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1632    e. wcel 1696   ` cfv 5271  (class class class)co 5874   Basecbs 13164   .rcmulr 13225   * rcstv 13226   Ringcrg 15353  opprcoppr 15420   RingHom crh 15510   * r fcstf 15624   *Ringcsr 15625
This theorem is referenced by:  ipassr  16566
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-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  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-tpos 6250  df-riota 6320  df-recs 6404  df-rdg 6439  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-nn 9763  df-2 9820  df-3 9821  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-plusg 13237  df-mulr 13238  df-0g 13420  df-mnd 14383  df-mhm 14431  df-ghm 14697  df-mgp 15342  df-rng 15356  df-ur 15358  df-oppr 15421  df-rnghom 15512  df-staf 15626  df-srng 15627
  Copyright terms: Public domain W3C validator