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Theorem srngadd 15721
Description: The involution function in a star ring distributes over addition. (Contributed by Mario Carneiro, 6-Oct-2015.)
Hypotheses
Ref Expression
srngcl.i  |-  .*  =  ( * r `  R )
srngcl.b  |-  B  =  ( Base `  R
)
srngadd.p  |-  .+  =  ( +g  `  R )
Assertion
Ref Expression
srngadd  |-  ( ( R  e.  *Ring  /\  X  e.  B  /\  Y  e.  B )  ->  (  .*  `  ( X  .+  Y ) )  =  ( (  .*  `  X )  .+  (  .*  `  Y ) ) )

Proof of Theorem srngadd
StepHypRef Expression
1 eqid 2358 . . . . 5  |-  (oppr `  R
)  =  (oppr `  R
)
2 eqid 2358 . . . . 5  |-  ( * r f `  R
)  =  ( * r f `  R
)
31, 2srngrhm 15715 . . . 4  |-  ( R  e.  *Ring  ->  ( * r f `  R )  e.  ( R RingHom  (oppr `  R
) ) )
4 rhmghm 15602 . . . 4  |-  ( ( * r f `  R )  e.  ( R RingHom  (oppr
`  R ) )  ->  ( * r f `  R )  e.  ( R  GrpHom  (oppr `  R ) ) )
53, 4syl 15 . . 3  |-  ( R  e.  *Ring  ->  ( * r f `  R )  e.  ( R  GrpHom  (oppr `  R ) ) )
6 srngcl.b . . . 4  |-  B  =  ( Base `  R
)
7 srngadd.p . . . 4  |-  .+  =  ( +g  `  R )
81, 7oppradd 15511 . . . 4  |-  .+  =  ( +g  `  (oppr `  R
) )
96, 7, 8ghmlin 14787 . . 3  |-  ( ( ( * r f `
 R )  e.  ( R  GrpHom  (oppr `  R
) )  /\  X  e.  B  /\  Y  e.  B )  ->  (
( * r f `
 R ) `  ( X  .+  Y ) )  =  ( ( ( * r f `
 R ) `  X )  .+  (
( * r f `
 R ) `  Y ) ) )
105, 9syl3an1 1215 . 2  |-  ( ( R  e.  *Ring  /\  X  e.  B  /\  Y  e.  B )  ->  (
( * r f `
 R ) `  ( X  .+  Y ) )  =  ( ( ( * r f `
 R ) `  X )  .+  (
( * r f `
 R ) `  Y ) ) )
11 srngrng 15716 . . . 4  |-  ( R  e.  *Ring  ->  R  e.  Ring )
126, 7rngacl 15467 . . . 4  |-  ( ( R  e.  Ring  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .+  Y )  e.  B )
1311, 12syl3an1 1215 . . 3  |-  ( ( R  e.  *Ring  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .+  Y )  e.  B )
14 srngcl.i . . . 4  |-  .*  =  ( * r `  R )
156, 14, 2stafval 15712 . . 3  |-  ( ( X  .+  Y )  e.  B  ->  (
( * r f `
 R ) `  ( X  .+  Y ) )  =  (  .* 
`  ( X  .+  Y ) ) )
1613, 15syl 15 . 2  |-  ( ( R  e.  *Ring  /\  X  e.  B  /\  Y  e.  B )  ->  (
( * r f `
 R ) `  ( X  .+  Y ) )  =  (  .* 
`  ( X  .+  Y ) ) )
176, 14, 2stafval 15712 . . . 4  |-  ( X  e.  B  ->  (
( * r f `
 R ) `  X )  =  (  .*  `  X ) )
18173ad2ant2 977 . . 3  |-  ( ( R  e.  *Ring  /\  X  e.  B  /\  Y  e.  B )  ->  (
( * r f `
 R ) `  X )  =  (  .*  `  X ) )
196, 14, 2stafval 15712 . . . 4  |-  ( Y  e.  B  ->  (
( * r f `
 R ) `  Y )  =  (  .*  `  Y ) )
20193ad2ant3 978 . . 3  |-  ( ( R  e.  *Ring  /\  X  e.  B  /\  Y  e.  B )  ->  (
( * r f `
 R ) `  Y )  =  (  .*  `  Y ) )
2118, 20oveq12d 5963 . 2  |-  ( ( R  e.  *Ring  /\  X  e.  B  /\  Y  e.  B )  ->  (
( ( * r f `  R ) `
 X )  .+  ( ( * r f `  R ) `
 Y ) )  =  ( (  .* 
`  X )  .+  (  .*  `  Y ) ) )
2210, 16, 213eqtr3d 2398 1  |-  ( ( R  e.  *Ring  /\  X  e.  B  /\  Y  e.  B )  ->  (  .*  `  ( X  .+  Y ) )  =  ( (  .*  `  X )  .+  (  .*  `  Y ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1642    e. wcel 1710   ` cfv 5337  (class class class)co 5945   Basecbs 13245   +g cplusg 13305   * rcstv 13307    GrpHom cghm 14779   Ringcrg 15436  opprcoppr 15503   RingHom crh 15593   * r fcstf 15707   *Ringcsr 15708
This theorem is referenced by:  ipdi  16650
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-cnex 8883  ax-resscn 8884  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-addrcl 8888  ax-mulcl 8889  ax-mulrcl 8890  ax-mulcom 8891  ax-addass 8892  ax-mulass 8893  ax-distr 8894  ax-i2m1 8895  ax-1ne0 8896  ax-1rid 8897  ax-rnegex 8898  ax-rrecex 8899  ax-cnre 8900  ax-pre-lttri 8901  ax-pre-lttrn 8902  ax-pre-ltadd 8903  ax-pre-mulgt0 8904
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-tpos 6321  df-riota 6391  df-recs 6475  df-rdg 6510  df-er 6747  df-map 6862  df-en 6952  df-dom 6953  df-sdom 6954  df-pnf 8959  df-mnf 8960  df-xr 8961  df-ltxr 8962  df-le 8963  df-sub 9129  df-neg 9130  df-nn 9837  df-2 9894  df-3 9895  df-ndx 13248  df-slot 13249  df-base 13250  df-sets 13251  df-plusg 13318  df-mulr 13319  df-0g 13503  df-mnd 14466  df-mhm 14514  df-grp 14588  df-ghm 14780  df-mgp 15425  df-rng 15439  df-ur 15441  df-oppr 15504  df-rnghom 15595  df-staf 15709  df-srng 15710
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