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Theorem sravsca 15935
Description: The scalar product operation of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.)
Hypotheses
Ref Expression
srapart.a  |-  ( ph  ->  A  =  ( ( subringAlg  `  W ) `  S
) )
srapart.s  |-  ( ph  ->  S  C_  ( Base `  W ) )
Assertion
Ref Expression
sravsca  |-  ( ph  ->  ( .r `  W
)  =  ( .s
`  A ) )

Proof of Theorem sravsca
StepHypRef Expression
1 srapart.a . . . . . 6  |-  ( ph  ->  A  =  ( ( subringAlg  `  W ) `  S
) )
21adantl 452 . . . . 5  |-  ( ( W  e.  _V  /\  ph )  ->  A  =  ( ( subringAlg  `  W ) `
 S ) )
3 srapart.s . . . . . 6  |-  ( ph  ->  S  C_  ( Base `  W ) )
4 sraval 15929 . . . . . 6  |-  ( ( W  e.  _V  /\  S  C_  ( Base `  W
) )  ->  (
( subringAlg  `  W ) `  S )  =  ( ( W sSet  <. (Scalar ` 
ndx ) ,  ( Ws  S ) >. ) sSet  <.
( .s `  ndx ) ,  ( .r `  W ) >. )
)
53, 4sylan2 460 . . . . 5  |-  ( ( W  e.  _V  /\  ph )  ->  ( ( subringAlg  `  W ) `  S
)  =  ( ( W sSet  <. (Scalar `  ndx ) ,  ( Ws  S
) >. ) sSet  <. ( .s `  ndx ) ,  ( .r `  W
) >. ) )
62, 5eqtrd 2315 . . . 4  |-  ( ( W  e.  _V  /\  ph )  ->  A  =  ( ( W sSet  <. (Scalar `  ndx ) ,  ( Ws  S ) >. ) sSet  <.
( .s `  ndx ) ,  ( .r `  W ) >. )
)
76fveq2d 5529 . . 3  |-  ( ( W  e.  _V  /\  ph )  ->  ( .s `  A )  =  ( .s `  ( ( W sSet  <. (Scalar `  ndx ) ,  ( Ws  S
) >. ) sSet  <. ( .s `  ndx ) ,  ( .r `  W
) >. ) ) )
8 ovex 5883 . . . 4  |-  ( W sSet  <. (Scalar `  ndx ) ,  ( Ws  S ) >. )  e.  _V
9 fvex 5539 . . . 4  |-  ( .r
`  W )  e. 
_V
10 vscaid 13271 . . . . 5  |-  .s  = Slot  ( .s `  ndx )
1110setsid 13187 . . . 4  |-  ( ( ( W sSet  <. (Scalar ` 
ndx ) ,  ( Ws  S ) >. )  e.  _V  /\  ( .r
`  W )  e. 
_V )  ->  ( .r `  W )  =  ( .s `  (
( W sSet  <. (Scalar `  ndx ) ,  ( Ws  S ) >. ) sSet  <. ( .s `  ndx ) ,  ( .r `  W
) >. ) ) )
128, 9, 11mp2an 653 . . 3  |-  ( .r
`  W )  =  ( .s `  (
( W sSet  <. (Scalar `  ndx ) ,  ( Ws  S ) >. ) sSet  <. ( .s `  ndx ) ,  ( .r `  W
) >. ) )
137, 12syl6reqr 2334 . 2  |-  ( ( W  e.  _V  /\  ph )  ->  ( .r `  W )  =  ( .s `  A ) )
1410str0 13184 . . 3  |-  (/)  =  ( .s `  (/) )
15 fvprc 5519 . . . 4  |-  ( -.  W  e.  _V  ->  ( .r `  W )  =  (/) )
1615adantr 451 . . 3  |-  ( ( -.  W  e.  _V  /\ 
ph )  ->  ( .r `  W )  =  (/) )
17 fvprc 5519 . . . . . . 7  |-  ( -.  W  e.  _V  ->  ( subringAlg  `  W )  =  (/) )
1817fveq1d 5527 . . . . . 6  |-  ( -.  W  e.  _V  ->  ( ( subringAlg  `  W ) `  S )  =  (
(/) `  S )
)
19 fv01 5559 . . . . . 6  |-  ( (/) `  S )  =  (/)
2018, 19syl6eq 2331 . . . . 5  |-  ( -.  W  e.  _V  ->  ( ( subringAlg  `  W ) `  S )  =  (/) )
211, 20sylan9eqr 2337 . . . 4  |-  ( ( -.  W  e.  _V  /\ 
ph )  ->  A  =  (/) )
2221fveq2d 5529 . . 3  |-  ( ( -.  W  e.  _V  /\ 
ph )  ->  ( .s `  A )  =  ( .s `  (/) ) )
2314, 16, 223eqtr4a 2341 . 2  |-  ( ( -.  W  e.  _V  /\ 
ph )  ->  ( .r `  W )  =  ( .s `  A
) )
2413, 23pm2.61ian 765 1  |-  ( ph  ->  ( .r `  W
)  =  ( .s
`  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788    C_ wss 3152   (/)c0 3455   <.cop 3643   ` cfv 5255  (class class class)co 5858   ndxcnx 13145   sSet csts 13146   Basecbs 13148   ↾s cress 13149   .rcmulr 13209  Scalarcsca 13211   .scvsca 13212   subringAlg csra 15921
This theorem is referenced by:  sralmod  15939  rlmvsca  15954  sraassa  16065  sranlm  18195
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-i2m1 8805  ax-1ne0 8806  ax-rrecex 8809  ax-cnre 8810
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-recs 6388  df-rdg 6423  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-ndx 13151  df-slot 13152  df-sets 13154  df-vsca 13225  df-sra 15925
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