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Theorem sravsca 17699
Description: The scalar product operation of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.)
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 466 . . . . 5  |-  ( ( W  e.  _V  /\  ph )  ->  A  =  ( (subringAlg  `  W ) `  S ) )
3 srapart.s . . . . . 6  |-  ( ph  ->  S  C_  ( Base `  W ) )
4 sraval 17693 . . . . . 6  |-  ( ( W  e.  _V  /\  S  C_  ( Base `  W
) )  ->  (
(subringAlg  `  W ) `  S )  =  ( ( ( W sSet  <. (Scalar `  ndx ) ,  ( Ws  S ) >. ) sSet  <.
( .s `  ndx ) ,  ( .r `  W ) >. ) sSet  <.
( .i `  ndx ) ,  ( .r `  W ) >. )
)
53, 4sylan2 474 . . . . 5  |-  ( ( W  e.  _V  /\  ph )  ->  ( (subringAlg  `  W ) `  S
)  =  ( ( ( W sSet  <. (Scalar ` 
ndx ) ,  ( Ws  S ) >. ) sSet  <.
( .s `  ndx ) ,  ( .r `  W ) >. ) sSet  <.
( .i `  ndx ) ,  ( .r `  W ) >. )
)
62, 5eqtrd 2508 . . . 4  |-  ( ( W  e.  _V  /\  ph )  ->  A  =  ( ( ( W sSet  <. (Scalar `  ndx ) ,  ( Ws  S ) >. ) sSet  <.
( .s `  ndx ) ,  ( .r `  W ) >. ) sSet  <.
( .i `  ndx ) ,  ( .r `  W ) >. )
)
76fveq2d 5876 . . 3  |-  ( ( W  e.  _V  /\  ph )  ->  ( .s `  A )  =  ( .s `  ( ( ( W sSet  <. (Scalar ` 
ndx ) ,  ( Ws  S ) >. ) sSet  <.
( .s `  ndx ) ,  ( .r `  W ) >. ) sSet  <.
( .i `  ndx ) ,  ( .r `  W ) >. )
) )
8 ovex 6320 . . . . 5  |-  ( W sSet  <. (Scalar `  ndx ) ,  ( Ws  S ) >. )  e.  _V
9 fvex 5882 . . . . 5  |-  ( .r
`  W )  e. 
_V
10 vscaid 14635 . . . . . 6  |-  .s  = Slot  ( .s `  ndx )
1110setsid 14548 . . . . 5  |-  ( ( ( 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 672 . . . 4  |-  ( .r
`  W )  =  ( .s `  (
( W sSet  <. (Scalar `  ndx ) ,  ( Ws  S ) >. ) sSet  <. ( .s `  ndx ) ,  ( .r `  W
) >. ) )
13 6re 10628 . . . . . . 7  |-  6  e.  RR
14 6lt8 10736 . . . . . . 7  |-  6  <  8
1513, 14ltneii 9709 . . . . . 6  |-  6  =/=  8
16 vscandx 14634 . . . . . . 7  |-  ( .s
`  ndx )  =  6
17 ipndx 14641 . . . . . . 7  |-  ( .i
`  ndx )  =  8
1816, 17neeq12i 2756 . . . . . 6  |-  ( ( .s `  ndx )  =/=  ( .i `  ndx ) 
<->  6  =/=  8 )
1915, 18mpbir 209 . . . . 5  |-  ( .s
`  ndx )  =/=  ( .i `  ndx )
2010, 19setsnid 14549 . . . 4  |-  ( .s
`  ( ( W sSet  <. (Scalar `  ndx ) ,  ( Ws  S ) >. ) sSet  <.
( .s `  ndx ) ,  ( .r `  W ) >. )
)  =  ( .s
`  ( ( ( W sSet  <. (Scalar `  ndx ) ,  ( Ws  S
) >. ) sSet  <. ( .s `  ndx ) ,  ( .r `  W
) >. ) sSet  <. ( .i `  ndx ) ,  ( .r `  W
) >. ) )
2112, 20eqtri 2496 . . 3  |-  ( .r
`  W )  =  ( .s `  (
( ( W sSet  <. (Scalar `  ndx ) ,  ( Ws  S ) >. ) sSet  <.
( .s `  ndx ) ,  ( .r `  W ) >. ) sSet  <.
( .i `  ndx ) ,  ( .r `  W ) >. )
)
227, 21syl6reqr 2527 . 2  |-  ( ( W  e.  _V  /\  ph )  ->  ( .r `  W )  =  ( .s `  A ) )
2310str0 14545 . . 3  |-  (/)  =  ( .s `  (/) )
24 fvprc 5866 . . . 4  |-  ( -.  W  e.  _V  ->  ( .r `  W )  =  (/) )
2524adantr 465 . . 3  |-  ( ( -.  W  e.  _V  /\ 
ph )  ->  ( .r `  W )  =  (/) )
26 fvprc 5866 . . . . . . 7  |-  ( -.  W  e.  _V  ->  (subringAlg  `  W )  =  (/) )
2726fveq1d 5874 . . . . . 6  |-  ( -.  W  e.  _V  ->  ( (subringAlg  `  W ) `  S )  =  (
(/) `  S )
)
28 0fv 5905 . . . . . 6  |-  ( (/) `  S )  =  (/)
2927, 28syl6eq 2524 . . . . 5  |-  ( -.  W  e.  _V  ->  ( (subringAlg  `  W ) `  S )  =  (/) )
301, 29sylan9eqr 2530 . . . 4  |-  ( ( -.  W  e.  _V  /\ 
ph )  ->  A  =  (/) )
3130fveq2d 5876 . . 3  |-  ( ( -.  W  e.  _V  /\ 
ph )  ->  ( .s `  A )  =  ( .s `  (/) ) )
3223, 25, 313eqtr4a 2534 . 2  |-  ( ( -.  W  e.  _V  /\ 
ph )  ->  ( .r `  W )  =  ( .s `  A
) )
3322, 32pm2.61ian 788 1  |-  ( ph  ->  ( .r `  W
)  =  ( .s
`  A ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   _Vcvv 3118    C_ wss 3481   (/)c0 3790   <.cop 4039   ` cfv 5594  (class class class)co 6295   6c6 10601   8c8 10603   ndxcnx 14504   sSet csts 14505   Basecbs 14507   ↾s cress 14508   .rcmulr 14573  Scalarcsca 14575   .scvsca 14576   .icip 14577  subringAlg csra 17685
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-recs 7054  df-rdg 7088  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-ndx 14510  df-slot 14511  df-sets 14513  df-vsca 14589  df-ip 14590  df-sra 17689
This theorem is referenced by:  sralmod  17704  rlmvsca  17719  sraassa  17844  sranlm  21061
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