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Theorem sravsca 18348
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 467 . . . . 5  |-  ( ( W  e.  _V  /\  ph )  ->  A  =  ( (subringAlg  `  W ) `  S ) )
3 srapart.s . . . . . 6  |-  ( ph  ->  S  C_  ( Base `  W ) )
4 sraval 18342 . . . . . 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 476 . . . . 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 2462 . . . 4  |-  ( ( W  e.  _V  /\  ph )  ->  A  =  ( ( ( W sSet  <. (Scalar `  ndx ) ,  ( Ws  S ) >. ) sSet  <.
( .s `  ndx ) ,  ( .r `  W ) >. ) sSet  <.
( .i `  ndx ) ,  ( .r `  W ) >. )
)
76fveq2d 5829 . . 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 6277 . . . . 5  |-  ( W sSet  <. (Scalar `  ndx ) ,  ( Ws  S ) >. )  e.  _V
9 fvex 5835 . . . . 5  |-  ( .r
`  W )  e. 
_V
10 vscaid 15203 . . . . . 6  |-  .s  = Slot  ( .s `  ndx )
1110setsid 15107 . . . . 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 676 . . . 4  |-  ( .r
`  W )  =  ( .s `  (
( W sSet  <. (Scalar `  ndx ) ,  ( Ws  S ) >. ) sSet  <. ( .s `  ndx ) ,  ( .r `  W
) >. ) )
13 6re 10641 . . . . . . 7  |-  6  e.  RR
14 6lt8 10749 . . . . . . 7  |-  6  <  8
1513, 14ltneii 9698 . . . . . 6  |-  6  =/=  8
16 vscandx 15202 . . . . . . 7  |-  ( .s
`  ndx )  =  6
17 ipndx 15209 . . . . . . 7  |-  ( .i
`  ndx )  =  8
1816, 17neeq12i 2667 . . . . . 6  |-  ( ( .s `  ndx )  =/=  ( .i `  ndx ) 
<->  6  =/=  8 )
1915, 18mpbir 212 . . . . 5  |-  ( .s
`  ndx )  =/=  ( .i `  ndx )
2010, 19setsnid 15108 . . . 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 2450 . . 3  |-  ( .r
`  W )  =  ( .s `  (
( ( W sSet  <. (Scalar `  ndx ) ,  ( Ws  S ) >. ) sSet  <.
( .s `  ndx ) ,  ( .r `  W ) >. ) sSet  <.
( .i `  ndx ) ,  ( .r `  W ) >. )
)
227, 21syl6reqr 2481 . 2  |-  ( ( W  e.  _V  /\  ph )  ->  ( .r `  W )  =  ( .s `  A ) )
2310str0 15104 . . 3  |-  (/)  =  ( .s `  (/) )
24 fvprc 5819 . . . 4  |-  ( -.  W  e.  _V  ->  ( .r `  W )  =  (/) )
2524adantr 466 . . 3  |-  ( ( -.  W  e.  _V  /\ 
ph )  ->  ( .r `  W )  =  (/) )
26 fvprc 5819 . . . . . . 7  |-  ( -.  W  e.  _V  ->  (subringAlg  `  W )  =  (/) )
2726fveq1d 5827 . . . . . 6  |-  ( -.  W  e.  _V  ->  ( (subringAlg  `  W ) `  S )  =  (
(/) `  S )
)
28 0fv 5858 . . . . . 6  |-  ( (/) `  S )  =  (/)
2927, 28syl6eq 2478 . . . . 5  |-  ( -.  W  e.  _V  ->  ( (subringAlg  `  W ) `  S )  =  (/) )
301, 29sylan9eqr 2484 . . . 4  |-  ( ( -.  W  e.  _V  /\ 
ph )  ->  A  =  (/) )
3130fveq2d 5829 . . 3  |-  ( ( -.  W  e.  _V  /\ 
ph )  ->  ( .s `  A )  =  ( .s `  (/) ) )
3223, 25, 313eqtr4a 2488 . 2  |-  ( ( -.  W  e.  _V  /\ 
ph )  ->  ( .r `  W )  =  ( .s `  A
) )
3322, 32pm2.61ian 797 1  |-  ( ph  ->  ( .r `  W
)  =  ( .s
`  A ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1872    =/= wne 2599   _Vcvv 3022    C_ wss 3379   (/)c0 3704   <.cop 3947   ` cfv 5544  (class class class)co 6249   6c6 10614   8c8 10616   ndxcnx 15061   sSet csts 15062   Basecbs 15064   ↾s cress 15065   .rcmulr 15134  Scalarcsca 15136   .scvsca 15137   .icip 15138  subringAlg csra 18334
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-rep 4479  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-nel 2602  df-ral 2719  df-rex 2720  df-reu 2721  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-pss 3395  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-tp 3946  df-op 3948  df-uni 4163  df-iun 4244  df-br 4367  df-opab 4426  df-mpt 4427  df-tr 4462  df-eprel 4707  df-id 4711  df-po 4717  df-so 4718  df-fr 4755  df-we 4757  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-pred 5342  df-ord 5388  df-on 5389  df-lim 5390  df-suc 5391  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-f1 5549  df-fo 5550  df-f1o 5551  df-fv 5552  df-riota 6211  df-ov 6252  df-oprab 6253  df-mpt2 6254  df-om 6651  df-wrecs 6983  df-recs 7045  df-rdg 7083  df-er 7318  df-en 7525  df-dom 7526  df-sdom 7527  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9813  df-neg 9814  df-nn 10561  df-2 10619  df-3 10620  df-4 10621  df-5 10622  df-6 10623  df-7 10624  df-8 10625  df-ndx 15067  df-slot 15068  df-sets 15070  df-vsca 15150  df-ip 15151  df-sra 18338
This theorem is referenced by:  sralmod  18353  rlmvsca  18368  sraassa  18492  sranlm  21629
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