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Theorem srasca 15950
Description: The set of scalars 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
srasca  |-  ( ph  ->  ( Ws  S )  =  (Scalar `  A ) )

Proof of Theorem srasca
StepHypRef Expression
1 scaid 13285 . . . 4  |- Scalar  = Slot  (Scalar ` 
ndx )
2 5re 9837 . . . . . 6  |-  5  e.  RR
3 5lt6 9912 . . . . . 6  |-  5  <  6
42, 3ltneii 8947 . . . . 5  |-  5  =/=  6
5 scandx 13284 . . . . . 6  |-  (Scalar `  ndx )  =  5
6 vscandx 13286 . . . . . 6  |-  ( .s
`  ndx )  =  6
75, 6neeq12i 2471 . . . . 5  |-  ( (Scalar `  ndx )  =/=  ( .s `  ndx )  <->  5  =/=  6 )
84, 7mpbir 200 . . . 4  |-  (Scalar `  ndx )  =/=  ( .s `  ndx )
91, 8setsnid 13204 . . 3  |-  (Scalar `  ( W sSet  <. (Scalar `  ndx ) ,  ( Ws  S ) >. ) )  =  (Scalar `  ( ( W sSet  <. (Scalar `  ndx ) ,  ( Ws  S
) >. ) sSet  <. ( .s `  ndx ) ,  ( .r `  W
) >. ) )
10 ovex 5899 . . . . 5  |-  ( Ws  S )  e.  _V
1110a1i 10 . . . 4  |-  ( ph  ->  ( Ws  S )  e.  _V )
121setsid 13203 . . . 4  |-  ( ( W  e.  _V  /\  ( Ws  S )  e.  _V )  ->  ( Ws  S )  =  (Scalar `  ( W sSet  <. (Scalar `  ndx ) ,  ( Ws  S
) >. ) ) )
1311, 12sylan2 460 . . 3  |-  ( ( W  e.  _V  /\  ph )  ->  ( Ws  S
)  =  (Scalar `  ( W sSet  <. (Scalar `  ndx ) ,  ( Ws  S ) >. ) ) )
14 srapart.a . . . . . 6  |-  ( ph  ->  A  =  ( ( subringAlg  `  W ) `  S
) )
1514adantl 452 . . . . 5  |-  ( ( W  e.  _V  /\  ph )  ->  A  =  ( ( subringAlg  `  W ) `
 S ) )
16 srapart.s . . . . . 6  |-  ( ph  ->  S  C_  ( Base `  W ) )
17 sraval 15945 . . . . . 6  |-  ( ( W  e.  _V  /\  S  C_  ( Base `  W
) )  ->  (
( subringAlg  `  W ) `  S )  =  ( ( W sSet  <. (Scalar ` 
ndx ) ,  ( Ws  S ) >. ) sSet  <.
( .s `  ndx ) ,  ( .r `  W ) >. )
)
1816, 17sylan2 460 . . . . 5  |-  ( ( W  e.  _V  /\  ph )  ->  ( ( subringAlg  `  W ) `  S
)  =  ( ( W sSet  <. (Scalar `  ndx ) ,  ( Ws  S
) >. ) sSet  <. ( .s `  ndx ) ,  ( .r `  W
) >. ) )
1915, 18eqtrd 2328 . . . 4  |-  ( ( W  e.  _V  /\  ph )  ->  A  =  ( ( W sSet  <. (Scalar `  ndx ) ,  ( Ws  S ) >. ) sSet  <.
( .s `  ndx ) ,  ( .r `  W ) >. )
)
2019fveq2d 5545 . . 3  |-  ( ( W  e.  _V  /\  ph )  ->  (Scalar `  A
)  =  (Scalar `  ( ( W sSet  <. (Scalar `  ndx ) ,  ( Ws  S ) >. ) sSet  <.
( .s `  ndx ) ,  ( .r `  W ) >. )
) )
219, 13, 203eqtr4a 2354 . 2  |-  ( ( W  e.  _V  /\  ph )  ->  ( Ws  S
)  =  (Scalar `  A ) )
221str0 13200 . . 3  |-  (/)  =  (Scalar `  (/) )
23 reldmress 13210 . . . . 5  |-  Rel  doms
2423ovprc1 5902 . . . 4  |-  ( -.  W  e.  _V  ->  ( Ws  S )  =  (/) )
2524adantr 451 . . 3  |-  ( ( -.  W  e.  _V  /\ 
ph )  ->  ( Ws  S )  =  (/) )
26 fvprc 5535 . . . . . . 7  |-  ( -.  W  e.  _V  ->  ( subringAlg  `  W )  =  (/) )
2726fveq1d 5543 . . . . . 6  |-  ( -.  W  e.  _V  ->  ( ( subringAlg  `  W ) `  S )  =  (
(/) `  S )
)
28 fv01 5575 . . . . . 6  |-  ( (/) `  S )  =  (/)
2927, 28syl6eq 2344 . . . . 5  |-  ( -.  W  e.  _V  ->  ( ( subringAlg  `  W ) `  S )  =  (/) )
3014, 29sylan9eqr 2350 . . . 4  |-  ( ( -.  W  e.  _V  /\ 
ph )  ->  A  =  (/) )
3130fveq2d 5545 . . 3  |-  ( ( -.  W  e.  _V  /\ 
ph )  ->  (Scalar `  A )  =  (Scalar `  (/) ) )
3222, 25, 313eqtr4a 2354 . 2  |-  ( ( -.  W  e.  _V  /\ 
ph )  ->  ( Ws  S )  =  (Scalar `  A ) )
3321, 32pm2.61ian 765 1  |-  ( ph  ->  ( Ws  S )  =  (Scalar `  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   _Vcvv 2801    C_ wss 3165   (/)c0 3468   <.cop 3656   ` cfv 5271  (class class class)co 5874   5c5 9814   6c6 9815   ndxcnx 13161   sSet csts 13162   Basecbs 13164   ↾s cress 13165   .rcmulr 13225  Scalarcsca 13227   .scvsca 13228   subringAlg csra 15937
This theorem is referenced by:  sralmod  15955  rlmsca  15968  rlmsca2  15969  sraassa  16081  sranlm  18211  srabn  18793
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-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  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-4 9822  df-5 9823  df-6 9824  df-ndx 13167  df-slot 13168  df-sets 13170  df-ress 13171  df-sca 13240  df-vsca 13241  df-sra 15941
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