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Theorem srasca 17368
Description: The set of scalars 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
srasca  |-  ( ph  ->  ( Ws  S )  =  (Scalar `  A ) )

Proof of Theorem srasca
StepHypRef Expression
1 scaid 14401 . . . . 5  |- Scalar  = Slot  (Scalar ` 
ndx )
2 5re 10501 . . . . . . 7  |-  5  e.  RR
3 5lt6 10599 . . . . . . 7  |-  5  <  6
42, 3ltneii 9588 . . . . . 6  |-  5  =/=  6
5 scandx 14400 . . . . . . 7  |-  (Scalar `  ndx )  =  5
6 vscandx 14402 . . . . . . 7  |-  ( .s
`  ndx )  =  6
75, 6neeq12i 2737 . . . . . 6  |-  ( (Scalar `  ndx )  =/=  ( .s `  ndx )  <->  5  =/=  6 )
84, 7mpbir 209 . . . . 5  |-  (Scalar `  ndx )  =/=  ( .s `  ndx )
91, 8setsnid 14318 . . . 4  |-  (Scalar `  ( W sSet  <. (Scalar `  ndx ) ,  ( Ws  S ) >. ) )  =  (Scalar `  ( ( W sSet  <. (Scalar `  ndx ) ,  ( Ws  S
) >. ) sSet  <. ( .s `  ndx ) ,  ( .r `  W
) >. ) )
10 5lt8 10612 . . . . . . 7  |-  5  <  8
112, 10ltneii 9588 . . . . . 6  |-  5  =/=  8
12 ipndx 14409 . . . . . . 7  |-  ( .i
`  ndx )  =  8
135, 12neeq12i 2737 . . . . . 6  |-  ( (Scalar `  ndx )  =/=  ( .i `  ndx )  <->  5  =/=  8 )
1411, 13mpbir 209 . . . . 5  |-  (Scalar `  ndx )  =/=  ( .i `  ndx )
151, 14setsnid 14318 . . . 4  |-  (Scalar `  ( ( W sSet  <. (Scalar `  ndx ) ,  ( Ws  S ) >. ) sSet  <.
( .s `  ndx ) ,  ( .r `  W ) >. )
)  =  (Scalar `  ( ( ( W sSet  <. (Scalar `  ndx ) ,  ( Ws  S ) >. ) sSet  <.
( .s `  ndx ) ,  ( .r `  W ) >. ) sSet  <.
( .i `  ndx ) ,  ( .r `  W ) >. )
)
169, 15eqtri 2480 . . 3  |-  (Scalar `  ( W sSet  <. (Scalar `  ndx ) ,  ( Ws  S ) >. ) )  =  (Scalar `  ( (
( W sSet  <. (Scalar `  ndx ) ,  ( Ws  S ) >. ) sSet  <. ( .s `  ndx ) ,  ( .r `  W
) >. ) sSet  <. ( .i `  ndx ) ,  ( .r `  W
) >. ) )
17 ovex 6215 . . . . 5  |-  ( Ws  S )  e.  _V
1817a1i 11 . . . 4  |-  ( ph  ->  ( Ws  S )  e.  _V )
191setsid 14317 . . . 4  |-  ( ( W  e.  _V  /\  ( Ws  S )  e.  _V )  ->  ( Ws  S )  =  (Scalar `  ( W sSet  <. (Scalar `  ndx ) ,  ( Ws  S
) >. ) ) )
2018, 19sylan2 474 . . 3  |-  ( ( W  e.  _V  /\  ph )  ->  ( Ws  S
)  =  (Scalar `  ( W sSet  <. (Scalar `  ndx ) ,  ( Ws  S ) >. ) ) )
21 srapart.a . . . . . 6  |-  ( ph  ->  A  =  ( (subringAlg  `  W ) `  S
) )
2221adantl 466 . . . . 5  |-  ( ( W  e.  _V  /\  ph )  ->  A  =  ( (subringAlg  `  W ) `  S ) )
23 srapart.s . . . . . 6  |-  ( ph  ->  S  C_  ( Base `  W ) )
24 sraval 17363 . . . . . 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 ) >. )
)
2523, 24sylan2 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 ) >. )
)
2622, 25eqtrd 2492 . . . 4  |-  ( ( W  e.  _V  /\  ph )  ->  A  =  ( ( ( W sSet  <. (Scalar `  ndx ) ,  ( Ws  S ) >. ) sSet  <.
( .s `  ndx ) ,  ( .r `  W ) >. ) sSet  <.
( .i `  ndx ) ,  ( .r `  W ) >. )
)
2726fveq2d 5793 . . 3  |-  ( ( W  e.  _V  /\  ph )  ->  (Scalar `  A
)  =  (Scalar `  ( ( ( W sSet  <. (Scalar `  ndx ) ,  ( Ws  S ) >. ) sSet  <.
( .s `  ndx ) ,  ( .r `  W ) >. ) sSet  <.
( .i `  ndx ) ,  ( .r `  W ) >. )
) )
2816, 20, 273eqtr4a 2518 . 2  |-  ( ( W  e.  _V  /\  ph )  ->  ( Ws  S
)  =  (Scalar `  A ) )
291str0 14314 . . 3  |-  (/)  =  (Scalar `  (/) )
30 reldmress 14326 . . . . 5  |-  Rel  doms
3130ovprc1 6218 . . . 4  |-  ( -.  W  e.  _V  ->  ( Ws  S )  =  (/) )
3231adantr 465 . . 3  |-  ( ( -.  W  e.  _V  /\ 
ph )  ->  ( Ws  S )  =  (/) )
33 fvprc 5783 . . . . . . 7  |-  ( -.  W  e.  _V  ->  (subringAlg  `  W )  =  (/) )
3433fveq1d 5791 . . . . . 6  |-  ( -.  W  e.  _V  ->  ( (subringAlg  `  W ) `  S )  =  (
(/) `  S )
)
35 0fv 5822 . . . . . 6  |-  ( (/) `  S )  =  (/)
3634, 35syl6eq 2508 . . . . 5  |-  ( -.  W  e.  _V  ->  ( (subringAlg  `  W ) `  S )  =  (/) )
3721, 36sylan9eqr 2514 . . . 4  |-  ( ( -.  W  e.  _V  /\ 
ph )  ->  A  =  (/) )
3837fveq2d 5793 . . 3  |-  ( ( -.  W  e.  _V  /\ 
ph )  ->  (Scalar `  A )  =  (Scalar `  (/) ) )
3929, 32, 383eqtr4a 2518 . 2  |-  ( ( -.  W  e.  _V  /\ 
ph )  ->  ( Ws  S )  =  (Scalar `  A ) )
4028, 39pm2.61ian 788 1  |-  ( ph  ->  ( Ws  S )  =  (Scalar `  A ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2644   _Vcvv 3068    C_ wss 3426   (/)c0 3735   <.cop 3981   ` cfv 5516  (class class class)co 6190   5c5 10475   6c6 10476   8c8 10478   ndxcnx 14273   sSet csts 14274   Basecbs 14276   ↾s cress 14277   .rcmulr 14341  Scalarcsca 14343   .scvsca 14344   .icip 14345  subringAlg csra 17355
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-cnex 9439  ax-resscn 9440  ax-1cn 9441  ax-icn 9442  ax-addcl 9443  ax-addrcl 9444  ax-mulcl 9445  ax-mulrcl 9446  ax-mulcom 9447  ax-addass 9448  ax-mulass 9449  ax-distr 9450  ax-i2m1 9451  ax-1ne0 9452  ax-1rid 9453  ax-rnegex 9454  ax-rrecex 9455  ax-cnre 9456  ax-pre-lttri 9457  ax-pre-lttrn 9458  ax-pre-ltadd 9459  ax-pre-mulgt0 9460
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-uni 4190  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-om 6577  df-recs 6932  df-rdg 6966  df-er 7201  df-en 7411  df-dom 7412  df-sdom 7413  df-pnf 9521  df-mnf 9522  df-xr 9523  df-ltxr 9524  df-le 9525  df-sub 9698  df-neg 9699  df-nn 10424  df-2 10481  df-3 10482  df-4 10483  df-5 10484  df-6 10485  df-7 10486  df-8 10487  df-ndx 14279  df-slot 14280  df-sets 14282  df-ress 14283  df-sca 14356  df-vsca 14357  df-ip 14358  df-sra 17359
This theorem is referenced by:  sralmod  17374  rlmsca  17387  rlmsca2  17388  sraassa  17502  frlmip  18312  sranlm  20381  srabn  20988  rrxprds  21009
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