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Theorem srasca 18482
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 15336 . . . . 5  |- Scalar  = Slot  (Scalar ` 
ndx )
2 5re 10710 . . . . . . 7  |-  5  e.  RR
3 5lt6 10809 . . . . . . 7  |-  5  <  6
42, 3ltneii 9765 . . . . . 6  |-  5  =/=  6
5 scandx 15335 . . . . . . 7  |-  (Scalar `  ndx )  =  5
6 vscandx 15337 . . . . . . 7  |-  ( .s
`  ndx )  =  6
75, 6neeq12i 2709 . . . . . 6  |-  ( (Scalar `  ndx )  =/=  ( .s `  ndx )  <->  5  =/=  6 )
84, 7mpbir 214 . . . . 5  |-  (Scalar `  ndx )  =/=  ( .s `  ndx )
91, 8setsnid 15243 . . . 4  |-  (Scalar `  ( W sSet  <. (Scalar `  ndx ) ,  ( Ws  S ) >. ) )  =  (Scalar `  ( ( W sSet  <. (Scalar `  ndx ) ,  ( Ws  S
) >. ) sSet  <. ( .s `  ndx ) ,  ( .r `  W
) >. ) )
10 5lt8 10822 . . . . . . 7  |-  5  <  8
112, 10ltneii 9765 . . . . . 6  |-  5  =/=  8
12 ipndx 15344 . . . . . . 7  |-  ( .i
`  ndx )  =  8
135, 12neeq12i 2709 . . . . . 6  |-  ( (Scalar `  ndx )  =/=  ( .i `  ndx )  <->  5  =/=  8 )
1411, 13mpbir 214 . . . . 5  |-  (Scalar `  ndx )  =/=  ( .i `  ndx )
151, 14setsnid 15243 . . . 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 2493 . . 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 6336 . . . . 5  |-  ( Ws  S )  e.  _V
1817a1i 11 . . . 4  |-  ( ph  ->  ( Ws  S )  e.  _V )
191setsid 15242 . . . 4  |-  ( ( W  e.  _V  /\  ( Ws  S )  e.  _V )  ->  ( Ws  S )  =  (Scalar `  ( W sSet  <. (Scalar `  ndx ) ,  ( Ws  S
) >. ) ) )
2018, 19sylan2 482 . . 3  |-  ( ( W  e.  _V  /\  ph )  ->  ( Ws  S
)  =  (Scalar `  ( W sSet  <. (Scalar `  ndx ) ,  ( Ws  S ) >. ) ) )
21 srapart.a . . . . . 6  |-  ( ph  ->  A  =  ( (subringAlg  `  W ) `  S
) )
2221adantl 473 . . . . 5  |-  ( ( W  e.  _V  /\  ph )  ->  A  =  ( (subringAlg  `  W ) `  S ) )
23 srapart.s . . . . . 6  |-  ( ph  ->  S  C_  ( Base `  W ) )
24 sraval 18477 . . . . . 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 482 . . . . 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 2505 . . . 4  |-  ( ( W  e.  _V  /\  ph )  ->  A  =  ( ( ( W sSet  <. (Scalar `  ndx ) ,  ( Ws  S ) >. ) sSet  <.
( .s `  ndx ) ,  ( .r `  W ) >. ) sSet  <.
( .i `  ndx ) ,  ( .r `  W ) >. )
)
2726fveq2d 5883 . . 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 2531 . 2  |-  ( ( W  e.  _V  /\  ph )  ->  ( Ws  S
)  =  (Scalar `  A ) )
291str0 15239 . . 3  |-  (/)  =  (Scalar `  (/) )
30 reldmress 15253 . . . . 5  |-  Rel  doms
3130ovprc1 6339 . . . 4  |-  ( -.  W  e.  _V  ->  ( Ws  S )  =  (/) )
3231adantr 472 . . 3  |-  ( ( -.  W  e.  _V  /\ 
ph )  ->  ( Ws  S )  =  (/) )
33 fvprc 5873 . . . . . . 7  |-  ( -.  W  e.  _V  ->  (subringAlg  `  W )  =  (/) )
3433fveq1d 5881 . . . . . 6  |-  ( -.  W  e.  _V  ->  ( (subringAlg  `  W ) `  S )  =  (
(/) `  S )
)
35 0fv 5912 . . . . . 6  |-  ( (/) `  S )  =  (/)
3634, 35syl6eq 2521 . . . . 5  |-  ( -.  W  e.  _V  ->  ( (subringAlg  `  W ) `  S )  =  (/) )
3721, 36sylan9eqr 2527 . . . 4  |-  ( ( -.  W  e.  _V  /\ 
ph )  ->  A  =  (/) )
3837fveq2d 5883 . . 3  |-  ( ( -.  W  e.  _V  /\ 
ph )  ->  (Scalar `  A )  =  (Scalar `  (/) ) )
3929, 32, 383eqtr4a 2531 . 2  |-  ( ( -.  W  e.  _V  /\ 
ph )  ->  ( Ws  S )  =  (Scalar `  A ) )
4028, 39pm2.61ian 807 1  |-  ( ph  ->  ( Ws  S )  =  (Scalar `  A ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 376    = wceq 1452    e. wcel 1904    =/= wne 2641   _Vcvv 3031    C_ wss 3390   (/)c0 3722   <.cop 3965   ` cfv 5589  (class class class)co 6308   5c5 10684   6c6 10685   8c8 10687   ndxcnx 15196   sSet csts 15197   Basecbs 15199   ↾s cress 15200   .rcmulr 15269  Scalarcsca 15271   .scvsca 15272   .icip 15273  subringAlg csra 18469
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-ndx 15202  df-slot 15203  df-sets 15205  df-ress 15206  df-sca 15284  df-vsca 15285  df-ip 15286  df-sra 18473
This theorem is referenced by:  sralmod  18488  rlmsca  18501  rlmsca2  18502  sraassa  18626  frlmip  19413  sranlm  21765  srabn  22405  rrxprds  22426
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