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Theorem sravsca 18023
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 464 . . . . 5  |-  ( ( W  e.  _V  /\  ph )  ->  A  =  ( (subringAlg  `  W ) `  S ) )
3 srapart.s . . . . . 6  |-  ( ph  ->  S  C_  ( Base `  W ) )
4 sraval 18017 . . . . . 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 472 . . . . 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 2495 . . . 4  |-  ( ( W  e.  _V  /\  ph )  ->  A  =  ( ( ( W sSet  <. (Scalar `  ndx ) ,  ( Ws  S ) >. ) sSet  <.
( .s `  ndx ) ,  ( .r `  W ) >. ) sSet  <.
( .i `  ndx ) ,  ( .r `  W ) >. )
)
76fveq2d 5852 . . 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 6298 . . . . 5  |-  ( W sSet  <. (Scalar `  ndx ) ,  ( Ws  S ) >. )  e.  _V
9 fvex 5858 . . . . 5  |-  ( .r
`  W )  e. 
_V
10 vscaid 14851 . . . . . 6  |-  .s  = Slot  ( .s `  ndx )
1110setsid 14759 . . . . 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 670 . . . 4  |-  ( .r
`  W )  =  ( .s `  (
( W sSet  <. (Scalar `  ndx ) ,  ( Ws  S ) >. ) sSet  <. ( .s `  ndx ) ,  ( .r `  W
) >. ) )
13 6re 10612 . . . . . . 7  |-  6  e.  RR
14 6lt8 10720 . . . . . . 7  |-  6  <  8
1513, 14ltneii 9686 . . . . . 6  |-  6  =/=  8
16 vscandx 14850 . . . . . . 7  |-  ( .s
`  ndx )  =  6
17 ipndx 14857 . . . . . . 7  |-  ( .i
`  ndx )  =  8
1816, 17neeq12i 2743 . . . . . 6  |-  ( ( .s `  ndx )  =/=  ( .i `  ndx ) 
<->  6  =/=  8 )
1915, 18mpbir 209 . . . . 5  |-  ( .s
`  ndx )  =/=  ( .i `  ndx )
2010, 19setsnid 14760 . . . 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 2483 . . 3  |-  ( .r
`  W )  =  ( .s `  (
( ( W sSet  <. (Scalar `  ndx ) ,  ( Ws  S ) >. ) sSet  <.
( .s `  ndx ) ,  ( .r `  W ) >. ) sSet  <.
( .i `  ndx ) ,  ( .r `  W ) >. )
)
227, 21syl6reqr 2514 . 2  |-  ( ( W  e.  _V  /\  ph )  ->  ( .r `  W )  =  ( .s `  A ) )
2310str0 14756 . . 3  |-  (/)  =  ( .s `  (/) )
24 fvprc 5842 . . . 4  |-  ( -.  W  e.  _V  ->  ( .r `  W )  =  (/) )
2524adantr 463 . . 3  |-  ( ( -.  W  e.  _V  /\ 
ph )  ->  ( .r `  W )  =  (/) )
26 fvprc 5842 . . . . . . 7  |-  ( -.  W  e.  _V  ->  (subringAlg  `  W )  =  (/) )
2726fveq1d 5850 . . . . . 6  |-  ( -.  W  e.  _V  ->  ( (subringAlg  `  W ) `  S )  =  (
(/) `  S )
)
28 0fv 5881 . . . . . 6  |-  ( (/) `  S )  =  (/)
2927, 28syl6eq 2511 . . . . 5  |-  ( -.  W  e.  _V  ->  ( (subringAlg  `  W ) `  S )  =  (/) )
301, 29sylan9eqr 2517 . . . 4  |-  ( ( -.  W  e.  _V  /\ 
ph )  ->  A  =  (/) )
3130fveq2d 5852 . . 3  |-  ( ( -.  W  e.  _V  /\ 
ph )  ->  ( .s `  A )  =  ( .s `  (/) ) )
3223, 25, 313eqtr4a 2521 . 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 367    = wceq 1398    e. wcel 1823    =/= wne 2649   _Vcvv 3106    C_ wss 3461   (/)c0 3783   <.cop 4022   ` cfv 5570  (class class class)co 6270   6c6 10585   8c8 10587   ndxcnx 14713   sSet csts 14714   Basecbs 14716   ↾s cress 14717   .rcmulr 14785  Scalarcsca 14787   .scvsca 14788   .icip 14789  subringAlg csra 18009
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-ndx 14719  df-slot 14720  df-sets 14722  df-vsca 14801  df-ip 14802  df-sra 18013
This theorem is referenced by:  sralmod  18028  rlmvsca  18043  sraassa  18169  sranlm  21359
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