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Theorem sravsca 17389
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 466 . . . . 5  |-  ( ( W  e.  _V  /\  ph )  ->  A  =  ( (subringAlg  `  W ) `  S ) )
3 srapart.s . . . . . 6  |-  ( ph  ->  S  C_  ( Base `  W ) )
4 sraval 17383 . . . . . 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 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 ) >. )
)
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 5806 . . 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 6228 . . . . 5  |-  ( W sSet  <. (Scalar `  ndx ) ,  ( Ws  S ) >. )  e.  _V
9 fvex 5812 . . . . 5  |-  ( .r
`  W )  e. 
_V
10 vscaid 14423 . . . . . 6  |-  .s  = Slot  ( .s `  ndx )
1110setsid 14336 . . . . 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 672 . . . 4  |-  ( .r
`  W )  =  ( .s `  (
( W sSet  <. (Scalar `  ndx ) ,  ( Ws  S ) >. ) sSet  <. ( .s `  ndx ) ,  ( .r `  W
) >. ) )
13 6re 10516 . . . . . . 7  |-  6  e.  RR
14 6lt8 10624 . . . . . . 7  |-  6  <  8
1513, 14ltneii 9601 . . . . . 6  |-  6  =/=  8
16 vscandx 14422 . . . . . . 7  |-  ( .s
`  ndx )  =  6
17 ipndx 14429 . . . . . . 7  |-  ( .i
`  ndx )  =  8
1816, 17neeq12i 2741 . . . . . 6  |-  ( ( .s `  ndx )  =/=  ( .i `  ndx ) 
<->  6  =/=  8 )
1915, 18mpbir 209 . . . . 5  |-  ( .s
`  ndx )  =/=  ( .i `  ndx )
2010, 19setsnid 14337 . . . 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 14333 . . 3  |-  (/)  =  ( .s `  (/) )
24 fvprc 5796 . . . 4  |-  ( -.  W  e.  _V  ->  ( .r `  W )  =  (/) )
2524adantr 465 . . 3  |-  ( ( -.  W  e.  _V  /\ 
ph )  ->  ( .r `  W )  =  (/) )
26 fvprc 5796 . . . . . . 7  |-  ( -.  W  e.  _V  ->  (subringAlg  `  W )  =  (/) )
2726fveq1d 5804 . . . . . 6  |-  ( -.  W  e.  _V  ->  ( (subringAlg  `  W ) `  S )  =  (
(/) `  S )
)
28 0fv 5835 . . . . . 6  |-  ( (/) `  S )  =  (/)
2927, 28syl6eq 2511 . . . . 5  |-  ( -.  W  e.  _V  ->  ( (subringAlg  `  W ) `  S )  =  (/) )
301, 29sylan9eqr 2517 . . . 4  |-  ( ( -.  W  e.  _V  /\ 
ph )  ->  A  =  (/) )
3130fveq2d 5806 . . 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 369    = wceq 1370    e. wcel 1758    =/= wne 2648   _Vcvv 3078    C_ wss 3439   (/)c0 3748   <.cop 3994   ` cfv 5529  (class class class)co 6203   6c6 10489   8c8 10491   ndxcnx 14292   sSet csts 14293   Basecbs 14295   ↾s cress 14296   .rcmulr 14361  Scalarcsca 14363   .scvsca 14364   .icip 14365  subringAlg csra 17375
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 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9452  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473
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 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-recs 6945  df-rdg 6979  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-nn 10437  df-2 10494  df-3 10495  df-4 10496  df-5 10497  df-6 10498  df-7 10499  df-8 10500  df-ndx 14298  df-slot 14299  df-sets 14301  df-vsca 14377  df-ip 14378  df-sra 17379
This theorem is referenced by:  sralmod  17394  rlmvsca  17409  sraassa  17522  sranlm  20400
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