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Theorem sralem 16280
Description: Lemma for srabase 16281 and similar theorems. (Contributed by Mario Carneiro, 4-Oct-2015.)
Hypotheses
Ref Expression
srapart.a  |-  ( ph  ->  A  =  ( ( subringAlg  `  W ) `  S
) )
srapart.s  |-  ( ph  ->  S  C_  ( Base `  W ) )
sralem.1  |-  E  = Slot 
N
sralem.2  |-  N  e.  NN
sralem.3  |-  ( N  <  5  \/  6  <  N )
Assertion
Ref Expression
sralem  |-  ( ph  ->  ( E `  W
)  =  ( E `
 A ) )

Proof of Theorem sralem
StepHypRef Expression
1 srapart.a . . . . . 6  |-  ( ph  ->  A  =  ( ( subringAlg  `  W ) `  S
) )
21adantl 454 . . . . 5  |-  ( ( W  e.  _V  /\  ph )  ->  A  =  ( ( subringAlg  `  W ) `
 S ) )
3 srapart.s . . . . . 6  |-  ( ph  ->  S  C_  ( Base `  W ) )
4 sraval 16279 . . . . . 6  |-  ( ( W  e.  _V  /\  S  C_  ( Base `  W
) )  ->  (
( subringAlg  `  W ) `  S )  =  ( ( W sSet  <. (Scalar ` 
ndx ) ,  ( Ws  S ) >. ) sSet  <.
( .s `  ndx ) ,  ( .r `  W ) >. )
)
53, 4sylan2 462 . . . . 5  |-  ( ( W  e.  _V  /\  ph )  ->  ( ( subringAlg  `  W ) `  S
)  =  ( ( W sSet  <. (Scalar `  ndx ) ,  ( Ws  S
) >. ) sSet  <. ( .s `  ndx ) ,  ( .r `  W
) >. ) )
62, 5eqtrd 2474 . . . 4  |-  ( ( W  e.  _V  /\  ph )  ->  A  =  ( ( W sSet  <. (Scalar `  ndx ) ,  ( Ws  S ) >. ) sSet  <.
( .s `  ndx ) ,  ( .r `  W ) >. )
)
76fveq2d 5761 . . 3  |-  ( ( W  e.  _V  /\  ph )  ->  ( E `  A )  =  ( E `  ( ( W sSet  <. (Scalar `  ndx ) ,  ( Ws  S
) >. ) sSet  <. ( .s `  ndx ) ,  ( .r `  W
) >. ) ) )
8 sralem.1 . . . . . 6  |-  E  = Slot 
N
9 sralem.2 . . . . . 6  |-  N  e.  NN
108, 9ndxid 13521 . . . . 5  |-  E  = Slot  ( E `  ndx )
11 sralem.3 . . . . . . 7  |-  ( N  <  5  \/  6  <  N )
129nnrei 10040 . . . . . . . . . 10  |-  N  e.  RR
13 5re 10106 . . . . . . . . . 10  |-  5  e.  RR
1412, 13ltnei 9228 . . . . . . . . 9  |-  ( N  <  5  ->  5  =/=  N )
1514necomd 2693 . . . . . . . 8  |-  ( N  <  5  ->  N  =/=  5 )
16 5lt6 10183 . . . . . . . . . 10  |-  5  <  6
17 6re 10107 . . . . . . . . . . 11  |-  6  e.  RR
1813, 17, 12lttri 9230 . . . . . . . . . 10  |-  ( ( 5  <  6  /\  6  <  N )  ->  5  <  N
)
1916, 18mpan 653 . . . . . . . . 9  |-  ( 6  <  N  ->  5  <  N )
2013, 12ltnei 9228 . . . . . . . . 9  |-  ( 5  <  N  ->  N  =/=  5 )
2119, 20syl 16 . . . . . . . 8  |-  ( 6  <  N  ->  N  =/=  5 )
2215, 21jaoi 370 . . . . . . 7  |-  ( ( N  <  5  \/  6  <  N )  ->  N  =/=  5
)
2311, 22ax-mp 5 . . . . . 6  |-  N  =/=  5
248, 9ndxarg 13520 . . . . . . 7  |-  ( E `
 ndx )  =  N
25 scandx 13620 . . . . . . 7  |-  (Scalar `  ndx )  =  5
2624, 25neeq12i 2619 . . . . . 6  |-  ( ( E `  ndx )  =/=  (Scalar `  ndx )  <->  N  =/=  5 )
2723, 26mpbir 202 . . . . 5  |-  ( E `
 ndx )  =/=  (Scalar `  ndx )
2810, 27setsnid 13540 . . . 4  |-  ( E `
 W )  =  ( E `  ( W sSet  <. (Scalar `  ndx ) ,  ( Ws  S
) >. ) )
2912, 13, 17lttri 9230 . . . . . . . . . . 11  |-  ( ( N  <  5  /\  5  <  6 )  ->  N  <  6
)
3016, 29mpan2 654 . . . . . . . . . 10  |-  ( N  <  5  ->  N  <  6 )
3112, 17ltnei 9228 . . . . . . . . . 10  |-  ( N  <  6  ->  6  =/=  N )
3230, 31syl 16 . . . . . . . . 9  |-  ( N  <  5  ->  6  =/=  N )
3332necomd 2693 . . . . . . . 8  |-  ( N  <  5  ->  N  =/=  6 )
3417, 12ltnei 9228 . . . . . . . 8  |-  ( 6  <  N  ->  N  =/=  6 )
3533, 34jaoi 370 . . . . . . 7  |-  ( ( N  <  5  \/  6  <  N )  ->  N  =/=  6
)
3611, 35ax-mp 5 . . . . . 6  |-  N  =/=  6
37 vscandx 13622 . . . . . . 7  |-  ( .s
`  ndx )  =  6
3824, 37neeq12i 2619 . . . . . 6  |-  ( ( E `  ndx )  =/=  ( .s `  ndx ) 
<->  N  =/=  6 )
3936, 38mpbir 202 . . . . 5  |-  ( E `
 ndx )  =/=  ( .s `  ndx )
4010, 39setsnid 13540 . . . 4  |-  ( E `
 ( W sSet  <. (Scalar `  ndx ) ,  ( Ws  S ) >. )
)  =  ( E `
 ( ( W sSet  <. (Scalar `  ndx ) ,  ( Ws  S ) >. ) sSet  <.
( .s `  ndx ) ,  ( .r `  W ) >. )
)
4128, 40eqtri 2462 . . 3  |-  ( E `
 W )  =  ( E `  (
( W sSet  <. (Scalar `  ndx ) ,  ( Ws  S ) >. ) sSet  <. ( .s `  ndx ) ,  ( .r `  W
) >. ) )
427, 41syl6reqr 2493 . 2  |-  ( ( W  e.  _V  /\  ph )  ->  ( E `  W )  =  ( E `  A ) )
438str0 13536 . . 3  |-  (/)  =  ( E `  (/) )
44 fvprc 5751 . . . 4  |-  ( -.  W  e.  _V  ->  ( E `  W )  =  (/) )
4544adantr 453 . . 3  |-  ( ( -.  W  e.  _V  /\ 
ph )  ->  ( E `  W )  =  (/) )
46 fvprc 5751 . . . . . . 7  |-  ( -.  W  e.  _V  ->  ( subringAlg  `  W )  =  (/) )
4746fveq1d 5759 . . . . . 6  |-  ( -.  W  e.  _V  ->  ( ( subringAlg  `  W ) `  S )  =  (
(/) `  S )
)
48 fv01 5792 . . . . . 6  |-  ( (/) `  S )  =  (/)
4947, 48syl6eq 2490 . . . . 5  |-  ( -.  W  e.  _V  ->  ( ( subringAlg  `  W ) `  S )  =  (/) )
501, 49sylan9eqr 2496 . . . 4  |-  ( ( -.  W  e.  _V  /\ 
ph )  ->  A  =  (/) )
5150fveq2d 5761 . . 3  |-  ( ( -.  W  e.  _V  /\ 
ph )  ->  ( E `  A )  =  ( E `  (/) ) )
5243, 45, 513eqtr4a 2500 . 2  |-  ( ( -.  W  e.  _V  /\ 
ph )  ->  ( E `  W )  =  ( E `  A ) )
5342, 52pm2.61ian 767 1  |-  ( ph  ->  ( E `  W
)  =  ( E `
 A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 359    /\ wa 360    = wceq 1653    e. wcel 1727    =/= wne 2605   _Vcvv 2962    C_ wss 3306   (/)c0 3613   <.cop 3841   class class class wbr 4237   ` cfv 5483  (class class class)co 6110    < clt 9151   NNcn 10031   5c5 10083   6c6 10084   ndxcnx 13497   sSet csts 13498  Slot cslot 13499   Basecbs 13500   ↾s cress 13501   .rcmulr 13561  Scalarcsca 13563   .scvsca 13564   subringAlg csra 16271
This theorem is referenced by:  srabase  16281  sraaddg  16282  sramulr  16283  sratset  16286  srads  16288
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-rep 4345  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730  ax-cnex 9077  ax-resscn 9078  ax-1cn 9079  ax-icn 9080  ax-addcl 9081  ax-addrcl 9082  ax-mulcl 9083  ax-mulrcl 9084  ax-mulcom 9085  ax-addass 9086  ax-mulass 9087  ax-distr 9088  ax-i2m1 9089  ax-1ne0 9090  ax-1rid 9091  ax-rnegex 9092  ax-rrecex 9093  ax-cnre 9094  ax-pre-lttri 9095  ax-pre-lttrn 9096  ax-pre-ltadd 9097  ax-pre-mulgt0 9098
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2716  df-rex 2717  df-reu 2718  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-pss 3322  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-tp 3846  df-op 3847  df-uni 4040  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-tr 4328  df-eprel 4523  df-id 4527  df-po 4532  df-so 4533  df-fr 4570  df-we 4572  df-ord 4613  df-on 4614  df-lim 4615  df-suc 4616  df-om 4875  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-riota 6578  df-recs 6662  df-rdg 6697  df-er 6934  df-en 7139  df-dom 7140  df-sdom 7141  df-pnf 9153  df-mnf 9154  df-xr 9155  df-ltxr 9156  df-le 9157  df-sub 9324  df-neg 9325  df-nn 10032  df-2 10089  df-3 10090  df-4 10091  df-5 10092  df-6 10093  df-ndx 13503  df-slot 13504  df-sets 13506  df-sca 13576  df-vsca 13577  df-sra 16275
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