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Theorem sralem 16212
Description: Lemma for srabase 16213 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 453 . . . . 5  |-  ( ( W  e.  _V  /\  ph )  ->  A  =  ( ( subringAlg  `  W ) `
 S ) )
3 srapart.s . . . . . 6  |-  ( ph  ->  S  C_  ( Base `  W ) )
4 sraval 16211 . . . . . 6  |-  ( ( W  e.  _V  /\  S  C_  ( Base `  W
) )  ->  (
( subringAlg  `  W ) `  S )  =  ( ( W sSet  <. (Scalar ` 
ndx ) ,  ( Ws  S ) >. ) sSet  <.
( .s `  ndx ) ,  ( .r `  W ) >. )
)
53, 4sylan2 461 . . . . 5  |-  ( ( W  e.  _V  /\  ph )  ->  ( ( subringAlg  `  W ) `  S
)  =  ( ( W sSet  <. (Scalar `  ndx ) ,  ( Ws  S
) >. ) sSet  <. ( .s `  ndx ) ,  ( .r `  W
) >. ) )
62, 5eqtrd 2444 . . . 4  |-  ( ( W  e.  _V  /\  ph )  ->  A  =  ( ( W sSet  <. (Scalar `  ndx ) ,  ( Ws  S ) >. ) sSet  <.
( .s `  ndx ) ,  ( .r `  W ) >. )
)
76fveq2d 5699 . . 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 13453 . . . . 5  |-  E  = Slot  ( E `  ndx )
11 sralem.3 . . . . . . 7  |-  ( N  <  5  \/  6  <  N )
129nnrei 9973 . . . . . . . . . 10  |-  N  e.  RR
13 5re 10039 . . . . . . . . . 10  |-  5  e.  RR
1412, 13ltnei 9161 . . . . . . . . 9  |-  ( N  <  5  ->  5  =/=  N )
1514necomd 2658 . . . . . . . 8  |-  ( N  <  5  ->  N  =/=  5 )
16 5lt6 10116 . . . . . . . . . 10  |-  5  <  6
17 6re 10040 . . . . . . . . . . 11  |-  6  e.  RR
1813, 17, 12lttri 9163 . . . . . . . . . 10  |-  ( ( 5  <  6  /\  6  <  N )  ->  5  <  N
)
1916, 18mpan 652 . . . . . . . . 9  |-  ( 6  <  N  ->  5  <  N )
2013, 12ltnei 9161 . . . . . . . . 9  |-  ( 5  <  N  ->  N  =/=  5 )
2119, 20syl 16 . . . . . . . 8  |-  ( 6  <  N  ->  N  =/=  5 )
2215, 21jaoi 369 . . . . . . 7  |-  ( ( N  <  5  \/  6  <  N )  ->  N  =/=  5
)
2311, 22ax-mp 8 . . . . . 6  |-  N  =/=  5
248, 9ndxarg 13452 . . . . . . 7  |-  ( E `
 ndx )  =  N
25 scandx 13552 . . . . . . 7  |-  (Scalar `  ndx )  =  5
2624, 25neeq12i 2587 . . . . . 6  |-  ( ( E `  ndx )  =/=  (Scalar `  ndx )  <->  N  =/=  5 )
2723, 26mpbir 201 . . . . 5  |-  ( E `
 ndx )  =/=  (Scalar `  ndx )
2810, 27setsnid 13472 . . . 4  |-  ( E `
 W )  =  ( E `  ( W sSet  <. (Scalar `  ndx ) ,  ( Ws  S
) >. ) )
2912, 13, 17lttri 9163 . . . . . . . . . . 11  |-  ( ( N  <  5  /\  5  <  6 )  ->  N  <  6
)
3016, 29mpan2 653 . . . . . . . . . 10  |-  ( N  <  5  ->  N  <  6 )
3112, 17ltnei 9161 . . . . . . . . . 10  |-  ( N  <  6  ->  6  =/=  N )
3230, 31syl 16 . . . . . . . . 9  |-  ( N  <  5  ->  6  =/=  N )
3332necomd 2658 . . . . . . . 8  |-  ( N  <  5  ->  N  =/=  6 )
3417, 12ltnei 9161 . . . . . . . 8  |-  ( 6  <  N  ->  N  =/=  6 )
3533, 34jaoi 369 . . . . . . 7  |-  ( ( N  <  5  \/  6  <  N )  ->  N  =/=  6
)
3611, 35ax-mp 8 . . . . . 6  |-  N  =/=  6
37 vscandx 13554 . . . . . . 7  |-  ( .s
`  ndx )  =  6
3824, 37neeq12i 2587 . . . . . 6  |-  ( ( E `  ndx )  =/=  ( .s `  ndx ) 
<->  N  =/=  6 )
3936, 38mpbir 201 . . . . 5  |-  ( E `
 ndx )  =/=  ( .s `  ndx )
4010, 39setsnid 13472 . . . 4  |-  ( E `
 ( W sSet  <. (Scalar `  ndx ) ,  ( Ws  S ) >. )
)  =  ( E `
 ( ( W sSet  <. (Scalar `  ndx ) ,  ( Ws  S ) >. ) sSet  <.
( .s `  ndx ) ,  ( .r `  W ) >. )
)
4128, 40eqtri 2432 . . 3  |-  ( E `
 W )  =  ( E `  (
( W sSet  <. (Scalar `  ndx ) ,  ( Ws  S ) >. ) sSet  <. ( .s `  ndx ) ,  ( .r `  W
) >. ) )
427, 41syl6reqr 2463 . 2  |-  ( ( W  e.  _V  /\  ph )  ->  ( E `  W )  =  ( E `  A ) )
438str0 13468 . . 3  |-  (/)  =  ( E `  (/) )
44 fvprc 5689 . . . 4  |-  ( -.  W  e.  _V  ->  ( E `  W )  =  (/) )
4544adantr 452 . . 3  |-  ( ( -.  W  e.  _V  /\ 
ph )  ->  ( E `  W )  =  (/) )
46 fvprc 5689 . . . . . . 7  |-  ( -.  W  e.  _V  ->  ( subringAlg  `  W )  =  (/) )
4746fveq1d 5697 . . . . . 6  |-  ( -.  W  e.  _V  ->  ( ( subringAlg  `  W ) `  S )  =  (
(/) `  S )
)
48 fv01 5730 . . . . . 6  |-  ( (/) `  S )  =  (/)
4947, 48syl6eq 2460 . . . . 5  |-  ( -.  W  e.  _V  ->  ( ( subringAlg  `  W ) `  S )  =  (/) )
501, 49sylan9eqr 2466 . . . 4  |-  ( ( -.  W  e.  _V  /\ 
ph )  ->  A  =  (/) )
5150fveq2d 5699 . . 3  |-  ( ( -.  W  e.  _V  /\ 
ph )  ->  ( E `  A )  =  ( E `  (/) ) )
5243, 45, 513eqtr4a 2470 . 2  |-  ( ( -.  W  e.  _V  /\ 
ph )  ->  ( E `  W )  =  ( E `  A ) )
5342, 52pm2.61ian 766 1  |-  ( ph  ->  ( E `  W
)  =  ( E `
 A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2575   _Vcvv 2924    C_ wss 3288   (/)c0 3596   <.cop 3785   class class class wbr 4180   ` cfv 5421  (class class class)co 6048    < clt 9084   NNcn 9964   5c5 10016   6c6 10017   ndxcnx 13429   sSet csts 13430  Slot cslot 13431   Basecbs 13432   ↾s cress 13433   .rcmulr 13493  Scalarcsca 13495   .scvsca 13496   subringAlg csra 16203
This theorem is referenced by:  srabase  16213  sraaddg  16214  sramulr  16215  sratset  16218  srads  16220
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-riota 6516  df-recs 6600  df-rdg 6635  df-er 6872  df-en 7077  df-dom 7078  df-sdom 7079  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-nn 9965  df-2 10022  df-3 10023  df-4 10024  df-5 10025  df-6 10026  df-ndx 13435  df-slot 13436  df-sets 13438  df-sca 13508  df-vsca 13509  df-sra 16207
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