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Theorem sralem 18336
Description: Lemma for srabase 18337 and similar theorems. (Contributed 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 ) )
sralem.1  |-  E  = Slot 
N
sralem.2  |-  N  e.  NN
sralem.3  |-  ( N  <  5  \/  8  <  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 467 . . . . 5  |-  ( ( W  e.  _V  /\  ph )  ->  A  =  ( (subringAlg  `  W ) `  S ) )
3 srapart.s . . . . . 6  |-  ( ph  ->  S  C_  ( Base `  W ) )
4 sraval 18335 . . . . . 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 476 . . . . 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 2456 . . . 4  |-  ( ( W  e.  _V  /\  ph )  ->  A  =  ( ( ( W sSet  <. (Scalar `  ndx ) ,  ( Ws  S ) >. ) sSet  <.
( .s `  ndx ) ,  ( .r `  W ) >. ) sSet  <.
( .i `  ndx ) ,  ( .r `  W ) >. )
)
76fveq2d 5822 . . 3  |-  ( ( W  e.  _V  /\  ph )  ->  ( E `  A )  =  ( E `  ( ( ( W sSet  <. (Scalar ` 
ndx ) ,  ( Ws  S ) >. ) sSet  <.
( .s `  ndx ) ,  ( .r `  W ) >. ) sSet  <.
( .i `  ndx ) ,  ( .r `  W ) >. )
) )
8 sralem.1 . . . . . 6  |-  E  = Slot 
N
9 sralem.2 . . . . . 6  |-  N  e.  NN
108, 9ndxid 15078 . . . . 5  |-  E  = Slot  ( E `  ndx )
11 sralem.3 . . . . . . 7  |-  ( N  <  5  \/  8  <  N )
129nnrei 10562 . . . . . . . . . 10  |-  N  e.  RR
13 5re 10632 . . . . . . . . . 10  |-  5  e.  RR
1412, 13ltnei 9702 . . . . . . . . 9  |-  ( N  <  5  ->  5  =/=  N )
1514necomd 2650 . . . . . . . 8  |-  ( N  <  5  ->  N  =/=  5 )
16 5lt8 10743 . . . . . . . . . 10  |-  5  <  8
17 8re 10638 . . . . . . . . . . 11  |-  8  e.  RR
1813, 17, 12lttri 9704 . . . . . . . . . 10  |-  ( ( 5  <  8  /\  8  <  N )  ->  5  <  N
)
1916, 18mpan 674 . . . . . . . . 9  |-  ( 8  <  N  ->  5  <  N )
2013, 12ltnei 9702 . . . . . . . . 9  |-  ( 5  <  N  ->  N  =/=  5 )
2119, 20syl 17 . . . . . . . 8  |-  ( 8  <  N  ->  N  =/=  5 )
2215, 21jaoi 380 . . . . . . 7  |-  ( ( N  <  5  \/  8  <  N )  ->  N  =/=  5
)
2311, 22ax-mp 5 . . . . . 6  |-  N  =/=  5
248, 9ndxarg 15077 . . . . . . 7  |-  ( E `
 ndx )  =  N
25 scandx 15193 . . . . . . 7  |-  (Scalar `  ndx )  =  5
2624, 25neeq12i 2661 . . . . . 6  |-  ( ( E `  ndx )  =/=  (Scalar `  ndx )  <->  N  =/=  5 )
2723, 26mpbir 212 . . . . 5  |-  ( E `
 ndx )  =/=  (Scalar `  ndx )
2810, 27setsnid 15101 . . . 4  |-  ( E `
 W )  =  ( E `  ( W sSet  <. (Scalar `  ndx ) ,  ( Ws  S
) >. ) )
29 5lt6 10730 . . . . . . . . . . 11  |-  5  <  6
30 6re 10634 . . . . . . . . . . . 12  |-  6  e.  RR
3112, 13, 30lttri 9704 . . . . . . . . . . 11  |-  ( ( N  <  5  /\  5  <  6 )  ->  N  <  6
)
3229, 31mpan2 675 . . . . . . . . . 10  |-  ( N  <  5  ->  N  <  6 )
3312, 30ltnei 9702 . . . . . . . . . 10  |-  ( N  <  6  ->  6  =/=  N )
3432, 33syl 17 . . . . . . . . 9  |-  ( N  <  5  ->  6  =/=  N )
3534necomd 2650 . . . . . . . 8  |-  ( N  <  5  ->  N  =/=  6 )
36 6lt8 10742 . . . . . . . . . 10  |-  6  <  8
3730, 17, 12lttri 9704 . . . . . . . . . 10  |-  ( ( 6  <  8  /\  8  <  N )  ->  6  <  N
)
3836, 37mpan 674 . . . . . . . . 9  |-  ( 8  <  N  ->  6  <  N )
3930, 12ltnei 9702 . . . . . . . . 9  |-  ( 6  <  N  ->  N  =/=  6 )
4038, 39syl 17 . . . . . . . 8  |-  ( 8  <  N  ->  N  =/=  6 )
4135, 40jaoi 380 . . . . . . 7  |-  ( ( N  <  5  \/  8  <  N )  ->  N  =/=  6
)
4211, 41ax-mp 5 . . . . . 6  |-  N  =/=  6
43 vscandx 15195 . . . . . . 7  |-  ( .s
`  ndx )  =  6
4424, 43neeq12i 2661 . . . . . 6  |-  ( ( E `  ndx )  =/=  ( .s `  ndx ) 
<->  N  =/=  6 )
4542, 44mpbir 212 . . . . 5  |-  ( E `
 ndx )  =/=  ( .s `  ndx )
4610, 45setsnid 15101 . . . 4  |-  ( E `
 ( W sSet  <. (Scalar `  ndx ) ,  ( Ws  S ) >. )
)  =  ( E `
 ( ( W sSet  <. (Scalar `  ndx ) ,  ( Ws  S ) >. ) sSet  <.
( .s `  ndx ) ,  ( .r `  W ) >. )
)
4712, 13, 17lttri 9704 . . . . . . . . . . 11  |-  ( ( N  <  5  /\  5  <  8 )  ->  N  <  8
)
4816, 47mpan2 675 . . . . . . . . . 10  |-  ( N  <  5  ->  N  <  8 )
4912, 17ltnei 9702 . . . . . . . . . 10  |-  ( N  <  8  ->  8  =/=  N )
5048, 49syl 17 . . . . . . . . 9  |-  ( N  <  5  ->  8  =/=  N )
5150necomd 2650 . . . . . . . 8  |-  ( N  <  5  ->  N  =/=  8 )
5217, 12ltnei 9702 . . . . . . . 8  |-  ( 8  <  N  ->  N  =/=  8 )
5351, 52jaoi 380 . . . . . . 7  |-  ( ( N  <  5  \/  8  <  N )  ->  N  =/=  8
)
5411, 53ax-mp 5 . . . . . 6  |-  N  =/=  8
55 ipndx 15202 . . . . . . 7  |-  ( .i
`  ndx )  =  8
5624, 55neeq12i 2661 . . . . . 6  |-  ( ( E `  ndx )  =/=  ( .i `  ndx ) 
<->  N  =/=  8 )
5754, 56mpbir 212 . . . . 5  |-  ( E `
 ndx )  =/=  ( .i `  ndx )
5810, 57setsnid 15101 . . . 4  |-  ( E `
 ( ( W sSet  <. (Scalar `  ndx ) ,  ( Ws  S ) >. ) sSet  <.
( .s `  ndx ) ,  ( .r `  W ) >. )
)  =  ( E `
 ( ( ( W sSet  <. (Scalar `  ndx ) ,  ( Ws  S
) >. ) sSet  <. ( .s `  ndx ) ,  ( .r `  W
) >. ) sSet  <. ( .i `  ndx ) ,  ( .r `  W
) >. ) )
5928, 46, 583eqtri 2448 . . 3  |-  ( E `
 W )  =  ( E `  (
( ( W sSet  <. (Scalar `  ndx ) ,  ( Ws  S ) >. ) sSet  <.
( .s `  ndx ) ,  ( .r `  W ) >. ) sSet  <.
( .i `  ndx ) ,  ( .r `  W ) >. )
)
607, 59syl6reqr 2475 . 2  |-  ( ( W  e.  _V  /\  ph )  ->  ( E `  W )  =  ( E `  A ) )
618str0 15097 . . 3  |-  (/)  =  ( E `  (/) )
62 fvprc 5812 . . . 4  |-  ( -.  W  e.  _V  ->  ( E `  W )  =  (/) )
6362adantr 466 . . 3  |-  ( ( -.  W  e.  _V  /\ 
ph )  ->  ( E `  W )  =  (/) )
64 fvprc 5812 . . . . . . 7  |-  ( -.  W  e.  _V  ->  (subringAlg  `  W )  =  (/) )
6564fveq1d 5820 . . . . . 6  |-  ( -.  W  e.  _V  ->  ( (subringAlg  `  W ) `  S )  =  (
(/) `  S )
)
66 0fv 5851 . . . . . 6  |-  ( (/) `  S )  =  (/)
6765, 66syl6eq 2472 . . . . 5  |-  ( -.  W  e.  _V  ->  ( (subringAlg  `  W ) `  S )  =  (/) )
681, 67sylan9eqr 2478 . . . 4  |-  ( ( -.  W  e.  _V  /\ 
ph )  ->  A  =  (/) )
6968fveq2d 5822 . . 3  |-  ( ( -.  W  e.  _V  /\ 
ph )  ->  ( E `  A )  =  ( E `  (/) ) )
7061, 63, 693eqtr4a 2482 . 2  |-  ( ( -.  W  e.  _V  /\ 
ph )  ->  ( E `  W )  =  ( E `  A ) )
7160, 70pm2.61ian 797 1  |-  ( ph  ->  ( E `  W
)  =  ( E `
 A ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 369    /\ wa 370    = wceq 1437    e. wcel 1872    =/= wne 2593   _Vcvv 3016    C_ wss 3372   (/)c0 3697   <.cop 3940   class class class wbr 4359   ` cfv 5537  (class class class)co 6242    < clt 9619   NNcn 10553   5c5 10606   6c6 10607   8c8 10609   ndxcnx 15054   sSet csts 15055  Slot cslot 15056   Basecbs 15057   ↾s cress 15058   .rcmulr 15127  Scalarcsca 15129   .scvsca 15130   .icip 15131  subringAlg csra 18327
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2058  ax-ext 2402  ax-rep 4472  ax-sep 4482  ax-nul 4491  ax-pow 4538  ax-pr 4596  ax-un 6534  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2274  df-mo 2275  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2552  df-ne 2595  df-nel 2596  df-ral 2713  df-rex 2714  df-reu 2715  df-rab 2717  df-v 3018  df-sbc 3236  df-csb 3332  df-dif 3375  df-un 3377  df-in 3379  df-ss 3386  df-pss 3388  df-nul 3698  df-if 3848  df-pw 3919  df-sn 3935  df-pr 3937  df-tp 3939  df-op 3941  df-uni 4156  df-iun 4237  df-br 4360  df-opab 4419  df-mpt 4420  df-tr 4455  df-eprel 4700  df-id 4704  df-po 4710  df-so 4711  df-fr 4748  df-we 4750  df-xp 4795  df-rel 4796  df-cnv 4797  df-co 4798  df-dm 4799  df-rn 4800  df-res 4801  df-ima 4802  df-pred 5335  df-ord 5381  df-on 5382  df-lim 5383  df-suc 5384  df-iota 5501  df-fun 5539  df-fn 5540  df-f 5541  df-f1 5542  df-fo 5543  df-f1o 5544  df-fv 5545  df-riota 6204  df-ov 6245  df-oprab 6246  df-mpt2 6247  df-om 6644  df-wrecs 6976  df-recs 7038  df-rdg 7076  df-er 7311  df-en 7518  df-dom 7519  df-sdom 7520  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9806  df-neg 9807  df-nn 10554  df-2 10612  df-3 10613  df-4 10614  df-5 10615  df-6 10616  df-7 10617  df-8 10618  df-ndx 15060  df-slot 15061  df-sets 15063  df-sca 15142  df-vsca 15143  df-ip 15144  df-sra 18331
This theorem is referenced by:  srabase  18337  sraaddg  18338  sramulr  18339  sratset  18343  srads  18345  cchhllem  24852
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