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Theorem sralem 17382
Description: Lemma for srabase 17383 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 466 . . . . 5  |-  ( ( W  e.  _V  /\  ph )  ->  A  =  ( (subringAlg  `  W ) `  S ) )
3 srapart.s . . . . . 6  |-  ( ph  ->  S  C_  ( Base `  W ) )
4 sraval 17381 . . . . . 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 5804 . . 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 14314 . . . . 5  |-  E  = Slot  ( E `  ndx )
11 sralem.3 . . . . . . 7  |-  ( N  <  5  \/  8  <  N )
129nnrei 10443 . . . . . . . . . 10  |-  N  e.  RR
13 5re 10512 . . . . . . . . . 10  |-  5  e.  RR
1412, 13ltnei 9610 . . . . . . . . 9  |-  ( N  <  5  ->  5  =/=  N )
1514necomd 2723 . . . . . . . 8  |-  ( N  <  5  ->  N  =/=  5 )
16 5lt8 10623 . . . . . . . . . 10  |-  5  <  8
17 8re 10518 . . . . . . . . . . 11  |-  8  e.  RR
1813, 17, 12lttri 9612 . . . . . . . . . 10  |-  ( ( 5  <  8  /\  8  <  N )  ->  5  <  N
)
1916, 18mpan 670 . . . . . . . . 9  |-  ( 8  <  N  ->  5  <  N )
2013, 12ltnei 9610 . . . . . . . . 9  |-  ( 5  <  N  ->  N  =/=  5 )
2119, 20syl 16 . . . . . . . 8  |-  ( 8  <  N  ->  N  =/=  5 )
2215, 21jaoi 379 . . . . . . 7  |-  ( ( N  <  5  \/  8  <  N )  ->  N  =/=  5
)
2311, 22ax-mp 5 . . . . . 6  |-  N  =/=  5
248, 9ndxarg 14313 . . . . . . 7  |-  ( E `
 ndx )  =  N
25 scandx 14418 . . . . . . 7  |-  (Scalar `  ndx )  =  5
2624, 25neeq12i 2741 . . . . . 6  |-  ( ( E `  ndx )  =/=  (Scalar `  ndx )  <->  N  =/=  5 )
2723, 26mpbir 209 . . . . 5  |-  ( E `
 ndx )  =/=  (Scalar `  ndx )
2810, 27setsnid 14335 . . . 4  |-  ( E `
 W )  =  ( E `  ( W sSet  <. (Scalar `  ndx ) ,  ( Ws  S
) >. ) )
29 5lt6 10610 . . . . . . . . . . 11  |-  5  <  6
30 6re 10514 . . . . . . . . . . . 12  |-  6  e.  RR
3112, 13, 30lttri 9612 . . . . . . . . . . 11  |-  ( ( N  <  5  /\  5  <  6 )  ->  N  <  6
)
3229, 31mpan2 671 . . . . . . . . . 10  |-  ( N  <  5  ->  N  <  6 )
3312, 30ltnei 9610 . . . . . . . . . 10  |-  ( N  <  6  ->  6  =/=  N )
3432, 33syl 16 . . . . . . . . 9  |-  ( N  <  5  ->  6  =/=  N )
3534necomd 2723 . . . . . . . 8  |-  ( N  <  5  ->  N  =/=  6 )
36 6lt8 10622 . . . . . . . . . 10  |-  6  <  8
3730, 17, 12lttri 9612 . . . . . . . . . 10  |-  ( ( 6  <  8  /\  8  <  N )  ->  6  <  N
)
3836, 37mpan 670 . . . . . . . . 9  |-  ( 8  <  N  ->  6  <  N )
3930, 12ltnei 9610 . . . . . . . . 9  |-  ( 6  <  N  ->  N  =/=  6 )
4038, 39syl 16 . . . . . . . 8  |-  ( 8  <  N  ->  N  =/=  6 )
4135, 40jaoi 379 . . . . . . 7  |-  ( ( N  <  5  \/  8  <  N )  ->  N  =/=  6
)
4211, 41ax-mp 5 . . . . . 6  |-  N  =/=  6
43 vscandx 14420 . . . . . . 7  |-  ( .s
`  ndx )  =  6
4424, 43neeq12i 2741 . . . . . 6  |-  ( ( E `  ndx )  =/=  ( .s `  ndx ) 
<->  N  =/=  6 )
4542, 44mpbir 209 . . . . 5  |-  ( E `
 ndx )  =/=  ( .s `  ndx )
4610, 45setsnid 14335 . . . 4  |-  ( E `
 ( W sSet  <. (Scalar `  ndx ) ,  ( Ws  S ) >. )
)  =  ( E `
 ( ( W sSet  <. (Scalar `  ndx ) ,  ( Ws  S ) >. ) sSet  <.
( .s `  ndx ) ,  ( .r `  W ) >. )
)
4712, 13, 17lttri 9612 . . . . . . . . . . 11  |-  ( ( N  <  5  /\  5  <  8 )  ->  N  <  8
)
4816, 47mpan2 671 . . . . . . . . . 10  |-  ( N  <  5  ->  N  <  8 )
4912, 17ltnei 9610 . . . . . . . . . 10  |-  ( N  <  8  ->  8  =/=  N )
5048, 49syl 16 . . . . . . . . 9  |-  ( N  <  5  ->  8  =/=  N )
5150necomd 2723 . . . . . . . 8  |-  ( N  <  5  ->  N  =/=  8 )
5217, 12ltnei 9610 . . . . . . . 8  |-  ( 8  <  N  ->  N  =/=  8 )
5351, 52jaoi 379 . . . . . . 7  |-  ( ( N  <  5  \/  8  <  N )  ->  N  =/=  8
)
5411, 53ax-mp 5 . . . . . 6  |-  N  =/=  8
55 ipndx 14427 . . . . . . 7  |-  ( .i
`  ndx )  =  8
5624, 55neeq12i 2741 . . . . . 6  |-  ( ( E `  ndx )  =/=  ( .i `  ndx ) 
<->  N  =/=  8 )
5754, 56mpbir 209 . . . . 5  |-  ( E `
 ndx )  =/=  ( .i `  ndx )
5810, 57setsnid 14335 . . . 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 2487 . . 3  |-  ( E `
 W )  =  ( E `  (
( ( W sSet  <. (Scalar `  ndx ) ,  ( Ws  S ) >. ) sSet  <.
( .s `  ndx ) ,  ( .r `  W ) >. ) sSet  <.
( .i `  ndx ) ,  ( .r `  W ) >. )
)
607, 59syl6reqr 2514 . 2  |-  ( ( W  e.  _V  /\  ph )  ->  ( E `  W )  =  ( E `  A ) )
618str0 14331 . . 3  |-  (/)  =  ( E `  (/) )
62 fvprc 5794 . . . 4  |-  ( -.  W  e.  _V  ->  ( E `  W )  =  (/) )
6362adantr 465 . . 3  |-  ( ( -.  W  e.  _V  /\ 
ph )  ->  ( E `  W )  =  (/) )
64 fvprc 5794 . . . . . . 7  |-  ( -.  W  e.  _V  ->  (subringAlg  `  W )  =  (/) )
6564fveq1d 5802 . . . . . 6  |-  ( -.  W  e.  _V  ->  ( (subringAlg  `  W ) `  S )  =  (
(/) `  S )
)
66 0fv 5833 . . . . . 6  |-  ( (/) `  S )  =  (/)
6765, 66syl6eq 2511 . . . . 5  |-  ( -.  W  e.  _V  ->  ( (subringAlg  `  W ) `  S )  =  (/) )
681, 67sylan9eqr 2517 . . . 4  |-  ( ( -.  W  e.  _V  /\ 
ph )  ->  A  =  (/) )
6968fveq2d 5804 . . 3  |-  ( ( -.  W  e.  _V  /\ 
ph )  ->  ( E `  A )  =  ( E `  (/) ) )
7061, 63, 693eqtr4a 2521 . 2  |-  ( ( -.  W  e.  _V  /\ 
ph )  ->  ( E `  W )  =  ( E `  A ) )
7160, 70pm2.61ian 788 1  |-  ( ph  ->  ( E `  W
)  =  ( E `
 A ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2648   _Vcvv 3078    C_ wss 3437   (/)c0 3746   <.cop 3992   class class class wbr 4401   ` cfv 5527  (class class class)co 6201    < clt 9530   NNcn 10434   5c5 10486   6c6 10487   8c8 10489   ndxcnx 14290   sSet csts 14291  Slot cslot 14292   Basecbs 14293   ↾s cress 14294   .rcmulr 14359  Scalarcsca 14361   .scvsca 14362   .icip 14363  subringAlg csra 17373
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 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-cnex 9450  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470  ax-pre-mulgt0 9471
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 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-om 6588  df-recs 6943  df-rdg 6977  df-er 7212  df-en 7422  df-dom 7423  df-sdom 7424  df-pnf 9532  df-mnf 9533  df-xr 9534  df-ltxr 9535  df-le 9536  df-sub 9709  df-neg 9710  df-nn 10435  df-2 10492  df-3 10493  df-4 10494  df-5 10495  df-6 10496  df-7 10497  df-8 10498  df-ndx 14296  df-slot 14297  df-sets 14299  df-sca 14374  df-vsca 14375  df-ip 14376  df-sra 17377
This theorem is referenced by:  srabase  17383  sraaddg  17384  sramulr  17385  sratset  17389  srads  17391  cchhllem  23286
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