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Theorem sralem 17599
Description: Lemma for srabase 17600 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 17598 . . . . . 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 2501 . . . 4  |-  ( ( W  e.  _V  /\  ph )  ->  A  =  ( ( ( W sSet  <. (Scalar `  ndx ) ,  ( Ws  S ) >. ) sSet  <.
( .s `  ndx ) ,  ( .r `  W ) >. ) sSet  <.
( .i `  ndx ) ,  ( .r `  W ) >. )
)
76fveq2d 5861 . . 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 14500 . . . . 5  |-  E  = Slot  ( E `  ndx )
11 sralem.3 . . . . . . 7  |-  ( N  <  5  \/  8  <  N )
129nnrei 10534 . . . . . . . . . 10  |-  N  e.  RR
13 5re 10603 . . . . . . . . . 10  |-  5  e.  RR
1412, 13ltnei 9697 . . . . . . . . 9  |-  ( N  <  5  ->  5  =/=  N )
1514necomd 2731 . . . . . . . 8  |-  ( N  <  5  ->  N  =/=  5 )
16 5lt8 10714 . . . . . . . . . 10  |-  5  <  8
17 8re 10609 . . . . . . . . . . 11  |-  8  e.  RR
1813, 17, 12lttri 9699 . . . . . . . . . 10  |-  ( ( 5  <  8  /\  8  <  N )  ->  5  <  N
)
1916, 18mpan 670 . . . . . . . . 9  |-  ( 8  <  N  ->  5  <  N )
2013, 12ltnei 9697 . . . . . . . . 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 14499 . . . . . . 7  |-  ( E `
 ndx )  =  N
25 scandx 14604 . . . . . . 7  |-  (Scalar `  ndx )  =  5
2624, 25neeq12i 2749 . . . . . 6  |-  ( ( E `  ndx )  =/=  (Scalar `  ndx )  <->  N  =/=  5 )
2723, 26mpbir 209 . . . . 5  |-  ( E `
 ndx )  =/=  (Scalar `  ndx )
2810, 27setsnid 14521 . . . 4  |-  ( E `
 W )  =  ( E `  ( W sSet  <. (Scalar `  ndx ) ,  ( Ws  S
) >. ) )
29 5lt6 10701 . . . . . . . . . . 11  |-  5  <  6
30 6re 10605 . . . . . . . . . . . 12  |-  6  e.  RR
3112, 13, 30lttri 9699 . . . . . . . . . . 11  |-  ( ( N  <  5  /\  5  <  6 )  ->  N  <  6
)
3229, 31mpan2 671 . . . . . . . . . 10  |-  ( N  <  5  ->  N  <  6 )
3312, 30ltnei 9697 . . . . . . . . . 10  |-  ( N  <  6  ->  6  =/=  N )
3432, 33syl 16 . . . . . . . . 9  |-  ( N  <  5  ->  6  =/=  N )
3534necomd 2731 . . . . . . . 8  |-  ( N  <  5  ->  N  =/=  6 )
36 6lt8 10713 . . . . . . . . . 10  |-  6  <  8
3730, 17, 12lttri 9699 . . . . . . . . . 10  |-  ( ( 6  <  8  /\  8  <  N )  ->  6  <  N
)
3836, 37mpan 670 . . . . . . . . 9  |-  ( 8  <  N  ->  6  <  N )
3930, 12ltnei 9697 . . . . . . . . 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 14606 . . . . . . 7  |-  ( .s
`  ndx )  =  6
4424, 43neeq12i 2749 . . . . . 6  |-  ( ( E `  ndx )  =/=  ( .s `  ndx ) 
<->  N  =/=  6 )
4542, 44mpbir 209 . . . . 5  |-  ( E `
 ndx )  =/=  ( .s `  ndx )
4610, 45setsnid 14521 . . . 4  |-  ( E `
 ( W sSet  <. (Scalar `  ndx ) ,  ( Ws  S ) >. )
)  =  ( E `
 ( ( W sSet  <. (Scalar `  ndx ) ,  ( Ws  S ) >. ) sSet  <.
( .s `  ndx ) ,  ( .r `  W ) >. )
)
4712, 13, 17lttri 9699 . . . . . . . . . . 11  |-  ( ( N  <  5  /\  5  <  8 )  ->  N  <  8
)
4816, 47mpan2 671 . . . . . . . . . 10  |-  ( N  <  5  ->  N  <  8 )
4912, 17ltnei 9697 . . . . . . . . . 10  |-  ( N  <  8  ->  8  =/=  N )
5048, 49syl 16 . . . . . . . . 9  |-  ( N  <  5  ->  8  =/=  N )
5150necomd 2731 . . . . . . . 8  |-  ( N  <  5  ->  N  =/=  8 )
5217, 12ltnei 9697 . . . . . . . 8  |-  ( 8  <  N  ->  N  =/=  8 )
5351, 52jaoi 379 . . . . . . 7  |-  ( ( N  <  5  \/  8  <  N )  ->  N  =/=  8
)
5411, 53ax-mp 5 . . . . . 6  |-  N  =/=  8
55 ipndx 14613 . . . . . . 7  |-  ( .i
`  ndx )  =  8
5624, 55neeq12i 2749 . . . . . 6  |-  ( ( E `  ndx )  =/=  ( .i `  ndx ) 
<->  N  =/=  8 )
5754, 56mpbir 209 . . . . 5  |-  ( E `
 ndx )  =/=  ( .i `  ndx )
5810, 57setsnid 14521 . . . 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 2493 . . 3  |-  ( E `
 W )  =  ( E `  (
( ( W sSet  <. (Scalar `  ndx ) ,  ( Ws  S ) >. ) sSet  <.
( .s `  ndx ) ,  ( .r `  W ) >. ) sSet  <.
( .i `  ndx ) ,  ( .r `  W ) >. )
)
607, 59syl6reqr 2520 . 2  |-  ( ( W  e.  _V  /\  ph )  ->  ( E `  W )  =  ( E `  A ) )
618str0 14517 . . 3  |-  (/)  =  ( E `  (/) )
62 fvprc 5851 . . . 4  |-  ( -.  W  e.  _V  ->  ( E `  W )  =  (/) )
6362adantr 465 . . 3  |-  ( ( -.  W  e.  _V  /\ 
ph )  ->  ( E `  W )  =  (/) )
64 fvprc 5851 . . . . . . 7  |-  ( -.  W  e.  _V  ->  (subringAlg  `  W )  =  (/) )
6564fveq1d 5859 . . . . . 6  |-  ( -.  W  e.  _V  ->  ( (subringAlg  `  W ) `  S )  =  (
(/) `  S )
)
66 0fv 5890 . . . . . 6  |-  ( (/) `  S )  =  (/)
6765, 66syl6eq 2517 . . . . 5  |-  ( -.  W  e.  _V  ->  ( (subringAlg  `  W ) `  S )  =  (/) )
681, 67sylan9eqr 2523 . . . 4  |-  ( ( -.  W  e.  _V  /\ 
ph )  ->  A  =  (/) )
6968fveq2d 5861 . . 3  |-  ( ( -.  W  e.  _V  /\ 
ph )  ->  ( E `  A )  =  ( E `  (/) ) )
7061, 63, 693eqtr4a 2527 . 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 1374    e. wcel 1762    =/= wne 2655   _Vcvv 3106    C_ wss 3469   (/)c0 3778   <.cop 4026   class class class wbr 4440   ` cfv 5579  (class class class)co 6275    < clt 9617   NNcn 10525   5c5 10577   6c6 10578   8c8 10580   ndxcnx 14476   sSet csts 14477  Slot cslot 14478   Basecbs 14479   ↾s cress 14480   .rcmulr 14545  Scalarcsca 14547   .scvsca 14548   .icip 14549  subringAlg csra 17590
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-recs 7032  df-rdg 7066  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-2 10583  df-3 10584  df-4 10585  df-5 10586  df-6 10587  df-7 10588  df-8 10589  df-ndx 14482  df-slot 14483  df-sets 14485  df-sca 14560  df-vsca 14561  df-ip 14562  df-sra 17594
This theorem is referenced by:  srabase  17600  sraaddg  17601  sramulr  17602  sratset  17606  srads  17608  cchhllem  23859
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