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Theorem 1elcpmat 19011
Description: The identity of the ring of all polynomial matrices over the ring  R is a constant polynomial matrix. (Contributed by AV, 16-Nov-2019.)
Hypotheses
Ref Expression
cpmatsrngpmat.s  |-  S  =  ( N ConstPolyMat  R )
cpmatsrngpmat.p  |-  P  =  (Poly1 `  R )
cpmatsrngpmat.c  |-  C  =  ( N Mat  P )
Assertion
Ref Expression
1elcpmat  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  -> 
( 1r `  C
)  e.  S )

Proof of Theorem 1elcpmat
Dummy variables  i 
j  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2467 . . . . . . . . . . 11  |-  ( Base `  R )  =  (
Base `  R )
2 eqid 2467 . . . . . . . . . . 11  |-  ( 1r
`  R )  =  ( 1r `  R
)
31, 2rngidcl 17020 . . . . . . . . . 10  |-  ( R  e.  Ring  ->  ( 1r
`  R )  e.  ( Base `  R
) )
43ancli 551 . . . . . . . . 9  |-  ( R  e.  Ring  ->  ( R  e.  Ring  /\  ( 1r `  R )  e.  ( Base `  R
) ) )
54adantl 466 . . . . . . . 8  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  -> 
( R  e.  Ring  /\  ( 1r `  R
)  e.  ( Base `  R ) ) )
65ad2antrl 727 . . . . . . 7  |-  ( ( i  =  j  /\  ( ( N  e. 
Fin  /\  R  e.  Ring )  /\  ( i  e.  N  /\  j  e.  N ) ) )  ->  ( R  e. 
Ring  /\  ( 1r `  R )  e.  (
Base `  R )
) )
7 eqid 2467 . . . . . . . 8  |-  ( 0g
`  R )  =  ( 0g `  R
)
8 cpmatsrngpmat.p . . . . . . . 8  |-  P  =  (Poly1 `  R )
9 eqid 2467 . . . . . . . 8  |-  ( Base `  P )  =  (
Base `  P )
10 eqid 2467 . . . . . . . 8  |-  (algSc `  P )  =  (algSc `  P )
111, 7, 8, 9, 10cply1coe0 18140 . . . . . . 7  |-  ( ( R  e.  Ring  /\  ( 1r `  R )  e.  ( Base `  R
) )  ->  A. n  e.  NN  ( (coe1 `  (
(algSc `  P ) `  ( 1r `  R
) ) ) `  n )  =  ( 0g `  R ) )
126, 11syl 16 . . . . . 6  |-  ( ( i  =  j  /\  ( ( N  e. 
Fin  /\  R  e.  Ring )  /\  ( i  e.  N  /\  j  e.  N ) ) )  ->  A. n  e.  NN  ( (coe1 `  ( (algSc `  P ) `  ( 1r `  R ) ) ) `  n )  =  ( 0g `  R ) )
13 iftrue 3945 . . . . . . . . . . 11  |-  ( i  =  j  ->  if ( i  =  j ,  ( (algSc `  P ) `  ( 1r `  R ) ) ,  ( (algSc `  P ) `  ( 0g `  R ) ) )  =  ( (algSc `  P ) `  ( 1r `  R ) ) )
1413fveq2d 5870 . . . . . . . . . 10  |-  ( i  =  j  ->  (coe1 `  if ( i  =  j ,  ( (algSc `  P ) `  ( 1r `  R ) ) ,  ( (algSc `  P ) `  ( 0g `  R ) ) ) )  =  (coe1 `  ( (algSc `  P
) `  ( 1r `  R ) ) ) )
1514fveq1d 5868 . . . . . . . . 9  |-  ( i  =  j  ->  (
(coe1 `  if ( i  =  j ,  ( (algSc `  P ) `  ( 1r `  R
) ) ,  ( (algSc `  P ) `  ( 0g `  R
) ) ) ) `
 n )  =  ( (coe1 `  ( (algSc `  P ) `  ( 1r `  R ) ) ) `  n ) )
1615eqeq1d 2469 . . . . . . . 8  |-  ( i  =  j  ->  (
( (coe1 `  if ( i  =  j ,  ( (algSc `  P ) `  ( 1r `  R
) ) ,  ( (algSc `  P ) `  ( 0g `  R
) ) ) ) `
 n )  =  ( 0g `  R
)  <->  ( (coe1 `  (
(algSc `  P ) `  ( 1r `  R
) ) ) `  n )  =  ( 0g `  R ) ) )
1716ralbidv 2903 . . . . . . 7  |-  ( i  =  j  ->  ( A. n  e.  NN  ( (coe1 `  if ( i  =  j ,  ( (algSc `  P ) `  ( 1r `  R
) ) ,  ( (algSc `  P ) `  ( 0g `  R
) ) ) ) `
 n )  =  ( 0g `  R
)  <->  A. n  e.  NN  ( (coe1 `  ( (algSc `  P ) `  ( 1r `  R ) ) ) `  n )  =  ( 0g `  R ) ) )
1817adantr 465 . . . . . 6  |-  ( ( i  =  j  /\  ( ( N  e. 
Fin  /\  R  e.  Ring )  /\  ( i  e.  N  /\  j  e.  N ) ) )  ->  ( A. n  e.  NN  ( (coe1 `  if ( i  =  j ,  ( (algSc `  P ) `  ( 1r `  R ) ) ,  ( (algSc `  P ) `  ( 0g `  R ) ) ) ) `  n
)  =  ( 0g
`  R )  <->  A. n  e.  NN  ( (coe1 `  (
(algSc `  P ) `  ( 1r `  R
) ) ) `  n )  =  ( 0g `  R ) ) )
1912, 18mpbird 232 . . . . 5  |-  ( ( i  =  j  /\  ( ( N  e. 
Fin  /\  R  e.  Ring )  /\  ( i  e.  N  /\  j  e.  N ) ) )  ->  A. n  e.  NN  ( (coe1 `  if ( i  =  j ,  ( (algSc `  P ) `  ( 1r `  R
) ) ,  ( (algSc `  P ) `  ( 0g `  R
) ) ) ) `
 n )  =  ( 0g `  R
) )
201, 7rng0cl 17021 . . . . . . . . . 10  |-  ( R  e.  Ring  ->  ( 0g
`  R )  e.  ( Base `  R
) )
2120ancli 551 . . . . . . . . 9  |-  ( R  e.  Ring  ->  ( R  e.  Ring  /\  ( 0g `  R )  e.  ( Base `  R
) ) )
2221adantl 466 . . . . . . . 8  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  -> 
( R  e.  Ring  /\  ( 0g `  R
)  e.  ( Base `  R ) ) )
231, 7, 8, 9, 10cply1coe0 18140 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  ( 0g `  R )  e.  ( Base `  R
) )  ->  A. n  e.  NN  ( (coe1 `  (
(algSc `  P ) `  ( 0g `  R
) ) ) `  n )  =  ( 0g `  R ) )
2422, 23syl 16 . . . . . . 7  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  A. n  e.  NN  ( (coe1 `  ( (algSc `  P ) `  ( 0g `  R ) ) ) `  n )  =  ( 0g `  R ) )
2524ad2antrl 727 . . . . . 6  |-  ( ( -.  i  =  j  /\  ( ( N  e.  Fin  /\  R  e.  Ring )  /\  (
i  e.  N  /\  j  e.  N )
) )  ->  A. n  e.  NN  ( (coe1 `  (
(algSc `  P ) `  ( 0g `  R
) ) ) `  n )  =  ( 0g `  R ) )
26 iffalse 3948 . . . . . . . . . . 11  |-  ( -.  i  =  j  ->  if ( i  =  j ,  ( (algSc `  P ) `  ( 1r `  R ) ) ,  ( (algSc `  P ) `  ( 0g `  R ) ) )  =  ( (algSc `  P ) `  ( 0g `  R ) ) )
2726adantr 465 . . . . . . . . . 10  |-  ( ( -.  i  =  j  /\  ( ( N  e.  Fin  /\  R  e.  Ring )  /\  (
i  e.  N  /\  j  e.  N )
) )  ->  if ( i  =  j ,  ( (algSc `  P ) `  ( 1r `  R ) ) ,  ( (algSc `  P ) `  ( 0g `  R ) ) )  =  ( (algSc `  P ) `  ( 0g `  R ) ) )
2827fveq2d 5870 . . . . . . . . 9  |-  ( ( -.  i  =  j  /\  ( ( N  e.  Fin  /\  R  e.  Ring )  /\  (
i  e.  N  /\  j  e.  N )
) )  ->  (coe1 `  if ( i  =  j ,  ( (algSc `  P ) `  ( 1r `  R ) ) ,  ( (algSc `  P ) `  ( 0g `  R ) ) ) )  =  (coe1 `  ( (algSc `  P
) `  ( 0g `  R ) ) ) )
2928fveq1d 5868 . . . . . . . 8  |-  ( ( -.  i  =  j  /\  ( ( N  e.  Fin  /\  R  e.  Ring )  /\  (
i  e.  N  /\  j  e.  N )
) )  ->  (
(coe1 `  if ( i  =  j ,  ( (algSc `  P ) `  ( 1r `  R
) ) ,  ( (algSc `  P ) `  ( 0g `  R
) ) ) ) `
 n )  =  ( (coe1 `  ( (algSc `  P ) `  ( 0g `  R ) ) ) `  n ) )
3029eqeq1d 2469 . . . . . . 7  |-  ( ( -.  i  =  j  /\  ( ( N  e.  Fin  /\  R  e.  Ring )  /\  (
i  e.  N  /\  j  e.  N )
) )  ->  (
( (coe1 `  if ( i  =  j ,  ( (algSc `  P ) `  ( 1r `  R
) ) ,  ( (algSc `  P ) `  ( 0g `  R
) ) ) ) `
 n )  =  ( 0g `  R
)  <->  ( (coe1 `  (
(algSc `  P ) `  ( 0g `  R
) ) ) `  n )  =  ( 0g `  R ) ) )
3130ralbidv 2903 . . . . . 6  |-  ( ( -.  i  =  j  /\  ( ( N  e.  Fin  /\  R  e.  Ring )  /\  (
i  e.  N  /\  j  e.  N )
) )  ->  ( A. n  e.  NN  ( (coe1 `  if ( i  =  j ,  ( (algSc `  P ) `  ( 1r `  R
) ) ,  ( (algSc `  P ) `  ( 0g `  R
) ) ) ) `
 n )  =  ( 0g `  R
)  <->  A. n  e.  NN  ( (coe1 `  ( (algSc `  P ) `  ( 0g `  R ) ) ) `  n )  =  ( 0g `  R ) ) )
3225, 31mpbird 232 . . . . 5  |-  ( ( -.  i  =  j  /\  ( ( N  e.  Fin  /\  R  e.  Ring )  /\  (
i  e.  N  /\  j  e.  N )
) )  ->  A. n  e.  NN  ( (coe1 `  if ( i  =  j ,  ( (algSc `  P ) `  ( 1r `  R ) ) ,  ( (algSc `  P ) `  ( 0g `  R ) ) ) ) `  n
)  =  ( 0g
`  R ) )
3319, 32pm2.61ian 788 . . . 4  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( i  e.  N  /\  j  e.  N
) )  ->  A. n  e.  NN  ( (coe1 `  if ( i  =  j ,  ( (algSc `  P ) `  ( 1r `  R ) ) ,  ( (algSc `  P ) `  ( 0g `  R ) ) ) ) `  n
)  =  ( 0g
`  R ) )
3433ralrimivva 2885 . . 3  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  A. i  e.  N  A. j  e.  N  A. n  e.  NN  ( (coe1 `  if ( i  =  j ,  ( (algSc `  P ) `  ( 1r `  R
) ) ,  ( (algSc `  P ) `  ( 0g `  R
) ) ) ) `
 n )  =  ( 0g `  R
) )
35 cpmatsrngpmat.c . . . . . . . . 9  |-  C  =  ( N Mat  P )
36 simpll 753 . . . . . . . . 9  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( i  e.  N  /\  j  e.  N
) )  ->  N  e.  Fin )
37 simplr 754 . . . . . . . . 9  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( i  e.  N  /\  j  e.  N
) )  ->  R  e.  Ring )
38 simprl 755 . . . . . . . . 9  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( i  e.  N  /\  j  e.  N
) )  ->  i  e.  N )
39 simprr 756 . . . . . . . . 9  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( i  e.  N  /\  j  e.  N
) )  ->  j  e.  N )
40 eqid 2467 . . . . . . . . 9  |-  ( 1r
`  C )  =  ( 1r `  C
)
418, 35, 10, 7, 2, 36, 37, 38, 39, 40pmat1ovscd 18996 . . . . . . . 8  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( i  e.  N  /\  j  e.  N
) )  ->  (
i ( 1r `  C ) j )  =  if ( i  =  j ,  ( (algSc `  P ) `  ( 1r `  R
) ) ,  ( (algSc `  P ) `  ( 0g `  R
) ) ) )
4241fveq2d 5870 . . . . . . 7  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( i  e.  N  /\  j  e.  N
) )  ->  (coe1 `  ( i ( 1r
`  C ) j ) )  =  (coe1 `  if ( i  =  j ,  ( (algSc `  P ) `  ( 1r `  R ) ) ,  ( (algSc `  P ) `  ( 0g `  R ) ) ) ) )
4342fveq1d 5868 . . . . . 6  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( i  e.  N  /\  j  e.  N
) )  ->  (
(coe1 `  ( i ( 1r `  C ) j ) ) `  n )  =  ( (coe1 `  if ( i  =  j ,  ( (algSc `  P ) `  ( 1r `  R
) ) ,  ( (algSc `  P ) `  ( 0g `  R
) ) ) ) `
 n ) )
4443eqeq1d 2469 . . . . 5  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( i  e.  N  /\  j  e.  N
) )  ->  (
( (coe1 `  ( i ( 1r `  C ) j ) ) `  n )  =  ( 0g `  R )  <-> 
( (coe1 `  if ( i  =  j ,  ( (algSc `  P ) `  ( 1r `  R
) ) ,  ( (algSc `  P ) `  ( 0g `  R
) ) ) ) `
 n )  =  ( 0g `  R
) ) )
4544ralbidv 2903 . . . 4  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( i  e.  N  /\  j  e.  N
) )  ->  ( A. n  e.  NN  ( (coe1 `  ( i ( 1r `  C ) j ) ) `  n )  =  ( 0g `  R )  <->  A. n  e.  NN  ( (coe1 `  if ( i  =  j ,  ( (algSc `  P ) `  ( 1r `  R
) ) ,  ( (algSc `  P ) `  ( 0g `  R
) ) ) ) `
 n )  =  ( 0g `  R
) ) )
46452ralbidva 2906 . . 3  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  -> 
( A. i  e.  N  A. j  e.  N  A. n  e.  NN  ( (coe1 `  (
i ( 1r `  C ) j ) ) `  n )  =  ( 0g `  R )  <->  A. i  e.  N  A. j  e.  N  A. n  e.  NN  ( (coe1 `  if ( i  =  j ,  ( (algSc `  P ) `  ( 1r `  R ) ) ,  ( (algSc `  P ) `  ( 0g `  R ) ) ) ) `  n
)  =  ( 0g
`  R ) ) )
4734, 46mpbird 232 . 2  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  A. i  e.  N  A. j  e.  N  A. n  e.  NN  ( (coe1 `  ( i ( 1r `  C ) j ) ) `  n )  =  ( 0g `  R ) )
488, 35pmatrng 18989 . . . 4  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  C  e.  Ring )
49 eqid 2467 . . . . 5  |-  ( Base `  C )  =  (
Base `  C )
5049, 40rngidcl 17020 . . . 4  |-  ( C  e.  Ring  ->  ( 1r
`  C )  e.  ( Base `  C
) )
5148, 50syl 16 . . 3  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  -> 
( 1r `  C
)  e.  ( Base `  C ) )
52 cpmatsrngpmat.s . . . 4  |-  S  =  ( N ConstPolyMat  R )
5352, 8, 35, 49cpmatel 19007 . . 3  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  ( 1r `  C )  e.  ( Base `  C
) )  ->  (
( 1r `  C
)  e.  S  <->  A. i  e.  N  A. j  e.  N  A. n  e.  NN  ( (coe1 `  (
i ( 1r `  C ) j ) ) `  n )  =  ( 0g `  R ) ) )
5451, 53mpd3an3 1325 . 2  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  -> 
( ( 1r `  C )  e.  S  <->  A. i  e.  N  A. j  e.  N  A. n  e.  NN  (
(coe1 `  ( i ( 1r `  C ) j ) ) `  n )  =  ( 0g `  R ) ) )
5547, 54mpbird 232 1  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  -> 
( 1r `  C
)  e.  S )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2814   ifcif 3939   ` cfv 5588  (class class class)co 6284   Fincfn 7516   NNcn 10536   Basecbs 14490   0gc0g 14695   1rcur 16955   Ringcrg 17000  algSccascl 17759  Poly1cpl1 18015  coe1cco1 18016   Mat cmat 18704   ConstPolyMat ccpmat 18999
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-inf2 8058  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-ot 4036  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-of 6524  df-ofr 6525  df-om 6685  df-1st 6784  df-2nd 6785  df-supp 6902  df-recs 7042  df-rdg 7076  df-1o 7130  df-2o 7131  df-oadd 7134  df-er 7311  df-map 7422  df-pm 7423  df-ixp 7470  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-fsupp 7830  df-sup 7901  df-oi 7935  df-card 8320  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-nn 10537  df-2 10594  df-3 10595  df-4 10596  df-5 10597  df-6 10598  df-7 10599  df-8 10600  df-9 10601  df-10 10602  df-n0 10796  df-z 10865  df-dec 10977  df-uz 11083  df-fz 11673  df-fzo 11793  df-seq 12076  df-hash 12374  df-struct 14492  df-ndx 14493  df-slot 14494  df-base 14495  df-sets 14496  df-ress 14497  df-plusg 14568  df-mulr 14569  df-sca 14571  df-vsca 14572  df-ip 14573  df-tset 14574  df-ple 14575  df-ds 14577  df-hom 14579  df-cco 14580  df-0g 14697  df-gsum 14698  df-prds 14703  df-pws 14705  df-mre 14841  df-mrc 14842  df-acs 14844  df-mnd 15732  df-mhm 15786  df-submnd 15787  df-grp 15867  df-minusg 15868  df-sbg 15869  df-mulg 15870  df-subg 16003  df-ghm 16070  df-cntz 16160  df-cmn 16606  df-abl 16607  df-mgp 16944  df-ur 16956  df-rng 17002  df-subrg 17227  df-lmod 17314  df-lss 17379  df-sra 17618  df-rgmod 17619  df-ascl 17762  df-psr 17804  df-mvr 17805  df-mpl 17806  df-opsr 17808  df-psr1 18018  df-vr1 18019  df-ply1 18020  df-coe1 18021  df-dsmm 18558  df-frlm 18573  df-mamu 18681  df-mat 18705  df-cpmat 19002
This theorem is referenced by:  cpmatsubgpmat  19016  cpmatsrgpmat  19017
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