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Theorem evls1var 18487
Description: Univariate polynomial evaluation for subrings maps the variable to the identity function. (Contributed by AV, 13-Sep-2019.)
Hypotheses
Ref Expression
evls1var.q  |-  Q  =  ( S evalSub1  R )
evls1var.x  |-  X  =  (var1 `  U )
evls1var.u  |-  U  =  ( Ss  R )
evls1var.b  |-  B  =  ( Base `  S
)
evls1var.s  |-  ( ph  ->  S  e.  CRing )
evls1var.r  |-  ( ph  ->  R  e.  (SubRing `  S
) )
Assertion
Ref Expression
evls1var  |-  ( ph  ->  ( Q `  X
)  =  (  _I  |`  B ) )

Proof of Theorem evls1var
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 eqid 2382 . . . . 5  |-  (var1 `  S
)  =  (var1 `  S
)
2 evls1var.r . . . . 5  |-  ( ph  ->  R  e.  (SubRing `  S
) )
3 evls1var.u . . . . 5  |-  U  =  ( Ss  R )
41, 2, 3subrgvr1 18415 . . . 4  |-  ( ph  ->  (var1 `  S )  =  (var1 `  U ) )
5 evls1var.x . . . 4  |-  X  =  (var1 `  U )
64, 5syl6reqr 2442 . . 3  |-  ( ph  ->  X  =  (var1 `  S
) )
76fveq2d 5778 . 2  |-  ( ph  ->  ( Q `  X
)  =  ( Q `
 (var1 `  S ) ) )
8 eqid 2382 . . . . . 6  |-  ( ( 1o evalSub  S ) `  R
)  =  ( ( 1o evalSub  S ) `  R
)
9 eqid 2382 . . . . . 6  |-  ( 1o eval  S )  =  ( 1o eval  S )
10 eqid 2382 . . . . . 6  |-  ( 1o mVar  U )  =  ( 1o mVar  U )
11 evls1var.b . . . . . 6  |-  B  =  ( Base `  S
)
12 1on 7055 . . . . . . 7  |-  1o  e.  On
1312a1i 11 . . . . . 6  |-  ( ph  ->  1o  e.  On )
14 evls1var.s . . . . . 6  |-  ( ph  ->  S  e.  CRing )
15 0lt1o 7072 . . . . . . 7  |-  (/)  e.  1o
1615a1i 11 . . . . . 6  |-  ( ph  -> 
(/)  e.  1o )
178, 9, 10, 3, 11, 13, 14, 2, 16evlsvarsrng 18310 . . . . 5  |-  ( ph  ->  ( ( ( 1o evalSub  S ) `  R
) `  ( ( 1o mVar  U ) `  (/) ) )  =  ( ( 1o eval  S ) `  (
( 1o mVar  U ) `  (/) ) ) )
181vr1val 18344 . . . . . . 7  |-  (var1 `  S
)  =  ( ( 1o mVar  S ) `  (/) )
19 eqid 2382 . . . . . . . . 9  |-  ( 1o mVar  S )  =  ( 1o mVar  S )
2019, 13, 2, 3subrgmvr 18236 . . . . . . . 8  |-  ( ph  ->  ( 1o mVar  S )  =  ( 1o mVar  U
) )
2120fveq1d 5776 . . . . . . 7  |-  ( ph  ->  ( ( 1o mVar  S
) `  (/) )  =  ( ( 1o mVar  U
) `  (/) ) )
2218, 21syl5eq 2435 . . . . . 6  |-  ( ph  ->  (var1 `  S )  =  ( ( 1o mVar  U
) `  (/) ) )
2322fveq2d 5778 . . . . 5  |-  ( ph  ->  ( ( ( 1o evalSub  S ) `  R
) `  (var1 `  S
) )  =  ( ( ( 1o evalSub  S ) `
 R ) `  ( ( 1o mVar  U
) `  (/) ) ) )
2422fveq2d 5778 . . . . 5  |-  ( ph  ->  ( ( 1o eval  S
) `  (var1 `  S
) )  =  ( ( 1o eval  S ) `
 ( ( 1o mVar  U ) `  (/) ) ) )
2517, 23, 243eqtr4d 2433 . . . 4  |-  ( ph  ->  ( ( ( 1o evalSub  S ) `  R
) `  (var1 `  S
) )  =  ( ( 1o eval  S ) `
 (var1 `  S ) ) )
2625coeq1d 5077 . . 3  |-  ( ph  ->  ( ( ( ( 1o evalSub  S ) `  R
) `  (var1 `  S
) )  o.  (
y  e.  B  |->  ( 1o  X.  { y } ) ) )  =  ( ( ( 1o eval  S ) `  (var1 `  S ) )  o.  ( y  e.  B  |->  ( 1o  X.  {
y } ) ) ) )
27 eqid 2382 . . . . 5  |-  (Poly1 `  U
)  =  (Poly1 `  U
)
28 eqid 2382 . . . . . . 7  |-  (Poly1 `  ( Ss  R ) )  =  (Poly1 `  ( Ss  R ) )
29 eqid 2382 . . . . . . 7  |-  (PwSer1 `  ( Ss  R ) )  =  (PwSer1 `  ( Ss  R ) )
303fveq2i 5777 . . . . . . . 8  |-  (Poly1 `  U
)  =  (Poly1 `  ( Ss  R ) )
3130fveq2i 5777 . . . . . . 7  |-  ( Base `  (Poly1 `  U ) )  =  ( Base `  (Poly1 `  ( Ss  R ) ) )
3228, 29, 31ply1bas 18347 . . . . . 6  |-  ( Base `  (Poly1 `  U ) )  =  ( Base `  ( 1o mPoly  ( Ss  R ) ) )
3332eqcomi 2395 . . . . 5  |-  ( Base `  ( 1o mPoly  ( Ss  R
) ) )  =  ( Base `  (Poly1 `  U ) )
341, 2, 3, 27, 33subrgvr1cl 18416 . . . 4  |-  ( ph  ->  (var1 `  S )  e.  ( Base `  ( 1o mPoly  ( Ss  R ) ) ) )
35 evls1var.q . . . . 5  |-  Q  =  ( S evalSub1  R )
36 eqid 2382 . . . . 5  |-  ( 1o evalSub  S )  =  ( 1o evalSub  S )
37 eqid 2382 . . . . 5  |-  ( 1o mPoly 
( Ss  R ) )  =  ( 1o mPoly  ( Ss  R
) )
38 eqid 2382 . . . . 5  |-  ( Base `  ( 1o mPoly  ( Ss  R
) ) )  =  ( Base `  ( 1o mPoly  ( Ss  R ) ) )
3935, 36, 11, 37, 38evls1val 18470 . . . 4  |-  ( ( S  e.  CRing  /\  R  e.  (SubRing `  S )  /\  (var1 `  S )  e.  ( Base `  ( 1o mPoly  ( Ss  R ) ) ) )  ->  ( Q `  (var1 `  S ) )  =  ( ( ( ( 1o evalSub  S ) `  R ) `  (var1 `  S ) )  o.  ( y  e.  B  |->  ( 1o  X.  {
y } ) ) ) )
4014, 2, 34, 39syl3anc 1226 . . 3  |-  ( ph  ->  ( Q `  (var1 `  S ) )  =  ( ( ( ( 1o evalSub  S ) `  R
) `  (var1 `  S
) )  o.  (
y  e.  B  |->  ( 1o  X.  { y } ) ) ) )
41 crngring 17322 . . . . 5  |-  ( S  e.  CRing  ->  S  e.  Ring )
42 eqid 2382 . . . . . 6  |-  (Poly1 `  S
)  =  (Poly1 `  S
)
43 eqid 2382 . . . . . . . 8  |-  (PwSer1 `  S
)  =  (PwSer1 `  S
)
44 eqid 2382 . . . . . . . 8  |-  ( Base `  (Poly1 `  S ) )  =  ( Base `  (Poly1 `  S ) )
4542, 43, 44ply1bas 18347 . . . . . . 7  |-  ( Base `  (Poly1 `  S ) )  =  ( Base `  ( 1o mPoly  S ) )
4645eqcomi 2395 . . . . . 6  |-  ( Base `  ( 1o mPoly  S )
)  =  ( Base `  (Poly1 `  S ) )
471, 42, 46vr1cl 18371 . . . . 5  |-  ( S  e.  Ring  ->  (var1 `  S
)  e.  ( Base `  ( 1o mPoly  S )
) )
4814, 41, 473syl 20 . . . 4  |-  ( ph  ->  (var1 `  S )  e.  ( Base `  ( 1o mPoly  S ) ) )
49 eqid 2382 . . . . 5  |-  (eval1 `  S
)  =  (eval1 `  S
)
50 eqid 2382 . . . . 5  |-  ( 1o mPoly  S )  =  ( 1o mPoly  S )
51 eqid 2382 . . . . 5  |-  ( Base `  ( 1o mPoly  S )
)  =  ( Base `  ( 1o mPoly  S )
)
5249, 9, 11, 50, 51evl1val 18478 . . . 4  |-  ( ( S  e.  CRing  /\  (var1 `  S )  e.  (
Base `  ( 1o mPoly  S ) ) )  -> 
( (eval1 `  S ) `  (var1 `  S ) )  =  ( ( ( 1o eval  S ) `  (var1 `  S ) )  o.  ( y  e.  B  |->  ( 1o  X.  {
y } ) ) ) )
5314, 48, 52syl2anc 659 . . 3  |-  ( ph  ->  ( (eval1 `  S ) `  (var1 `  S ) )  =  ( ( ( 1o eval  S ) `  (var1 `  S ) )  o.  ( y  e.  B  |->  ( 1o  X.  {
y } ) ) ) )
5426, 40, 533eqtr4d 2433 . 2  |-  ( ph  ->  ( Q `  (var1 `  S ) )  =  ( (eval1 `  S ) `  (var1 `  S ) ) )
5549, 1, 11evl1var 18485 . . 3  |-  ( S  e.  CRing  ->  ( (eval1 `  S ) `  (var1 `  S ) )  =  (  _I  |`  B ) )
5614, 55syl 16 . 2  |-  ( ph  ->  ( (eval1 `  S ) `  (var1 `  S ) )  =  (  _I  |`  B ) )
577, 54, 563eqtrd 2427 1  |-  ( ph  ->  ( Q `  X
)  =  (  _I  |`  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1399    e. wcel 1826   (/)c0 3711   {csn 3944    |-> cmpt 4425    _I cid 4704   Oncon0 4792    X. cxp 4911    |` cres 4915    o. ccom 4917   ` cfv 5496  (class class class)co 6196   1oc1o 7041   Basecbs 14634   ↾s cress 14635   Ringcrg 17311   CRingccrg 17312  SubRingcsubrg 17538   mVar cmvr 18114   mPoly cmpl 18115   evalSub ces 18282   eval cevl 18283  PwSer1cps1 18327  var1cv1 18328  Poly1cpl1 18329   evalSub1 ces1 18463  eval1ce1 18464
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-inf2 7972  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-int 4200  df-iun 4245  df-iin 4246  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-se 4753  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-isom 5505  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-of 6439  df-ofr 6440  df-om 6600  df-1st 6699  df-2nd 6700  df-supp 6818  df-recs 6960  df-rdg 6994  df-1o 7048  df-2o 7049  df-oadd 7052  df-er 7229  df-map 7340  df-pm 7341  df-ixp 7389  df-en 7436  df-dom 7437  df-sdom 7438  df-fin 7439  df-fsupp 7745  df-sup 7816  df-oi 7850  df-card 8233  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-nn 10453  df-2 10511  df-3 10512  df-4 10513  df-5 10514  df-6 10515  df-7 10516  df-8 10517  df-9 10518  df-10 10519  df-n0 10713  df-z 10782  df-dec 10896  df-uz 11002  df-fz 11594  df-fzo 11718  df-seq 12011  df-hash 12308  df-struct 14636  df-ndx 14637  df-slot 14638  df-base 14639  df-sets 14640  df-ress 14641  df-plusg 14715  df-mulr 14716  df-sca 14718  df-vsca 14719  df-ip 14720  df-tset 14721  df-ple 14722  df-ds 14724  df-hom 14726  df-cco 14727  df-0g 14849  df-gsum 14850  df-prds 14855  df-pws 14857  df-mre 14993  df-mrc 14994  df-acs 14996  df-mgm 15989  df-sgrp 16028  df-mnd 16038  df-mhm 16083  df-submnd 16084  df-grp 16174  df-minusg 16175  df-sbg 16176  df-mulg 16177  df-subg 16315  df-ghm 16382  df-cntz 16472  df-cmn 16917  df-abl 16918  df-mgp 17255  df-ur 17267  df-srg 17271  df-ring 17313  df-cring 17314  df-rnghom 17477  df-subrg 17540  df-lmod 17627  df-lss 17692  df-lsp 17731  df-assa 18074  df-asp 18075  df-ascl 18076  df-psr 18118  df-mvr 18119  df-mpl 18120  df-opsr 18122  df-evls 18284  df-evl 18285  df-psr1 18332  df-vr1 18333  df-ply1 18334  df-evls1 18465  df-evl1 18466
This theorem is referenced by:  evls1varsrng  18489
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