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Theorem evls1sca 18989
Description: Univariate polynomial evaluation maps scalars to constant functions. (Contributed by AV, 8-Sep-2019.)
Hypotheses
Ref Expression
evls1sca.q |- Q = ( S evalSub1 R )
evls1sca.w |- W = (Poly1 ` U )
evls1sca.u |- U = ( S ↾s R )
evls1sca.b |- B = ( Base ` S )
evls1sca.a |- A = (algSc ` W )
evls1sca.s |- ( ph -> S e. CRing )
evls1sca.r |- ( ph -> R e. (SubRing ` S ) )
evls1sca.x |- ( ph -> X e. R )
Assertion
Ref Expression
evls1sca |- ( ph -> ( Q ` ( A ` X ) ) = ( B X. { X } ) )
Proof of Theorem evls1sca
Dummy variables x y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 1on 7207 . . . . . . 7 |- 1o e. On
21a1i 11 . . . . . 6 |- ( ph -> 1o e. On )
3 evls1sca.s . . . . . 6 |- ( ph -> S e. CRing )
4 evls1sca.r . . . . . 6 |- ( ph -> R e. (SubRing ` S ) )
5 eqid 2471 . . . . . . 7 |- ( ( 1o evalSub S ) ` R ) = ( ( 1o evalSub S ) ` R )
6 eqid 2471 . . . . . . 7 |- ( 1o mPoly U ) = ( 1o mPoly U )
7 evls1sca.u . . . . . . 7 |- U = ( S ↾s R )
8 eqid 2471 . . . . . . 7 |- ( S ^s ( B ^m 1o ) ) = ( S ^s ( B ^m 1o ) )
9 evls1sca.b . . . . . . 7 |- B = ( Base ` S )
105, 6, 7, 8, 9evlsrhm 18821 . . . . . 6 |- ( ( 1o e. On /\ S e. CRing /\ R e. (SubRing ` S ) ) -> ( ( 1o evalSub S ) ` R ) e. ( ( 1o mPoly U ) RingHom ( S ^s ( B ^m 1o ) ) ) )
112, 3, 4, 10syl3anc 1292 . . . . 5 |- ( ph -> ( ( 1o evalSub S ) ` R ) e. ( ( 1o mPoly U ) RingHom ( S ^s ( B ^m 1o ) ) ) )
12 eqid 2471 . . . . . 6 |- ( Base ` ( 1o mPoly U ) ) = ( Base ` ( 1o mPoly U ) )
13 eqid 2471 . . . . . 6 |- ( Base ` ( S ^s ( B ^m 1o ) ) ) = ( Base ` ( S ^s ( B ^m 1o ) ) )
1412, 13rhmf 18032 . . . . 5 |- ( ( ( 1o evalSub S ) ` R ) e. ( ( 1o mPoly U ) RingHom ( S ^s ( B ^m 1o ) ) ) -> ( ( 1o evalSub S ) ` R ) : ( Base ` ( 1o mPoly U ) ) --> ( Base ` ( S ^s ( B ^m 1o ) ) ) )
1511, 14syl 17 . . . 4 |- ( ph -> ( ( 1o evalSub S ) ` R ) : ( Base ` ( 1o mPoly U ) ) --> ( Base ` ( S ^s ( B ^m 1o ) ) ) )
16 evls1sca.a . . . . . . 7 |- A = (algSc ` W )
17 eqid 2471 . . . . . . 7 |- (Scalar ` W ) = (Scalar ` W )
187subrgring 18089 . . . . . . . . 9 |- ( R e. (SubRing ` S ) -> U e. Ring )
194, 18syl 17 . . . . . . . 8 |- ( ph -> U e. Ring )
20 evls1sca.w . . . . . . . . 9 |- W = (Poly1 ` U )
2120ply1ring 18918 . . . . . . . 8 |- ( U e. Ring -> W e. Ring )
2219, 21syl 17 . . . . . . 7 |- ( ph -> W e. Ring )
2320ply1lmod 18922 . . . . . . . 8 |- ( U e. Ring -> W e. LMod )
2419, 23syl 17 . . . . . . 7 |- ( ph -> W e. LMod )
25 eqid 2471 . . . . . . 7 |- ( Base ` (Scalar ` W ) ) = ( Base ` (Scalar ` W ) )
26 eqid 2471 . . . . . . 7 |- ( Base ` W ) = ( Base ` W )
2716, 17, 22, 24, 25, 26asclf 18638 . . . . . 6 |- ( ph -> A : ( Base ` (Scalar ` W ) ) --> ( Base ` W ) )
289subrgss 18087 . . . . . . . . . 10 |- ( R e. (SubRing ` S ) -> R C_ B )
294, 28syl 17 . . . . . . . . 9 |- ( ph -> R C_ B )
307, 9ressbas2 15258 . . . . . . . . 9 |- ( R C_ B -> R = ( Base ` U ) )
3129, 30syl 17 . . . . . . . 8 |- ( ph -> R = ( Base ` U ) )
3220ply1sca 18923 . . . . . . . . . 10 |- ( U e. Ring -> U = (Scalar ` W ) )
3319, 32syl 17 . . . . . . . . 9 |- ( ph -> U = (Scalar ` W ) )
3433fveq2d 5883 . . . . . . . 8 |- ( ph -> ( Base ` U ) = ( Base ` (Scalar ` W ) ) )
3531, 34eqtrd 2505 . . . . . . 7 |- ( ph -> R = ( Base ` (Scalar ` W ) ) )
36 eqid 2471 . . . . . . . . . 10 |- (PwSer1 ` U ) = (PwSer1 ` U )
3720, 36, 26ply1bas 18865 . . . . . . . . 9 |- ( Base ` W ) = ( Base ` ( 1o mPoly U ) )
3837a1i 11 . . . . . . . 8 |- ( ph -> ( Base ` W ) = ( Base ` ( 1o mPoly U ) ) )
3938eqcomd 2477 . . . . . . 7 |- ( ph -> ( Base ` ( 1o mPoly U ) ) = ( Base ` W ) )
4035, 39feq23d 5734 . . . . . 6 |- ( ph -> ( A : R --> ( Base ` ( 1o mPoly U ) ) <-> A : ( Base ` (Scalar ` W ) ) --> ( Base ` W ) ) )
4127, 40mpbird 240 . . . . 5 |- ( ph -> A : R --> ( Base ` ( 1o mPoly U ) ) )
42 evls1sca.x . . . . 5 |- ( ph -> X e. R )
4341, 42ffvelrnd 6038 . . . 4 |- ( ph -> ( A ` X ) e. ( Base ` ( 1o mPoly U ) ) )
44 fvco3 5957 . . . 4 |- ( ( ( ( 1o evalSub S ) ` R ) : ( Base ` ( 1o mPoly U ) ) --> ( Base ` ( S ^s ( B ^m 1o ) ) ) /\ ( A ` X ) e. ( Base ` ( 1o mPoly U ) ) ) -> ( ( ( x e. ( B ^m ( B ^m 1o ) ) |-> ( x o. ( y e. B |-> ( 1o X. { y } ) ) ) ) o. ( ( 1o evalSub S ) ` R ) ) ` ( A ` X ) ) = ( ( x e. ( B ^m ( B ^m 1o ) ) |-> ( x o. ( y e. B |-> ( 1o X. { y } ) ) ) ) ` ( ( ( 1o evalSub S ) ` R ) ` ( A ` X ) ) ) )
4515, 43, 44syl2anc 673 . . 3 |- ( ph -> ( ( ( x e. ( B ^m ( B ^m 1o ) ) |-> ( x o. ( y e. B |-> ( 1o X. { y } ) ) ) ) o. ( ( 1o evalSub S ) ` R ) ) ` ( A ` X ) ) = ( ( x e. ( B ^m ( B ^m 1o ) ) |-> ( x o. ( y e. B |-> ( 1o X. { y } ) ) ) ) ` ( ( ( 1o evalSub S ) ` R ) ` ( A ` X ) ) ) )
4616a1i 11 . . . . . . . 8 |- ( ph -> A = (algSc ` W ) )
47 eqid 2471 . . . . . . . . 9 |- (algSc ` W ) = (algSc ` W )
4820, 47ply1ascl 18928 . . . . . . . 8 |- (algSc ` W ) = (algSc ` ( 1o mPoly U ) )
4946, 48syl6eq 2521 . . . . . . 7 |- ( ph -> A = (algSc ` ( 1o mPoly U ) ) )
5049fveq1d 5881 . . . . . 6 |- ( ph -> ( A ` X ) = ( (algSc ` ( 1o mPoly U ) ) ` X ) )
5150fveq2d 5883 . . . . 5 |- ( ph -> ( ( ( 1o evalSub S ) ` R ) ` ( A ` X ) ) = ( ( ( 1o evalSub S ) ` R ) ` ( (algSc ` ( 1o mPoly U ) ) ` X ) ) )
52 eqid 2471 . . . . . 6 |- (algSc ` ( 1o mPoly U ) ) = (algSc ` ( 1o mPoly U ) )
535, 6, 7, 9, 52, 2, 3, 4, 42evlssca 18822 . . . . 5 |- ( ph -> ( ( ( 1o evalSub S ) ` R ) ` ( (algSc ` ( 1o mPoly U ) ) ` X ) ) = ( ( B ^m 1o ) X. { X } ) )
5451, 53eqtrd 2505 . . . 4 |- ( ph -> ( ( ( 1o evalSub S ) ` R ) ` ( A ` X ) ) = ( ( B ^m 1o ) X. { X } ) )
5554fveq2d 5883 . . 3 |- ( ph -> ( ( x e. ( B ^m ( B ^m 1o ) ) |-> ( x o. ( y e. B |-> ( 1o X. { y } ) ) ) ) ` ( ( ( 1o evalSub S ) ` R ) ` ( A ` X ) ) ) = ( ( x e. ( B ^m ( B ^m 1o ) ) |-> ( x o. ( y e. B |-> ( 1o X. { y } ) ) ) ) ` ( ( B ^m 1o ) X. { X } ) ) )
56 eqidd 2472 . . . . 5 |- ( ph -> ( x e. ( B ^m ( B ^m 1o ) ) |-> ( x o. ( y e. B |-> ( 1o X. { y } ) ) ) ) = ( x e. ( B ^m ( B ^m 1o ) ) |-> ( x o. ( y e. B |-> ( 1o X. { y } ) ) ) ) )
57 coeq1 4997 . . . . . 6 |- ( x = ( ( B ^m 1o ) X. { X } ) -> ( x o. ( y e. B |-> ( 1o X. { y } ) ) ) = ( ( ( B ^m 1o ) X. { X } ) o. ( y e. B |-> ( 1o X. { y } ) ) ) )
5857adantl 473 . . . . 5 |- ( ( ph /\ x = ( ( B ^m 1o ) X. { X } ) ) -> ( x o. ( y e. B |-> ( 1o X. { y } ) ) ) = ( ( ( B ^m 1o ) X. { X } ) o. ( y e. B |-> ( 1o X. { y } ) ) ) )
5929, 42sseldd 3419 . . . . . . 7 |- ( ph -> X e. B )
60 fconst6g 5785 . . . . . . 7 |- ( X e. B -> ( ( B ^m 1o ) X. { X } ) : ( B ^m 1o ) --> B )
6159, 60syl 17 . . . . . 6 |- ( ph -> ( ( B ^m 1o ) X. { X } ) : ( B ^m 1o ) --> B )
62 fvex 5889 . . . . . . . . 9 |- ( Base ` S ) e. _V
639, 62eqeltri 2545 . . . . . . . 8 |- B e. _V
6463a1i 11 . . . . . . 7 |- ( ph -> B e. _V )
65 ovex 6336 . . . . . . . 8 |- ( B ^m 1o ) e. _V
6665a1i 11 . . . . . . 7 |- ( ph -> ( B ^m 1o ) e. _V )
6764, 66elmapd 7504 . . . . . 6 |- ( ph -> ( ( ( B ^m 1o ) X. { X } ) e. ( B ^m ( B ^m 1o ) ) <-> ( ( B ^m 1o ) X. { X } ) : ( B ^m 1o ) --> B ) )
6861, 67mpbird 240 . . . . 5 |- ( ph -> ( ( B ^m 1o ) X. { X } ) e. ( B ^m ( B ^m 1o ) ) )
69 snex 4641 . . . . . . . 8 |- { X } e. _V
7065, 69xpex 6614 . . . . . . 7 |- ( ( B ^m 1o ) X. { X } ) e. _V
7170a1i 11 . . . . . 6 |- ( ph -> ( ( B ^m 1o ) X. { X } ) e. _V )
72 mptexg 6151 . . . . . . 7 |- ( B e. _V -> ( y e. B |-> ( 1o X. { y } ) ) e. _V )
7364, 72syl 17 . . . . . 6 |- ( ph -> ( y e. B |-> ( 1o X. { y } ) ) e. _V )
74 coexg 6763 . . . . . 6 |- ( ( ( ( B ^m 1o ) X. { X } ) e. _V /\ ( y e. B |-> ( 1o X. { y } ) ) e. _V ) -> ( ( ( B ^m 1o ) X. { X } ) o. ( y e. B |-> ( 1o X. { y } ) ) ) e. _V )
7571, 73, 74syl2anc 673 . . . . 5 |- ( ph -> ( ( ( B ^m 1o ) X. { X } ) o. ( y e. B |-> ( 1o X. { y } ) ) ) e. _V )
7656, 58, 68, 75fvmptd 5969 . . . 4 |- ( ph -> ( ( x e. ( B ^m ( B ^m 1o ) ) |-> ( x o. ( y e. B |-> ( 1o X. { y } ) ) ) ) ` ( ( B ^m 1o ) X. { X } ) ) = ( ( ( B ^m 1o ) X. { X } ) o. ( y e. B |-> ( 1o X. { y } ) ) ) )
77 fconst6g 5785 . . . . . . 7 |- ( y e. B -> ( 1o X. { y } ) : 1o --> B )
7877adantl 473 . . . . . 6 |- ( ( ph /\ y e. B ) -> ( 1o X. { y } ) : 1o --> B )
7963, 1pm3.2i 462 . . . . . . . 8 |- ( B e. _V /\ 1o e. On )
8079a1i 11 . . . . . . 7 |- ( ( ph /\ y e. B ) -> ( B e. _V /\ 1o e. On ) )
81 elmapg 7503 . . . . . . 7 |- ( ( B e. _V /\ 1o e. On ) -> ( ( 1o X. { y } ) e. ( B ^m 1o ) <-> ( 1o X. { y } ) : 1o --> B ) )
8280, 81syl 17 . . . . . 6 |- ( ( ph /\ y e. B ) -> ( ( 1o X. { y } ) e. ( B ^m 1o ) <-> ( 1o X. { y } ) : 1o --> B ) )
8378, 82mpbird 240 . . . . 5 |- ( ( ph /\ y e. B ) -> ( 1o X. { y } ) e. ( B ^m 1o ) )
84 eqidd 2472 . . . . 5 |- ( ph -> ( y e. B |-> ( 1o X. { y } ) ) = ( y e. B |-> ( 1o X. { y } ) ) )
85 fconstmpt 4883 . . . . . 6 |- ( ( B ^m 1o ) X. { X } ) = ( z e. ( B ^m 1o ) |-> X )
8685a1i 11 . . . . 5 |- ( ph -> ( ( B ^m 1o ) X. { X } ) = ( z e. ( B ^m 1o ) |-> X ) )
87 eqidd 2472 . . . . 5 |- ( z = ( 1o X. { y } ) -> X = X )
8883, 84, 86, 87fmptco 6072 . . . 4 |- ( ph -> ( ( ( B ^m 1o ) X. { X } ) o. ( y e. B |-> ( 1o X. { y } ) ) ) = ( y e. B |-> X ) )
8976, 88eqtrd 2505 . . 3 |- ( ph -> ( ( x e. ( B ^m ( B ^m 1o ) ) |-> ( x o. ( y e. B |-> ( 1o X. { y } ) ) ) ) ` ( ( B ^m 1o ) X. { X } ) ) = ( y e. B |-> X ) )
9045, 55, 893eqtrd 2509 . 2 |- ( ph -> ( ( ( x e. ( B ^m ( B ^m 1o ) ) |-> ( x o. ( y e. B |-> ( 1o X. { y } ) ) ) ) o. ( ( 1o evalSub S ) ` R ) ) ` ( A ` X ) ) = ( y e. B |-> X ) )
91 elpwg 3950 . . . . . 6 |- ( R e. (SubRing ` S ) -> ( R e. ~P B <-> R C_ B ) )
9228, 91mpbird 240 . . . . 5 |- ( R e. (SubRing ` S ) -> R e. ~P B )
934, 92syl 17 . . . 4 |- ( ph -> R e. ~P B )
94 evls1sca.q . . . . 5 |- Q = ( S evalSub1 R )
95 eqid 2471 . . . . 5 |- ( 1o evalSub S ) = ( 1o evalSub S )
9694, 95, 9evls1fval 18985 . . . 4 |- ( ( S e. CRing /\ R e. ~P B ) -> Q = ( ( x e. ( B ^m ( B ^m 1o ) ) |-> ( x o. ( y e. B |-> ( 1o X. { y } ) ) ) ) o. ( ( 1o evalSub S ) ` R ) ) )
973, 93, 96syl2anc 673 . . 3 |- ( ph -> Q = ( ( x e. ( B ^m ( B ^m 1o ) ) |-> ( x o. ( y e. B |-> ( 1o X. { y } ) ) ) ) o. ( ( 1o evalSub S ) ` R ) ) )
9897fveq1d 5881 . 2 |- ( ph -> ( Q ` ( A ` X ) ) = ( ( ( x e. ( B ^m ( B ^m 1o ) ) |-> ( x o. ( y e. B |-> ( 1o X. { y } ) ) ) ) o. ( ( 1o evalSub S ) ` R ) ) ` ( A ` X ) ) )
99 fconstmpt 4883 . . 3 |- ( B X. { X } ) = ( y e. B |-> X )
10099a1i 11 . 2 |- ( ph -> ( B X. { X } ) = ( y e. B |-> X ) )
10190, 98, 1003eqtr4d 2515 1 |- ( ph -> ( Q ` ( A ` X ) ) = ( B X. { X } ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   <-> wb 189   /\ wa 376   = wceq 1452   e. wcel 1904   _Vcvv 3031   C_ wss 3390   ~Pcpw 3942   {csn 3959   |-> cmpt 4454   X. cxp 4837   o. ccom 4843   Oncon0 5430   -->wf 5585   ` cfv 5589  (class class class)co 6308   1oc1o 7193   ^m cmap 7490   Basecbs 15199   ↾s cress 15200  Scalarcsca 15271   ^s cpws 15423   Ringcrg 17858   CRingccrg 17859   RingHom crh 18018  SubRingcsubrg 18082   LModclmod 18169  algSccascl 18612   mPoly cmpl 18654   evalSub ces 18804  PwSer1cps1 18845  Poly1cpl1 18847   evalSub1 ces1 18979
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-ofr 6551  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-ixp 7541  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-sup 7974  df-oi 8043  df-card 8391  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-fz 11811  df-fzo 11943  df-seq 12252  df-hash 12554  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-sca 15284  df-vsca 15285  df-ip 15286  df-tset 15287  df-ple 15288  df-ds 15290  df-hom 15292  df-cco 15293  df-0g 15418  df-gsum 15419  df-prds 15424  df-pws 15426  df-mre 15570  df-mrc 15571  df-acs 15573  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-mhm 16660  df-submnd 16661  df-grp 16751  df-minusg 16752  df-sbg 16753  df-mulg 16754  df-subg 16892  df-ghm 16959  df-cntz 17049  df-cmn 17510  df-abl 17511  df-mgp 17802  df-ur 17814  df-srg 17818  df-ring 17860  df-cring 17861  df-rnghom 18021  df-subrg 18084  df-lmod 18171  df-lss 18234  df-lsp 18273  df-assa 18613  df-asp 18614  df-ascl 18615  df-psr 18657  df-mvr 18658  df-mpl 18659  df-opsr 18661  df-evls 18806  df-psr1 18850  df-ply1 18852  df-evls1 18981
This theorem is referenced by:  evls1scasrng  19004
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