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Theorem evls1sca 19688
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 7567 . . . . . . 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 2622 . . . . . . 7 |- ( ( 1o evalSub S ) ` R ) = ( ( 1o evalSub S ) ` R )
6 eqid 2622 . . . . . . 7 |- ( 1o mPoly U ) = ( 1o mPoly U )
7 evls1sca.u . . . . . . 7 |- U = ( S ↾s R )
8 eqid 2622 . . . . . . 7 |- ( S ^s ( B ^m 1o ) ) = ( S ^s ( B ^m 1o ) )
9 evls1sca.b . . . . . . 7 |- B = ( Base ` S )
105, 6, 7, 8, 9evlsrhm 19521 . . . . . 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 1326 . . . . 5 |- ( ph -> ( ( 1o evalSub S ) ` R ) e. ( ( 1o mPoly U ) RingHom ( S ^s ( B ^m 1o ) ) ) )
12 eqid 2622 . . . . . 6 |- ( Base ` ( 1o mPoly U ) ) = ( Base ` ( 1o mPoly U ) )
13 eqid 2622 . . . . . 6 |- ( Base ` ( S ^s ( B ^m 1o ) ) ) = ( Base ` ( S ^s ( B ^m 1o ) ) )
1412, 13rhmf 18726 . . . . 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 2622 . . . . . . 7 |- (Scalar ` W ) = (Scalar ` W )
187subrgring 18783 . . . . . . . . 9 |- ( R e. (SubRing ` S ) -> U e. Ring )
194, 18syl 17 . . . . . . . 8 |- ( ph -> U e. Ring )
20 evls1sca.w . . . . . . . . 9 |- W = (Poly1 ` U )
2120ply1ring 19618 . . . . . . . 8 |- ( U e. Ring -> W e. Ring )
2219, 21syl 17 . . . . . . 7 |- ( ph -> W e. Ring )
2320ply1lmod 19622 . . . . . . . 8 |- ( U e. Ring -> W e. LMod )
2419, 23syl 17 . . . . . . 7 |- ( ph -> W e. LMod )
25 eqid 2622 . . . . . . 7 |- ( Base ` (Scalar ` W ) ) = ( Base ` (Scalar ` W ) )
26 eqid 2622 . . . . . . 7 |- ( Base ` W ) = ( Base ` W )
2716, 17, 22, 24, 25, 26asclf 19337 . . . . . 6 |- ( ph -> A : ( Base ` (Scalar ` W ) ) --> ( Base ` W ) )
289subrgss 18781 . . . . . . . . . 10 |- ( R e. (SubRing ` S ) -> R C_ B )
294, 28syl 17 . . . . . . . . 9 |- ( ph -> R C_ B )
307, 9ressbas2 15931 . . . . . . . . 9 |- ( R C_ B -> R = ( Base ` U ) )
3129, 30syl 17 . . . . . . . 8 |- ( ph -> R = ( Base ` U ) )
3220ply1sca 19623 . . . . . . . . . 10 |- ( U e. Ring -> U = (Scalar ` W ) )
3319, 32syl 17 . . . . . . . . 9 |- ( ph -> U = (Scalar ` W ) )
3433fveq2d 6195 . . . . . . . 8 |- ( ph -> ( Base ` U ) = ( Base ` (Scalar ` W ) ) )
3531, 34eqtrd 2656 . . . . . . 7 |- ( ph -> R = ( Base ` (Scalar ` W ) ) )
36 eqid 2622 . . . . . . . . . 10 |- (PwSer1 ` U ) = (PwSer1 ` U )
3720, 36, 26ply1bas 19565 . . . . . . . . 9 |- ( Base ` W ) = ( Base ` ( 1o mPoly U ) )
3837a1i 11 . . . . . . . 8 |- ( ph -> ( Base ` W ) = ( Base ` ( 1o mPoly U ) ) )
3938eqcomd 2628 . . . . . . 7 |- ( ph -> ( Base ` ( 1o mPoly U ) ) = ( Base ` W ) )
4035, 39feq23d 6040 . . . . . 6 |- ( ph -> ( A : R --> ( Base ` ( 1o mPoly U ) ) <-> A : ( Base ` (Scalar ` W ) ) --> ( Base ` W ) ) )
4127, 40mpbird 247 . . . . 5 |- ( ph -> A : R --> ( Base ` ( 1o mPoly U ) ) )
42 evls1sca.x . . . . 5 |- ( ph -> X e. R )
4341, 42ffvelrnd 6360 . . . 4 |- ( ph -> ( A ` X ) e. ( Base ` ( 1o mPoly U ) ) )
44 fvco3 6275 . . . 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 693 . . 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 2622 . . . . . . . . 9 |- (algSc ` W ) = (algSc ` W )
4820, 47ply1ascl 19628 . . . . . . . 8 |- (algSc ` W ) = (algSc ` ( 1o mPoly U ) )
4946, 48syl6eq 2672 . . . . . . 7 |- ( ph -> A = (algSc ` ( 1o mPoly U ) ) )
5049fveq1d 6193 . . . . . 6 |- ( ph -> ( A ` X ) = ( (algSc ` ( 1o mPoly U ) ) ` X ) )
5150fveq2d 6195 . . . . 5 |- ( ph -> ( ( ( 1o evalSub S ) ` R ) ` ( A ` X ) ) = ( ( ( 1o evalSub S ) ` R ) ` ( (algSc ` ( 1o mPoly U ) ) ` X ) ) )
52 eqid 2622 . . . . . 6 |- (algSc ` ( 1o mPoly U ) ) = (algSc ` ( 1o mPoly U ) )
535, 6, 7, 9, 52, 2, 3, 4, 42evlssca 19522 . . . . 5 |- ( ph -> ( ( ( 1o evalSub S ) ` R ) ` ( (algSc ` ( 1o mPoly U ) ) ` X ) ) = ( ( B ^m 1o ) X. { X } ) )
5451, 53eqtrd 2656 . . . 4 |- ( ph -> ( ( ( 1o evalSub S ) ` R ) ` ( A ` X ) ) = ( ( B ^m 1o ) X. { X } ) )
5554fveq2d 6195 . . 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 2623 . . . . 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 5279 . . . . . 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 482 . . . . 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 3604 . . . . . . 7 |- ( ph -> X e. B )
60 fconst6g 6094 . . . . . . 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 6201 . . . . . . . . 9 |- ( Base ` S ) e. _V
639, 62eqeltri 2697 . . . . . . . 8 |- B e. _V
6463a1i 11 . . . . . . 7 |- ( ph -> B e. _V )
65 ovex 6678 . . . . . . . 8 |- ( B ^m 1o ) e. _V
6665a1i 11 . . . . . . 7 |- ( ph -> ( B ^m 1o ) e. _V )
6764, 66elmapd 7871 . . . . . 6 |- ( ph -> ( ( ( B ^m 1o ) X. { X } ) e. ( B ^m ( B ^m 1o ) ) <-> ( ( B ^m 1o ) X. { X } ) : ( B ^m 1o ) --> B ) )
6861, 67mpbird 247 . . . . 5 |- ( ph -> ( ( B ^m 1o ) X. { X } ) e. ( B ^m ( B ^m 1o ) ) )
69 snex 4908 . . . . . . . 8 |- { X } e. _V
7065, 69xpex 6962 . . . . . . 7 |- ( ( B ^m 1o ) X. { X } ) e. _V
7170a1i 11 . . . . . 6 |- ( ph -> ( ( B ^m 1o ) X. { X } ) e. _V )
72 mptexg 6484 . . . . . . 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 7117 . . . . . 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 693 . . . . 5 |- ( ph -> ( ( ( B ^m 1o ) X. { X } ) o. ( y e. B |-> ( 1o X. { y } ) ) ) e. _V )
7656, 58, 68, 75fvmptd 6288 . . . 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 6094 . . . . . . 7 |- ( y e. B -> ( 1o X. { y } ) : 1o --> B )
7877adantl 482 . . . . . 6 |- ( ( ph /\ y e. B ) -> ( 1o X. { y } ) : 1o --> B )
7963, 1pm3.2i 471 . . . . . . . 8 |- ( B e. _V /\ 1o e. On )
8079a1i 11 . . . . . . 7 |- ( ( ph /\ y e. B ) -> ( B e. _V /\ 1o e. On ) )
81 elmapg 7870 . . . . . . 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 247 . . . . 5 |- ( ( ph /\ y e. B ) -> ( 1o X. { y } ) e. ( B ^m 1o ) )
84 eqidd 2623 . . . . 5 |- ( ph -> ( y e. B |-> ( 1o X. { y } ) ) = ( y e. B |-> ( 1o X. { y } ) ) )
85 fconstmpt 5163 . . . . . 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 2623 . . . . 5 |- ( z = ( 1o X. { y } ) -> X = X )
8883, 84, 86, 87fmptco 6396 . . . 4 |- ( ph -> ( ( ( B ^m 1o ) X. { X } ) o. ( y e. B |-> ( 1o X. { y } ) ) ) = ( y e. B |-> X ) )
8976, 88eqtrd 2656 . . 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 2660 . 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 4166 . . . . . 6 |- ( R e. (SubRing ` S ) -> ( R e. ~P B <-> R C_ B ) )
9228, 91mpbird 247 . . . . 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 2622 . . . . 5 |- ( 1o evalSub S ) = ( 1o evalSub S )
9694, 95, 9evls1fval 19684 . . . 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 693 . . 3 |- ( ph -> Q = ( ( x e. ( B ^m ( B ^m 1o ) ) |-> ( x o. ( y e. B |-> ( 1o X. { y } ) ) ) ) o. ( ( 1o evalSub S ) ` R ) ) )
9897fveq1d 6193 . 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 5163 . . 3 |- ( B X. { X } ) = ( y e. B |-> X )
10099a1i 11 . 2 |- ( ph -> ( B X. { X } ) = ( y e. B |-> X ) )
10190, 98, 1003eqtr4d 2666 1 |- ( ph -> ( Q ` ( A ` X ) ) = ( B X. { X } ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   C_ wss 3574   ~Pcpw 4158   {csn 4177   |-> cmpt 4729   X. cxp 5112   o. ccom 5118   Oncon0 5723   -->wf 5884   ` cfv 5888  (class class class)co 6650   1oc1o 7553   ^m cmap 7857   Basecbs 15857   ↾s cress 15858  Scalarcsca 15944   ^s cpws 16107   Ringcrg 18547   CRingccrg 18548   RingHom crh 18712  SubRingcsubrg 18776   LModclmod 18863  algSccascl 19311   mPoly cmpl 19353   evalSub ces 19504  PwSer1cps1 19545  Poly1cpl1 19547   evalSub1 ces1 19678
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-ofr 6898  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-sup 8348  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-fzo 12466  df-seq 12802  df-hash 13118  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-hom 15966  df-cco 15967  df-0g 16102  df-gsum 16103  df-prds 16108  df-pws 16110  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-submnd 17336  df-grp 17425  df-minusg 17426  df-sbg 17427  df-mulg 17541  df-subg 17591  df-ghm 17658  df-cntz 17750  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-srg 18506  df-ring 18549  df-cring 18550  df-rnghom 18715  df-subrg 18778  df-lmod 18865  df-lss 18933  df-lsp 18972  df-assa 19312  df-asp 19313  df-ascl 19314  df-psr 19356  df-mvr 19357  df-mpl 19358  df-opsr 19360  df-evls 19506  df-psr1 19550  df-ply1 19552  df-evls1 19680
This theorem is referenced by:  evls1scasrng  19703
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