Theorem mpfpf1 19715

Description: Convert a multivariate polynomial function to univariate. (Contributed by Mario Carneiro, 12-Jun-2015.)

Hypotheses

Ref	Expression
pf1rcl.q	|- Q = ran (eval1 ` R )
pf1f.b	|- B = ( Base ` R )
mpfpf1.q	|- E = ran ( 1o eval R )

Assertion

Ref	Expression
mpfpf1	|- ( F e. E -> ( F o. ( y e. B |-> ( 1o X. { y } ) ) ) e. Q )

Distinct variable groups:   y, B   y, E   y, F   y, R

Allowed substitution hint:   Q( y)

Proof of Theorem mpfpf1

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mpfpf1.q	. . . . 5 |- E = ran ( 1o eval R )
2		eqid 2622	. . . . . . 7 |- ( 1o eval R ) = ( 1o eval R )
3		pf1f.b	. . . . . . 7 |- B = ( Base ` R )
4	2, 3	evlval 19524	. . . . . 6 |- ( 1o eval R ) = ( ( 1o evalSub R ) ` B )
5	4	rneqi 5352	. . . . 5 |- ran ( 1o eval R ) = ran ( ( 1o evalSub R ) ` B )
6	1, 5	eqtri 2644	. . . 4 |- E = ran ( ( 1o evalSub R ) ` B )
7	6	mpfrcl 19518	. . 3 |- ( F e. E -> ( 1o e. _V /\ R e. CRing /\ B e. (SubRing ` R ) ) )
8	7	simp2d 1074	. 2 |- ( F e. E -> R e. CRing )
9		id 22	. . . 4 |- ( F e. E -> F e. E )
10	9, 1	syl6eleq 2711	. . 3 |- ( F e. E -> F e. ran ( 1o eval R ) )
11		1on 7567	. . . . 5 |- 1o e. On
12		eqid 2622	. . . . . 6 |- ( 1o mPoly R ) = ( 1o mPoly R )
13		eqid 2622	. . . . . 6 |- ( R ^s ( B ^m 1o ) ) = ( R ^s ( B ^m 1o ) )
14	2, 3, 12, 13	evlrhm 19525	. . . . 5 |- ( ( 1o e. On /\ R e. CRing ) -> ( 1o eval R ) e. ( ( 1o mPoly R ) RingHom ( R ^s ( B ^m 1o ) ) ) )
15	11, 8, 14	sylancr 695	. . . 4 |- ( F e. E -> ( 1o eval R ) e. ( ( 1o mPoly R ) RingHom ( R ^s ( B ^m 1o ) ) ) )
16		eqid 2622	. . . . . 6 |- (Poly1 ` R ) = (Poly1 ` R )
17		eqid 2622	. . . . . 6 |- (PwSer1 ` R ) = (PwSer1 ` R )
18		eqid 2622	. . . . . 6 |- ( Base ` (Poly1 ` R ) ) = ( Base ` (Poly1 ` R ) )
19	16, 17, 18	ply1bas 19565	. . . . 5 |- ( Base ` (Poly1 ` R ) ) = ( Base ` ( 1o mPoly R ) )
20		eqid 2622	. . . . 5 |- ( Base ` ( R ^s ( B ^m 1o ) ) ) = ( Base ` ( R ^s ( B ^m 1o ) ) )
21	19, 20	rhmf 18726	. . . 4 |- ( ( 1o eval R ) e. ( ( 1o mPoly R ) RingHom ( R ^s ( B ^m 1o ) ) ) -> ( 1o eval R ) : ( Base ` (Poly1 ` R ) ) --> ( Base ` ( R ^s ( B ^m 1o ) ) ) )
22		ffn 6045	. . . 4 |- ( ( 1o eval R ) : ( Base ` (Poly1 ` R ) ) --> ( Base ` ( R ^s ( B ^m 1o ) ) ) -> ( 1o eval R ) Fn ( Base ` (Poly1 ` R ) ) )
23		fvelrnb 6243	. . . 4 |- ( ( 1o eval R ) Fn ( Base ` (Poly1 ` R ) ) -> ( F e. ran ( 1o eval R ) <-> E. x e. ( Base ` (Poly1 ` R ) ) ( ( 1o eval R ) ` x ) = F ) )
24	15, 21, 22, 23	4syl 19	. . 3 |- ( F e. E -> ( F e. ran ( 1o eval R ) <-> E. x e. ( Base ` (Poly1 ` R ) ) ( ( 1o eval R ) ` x ) = F ) )
25	10, 24	mpbid 222	. 2 |- ( F e. E -> E. x e. ( Base ` (Poly1 ` R ) ) ( ( 1o eval R ) ` x ) = F )
26		eqid 2622	. . . . . 6 |- (eval1 ` R ) = (eval1 ` R )
27	26, 2, 3, 12, 19	evl1val 19693	. . . . 5 |- ( ( R e. CRing /\ x e. ( Base ` (Poly1 ` R ) ) ) -> ( (eval1 ` R ) ` x ) = ( ( ( 1o eval R ) ` x ) o. ( y e. B |-> ( 1o X. { y } ) ) ) )
28		eqid 2622	. . . . . . . . 9 |- ( R ^s B ) = ( R ^s B )
29	26, 16, 28, 3	evl1rhm 19696	. . . . . . . 8 |- ( R e. CRing -> (eval1 ` R ) e. ( (Poly1 ` R ) RingHom ( R ^s B ) ) )
30		eqid 2622	. . . . . . . . 9 |- ( Base ` ( R ^s B ) ) = ( Base ` ( R ^s B ) )
31	18, 30	rhmf 18726	. . . . . . . 8 |- ( (eval1 ` R ) e. ( (Poly1 ` R ) RingHom ( R ^s B ) ) -> (eval1 ` R ) : ( Base ` (Poly1 ` R ) ) --> ( Base ` ( R ^s B ) ) )
32		ffn 6045	. . . . . . . 8 |- ( (eval1 ` R ) : ( Base ` (Poly1 ` R ) ) --> ( Base ` ( R ^s B ) ) -> (eval1 ` R ) Fn ( Base ` (Poly1 ` R ) ) )
33	29, 31, 32	3syl 18	. . . . . . 7 |- ( R e. CRing -> (eval1 ` R ) Fn ( Base ` (Poly1 ` R ) ) )
34		fnfvelrn 6356	. . . . . . 7 |- ( ( (eval1 ` R ) Fn ( Base ` (Poly1 ` R ) ) /\ x e. ( Base ` (Poly1 ` R ) ) ) -> ( (eval1 ` R ) ` x ) e. ran (eval1 ` R ) )
35	33, 34	sylan 488	. . . . . 6 |- ( ( R e. CRing /\ x e. ( Base ` (Poly1 ` R ) ) ) -> ( (eval1 ` R ) ` x ) e. ran (eval1 ` R ) )
36		pf1rcl.q	. . . . . 6 |- Q = ran (eval1 ` R )
37	35, 36	syl6eleqr 2712	. . . . 5 |- ( ( R e. CRing /\ x e. ( Base ` (Poly1 ` R ) ) ) -> ( (eval1 ` R ) ` x ) e. Q )
38	27, 37	eqeltrrd 2702	. . . 4 |- ( ( R e. CRing /\ x e. ( Base ` (Poly1 ` R ) ) ) -> ( ( ( 1o eval R ) ` x ) o. ( y e. B |-> ( 1o X. { y } ) ) ) e. Q )
39		coeq1 5279	. . . . 5 |- ( ( ( 1o eval R ) ` x ) = F -> ( ( ( 1o eval R ) ` x ) o. ( y e. B |-> ( 1o X. { y } ) ) ) = ( F o. ( y e. B |-> ( 1o X. { y } ) ) ) )
40	39	eleq1d 2686	. . . 4 |- ( ( ( 1o eval R ) ` x ) = F -> ( ( ( ( 1o eval R ) ` x ) o. ( y e. B |-> ( 1o X. { y } ) ) ) e. Q <-> ( F o. ( y e. B |-> ( 1o X. { y } ) ) ) e. Q ) )
41	38, 40	syl5ibcom 235	. . 3 |- ( ( R e. CRing /\ x e. ( Base ` (Poly1 ` R ) ) ) -> ( ( ( 1o eval R ) ` x ) = F -> ( F o. ( y e. B |-> ( 1o X. { y } ) ) ) e. Q ) )
42	41	rexlimdva 3031	. 2 |- ( R e. CRing -> ( E. x e. ( Base ` (Poly1 ` R ) ) ( ( 1o eval R ) ` x ) = F -> ( F o. ( y e. B |-> ( 1o X. { y } ) ) ) e. Q ) )
43	8, 25, 42	sylc 65	1 |- ( F e. E -> ( F o. ( y e. B |-> ( 1o X. { y } ) ) ) e. Q )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   _Vcvv 3200   {csn 4177   |-> cmpt 4729   X. cxp 5112   ran crn 5115   o. ccom 5118   Oncon0 5723   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   1oc1o 7553   ^m cmap 7857   Basecbs 15857   ^_s cpws 16107   CRingccrg 18548   RingHom crh 18712  SubRingcsubrg 18776   mPoly cmpl 19353   evalSub ces 19504   eval cevl 19505  PwSer₁cps1 19545  Poly₁cpl1 19547  eval₁ce1 19679

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-evl 19507  df-psr1 19550  df-ply1 19552  df-evl1 19681

This theorem is referenced by:  pf1ind  19719