Theorem mzpmfp 37310

Description: Relationship between multivariate Z-polynomials and general multivariate polynomial functions. (Contributed by Stefan O'Rear, 20-Mar-2015.) (Revised by AV, 13-Jun-2019.)

Assertion

Ref	Expression
mzpmfp	|- (mzPoly ` I ) = ran ( I eval ℤring )

Proof of Theorem mzpmfp

Dummy variables a b x y f g are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		zringbas 19824	. . . . . 6 |- ZZ = ( Base ` ℤring )
2		eqid 2622	. . . . . . . 8 |- ( I eval ℤring ) = ( I eval ℤring )
3	2, 1	evlval 19524	. . . . . . 7 |- ( I eval ℤring ) = ( ( I evalSub ℤring ) ` ZZ )
4	3	rneqi 5352	. . . . . 6 |- ran ( I eval ℤring ) = ran ( ( I evalSub ℤring ) ` ZZ )
5		simpl 473	. . . . . 6 |- ( ( I e. _V /\ f e. ZZ ) -> I e. _V )
6		zringcrng 19820	. . . . . . 7 |- ℤring e. CRing
7	6	a1i 11	. . . . . 6 |- ( ( I e. _V /\ f e. ZZ ) -> ℤring e. CRing )
8		zringring 19821	. . . . . . . 8 |- ℤring e. Ring
9	1	subrgid 18782	. . . . . . . 8 |- (ℤring e. Ring -> ZZ e. (SubRing ` ℤring ) )
10	8, 9	ax-mp 5	. . . . . . 7 |- ZZ e. (SubRing ` ℤring )
11	10	a1i 11	. . . . . 6 |- ( ( I e. _V /\ f e. ZZ ) -> ZZ e. (SubRing ` ℤring ) )
12		simpr 477	. . . . . 6 |- ( ( I e. _V /\ f e. ZZ ) -> f e. ZZ )
13	1, 4, 5, 7, 11, 12	mpfconst 19530	. . . . 5 |- ( ( I e. _V /\ f e. ZZ ) -> ( ( ZZ ^m I ) X. { f } ) e. ran ( I eval ℤring ) )
14		simpl 473	. . . . . 6 |- ( ( I e. _V /\ f e. I ) -> I e. _V )
15	6	a1i 11	. . . . . 6 |- ( ( I e. _V /\ f e. I ) -> ℤring e. CRing )
16	10	a1i 11	. . . . . 6 |- ( ( I e. _V /\ f e. I ) -> ZZ e. (SubRing ` ℤring ) )
17		simpr 477	. . . . . 6 |- ( ( I e. _V /\ f e. I ) -> f e. I )
18	1, 4, 14, 15, 16, 17	mpfproj 19531	. . . . 5 |- ( ( I e. _V /\ f e. I ) -> ( g e. ( ZZ ^m I ) |-> ( g ` f ) ) e. ran ( I eval ℤring ) )
19		simp2r 1088	. . . . . 6 |- ( ( I e. _V /\ ( f : ( ZZ ^m I ) --> ZZ /\ f e. ran ( I eval ℤring ) ) /\ ( g : ( ZZ ^m I ) --> ZZ /\ g e. ran ( I eval ℤring ) ) ) -> f e. ran ( I eval ℤring ) )
20		simp3r 1090	. . . . . 6 |- ( ( I e. _V /\ ( f : ( ZZ ^m I ) --> ZZ /\ f e. ran ( I eval ℤring ) ) /\ ( g : ( ZZ ^m I ) --> ZZ /\ g e. ran ( I eval ℤring ) ) ) -> g e. ran ( I eval ℤring ) )
21		zringplusg 19825	. . . . . . 7 |- + = ( +g ` ℤring )
22	4, 21	mpfaddcl 19534	. . . . . 6 |- ( ( f e. ran ( I eval ℤring ) /\ g e. ran ( I eval ℤring ) ) -> ( f oF + g ) e. ran ( I eval ℤring ) )
23	19, 20, 22	syl2anc 693	. . . . 5 |- ( ( I e. _V /\ ( f : ( ZZ ^m I ) --> ZZ /\ f e. ran ( I eval ℤring ) ) /\ ( g : ( ZZ ^m I ) --> ZZ /\ g e. ran ( I eval ℤring ) ) ) -> ( f oF + g ) e. ran ( I eval ℤring ) )
24		zringmulr 19827	. . . . . . 7 |- x. = ( .r ` ℤring )
25	4, 24	mpfmulcl 19535	. . . . . 6 |- ( ( f e. ran ( I eval ℤring ) /\ g e. ran ( I eval ℤring ) ) -> ( f oF x. g ) e. ran ( I eval ℤring ) )
26	19, 20, 25	syl2anc 693	. . . . 5 |- ( ( I e. _V /\ ( f : ( ZZ ^m I ) --> ZZ /\ f e. ran ( I eval ℤring ) ) /\ ( g : ( ZZ ^m I ) --> ZZ /\ g e. ran ( I eval ℤring ) ) ) -> ( f oF x. g ) e. ran ( I eval ℤring ) )
27		eleq1 2689	. . . . 5 |- ( b = ( ( ZZ ^m I ) X. { f } ) -> ( b e. ran ( I eval ℤring ) <-> ( ( ZZ ^m I ) X. { f } ) e. ran ( I eval ℤring ) ) )
28		eleq1 2689	. . . . 5 |- ( b = ( g e. ( ZZ ^m I ) |-> ( g ` f ) ) -> ( b e. ran ( I eval ℤring ) <-> ( g e. ( ZZ ^m I ) |-> ( g ` f ) ) e. ran ( I eval ℤring ) ) )
29		eleq1 2689	. . . . 5 |- ( b = f -> ( b e. ran ( I eval ℤring ) <-> f e. ran ( I eval ℤring ) ) )
30		eleq1 2689	. . . . 5 |- ( b = g -> ( b e. ran ( I eval ℤring ) <-> g e. ran ( I eval ℤring ) ) )
31		eleq1 2689	. . . . 5 |- ( b = ( f oF + g ) -> ( b e. ran ( I eval ℤring ) <-> ( f oF + g ) e. ran ( I eval ℤring ) ) )
32		eleq1 2689	. . . . 5 |- ( b = ( f oF x. g ) -> ( b e. ran ( I eval ℤring ) <-> ( f oF x. g ) e. ran ( I eval ℤring ) ) )
33		eleq1 2689	. . . . 5 |- ( b = a -> ( b e. ran ( I eval ℤring ) <-> a e. ran ( I eval ℤring ) ) )
34	13, 18, 23, 26, 27, 28, 29, 30, 31, 32, 33	mzpindd 37309	. . . 4 |- ( ( I e. _V /\ a e. (mzPoly ` I ) ) -> a e. ran ( I eval ℤring ) )
35		simprlr 803	. . . . . 6 |- ( ( ( I e. _V /\ a e. ran ( I eval ℤring ) ) /\ ( ( x e. ran ( I eval ℤring ) /\ x e. (mzPoly ` I ) ) /\ ( y e. ran ( I eval ℤring ) /\ y e. (mzPoly ` I ) ) ) ) -> x e. (mzPoly ` I ) )
36		simprrr 805	. . . . . 6 |- ( ( ( I e. _V /\ a e. ran ( I eval ℤring ) ) /\ ( ( x e. ran ( I eval ℤring ) /\ x e. (mzPoly ` I ) ) /\ ( y e. ran ( I eval ℤring ) /\ y e. (mzPoly ` I ) ) ) ) -> y e. (mzPoly ` I ) )
37		mzpadd 37301	. . . . . 6 |- ( ( x e. (mzPoly ` I ) /\ y e. (mzPoly ` I ) ) -> ( x oF + y ) e. (mzPoly ` I ) )
38	35, 36, 37	syl2anc 693	. . . . 5 |- ( ( ( I e. _V /\ a e. ran ( I eval ℤring ) ) /\ ( ( x e. ran ( I eval ℤring ) /\ x e. (mzPoly ` I ) ) /\ ( y e. ran ( I eval ℤring ) /\ y e. (mzPoly ` I ) ) ) ) -> ( x oF + y ) e. (mzPoly ` I ) )
39		mzpmul 37302	. . . . . 6 |- ( ( x e. (mzPoly ` I ) /\ y e. (mzPoly ` I ) ) -> ( x oF x. y ) e. (mzPoly ` I ) )
40	35, 36, 39	syl2anc 693	. . . . 5 |- ( ( ( I e. _V /\ a e. ran ( I eval ℤring ) ) /\ ( ( x e. ran ( I eval ℤring ) /\ x e. (mzPoly ` I ) ) /\ ( y e. ran ( I eval ℤring ) /\ y e. (mzPoly ` I ) ) ) ) -> ( x oF x. y ) e. (mzPoly ` I ) )
41		eleq1 2689	. . . . 5 |- ( b = ( ( ZZ ^m I ) X. { x } ) -> ( b e. (mzPoly ` I ) <-> ( ( ZZ ^m I ) X. { x } ) e. (mzPoly ` I ) ) )
42		eleq1 2689	. . . . 5 |- ( b = ( y e. ( ZZ ^m I ) |-> ( y ` x ) ) -> ( b e. (mzPoly ` I ) <-> ( y e. ( ZZ ^m I ) |-> ( y ` x ) ) e. (mzPoly ` I ) ) )
43		eleq1 2689	. . . . 5 |- ( b = x -> ( b e. (mzPoly ` I ) <-> x e. (mzPoly ` I ) ) )
44		eleq1 2689	. . . . 5 |- ( b = y -> ( b e. (mzPoly ` I ) <-> y e. (mzPoly ` I ) ) )
45		eleq1 2689	. . . . 5 |- ( b = ( x oF + y ) -> ( b e. (mzPoly ` I ) <-> ( x oF + y ) e. (mzPoly ` I ) ) )
46		eleq1 2689	. . . . 5 |- ( b = ( x oF x. y ) -> ( b e. (mzPoly ` I ) <-> ( x oF x. y ) e. (mzPoly ` I ) ) )
47		eleq1 2689	. . . . 5 |- ( b = a -> ( b e. (mzPoly ` I ) <-> a e. (mzPoly ` I ) ) )
48		mzpconst 37298	. . . . . 6 |- ( ( I e. _V /\ x e. ZZ ) -> ( ( ZZ ^m I ) X. { x } ) e. (mzPoly ` I ) )
49	48	adantlr 751	. . . . 5 |- ( ( ( I e. _V /\ a e. ran ( I eval ℤring ) ) /\ x e. ZZ ) -> ( ( ZZ ^m I ) X. { x } ) e. (mzPoly ` I ) )
50		mzpproj 37300	. . . . . 6 |- ( ( I e. _V /\ x e. I ) -> ( y e. ( ZZ ^m I ) |-> ( y ` x ) ) e. (mzPoly ` I ) )
51	50	adantlr 751	. . . . 5 |- ( ( ( I e. _V /\ a e. ran ( I eval ℤring ) ) /\ x e. I ) -> ( y e. ( ZZ ^m I ) |-> ( y ` x ) ) e. (mzPoly ` I ) )
52		simpr 477	. . . . 5 |- ( ( I e. _V /\ a e. ran ( I eval ℤring ) ) -> a e. ran ( I eval ℤring ) )
53	1, 21, 24, 4, 38, 40, 41, 42, 43, 44, 45, 46, 47, 49, 51, 52	mpfind 19536	. . . 4 |- ( ( I e. _V /\ a e. ran ( I eval ℤring ) ) -> a e. (mzPoly ` I ) )
54	34, 53	impbida 877	. . 3 |- ( I e. _V -> ( a e. (mzPoly ` I ) <-> a e. ran ( I eval ℤring ) ) )
55	54	eqrdv 2620	. 2 |- ( I e. _V -> (mzPoly ` I ) = ran ( I eval ℤring ) )
56		fvprc 6185	. . 3 |- ( -. I e. _V -> (mzPoly ` I ) = (/) )
57		df-evl 19507	. . . . . . 7 |- eval = ( a e. _V , b e. _V |-> ( ( a evalSub b ) ` ( Base ` b ) ) )
58	57	reldmmpt2 6771	. . . . . 6 |- Rel dom eval
59	58	ovprc1 6684	. . . . 5 |- ( -. I e. _V -> ( I eval ℤring ) = (/) )
60	59	rneqd 5353	. . . 4 |- ( -. I e. _V -> ran ( I eval ℤring ) = ran (/) )
61		rn0 5377	. . . 4 |- ran (/) = (/)
62	60, 61	syl6eq 2672	. . 3 |- ( -. I e. _V -> ran ( I eval ℤring ) = (/) )
63	56, 62	eqtr4d 2659	. 2 |- ( -. I e. _V -> (mzPoly ` I ) = ran ( I eval ℤring ) )
64	55, 63	pm2.61i 176	1 |- (mzPoly ` I ) = ran ( I eval ℤring )

Syntax hints:   -. wn 3   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   (/)c0 3915   {csn 4177   |-> cmpt 4729   X. cxp 5112   ran crn 5115   -->wf 5884   ` cfv 5888  (class class class)co 6650   oFcof 6895   ^m cmap 7857   + caddc 9939   x. cmul 9941   ZZcz 11377   Basecbs 15857   Ringcrg 18547   CRingccrg 18548  SubRingcsubrg 18776   evalSub ces 19504   eval cevl 19505  ℤ_ringzring 19818  mzPolycmzp 37285

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  ax-addf 10015  ax-mulf 10016

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-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  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-evls 19506  df-evl 19507  df-cnfld 19747  df-zring 19819  df-mzpcl 37286  df-mzp 37287

This theorem is referenced by: (None)