Theorem mdegfval 23822

Description: Value of the multivariate degree function. (Contributed by Stefan O'Rear, 19-Mar-2015.) (Revised by AV, 25-Jun-2019.)

Hypotheses

Ref	Expression
mdegval.d	|- D = ( I mDeg R )
mdegval.p	|- P = ( I mPoly R )
mdegval.b	|- B = ( Base ` P )
mdegval.z	|- .0. = ( 0g ` R )
mdegval.a	|- A = { m e. ( NN0 ^m I ) | ( `' m " NN ) e. Fin }
mdegval.h	|- H = ( h e. A |-> (ℂfld gsumg h ) )

Assertion

Ref	Expression
mdegfval	|- D = ( f e. B |-> sup ( ( H " ( f supp .0. ) ) , RR* , < ) )

Distinct variable groups:   A, h   B, f   f, I   m, I   R, f   .0. , h   f, h

Allowed substitution hints:   A( f, m)   B( h, m)   D( f, h, m)   P( f, h, m)   R( h, m)   H( f, h, m)   I( h)   .0. ( f, m)

Proof of Theorem mdegfval

Dummy variables i r are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mdegval.d	. 2 |- D = ( I mDeg R )
2		oveq12 6659	. . . . . . . . 9 |- ( ( i = I /\ r = R ) -> ( i mPoly r ) = ( I mPoly R ) )
3		mdegval.p	. . . . . . . . 9 |- P = ( I mPoly R )
4	2, 3	syl6eqr 2674	. . . . . . . 8 |- ( ( i = I /\ r = R ) -> ( i mPoly r ) = P )
5	4	fveq2d 6195	. . . . . . 7 |- ( ( i = I /\ r = R ) -> ( Base ` ( i mPoly r ) ) = ( Base ` P ) )
6		mdegval.b	. . . . . . 7 |- B = ( Base ` P )
7	5, 6	syl6eqr 2674	. . . . . 6 |- ( ( i = I /\ r = R ) -> ( Base ` ( i mPoly r ) ) = B )
8		fveq2 6191	. . . . . . . . . . . 12 |- ( r = R -> ( 0g ` r ) = ( 0g ` R ) )
9		mdegval.z	. . . . . . . . . . . 12 |- .0. = ( 0g ` R )
10	8, 9	syl6eqr 2674	. . . . . . . . . . 11 |- ( r = R -> ( 0g ` r ) = .0. )
11	10	oveq2d 6666	. . . . . . . . . 10 |- ( r = R -> ( f supp ( 0g ` r ) ) = ( f supp .0. ) )
12	11	mpteq1d 4738	. . . . . . . . 9 |- ( r = R -> ( h e. ( f supp ( 0g ` r ) ) |-> (ℂfld gsumg h ) ) = ( h e. ( f supp .0. ) |-> (ℂfld gsumg h ) ) )
13	12	rneqd 5353	. . . . . . . 8 |- ( r = R -> ran ( h e. ( f supp ( 0g ` r ) ) |-> (ℂfld gsumg h ) ) = ran ( h e. ( f supp .0. ) |-> (ℂfld gsumg h ) ) )
14	13	supeq1d 8352	. . . . . . 7 |- ( r = R -> sup ( ran ( h e. ( f supp ( 0g ` r ) ) |-> (ℂfld gsumg h ) ) , RR* , < ) = sup ( ran ( h e. ( f supp .0. ) |-> (ℂfld gsumg h ) ) , RR* , < ) )
15	14	adantl 482	. . . . . 6 |- ( ( i = I /\ r = R ) -> sup ( ran ( h e. ( f supp ( 0g ` r ) ) |-> (ℂfld gsumg h ) ) , RR* , < ) = sup ( ran ( h e. ( f supp .0. ) |-> (ℂfld gsumg h ) ) , RR* , < ) )
16	7, 15	mpteq12dv 4733	. . . . 5 |- ( ( i = I /\ r = R ) -> ( f e. ( Base ` ( i mPoly r ) ) |-> sup ( ran ( h e. ( f supp ( 0g ` r ) ) |-> (ℂfld gsumg h ) ) , RR* , < ) ) = ( f e. B |-> sup ( ran ( h e. ( f supp .0. ) |-> (ℂfld gsumg h ) ) , RR* , < ) ) )
17		df-mdeg 23815	. . . . 5 |- mDeg = ( i e. _V , r e. _V |-> ( f e. ( Base ` ( i mPoly r ) ) |-> sup ( ran ( h e. ( f supp ( 0g ` r ) ) |-> (ℂfld gsumg h ) ) , RR* , < ) ) )
18		fvex 6201	. . . . . . 7 |- ( Base ` P ) e. _V
19	6, 18	eqeltri 2697	. . . . . 6 |- B e. _V
20	19	mptex 6486	. . . . 5 |- ( f e. B |-> sup ( ran ( h e. ( f supp .0. ) |-> (ℂfld gsumg h ) ) , RR* , < ) ) e. _V
21	16, 17, 20	ovmpt2a 6791	. . . 4 |- ( ( I e. _V /\ R e. _V ) -> ( I mDeg R ) = ( f e. B |-> sup ( ran ( h e. ( f supp .0. ) |-> (ℂfld gsumg h ) ) , RR* , < ) ) )
22		mdegval.h	. . . . . . . . . 10 |- H = ( h e. A |-> (ℂfld gsumg h ) )
23	22	reseq1i 5392	. . . . . . . . 9 |- ( H |` ( f supp .0. ) ) = ( ( h e. A |-> (ℂfld gsumg h ) ) |` ( f supp .0. ) )
24		suppssdm 7308	. . . . . . . . . . 11 |- ( f supp .0. ) C_ dom f
25		eqid 2622	. . . . . . . . . . . . 13 |- ( Base ` R ) = ( Base ` R )
26		mdegval.a	. . . . . . . . . . . . 13 |- A = { m e. ( NN0 ^m I ) | ( `' m " NN ) e. Fin }
27		simpr 477	. . . . . . . . . . . . 13 |- ( ( ( I e. _V /\ R e. _V ) /\ f e. B ) -> f e. B )
28	3, 25, 6, 26, 27	mplelf 19433	. . . . . . . . . . . 12 |- ( ( ( I e. _V /\ R e. _V ) /\ f e. B ) -> f : A --> ( Base ` R ) )
29		fdm 6051	. . . . . . . . . . . 12 |- ( f : A --> ( Base ` R ) -> dom f = A )
30	28, 29	syl 17	. . . . . . . . . . 11 |- ( ( ( I e. _V /\ R e. _V ) /\ f e. B ) -> dom f = A )
31	24, 30	syl5sseq 3653	. . . . . . . . . 10 |- ( ( ( I e. _V /\ R e. _V ) /\ f e. B ) -> ( f supp .0. ) C_ A )
32	31	resmptd 5452	. . . . . . . . 9 |- ( ( ( I e. _V /\ R e. _V ) /\ f e. B ) -> ( ( h e. A |-> (ℂfld gsumg h ) ) |` ( f supp .0. ) ) = ( h e. ( f supp .0. ) |-> (ℂfld gsumg h ) ) )
33	23, 32	syl5req 2669	. . . . . . . 8 |- ( ( ( I e. _V /\ R e. _V ) /\ f e. B ) -> ( h e. ( f supp .0. ) |-> (ℂfld gsumg h ) ) = ( H |` ( f supp .0. ) ) )
34	33	rneqd 5353	. . . . . . 7 |- ( ( ( I e. _V /\ R e. _V ) /\ f e. B ) -> ran ( h e. ( f supp .0. ) |-> (ℂfld gsumg h ) ) = ran ( H |` ( f supp .0. ) ) )
35		df-ima 5127	. . . . . . 7 |- ( H " ( f supp .0. ) ) = ran ( H |` ( f supp .0. ) )
36	34, 35	syl6eqr 2674	. . . . . 6 |- ( ( ( I e. _V /\ R e. _V ) /\ f e. B ) -> ran ( h e. ( f supp .0. ) |-> (ℂfld gsumg h ) ) = ( H " ( f supp .0. ) ) )
37	36	supeq1d 8352	. . . . 5 |- ( ( ( I e. _V /\ R e. _V ) /\ f e. B ) -> sup ( ran ( h e. ( f supp .0. ) |-> (ℂfld gsumg h ) ) , RR* , < ) = sup ( ( H " ( f supp .0. ) ) , RR* , < ) )
38	37	mpteq2dva 4744	. . . 4 |- ( ( I e. _V /\ R e. _V ) -> ( f e. B |-> sup ( ran ( h e. ( f supp .0. ) |-> (ℂfld gsumg h ) ) , RR* , < ) ) = ( f e. B |-> sup ( ( H " ( f supp .0. ) ) , RR* , < ) ) )
39	21, 38	eqtrd 2656	. . 3 |- ( ( I e. _V /\ R e. _V ) -> ( I mDeg R ) = ( f e. B |-> sup ( ( H " ( f supp .0. ) ) , RR* , < ) ) )
40		reldmmdeg 23817	. . . . . 6 |- Rel dom mDeg
41	40	ovprc 6683	. . . . 5 |- ( -. ( I e. _V /\ R e. _V ) -> ( I mDeg R ) = (/) )
42		mpt0 6021	. . . . 5 |- ( f e. (/) |-> sup ( ( H " ( f supp .0. ) ) , RR* , < ) ) = (/)
43	41, 42	syl6eqr 2674	. . . 4 |- ( -. ( I e. _V /\ R e. _V ) -> ( I mDeg R ) = ( f e. (/) |-> sup ( ( H " ( f supp .0. ) ) , RR* , < ) ) )
44		reldmmpl 19427	. . . . . . . . 9 |- Rel dom mPoly
45	44	ovprc 6683	. . . . . . . 8 |- ( -. ( I e. _V /\ R e. _V ) -> ( I mPoly R ) = (/) )
46	3, 45	syl5eq 2668	. . . . . . 7 |- ( -. ( I e. _V /\ R e. _V ) -> P = (/) )
47	46	fveq2d 6195	. . . . . 6 |- ( -. ( I e. _V /\ R e. _V ) -> ( Base ` P ) = ( Base ` (/) ) )
48		base0 15912	. . . . . 6 |- (/) = ( Base ` (/) )
49	47, 6, 48	3eqtr4g 2681	. . . . 5 |- ( -. ( I e. _V /\ R e. _V ) -> B = (/) )
50	49	mpteq1d 4738	. . . 4 |- ( -. ( I e. _V /\ R e. _V ) -> ( f e. B |-> sup ( ( H " ( f supp .0. ) ) , RR* , < ) ) = ( f e. (/) |-> sup ( ( H " ( f supp .0. ) ) , RR* , < ) ) )
51	43, 50	eqtr4d 2659	. . 3 |- ( -. ( I e. _V /\ R e. _V ) -> ( I mDeg R ) = ( f e. B |-> sup ( ( H " ( f supp .0. ) ) , RR* , < ) ) )
52	39, 51	pm2.61i 176	. 2 |- ( I mDeg R ) = ( f e. B |-> sup ( ( H " ( f supp .0. ) ) , RR* , < ) )
53	1, 52	eqtri 2644	1 |- D = ( f e. B |-> sup ( ( H " ( f supp .0. ) ) , RR* , < ) )

Syntax hints:   -. wn 3   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   _Vcvv 3200   (/)c0 3915   |-> cmpt 4729   `'ccnv 5113   dom cdm 5114   ran crn 5115   |` cres 5116   "cima 5117   -->wf 5884   ` cfv 5888  (class class class)co 6650   supp csupp 7295   ^m cmap 7857   Fincfn 7955   supcsup 8346   RR*cxr 10073   < clt 10074   NNcn 11020   NN0cn0 11292   Basecbs 15857   0gc0g 16100   gsum_g cgsu 16101   mPoly cmpl 19353  ℂ_fldccnfld 19746   mDeg cmdg 23813

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-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-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-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-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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  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-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-sup 8348  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-uz 11688  df-fz 12327  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-tset 15960  df-psr 19356  df-mpl 19358  df-mdeg 23815

This theorem is referenced by:  mdegval  23823  mdegxrf  23828  mdegpropd  23844