Theorem mdegfval 26200

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 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑚 “ ℕ) ∈ Fin}
mdegval.h	⊢ 𝐻 = (ℎ ∈ 𝐴 ↦ (ℂfld Σg ℎ))

Assertion

Ref	Expression
mdegfval	⊢ 𝐷 = (𝑓 ∈ 𝐵 ↦ sup((𝐻 “ (𝑓 supp 0 )), ℝ*, < ))

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 7421	. . . . . . . . 9 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (𝑖 mPoly 𝑟) = (𝐼 mPoly 𝑅))
3		mdegval.p	. . . . . . . . 9 ⊢ 𝑃 = (𝐼 mPoly 𝑅)
4	2, 3	eqtr4di 2816	. . . . . . . 8 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (𝑖 mPoly 𝑟) = 𝑃)
5	4	fveq2d 6887	. . . . . . 7 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (Base‘(𝑖 mPoly 𝑟)) = (Base‘𝑃))
6		mdegval.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑃)
7	5, 6	eqtr4di 2816	. . . . . 6 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (Base‘(𝑖 mPoly 𝑟)) = 𝐵)
8		fveq2 6883	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = (0g‘𝑅))
9		mdegval.z	. . . . . . . . . . . 12 ⊢ 0 = (0g‘𝑅)
10	8, 9	eqtr4di 2816	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = 0 )
11	10	oveq2d 7428	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (𝑓 supp (0g‘𝑟)) = (𝑓 supp 0 ))
12	11	mpteq1d 5202	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (ℎ ∈ (𝑓 supp (0g‘𝑟)) ↦ (ℂfld Σg ℎ)) = (ℎ ∈ (𝑓 supp 0 ) ↦ (ℂfld Σg ℎ)))
13	12	rneqd 5930	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → ran (ℎ ∈ (𝑓 supp (0g‘𝑟)) ↦ (ℂfld Σg ℎ)) = ran (ℎ ∈ (𝑓 supp 0 ) ↦ (ℂfld Σg ℎ)))
14	13	supeq1d 9407	. . . . . . 7 ⊢ (𝑟 = 𝑅 → sup(ran (ℎ ∈ (𝑓 supp (0g‘𝑟)) ↦ (ℂfld Σg ℎ)), ℝ*, < ) = sup(ran (ℎ ∈ (𝑓 supp 0 ) ↦ (ℂfld Σg ℎ)), ℝ*, < ))
15	14	adantl 486	. . . . . 6 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → sup(ran (ℎ ∈ (𝑓 supp (0g‘𝑟)) ↦ (ℂfld Σg ℎ)), ℝ*, < ) = sup(ran (ℎ ∈ (𝑓 supp 0 ) ↦ (ℂfld Σg ℎ)), ℝ*, < ))
16	7, 15	mpteq12dv 5199	. . . . 5 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (𝑓 ∈ (Base‘(𝑖 mPoly 𝑟)) ↦ sup(ran (ℎ ∈ (𝑓 supp (0g‘𝑟)) ↦ (ℂfld Σg ℎ)), ℝ*, < )) = (𝑓 ∈ 𝐵 ↦ sup(ran (ℎ ∈ (𝑓 supp 0 ) ↦ (ℂfld Σg ℎ)), ℝ*, < )))
17		df-mdeg 26193	. . . . 5 ⊢ mDeg = (𝑖 ∈ V, 𝑟 ∈ V ↦ (𝑓 ∈ (Base‘(𝑖 mPoly 𝑟)) ↦ sup(ran (ℎ ∈ (𝑓 supp (0g‘𝑟)) ↦ (ℂfld Σg ℎ)), ℝ*, < )))
18	6	fvexi 6897	. . . . . 6 ⊢ 𝐵 ∈ V
19	18	mptex 7223	. . . . 5 ⊢ (𝑓 ∈ 𝐵 ↦ sup(ran (ℎ ∈ (𝑓 supp 0 ) ↦ (ℂfld Σg ℎ)), ℝ*, < )) ∈ V
20	16, 17, 19	ovmpoa 7567	. . . 4 ⊢ ((𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 mDeg 𝑅) = (𝑓 ∈ 𝐵 ↦ sup(ran (ℎ ∈ (𝑓 supp 0 ) ↦ (ℂfld Σg ℎ)), ℝ*, < )))
21		mdegval.h	. . . . . . . . . 10 ⊢ 𝐻 = (ℎ ∈ 𝐴 ↦ (ℂfld Σg ℎ))
22	21	reseq1i 5976	. . . . . . . . 9 ⊢ (𝐻 ↾ (𝑓 supp 0 )) = ((ℎ ∈ 𝐴 ↦ (ℂfld Σg ℎ)) ↾ (𝑓 supp 0 ))
23		suppssdm 8174	. . . . . . . . . . 11 ⊢ (𝑓 supp 0 ) ⊆ dom 𝑓
24		eqid 2763	. . . . . . . . . . . 12 ⊢ (Base‘𝑅) = (Base‘𝑅)
25		mdegval.a	. . . . . . . . . . . 12 ⊢ 𝐴 = {𝑚 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑚 “ ℕ) ∈ Fin}
26		simpr 489	. . . . . . . . . . . 12 ⊢ (((𝐼 ∈ V ∧ 𝑅 ∈ V) ∧ 𝑓 ∈ 𝐵) → 𝑓 ∈ 𝐵)
27	3, 24, 6, 25, 26	mplelf 22128	. . . . . . . . . . 11 ⊢ (((𝐼 ∈ V ∧ 𝑅 ∈ V) ∧ 𝑓 ∈ 𝐵) → 𝑓:𝐴⟶(Base‘𝑅))
28	23, 27	fssdm 6727	. . . . . . . . . 10 ⊢ (((𝐼 ∈ V ∧ 𝑅 ∈ V) ∧ 𝑓 ∈ 𝐵) → (𝑓 supp 0 ) ⊆ 𝐴)
29	28	resmptd 6044	. . . . . . . . 9 ⊢ (((𝐼 ∈ V ∧ 𝑅 ∈ V) ∧ 𝑓 ∈ 𝐵) → ((ℎ ∈ 𝐴 ↦ (ℂfld Σg ℎ)) ↾ (𝑓 supp 0 )) = (ℎ ∈ (𝑓 supp 0 ) ↦ (ℂfld Σg ℎ)))
30	22, 29	eqtr2id 2811	. . . . . . . 8 ⊢ (((𝐼 ∈ V ∧ 𝑅 ∈ V) ∧ 𝑓 ∈ 𝐵) → (ℎ ∈ (𝑓 supp 0 ) ↦ (ℂfld Σg ℎ)) = (𝐻 ↾ (𝑓 supp 0 )))
31	30	rneqd 5930	. . . . . . 7 ⊢ (((𝐼 ∈ V ∧ 𝑅 ∈ V) ∧ 𝑓 ∈ 𝐵) → ran (ℎ ∈ (𝑓 supp 0 ) ↦ (ℂfld Σg ℎ)) = ran (𝐻 ↾ (𝑓 supp 0 )))
32		df-ima 5676	. . . . . . 7 ⊢ (𝐻 “ (𝑓 supp 0 )) = ran (𝐻 ↾ (𝑓 supp 0 ))
33	31, 32	eqtr4di 2816	. . . . . 6 ⊢ (((𝐼 ∈ V ∧ 𝑅 ∈ V) ∧ 𝑓 ∈ 𝐵) → ran (ℎ ∈ (𝑓 supp 0 ) ↦ (ℂfld Σg ℎ)) = (𝐻 “ (𝑓 supp 0 )))
34	33	supeq1d 9407	. . . . 5 ⊢ (((𝐼 ∈ V ∧ 𝑅 ∈ V) ∧ 𝑓 ∈ 𝐵) → sup(ran (ℎ ∈ (𝑓 supp 0 ) ↦ (ℂfld Σg ℎ)), ℝ*, < ) = sup((𝐻 “ (𝑓 supp 0 )), ℝ*, < ))
35	34	mpteq2dva 5205	. . . 4 ⊢ ((𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝑓 ∈ 𝐵 ↦ sup(ran (ℎ ∈ (𝑓 supp 0 ) ↦ (ℂfld Σg ℎ)), ℝ*, < )) = (𝑓 ∈ 𝐵 ↦ sup((𝐻 “ (𝑓 supp 0 )), ℝ*, < )))
36	20, 35	eqtrd 2798	. . 3 ⊢ ((𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 mDeg 𝑅) = (𝑓 ∈ 𝐵 ↦ sup((𝐻 “ (𝑓 supp 0 )), ℝ*, < )))
37		reldmmdeg 26195	. . . . . 6 ⊢ Rel dom mDeg
38	37	ovprc 7450	. . . . 5 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 mDeg 𝑅) = ∅)
39		mpt0 6679	. . . . 5 ⊢ (𝑓 ∈ ∅ ↦ sup((𝐻 “ (𝑓 supp 0 )), ℝ*, < )) = ∅
40	38, 39	eqtr4di 2816	. . . 4 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 mDeg 𝑅) = (𝑓 ∈ ∅ ↦ sup((𝐻 “ (𝑓 supp 0 )), ℝ*, < )))
41		reldmmpl 22118	. . . . . . . . 9 ⊢ Rel dom mPoly
42	41	ovprc 7450	. . . . . . . 8 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 mPoly 𝑅) = ∅)
43	3, 42	eqtrid 2810	. . . . . . 7 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → 𝑃 = ∅)
44	43	fveq2d 6887	. . . . . 6 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (Base‘𝑃) = (Base‘∅))
45		base0 17275	. . . . . 6 ⊢ ∅ = (Base‘∅)
46	44, 6, 45	3eqtr4g 2823	. . . . 5 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → 𝐵 = ∅)
47	46	mpteq1d 5202	. . . 4 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝑓 ∈ 𝐵 ↦ sup((𝐻 “ (𝑓 supp 0 )), ℝ*, < )) = (𝑓 ∈ ∅ ↦ sup((𝐻 “ (𝑓 supp 0 )), ℝ*, < )))
48	40, 47	eqtr4d 2801	. . 3 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 mDeg 𝑅) = (𝑓 ∈ 𝐵 ↦ sup((𝐻 “ (𝑓 supp 0 )), ℝ*, < )))
49	36, 48	pm2.61i 184	. 2 ⊢ (𝐼 mDeg 𝑅) = (𝑓 ∈ 𝐵 ↦ sup((𝐻 “ (𝑓 supp 0 )), ℝ*, < ))
50	1, 49	eqtri 2786	1 ⊢ 𝐷 = (𝑓 ∈ 𝐵 ↦ sup((𝐻 “ (𝑓 supp 0 )), ℝ*, < ))

Syntax hints:  ¬ wn 3   ∧ wa 400   = wceq 1570   ∈ wcel 2143  {crab 3416  Vcvv 3455  ∅c0 4287   ↦ cmpt 5193  ^◡ccnv 5662  ran crn 5664   ↾ cres 5665   “ cima 5666  ‘cfv 6538  (class class class)co 7412   supp csupp 8157   ↑_m cmap 8825  Fincfn 8944  supcsup 9401  ℝ^*cxr 11243   < clt 11244  ℕcn 12234  ℕ₀cn0 12505  Basecbs 17270  0_gc0g 17493   Σ_g cgsu 17494  ℂ_fldccnfld 21503   mPoly cmpl 22037   mDeg cmdg 26191

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734  ax-cnex 11157  ax-resscn 11158  ax-1cn 11159  ax-icn 11160  ax-addcl 11161  ax-addrcl 11162  ax-mulcl 11163  ax-mulrcl 11164  ax-mulcom 11165  ax-addass 11166  ax-mulass 11167  ax-distr 11168  ax-i2m1 11169  ax-1ne0 11170  ax-1rid 11171  ax-rnegex 11172  ax-rrecex 11173  ax-cnre 11174  ax-pre-lttri 11175  ax-pre-lttrn 11176  ax-pre-ltadd 11177  ax-pre-mulgt0 11178

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-tp 4595  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-of 7676  df-om 7864  df-1st 7987  df-2nd 7988  df-supp 8158  df-frecs 8279  df-wrecs 8310  df-recs 8359  df-rdg 8398  df-1o 8454  df-er 8695  df-map 8827  df-en 8945  df-dom 8946  df-sdom 8947  df-fin 8948  df-fsupp 9323  df-sup 9403  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11444  df-neg 11445  df-nn 12235  df-2 12304  df-3 12305  df-4 12306  df-5 12307  df-6 12308  df-7 12309  df-8 12310  df-9 12311  df-n0 12506  df-z 12593  df-uz 12864  df-fz 13537  df-struct 17208  df-sets 17225  df-slot 17243  df-ndx 17255  df-base 17271  df-ress 17292  df-plusg 17324  df-mulr 17325  df-sca 17327  df-vsca 17328  df-tset 17330  df-psr 22040  df-mpl 22042  df-mdeg 26193

This theorem is referenced by:  mdegval  26201  mdegxrf  26206  mdegpropd  26222