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Theorem mdegfval 23743
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 = (0g𝑅)
mdegval.a 𝐴 = {𝑚 ∈ (ℕ0𝑚 𝐼) ∣ (𝑚 “ ℕ) ∈ Fin}
mdegval.h 𝐻 = (𝐴 ↦ (ℂfld Σg ))
Assertion
Ref Expression
mdegfval 𝐷 = (𝑓𝐵 ↦ sup((𝐻 “ (𝑓 supp 0 )), ℝ*, < ))
Distinct variable groups:   𝐴,   𝐵,𝑓   𝑓,𝐼   𝑚,𝐼   𝑅,𝑓   0 ,   𝑓,
Allowed substitution hints:   𝐴(𝑓,𝑚)   𝐵(,𝑚)   𝐷(𝑓,,𝑚)   𝑃(𝑓,,𝑚)   𝑅(,𝑚)   𝐻(𝑓,,𝑚)   𝐼()   0 (𝑓,𝑚)

Proof of Theorem mdegfval
Dummy variables 𝑖 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mdegval.d . 2 𝐷 = (𝐼 mDeg 𝑅)
2 oveq12 6619 . . . . . . . . 9 ((𝑖 = 𝐼𝑟 = 𝑅) → (𝑖 mPoly 𝑟) = (𝐼 mPoly 𝑅))
3 mdegval.p . . . . . . . . 9 𝑃 = (𝐼 mPoly 𝑅)
42, 3syl6eqr 2673 . . . . . . . 8 ((𝑖 = 𝐼𝑟 = 𝑅) → (𝑖 mPoly 𝑟) = 𝑃)
54fveq2d 6157 . . . . . . 7 ((𝑖 = 𝐼𝑟 = 𝑅) → (Base‘(𝑖 mPoly 𝑟)) = (Base‘𝑃))
6 mdegval.b . . . . . . 7 𝐵 = (Base‘𝑃)
75, 6syl6eqr 2673 . . . . . 6 ((𝑖 = 𝐼𝑟 = 𝑅) → (Base‘(𝑖 mPoly 𝑟)) = 𝐵)
8 fveq2 6153 . . . . . . . . . . . 12 (𝑟 = 𝑅 → (0g𝑟) = (0g𝑅))
9 mdegval.z . . . . . . . . . . . 12 0 = (0g𝑅)
108, 9syl6eqr 2673 . . . . . . . . . . 11 (𝑟 = 𝑅 → (0g𝑟) = 0 )
1110oveq2d 6626 . . . . . . . . . 10 (𝑟 = 𝑅 → (𝑓 supp (0g𝑟)) = (𝑓 supp 0 ))
1211mpteq1d 4703 . . . . . . . . 9 (𝑟 = 𝑅 → ( ∈ (𝑓 supp (0g𝑟)) ↦ (ℂfld Σg )) = ( ∈ (𝑓 supp 0 ) ↦ (ℂfld Σg )))
1312rneqd 5318 . . . . . . . 8 (𝑟 = 𝑅 → ran ( ∈ (𝑓 supp (0g𝑟)) ↦ (ℂfld Σg )) = ran ( ∈ (𝑓 supp 0 ) ↦ (ℂfld Σg )))
1413supeq1d 8304 . . . . . . 7 (𝑟 = 𝑅 → sup(ran ( ∈ (𝑓 supp (0g𝑟)) ↦ (ℂfld Σg )), ℝ*, < ) = sup(ran ( ∈ (𝑓 supp 0 ) ↦ (ℂfld Σg )), ℝ*, < ))
1514adantl 482 . . . . . 6 ((𝑖 = 𝐼𝑟 = 𝑅) → sup(ran ( ∈ (𝑓 supp (0g𝑟)) ↦ (ℂfld Σg )), ℝ*, < ) = sup(ran ( ∈ (𝑓 supp 0 ) ↦ (ℂfld Σg )), ℝ*, < ))
167, 15mpteq12dv 4698 . . . . 5 ((𝑖 = 𝐼𝑟 = 𝑅) → (𝑓 ∈ (Base‘(𝑖 mPoly 𝑟)) ↦ sup(ran ( ∈ (𝑓 supp (0g𝑟)) ↦ (ℂfld Σg )), ℝ*, < )) = (𝑓𝐵 ↦ sup(ran ( ∈ (𝑓 supp 0 ) ↦ (ℂfld Σg )), ℝ*, < )))
17 df-mdeg 23736 . . . . 5 mDeg = (𝑖 ∈ V, 𝑟 ∈ V ↦ (𝑓 ∈ (Base‘(𝑖 mPoly 𝑟)) ↦ sup(ran ( ∈ (𝑓 supp (0g𝑟)) ↦ (ℂfld Σg )), ℝ*, < )))
18 fvex 6163 . . . . . . 7 (Base‘𝑃) ∈ V
196, 18eqeltri 2694 . . . . . 6 𝐵 ∈ V
2019mptex 6446 . . . . 5 (𝑓𝐵 ↦ sup(ran ( ∈ (𝑓 supp 0 ) ↦ (ℂfld Σg )), ℝ*, < )) ∈ V
2116, 17, 20ovmpt2a 6751 . . . 4 ((𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 mDeg 𝑅) = (𝑓𝐵 ↦ sup(ran ( ∈ (𝑓 supp 0 ) ↦ (ℂfld Σg )), ℝ*, < )))
22 mdegval.h . . . . . . . . . 10 𝐻 = (𝐴 ↦ (ℂfld Σg ))
2322reseq1i 5357 . . . . . . . . 9 (𝐻 ↾ (𝑓 supp 0 )) = ((𝐴 ↦ (ℂfld Σg )) ↾ (𝑓 supp 0 ))
24 suppssdm 7260 . . . . . . . . . . 11 (𝑓 supp 0 ) ⊆ dom 𝑓
25 eqid 2621 . . . . . . . . . . . . 13 (Base‘𝑅) = (Base‘𝑅)
26 mdegval.a . . . . . . . . . . . . 13 𝐴 = {𝑚 ∈ (ℕ0𝑚 𝐼) ∣ (𝑚 “ ℕ) ∈ Fin}
27 simpr 477 . . . . . . . . . . . . 13 (((𝐼 ∈ V ∧ 𝑅 ∈ V) ∧ 𝑓𝐵) → 𝑓𝐵)
283, 25, 6, 26, 27mplelf 19365 . . . . . . . . . . . 12 (((𝐼 ∈ V ∧ 𝑅 ∈ V) ∧ 𝑓𝐵) → 𝑓:𝐴⟶(Base‘𝑅))
29 fdm 6013 . . . . . . . . . . . 12 (𝑓:𝐴⟶(Base‘𝑅) → dom 𝑓 = 𝐴)
3028, 29syl 17 . . . . . . . . . . 11 (((𝐼 ∈ V ∧ 𝑅 ∈ V) ∧ 𝑓𝐵) → dom 𝑓 = 𝐴)
3124, 30syl5sseq 3637 . . . . . . . . . 10 (((𝐼 ∈ V ∧ 𝑅 ∈ V) ∧ 𝑓𝐵) → (𝑓 supp 0 ) ⊆ 𝐴)
3231resmptd 5416 . . . . . . . . 9 (((𝐼 ∈ V ∧ 𝑅 ∈ V) ∧ 𝑓𝐵) → ((𝐴 ↦ (ℂfld Σg )) ↾ (𝑓 supp 0 )) = ( ∈ (𝑓 supp 0 ) ↦ (ℂfld Σg )))
3323, 32syl5req 2668 . . . . . . . 8 (((𝐼 ∈ V ∧ 𝑅 ∈ V) ∧ 𝑓𝐵) → ( ∈ (𝑓 supp 0 ) ↦ (ℂfld Σg )) = (𝐻 ↾ (𝑓 supp 0 )))
3433rneqd 5318 . . . . . . 7 (((𝐼 ∈ V ∧ 𝑅 ∈ V) ∧ 𝑓𝐵) → ran ( ∈ (𝑓 supp 0 ) ↦ (ℂfld Σg )) = ran (𝐻 ↾ (𝑓 supp 0 )))
35 df-ima 5092 . . . . . . 7 (𝐻 “ (𝑓 supp 0 )) = ran (𝐻 ↾ (𝑓 supp 0 ))
3634, 35syl6eqr 2673 . . . . . 6 (((𝐼 ∈ V ∧ 𝑅 ∈ V) ∧ 𝑓𝐵) → ran ( ∈ (𝑓 supp 0 ) ↦ (ℂfld Σg )) = (𝐻 “ (𝑓 supp 0 )))
3736supeq1d 8304 . . . . 5 (((𝐼 ∈ V ∧ 𝑅 ∈ V) ∧ 𝑓𝐵) → sup(ran ( ∈ (𝑓 supp 0 ) ↦ (ℂfld Σg )), ℝ*, < ) = sup((𝐻 “ (𝑓 supp 0 )), ℝ*, < ))
3837mpteq2dva 4709 . . . 4 ((𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝑓𝐵 ↦ sup(ran ( ∈ (𝑓 supp 0 ) ↦ (ℂfld Σg )), ℝ*, < )) = (𝑓𝐵 ↦ sup((𝐻 “ (𝑓 supp 0 )), ℝ*, < )))
3921, 38eqtrd 2655 . . 3 ((𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 mDeg 𝑅) = (𝑓𝐵 ↦ sup((𝐻 “ (𝑓 supp 0 )), ℝ*, < )))
40 reldmmdeg 23738 . . . . . 6 Rel dom mDeg
4140ovprc 6643 . . . . 5 (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 mDeg 𝑅) = ∅)
42 mpt0 5983 . . . . 5 (𝑓 ∈ ∅ ↦ sup((𝐻 “ (𝑓 supp 0 )), ℝ*, < )) = ∅
4341, 42syl6eqr 2673 . . . 4 (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 mDeg 𝑅) = (𝑓 ∈ ∅ ↦ sup((𝐻 “ (𝑓 supp 0 )), ℝ*, < )))
44 reldmmpl 19359 . . . . . . . . 9 Rel dom mPoly
4544ovprc 6643 . . . . . . . 8 (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 mPoly 𝑅) = ∅)
463, 45syl5eq 2667 . . . . . . 7 (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → 𝑃 = ∅)
4746fveq2d 6157 . . . . . 6 (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (Base‘𝑃) = (Base‘∅))
48 base0 15844 . . . . . 6 ∅ = (Base‘∅)
4947, 6, 483eqtr4g 2680 . . . . 5 (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → 𝐵 = ∅)
5049mpteq1d 4703 . . . 4 (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝑓𝐵 ↦ sup((𝐻 “ (𝑓 supp 0 )), ℝ*, < )) = (𝑓 ∈ ∅ ↦ sup((𝐻 “ (𝑓 supp 0 )), ℝ*, < )))
5143, 50eqtr4d 2658 . . 3 (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 mDeg 𝑅) = (𝑓𝐵 ↦ sup((𝐻 “ (𝑓 supp 0 )), ℝ*, < )))
5239, 51pm2.61i 176 . 2 (𝐼 mDeg 𝑅) = (𝑓𝐵 ↦ sup((𝐻 “ (𝑓 supp 0 )), ℝ*, < ))
531, 52eqtri 2643 1 𝐷 = (𝑓𝐵 ↦ sup((𝐻 “ (𝑓 supp 0 )), ℝ*, < ))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 384   = wceq 1480  wcel 1987  {crab 2911  Vcvv 3189  c0 3896  cmpt 4678  ccnv 5078  dom cdm 5079  ran crn 5080  cres 5081  cima 5082  wf 5848  cfv 5852  (class class class)co 6610   supp csupp 7247  𝑚 cmap 7809  Fincfn 7907  supcsup 8298  *cxr 10025   < clt 10026  cn 10972  0cn0 11244  Basecbs 15792  0gc0g 16032   Σg cgsu 16033   mPoly cmpl 19285  fldccnfld 19678   mDeg cmdg 23734
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-cnex 9944  ax-resscn 9945  ax-1cn 9946  ax-icn 9947  ax-addcl 9948  ax-addrcl 9949  ax-mulcl 9950  ax-mulrcl 9951  ax-mulcom 9952  ax-addass 9953  ax-mulass 9954  ax-distr 9955  ax-i2m1 9956  ax-1ne0 9957  ax-1rid 9958  ax-rnegex 9959  ax-rrecex 9960  ax-cnre 9961  ax-pre-lttri 9962  ax-pre-lttrn 9963  ax-pre-ltadd 9964  ax-pre-mulgt0 9965
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-of 6857  df-om 7020  df-1st 7120  df-2nd 7121  df-supp 7248  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-oadd 7516  df-er 7694  df-map 7811  df-en 7908  df-dom 7909  df-sdom 7910  df-fin 7911  df-fsupp 8228  df-sup 8300  df-pnf 10028  df-mnf 10029  df-xr 10030  df-ltxr 10031  df-le 10032  df-sub 10220  df-neg 10221  df-nn 10973  df-2 11031  df-3 11032  df-4 11033  df-5 11034  df-6 11035  df-7 11036  df-8 11037  df-9 11038  df-n0 11245  df-z 11330  df-uz 11640  df-fz 12277  df-struct 15794  df-ndx 15795  df-slot 15796  df-base 15797  df-sets 15798  df-ress 15799  df-plusg 15886  df-mulr 15887  df-sca 15889  df-vsca 15890  df-tset 15892  df-psr 19288  df-mpl 19290  df-mdeg 23736
This theorem is referenced by:  mdegval  23744  mdegxrf  23749  mdegpropd  23765
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