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Theorem mdegfval 23626
 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 6558 . . . . . . . . 9 ((𝑖 = 𝐼𝑟 = 𝑅) → (𝑖 mPoly 𝑟) = (𝐼 mPoly 𝑅))
3 mdegval.p . . . . . . . . 9 𝑃 = (𝐼 mPoly 𝑅)
42, 3syl6eqr 2662 . . . . . . . 8 ((𝑖 = 𝐼𝑟 = 𝑅) → (𝑖 mPoly 𝑟) = 𝑃)
54fveq2d 6107 . . . . . . 7 ((𝑖 = 𝐼𝑟 = 𝑅) → (Base‘(𝑖 mPoly 𝑟)) = (Base‘𝑃))
6 mdegval.b . . . . . . 7 𝐵 = (Base‘𝑃)
75, 6syl6eqr 2662 . . . . . 6 ((𝑖 = 𝐼𝑟 = 𝑅) → (Base‘(𝑖 mPoly 𝑟)) = 𝐵)
8 fveq2 6103 . . . . . . . . . . . 12 (𝑟 = 𝑅 → (0g𝑟) = (0g𝑅))
9 mdegval.z . . . . . . . . . . . 12 0 = (0g𝑅)
108, 9syl6eqr 2662 . . . . . . . . . . 11 (𝑟 = 𝑅 → (0g𝑟) = 0 )
1110oveq2d 6565 . . . . . . . . . 10 (𝑟 = 𝑅 → (𝑓 supp (0g𝑟)) = (𝑓 supp 0 ))
1211mpteq1d 4666 . . . . . . . . 9 (𝑟 = 𝑅 → ( ∈ (𝑓 supp (0g𝑟)) ↦ (ℂfld Σg )) = ( ∈ (𝑓 supp 0 ) ↦ (ℂfld Σg )))
1312rneqd 5274 . . . . . . . 8 (𝑟 = 𝑅 → ran ( ∈ (𝑓 supp (0g𝑟)) ↦ (ℂfld Σg )) = ran ( ∈ (𝑓 supp 0 ) ↦ (ℂfld Σg )))
1413supeq1d 8235 . . . . . . 7 (𝑟 = 𝑅 → sup(ran ( ∈ (𝑓 supp (0g𝑟)) ↦ (ℂfld Σg )), ℝ*, < ) = sup(ran ( ∈ (𝑓 supp 0 ) ↦ (ℂfld Σg )), ℝ*, < ))
1514adantl 481 . . . . . 6 ((𝑖 = 𝐼𝑟 = 𝑅) → sup(ran ( ∈ (𝑓 supp (0g𝑟)) ↦ (ℂfld Σg )), ℝ*, < ) = sup(ran ( ∈ (𝑓 supp 0 ) ↦ (ℂfld Σg )), ℝ*, < ))
167, 15mpteq12dv 4663 . . . . 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 23619 . . . . 5 mDeg = (𝑖 ∈ V, 𝑟 ∈ V ↦ (𝑓 ∈ (Base‘(𝑖 mPoly 𝑟)) ↦ sup(ran ( ∈ (𝑓 supp (0g𝑟)) ↦ (ℂfld Σg )), ℝ*, < )))
18 fvex 6113 . . . . . . 7 (Base‘𝑃) ∈ V
196, 18eqeltri 2684 . . . . . 6 𝐵 ∈ V
2019mptex 6390 . . . . 5 (𝑓𝐵 ↦ sup(ran ( ∈ (𝑓 supp 0 ) ↦ (ℂfld Σg )), ℝ*, < )) ∈ V
2116, 17, 20ovmpt2a 6689 . . . 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 5313 . . . . . . . . 9 (𝐻 ↾ (𝑓 supp 0 )) = ((𝐴 ↦ (ℂfld Σg )) ↾ (𝑓 supp 0 ))
24 suppssdm 7195 . . . . . . . . . . 11 (𝑓 supp 0 ) ⊆ dom 𝑓
25 eqid 2610 . . . . . . . . . . . . 13 (Base‘𝑅) = (Base‘𝑅)
26 mdegval.a . . . . . . . . . . . . 13 𝐴 = {𝑚 ∈ (ℕ0𝑚 𝐼) ∣ (𝑚 “ ℕ) ∈ Fin}
27 simpr 476 . . . . . . . . . . . . 13 (((𝐼 ∈ V ∧ 𝑅 ∈ V) ∧ 𝑓𝐵) → 𝑓𝐵)
283, 25, 6, 26, 27mplelf 19254 . . . . . . . . . . . 12 (((𝐼 ∈ V ∧ 𝑅 ∈ V) ∧ 𝑓𝐵) → 𝑓:𝐴⟶(Base‘𝑅))
29 fdm 5964 . . . . . . . . . . . 12 (𝑓:𝐴⟶(Base‘𝑅) → dom 𝑓 = 𝐴)
3028, 29syl 17 . . . . . . . . . . 11 (((𝐼 ∈ V ∧ 𝑅 ∈ V) ∧ 𝑓𝐵) → dom 𝑓 = 𝐴)
3124, 30syl5sseq 3616 . . . . . . . . . 10 (((𝐼 ∈ V ∧ 𝑅 ∈ V) ∧ 𝑓𝐵) → (𝑓 supp 0 ) ⊆ 𝐴)
3231resmptd 5371 . . . . . . . . 9 (((𝐼 ∈ V ∧ 𝑅 ∈ V) ∧ 𝑓𝐵) → ((𝐴 ↦ (ℂfld Σg )) ↾ (𝑓 supp 0 )) = ( ∈ (𝑓 supp 0 ) ↦ (ℂfld Σg )))
3323, 32syl5req 2657 . . . . . . . 8 (((𝐼 ∈ V ∧ 𝑅 ∈ V) ∧ 𝑓𝐵) → ( ∈ (𝑓 supp 0 ) ↦ (ℂfld Σg )) = (𝐻 ↾ (𝑓 supp 0 )))
3433rneqd 5274 . . . . . . 7 (((𝐼 ∈ V ∧ 𝑅 ∈ V) ∧ 𝑓𝐵) → ran ( ∈ (𝑓 supp 0 ) ↦ (ℂfld Σg )) = ran (𝐻 ↾ (𝑓 supp 0 )))
35 df-ima 5051 . . . . . . 7 (𝐻 “ (𝑓 supp 0 )) = ran (𝐻 ↾ (𝑓 supp 0 ))
3634, 35syl6eqr 2662 . . . . . 6 (((𝐼 ∈ V ∧ 𝑅 ∈ V) ∧ 𝑓𝐵) → ran ( ∈ (𝑓 supp 0 ) ↦ (ℂfld Σg )) = (𝐻 “ (𝑓 supp 0 )))
3736supeq1d 8235 . . . . 5 (((𝐼 ∈ V ∧ 𝑅 ∈ V) ∧ 𝑓𝐵) → sup(ran ( ∈ (𝑓 supp 0 ) ↦ (ℂfld Σg )), ℝ*, < ) = sup((𝐻 “ (𝑓 supp 0 )), ℝ*, < ))
3837mpteq2dva 4672 . . . 4 ((𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝑓𝐵 ↦ sup(ran ( ∈ (𝑓 supp 0 ) ↦ (ℂfld Σg )), ℝ*, < )) = (𝑓𝐵 ↦ sup((𝐻 “ (𝑓 supp 0 )), ℝ*, < )))
3921, 38eqtrd 2644 . . 3 ((𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 mDeg 𝑅) = (𝑓𝐵 ↦ sup((𝐻 “ (𝑓 supp 0 )), ℝ*, < )))
40 reldmmdeg 23621 . . . . . 6 Rel dom mDeg
4140ovprc 6581 . . . . 5 (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 mDeg 𝑅) = ∅)
42 mpt0 5934 . . . . 5 (𝑓 ∈ ∅ ↦ sup((𝐻 “ (𝑓 supp 0 )), ℝ*, < )) = ∅
4341, 42syl6eqr 2662 . . . 4 (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 mDeg 𝑅) = (𝑓 ∈ ∅ ↦ sup((𝐻 “ (𝑓 supp 0 )), ℝ*, < )))
44 reldmmpl 19248 . . . . . . . . 9 Rel dom mPoly
4544ovprc 6581 . . . . . . . 8 (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 mPoly 𝑅) = ∅)
463, 45syl5eq 2656 . . . . . . 7 (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → 𝑃 = ∅)
4746fveq2d 6107 . . . . . 6 (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (Base‘𝑃) = (Base‘∅))
48 base0 15740 . . . . . 6 ∅ = (Base‘∅)
4947, 6, 483eqtr4g 2669 . . . . 5 (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → 𝐵 = ∅)
5049mpteq1d 4666 . . . 4 (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝑓𝐵 ↦ sup((𝐻 “ (𝑓 supp 0 )), ℝ*, < )) = (𝑓 ∈ ∅ ↦ sup((𝐻 “ (𝑓 supp 0 )), ℝ*, < )))
5143, 50eqtr4d 2647 . . 3 (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 mDeg 𝑅) = (𝑓𝐵 ↦ sup((𝐻 “ (𝑓 supp 0 )), ℝ*, < )))
5239, 51pm2.61i 175 . 2 (𝐼 mDeg 𝑅) = (𝑓𝐵 ↦ sup((𝐻 “ (𝑓 supp 0 )), ℝ*, < ))
531, 52eqtri 2632 1 𝐷 = (𝑓𝐵 ↦ sup((𝐻 “ (𝑓 supp 0 )), ℝ*, < ))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {crab 2900  Vcvv 3173  ∅c0 3874   ↦ cmpt 4643  ◡ccnv 5037  dom cdm 5038  ran crn 5039   ↾ cres 5040   “ cima 5041  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   supp csupp 7182   ↑𝑚 cmap 7744  Fincfn 7841  supcsup 8229  ℝ*cxr 9952   < clt 9953  ℕcn 10897  ℕ0cn0 11169  Basecbs 15695  0gc0g 15923   Σg cgsu 15924   mPoly cmpl 19174  ℂfldccnfld 19567   mDeg cmdg 23617 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-sup 8231  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-mulr 15782  df-sca 15784  df-vsca 15785  df-tset 15787  df-psr 19177  df-mpl 19179  df-mdeg 23619 This theorem is referenced by:  mdegval  23627  mdegxrf  23632  mdegpropd  23648
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