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Theorem dgraaval 39174
 Description: Value of the degree function on an algebraic number. (Contributed by Stefan O'Rear, 25-Nov-2014.) (Revised by AV, 29-Sep-2020.)
Assertion
Ref Expression
dgraaval (𝐴 ∈ 𝔸 → (degAA𝐴) = inf({𝑑 ∈ ℕ ∣ ∃𝑝 ∈ ((Poly‘ℚ) ∖ {0𝑝})((deg‘𝑝) = 𝑑 ∧ (𝑝𝐴) = 0)}, ℝ, < ))
Distinct variable group:   𝐴,𝑑,𝑝

Proof of Theorem dgraaval
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 fveqeq2 6506 . . . . . 6 (𝑎 = 𝐴 → ((𝑝𝑎) = 0 ↔ (𝑝𝐴) = 0))
21anbi2d 620 . . . . 5 (𝑎 = 𝐴 → (((deg‘𝑝) = 𝑑 ∧ (𝑝𝑎) = 0) ↔ ((deg‘𝑝) = 𝑑 ∧ (𝑝𝐴) = 0)))
32rexbidv 3237 . . . 4 (𝑎 = 𝐴 → (∃𝑝 ∈ ((Poly‘ℚ) ∖ {0𝑝})((deg‘𝑝) = 𝑑 ∧ (𝑝𝑎) = 0) ↔ ∃𝑝 ∈ ((Poly‘ℚ) ∖ {0𝑝})((deg‘𝑝) = 𝑑 ∧ (𝑝𝐴) = 0)))
43rabbidv 3398 . . 3 (𝑎 = 𝐴 → {𝑑 ∈ ℕ ∣ ∃𝑝 ∈ ((Poly‘ℚ) ∖ {0𝑝})((deg‘𝑝) = 𝑑 ∧ (𝑝𝑎) = 0)} = {𝑑 ∈ ℕ ∣ ∃𝑝 ∈ ((Poly‘ℚ) ∖ {0𝑝})((deg‘𝑝) = 𝑑 ∧ (𝑝𝐴) = 0)})
54infeq1d 8735 . 2 (𝑎 = 𝐴 → inf({𝑑 ∈ ℕ ∣ ∃𝑝 ∈ ((Poly‘ℚ) ∖ {0𝑝})((deg‘𝑝) = 𝑑 ∧ (𝑝𝑎) = 0)}, ℝ, < ) = inf({𝑑 ∈ ℕ ∣ ∃𝑝 ∈ ((Poly‘ℚ) ∖ {0𝑝})((deg‘𝑝) = 𝑑 ∧ (𝑝𝐴) = 0)}, ℝ, < ))
6 df-dgraa 39172 . 2 degAA = (𝑎 ∈ 𝔸 ↦ inf({𝑑 ∈ ℕ ∣ ∃𝑝 ∈ ((Poly‘ℚ) ∖ {0𝑝})((deg‘𝑝) = 𝑑 ∧ (𝑝𝑎) = 0)}, ℝ, < ))
7 ltso 10520 . . 3 < Or ℝ
87infex 8751 . 2 inf({𝑑 ∈ ℕ ∣ ∃𝑝 ∈ ((Poly‘ℚ) ∖ {0𝑝})((deg‘𝑝) = 𝑑 ∧ (𝑝𝐴) = 0)}, ℝ, < ) ∈ V
95, 6, 8fvmpt 6594 1 (𝐴 ∈ 𝔸 → (degAA𝐴) = inf({𝑑 ∈ ℕ ∣ ∃𝑝 ∈ ((Poly‘ℚ) ∖ {0𝑝})((deg‘𝑝) = 𝑑 ∧ (𝑝𝐴) = 0)}, ℝ, < ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 387   = wceq 1508   ∈ wcel 2051  ∃wrex 3084  {crab 3087   ∖ cdif 3821  {csn 4436  ‘cfv 6186  infcinf 8699  ℝcr 10333  0cc0 10334   < clt 10473  ℕcn 11438  ℚcq 12161  0𝑝c0p 23989  Polycply 24493  degcdgr 24496  𝔸caa 24622  degAAcdgraa 39170 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2745  ax-sep 5057  ax-nul 5064  ax-pow 5116  ax-pr 5183  ax-un 7278  ax-resscn 10391  ax-pre-lttri 10408  ax-pre-lttrn 10409 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3or 1070  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2754  df-cleq 2766  df-clel 2841  df-nfc 2913  df-ne 2963  df-nel 3069  df-ral 3088  df-rex 3089  df-rmo 3091  df-rab 3092  df-v 3412  df-sbc 3677  df-csb 3782  df-dif 3827  df-un 3829  df-in 3831  df-ss 3838  df-nul 4174  df-if 4346  df-pw 4419  df-sn 4437  df-pr 4439  df-op 4443  df-uni 4710  df-br 4927  df-opab 4989  df-mpt 5006  df-id 5309  df-po 5323  df-so 5324  df-xp 5410  df-rel 5411  df-cnv 5412  df-co 5413  df-dm 5414  df-rn 5415  df-res 5416  df-ima 5417  df-iota 6150  df-fun 6188  df-fn 6189  df-f 6190  df-f1 6191  df-fo 6192  df-f1o 6193  df-fv 6194  df-er 8088  df-en 8306  df-dom 8307  df-sdom 8308  df-sup 8700  df-inf 8701  df-pnf 10475  df-mnf 10476  df-ltxr 10478  df-dgraa 39172 This theorem is referenced by:  dgraalem  39175  dgraaub  39178
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