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Theorem dgraaval 38233
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 fveq2 6332 . . . . . . 7 (𝑎 = 𝐴 → (𝑝𝑎) = (𝑝𝐴))
21eqeq1d 2772 . . . . . 6 (𝑎 = 𝐴 → ((𝑝𝑎) = 0 ↔ (𝑝𝐴) = 0))
32anbi2d 606 . . . . 5 (𝑎 = 𝐴 → (((deg‘𝑝) = 𝑑 ∧ (𝑝𝑎) = 0) ↔ ((deg‘𝑝) = 𝑑 ∧ (𝑝𝐴) = 0)))
43rexbidv 3199 . . . 4 (𝑎 = 𝐴 → (∃𝑝 ∈ ((Poly‘ℚ) ∖ {0𝑝})((deg‘𝑝) = 𝑑 ∧ (𝑝𝑎) = 0) ↔ ∃𝑝 ∈ ((Poly‘ℚ) ∖ {0𝑝})((deg‘𝑝) = 𝑑 ∧ (𝑝𝐴) = 0)))
54rabbidv 3338 . . 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)})
65infeq1d 8538 . 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, < ))
7 df-dgraa 38231 . 2 degAA = (𝑎 ∈ 𝔸 ↦ inf({𝑑 ∈ ℕ ∣ ∃𝑝 ∈ ((Poly‘ℚ) ∖ {0𝑝})((deg‘𝑝) = 𝑑 ∧ (𝑝𝑎) = 0)}, ℝ, < ))
8 ltso 10319 . . 3 < Or ℝ
98infex 8554 . 2 inf({𝑑 ∈ ℕ ∣ ∃𝑝 ∈ ((Poly‘ℚ) ∖ {0𝑝})((deg‘𝑝) = 𝑑 ∧ (𝑝𝐴) = 0)}, ℝ, < ) ∈ V
106, 7, 9fvmpt 6424 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 382   = wceq 1630  wcel 2144  wrex 3061  {crab 3064  cdif 3718  {csn 4314  cfv 6031  infcinf 8502  cr 10136  0cc0 10137   < clt 10275  cn 11221  cq 11990  0𝑝c0p 23655  Polycply 24159  degcdgr 24162  𝔸caa 24288  degAAcdgraa 38229
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095  ax-resscn 10194  ax-pre-lttri 10211  ax-pre-lttrn 10212
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-nel 3046  df-ral 3065  df-rex 3066  df-rmo 3068  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-po 5170  df-so 5171  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-er 7895  df-en 8109  df-dom 8110  df-sdom 8111  df-sup 8503  df-inf 8504  df-pnf 10277  df-mnf 10278  df-ltxr 10280  df-dgraa 38231
This theorem is referenced by:  dgraalem  38234  dgraaub  38237
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