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Theorem dgraaval 36733
 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 6103 . . . . . . 7 (𝑎 = 𝐴 → (𝑝𝑎) = (𝑝𝐴))
21eqeq1d 2612 . . . . . 6 (𝑎 = 𝐴 → ((𝑝𝑎) = 0 ↔ (𝑝𝐴) = 0))
32anbi2d 736 . . . . 5 (𝑎 = 𝐴 → (((deg‘𝑝) = 𝑑 ∧ (𝑝𝑎) = 0) ↔ ((deg‘𝑝) = 𝑑 ∧ (𝑝𝐴) = 0)))
43rexbidv 3034 . . . 4 (𝑎 = 𝐴 → (∃𝑝 ∈ ((Poly‘ℚ) ∖ {0𝑝})((deg‘𝑝) = 𝑑 ∧ (𝑝𝑎) = 0) ↔ ∃𝑝 ∈ ((Poly‘ℚ) ∖ {0𝑝})((deg‘𝑝) = 𝑑 ∧ (𝑝𝐴) = 0)))
54rabbidv 3164 . . 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 8266 . 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 36731 . 2 degAA = (𝑎 ∈ 𝔸 ↦ inf({𝑑 ∈ ℕ ∣ ∃𝑝 ∈ ((Poly‘ℚ) ∖ {0𝑝})((deg‘𝑝) = 𝑑 ∧ (𝑝𝑎) = 0)}, ℝ, < ))
8 ltso 9997 . . 3 < Or ℝ
98infex 8282 . 2 inf({𝑑 ∈ ℕ ∣ ∃𝑝 ∈ ((Poly‘ℚ) ∖ {0𝑝})((deg‘𝑝) = 𝑑 ∧ (𝑝𝐴) = 0)}, ℝ, < ) ∈ V
106, 7, 9fvmpt 6191 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 383   = wceq 1475   ∈ wcel 1977  ∃wrex 2897  {crab 2900   ∖ cdif 3537  {csn 4125  ‘cfv 5804  infcinf 8230  ℝcr 9814  0cc0 9815   < clt 9953  ℕcn 10897  ℚcq 11664  0𝑝c0p 23242  Polycply 23744  degcdgr 23747  𝔸caa 23873  degAAcdgraa 36729 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-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-resscn 9872  ax-pre-lttri 9889  ax-pre-lttrn 9890 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-rmo 2904  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-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-po 4959  df-so 4960  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-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-sup 8231  df-inf 8232  df-pnf 9955  df-mnf 9956  df-ltxr 9958  df-dgraa 36731 This theorem is referenced by:  dgraalem  36734  dgraaub  36737
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