Theorem dgraaval 43168

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 → (degAA‘𝐴) = inf({𝑑 ∈ ℕ ∣ ∃𝑝 ∈ ((Poly‘ℚ) ∖ {0𝑝})((deg‘𝑝) = 𝑑 ∧ (𝑝‘𝐴) = 0)}, ℝ, < ))

Distinct variable group:   𝐴,𝑑,𝑝

Proof of Theorem dgraaval

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveqeq2 6885	. . . . . 6 ⊢ (𝑎 = 𝐴 → ((𝑝‘𝑎) = 0 ↔ (𝑝‘𝐴) = 0))
2	1	anbi2d 630	. . . . 5 ⊢ (𝑎 = 𝐴 → (((deg‘𝑝) = 𝑑 ∧ (𝑝‘𝑎) = 0) ↔ ((deg‘𝑝) = 𝑑 ∧ (𝑝‘𝐴) = 0)))
3	2	rexbidv 3164	. . . 4 ⊢ (𝑎 = 𝐴 → (∃𝑝 ∈ ((Poly‘ℚ) ∖ {0𝑝})((deg‘𝑝) = 𝑑 ∧ (𝑝‘𝑎) = 0) ↔ ∃𝑝 ∈ ((Poly‘ℚ) ∖ {0𝑝})((deg‘𝑝) = 𝑑 ∧ (𝑝‘𝐴) = 0)))
4	3	rabbidv 3423	. . 3 ⊢ (𝑎 = 𝐴 → {𝑑 ∈ ℕ ∣ ∃𝑝 ∈ ((Poly‘ℚ) ∖ {0𝑝})((deg‘𝑝) = 𝑑 ∧ (𝑝‘𝑎) = 0)} = {𝑑 ∈ ℕ ∣ ∃𝑝 ∈ ((Poly‘ℚ) ∖ {0𝑝})((deg‘𝑝) = 𝑑 ∧ (𝑝‘𝐴) = 0)})
5	4	infeq1d 9490	. 2 ⊢ (𝑎 = 𝐴 → inf({𝑑 ∈ ℕ ∣ ∃𝑝 ∈ ((Poly‘ℚ) ∖ {0𝑝})((deg‘𝑝) = 𝑑 ∧ (𝑝‘𝑎) = 0)}, ℝ, < ) = inf({𝑑 ∈ ℕ ∣ ∃𝑝 ∈ ((Poly‘ℚ) ∖ {0𝑝})((deg‘𝑝) = 𝑑 ∧ (𝑝‘𝐴) = 0)}, ℝ, < ))
6		df-dgraa 43166	. 2 ⊢ degAA = (𝑎 ∈ 𝔸 ↦ inf({𝑑 ∈ ℕ ∣ ∃𝑝 ∈ ((Poly‘ℚ) ∖ {0𝑝})((deg‘𝑝) = 𝑑 ∧ (𝑝‘𝑎) = 0)}, ℝ, < ))
7		ltso 11315	. . 3 ⊢ < Or ℝ
8	7	infex 9507	. 2 ⊢ inf({𝑑 ∈ ℕ ∣ ∃𝑝 ∈ ((Poly‘ℚ) ∖ {0𝑝})((deg‘𝑝) = 𝑑 ∧ (𝑝‘𝐴) = 0)}, ℝ, < ) ∈ V
9	5, 6, 8	fvmpt 6986	1 ⊢ (𝐴 ∈ 𝔸 → (degAA‘𝐴) = inf({𝑑 ∈ ℕ ∣ ∃𝑝 ∈ ((Poly‘ℚ) ∖ {0𝑝})((deg‘𝑝) = 𝑑 ∧ (𝑝‘𝐴) = 0)}, ℝ, < ))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∃wrex 3060  {crab 3415   ∖ cdif 3923  {csn 4601  ‘cfv 6531  infcinf 9453  ℝcr 11128  0cc0 11129   < clt 11269  ℕcn 12240  ℚcq 12964  0_𝑝c0p 25622  Polycply 26141  degcdgr 26144  𝔸caa 26274  deg_AAcdgraa 43164

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729  ax-resscn 11186  ax-pre-lttri 11203  ax-pre-lttrn 11204

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3359  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-po 5561  df-so 5562  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-er 8719  df-en 8960  df-dom 8961  df-sdom 8962  df-sup 9454  df-inf 9455  df-pnf 11271  df-mnf 11272  df-ltxr 11274  df-dgraa 43166

This theorem is referenced by:  dgraalem  43169  dgraaub  43172