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Theorem dgraaval 42599
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 β†’ (degAAβ€˜π΄) = inf({𝑑 ∈ β„• ∣ βˆƒπ‘ ∈ ((Polyβ€˜β„š) βˆ– {0𝑝})((degβ€˜π‘) = 𝑑 ∧ (π‘β€˜π΄) = 0)}, ℝ, < ))
Distinct variable group:   𝐴,𝑑,𝑝

Proof of Theorem dgraaval
Dummy variable π‘Ž is distinct from all other variables.
StepHypRef Expression
1 fveqeq2 6911 . . . . . 6 (π‘Ž = 𝐴 β†’ ((π‘β€˜π‘Ž) = 0 ↔ (π‘β€˜π΄) = 0))
21anbi2d 628 . . . . 5 (π‘Ž = 𝐴 β†’ (((degβ€˜π‘) = 𝑑 ∧ (π‘β€˜π‘Ž) = 0) ↔ ((degβ€˜π‘) = 𝑑 ∧ (π‘β€˜π΄) = 0)))
32rexbidv 3176 . . . 4 (π‘Ž = 𝐴 β†’ (βˆƒπ‘ ∈ ((Polyβ€˜β„š) βˆ– {0𝑝})((degβ€˜π‘) = 𝑑 ∧ (π‘β€˜π‘Ž) = 0) ↔ βˆƒπ‘ ∈ ((Polyβ€˜β„š) βˆ– {0𝑝})((degβ€˜π‘) = 𝑑 ∧ (π‘β€˜π΄) = 0)))
43rabbidv 3438 . . 3 (π‘Ž = 𝐴 β†’ {𝑑 ∈ β„• ∣ βˆƒπ‘ ∈ ((Polyβ€˜β„š) βˆ– {0𝑝})((degβ€˜π‘) = 𝑑 ∧ (π‘β€˜π‘Ž) = 0)} = {𝑑 ∈ β„• ∣ βˆƒπ‘ ∈ ((Polyβ€˜β„š) βˆ– {0𝑝})((degβ€˜π‘) = 𝑑 ∧ (π‘β€˜π΄) = 0)})
54infeq1d 9508 . 2 (π‘Ž = 𝐴 β†’ inf({𝑑 ∈ β„• ∣ βˆƒπ‘ ∈ ((Polyβ€˜β„š) βˆ– {0𝑝})((degβ€˜π‘) = 𝑑 ∧ (π‘β€˜π‘Ž) = 0)}, ℝ, < ) = inf({𝑑 ∈ β„• ∣ βˆƒπ‘ ∈ ((Polyβ€˜β„š) βˆ– {0𝑝})((degβ€˜π‘) = 𝑑 ∧ (π‘β€˜π΄) = 0)}, ℝ, < ))
6 df-dgraa 42597 . 2 degAA = (π‘Ž ∈ 𝔸 ↦ inf({𝑑 ∈ β„• ∣ βˆƒπ‘ ∈ ((Polyβ€˜β„š) βˆ– {0𝑝})((degβ€˜π‘) = 𝑑 ∧ (π‘β€˜π‘Ž) = 0)}, ℝ, < ))
7 ltso 11332 . . 3 < Or ℝ
87infex 9524 . 2 inf({𝑑 ∈ β„• ∣ βˆƒπ‘ ∈ ((Polyβ€˜β„š) βˆ– {0𝑝})((degβ€˜π‘) = 𝑑 ∧ (π‘β€˜π΄) = 0)}, ℝ, < ) ∈ V
95, 6, 8fvmpt 7010 1 (𝐴 ∈ 𝔸 β†’ (degAAβ€˜π΄) = inf({𝑑 ∈ β„• ∣ βˆƒπ‘ ∈ ((Polyβ€˜β„š) βˆ– {0𝑝})((degβ€˜π‘) = 𝑑 ∧ (π‘β€˜π΄) = 0)}, ℝ, < ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098  βˆƒwrex 3067  {crab 3430   βˆ– cdif 3946  {csn 4632  β€˜cfv 6553  infcinf 9472  β„cr 11145  0cc0 11146   < clt 11286  β„•cn 12250  β„šcq 12970  0𝑝c0p 25618  Polycply 26138  degcdgr 26141  π”Έcaa 26269  degAAcdgraa 42595
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-resscn 11203  ax-pre-lttri 11220  ax-pre-lttrn 11221
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3374  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-po 5594  df-so 5595  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-er 8731  df-en 8971  df-dom 8972  df-sdom 8973  df-sup 9473  df-inf 9474  df-pnf 11288  df-mnf 11289  df-ltxr 11291  df-dgraa 42597
This theorem is referenced by:  dgraalem  42600  dgraaub  42603
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