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Theorem dgraaval 41232
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 6834 . . . . . 6 (π‘Ž = 𝐴 β†’ ((π‘β€˜π‘Ž) = 0 ↔ (π‘β€˜π΄) = 0))
21anbi2d 629 . . . . 5 (π‘Ž = 𝐴 β†’ (((degβ€˜π‘) = 𝑑 ∧ (π‘β€˜π‘Ž) = 0) ↔ ((degβ€˜π‘) = 𝑑 ∧ (π‘β€˜π΄) = 0)))
32rexbidv 3171 . . . 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 3411 . . 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 9334 . 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 41230 . 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 11156 . . 3 < Or ℝ
87infex 9350 . 2 inf({𝑑 ∈ β„• ∣ βˆƒπ‘ ∈ ((Polyβ€˜β„š) βˆ– {0𝑝})((degβ€˜π‘) = 𝑑 ∧ (π‘β€˜π΄) = 0)}, ℝ, < ) ∈ V
95, 6, 8fvmpt 6931 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 396   = wceq 1540   ∈ wcel 2105  βˆƒwrex 3070  {crab 3403   βˆ– cdif 3895  {csn 4573  β€˜cfv 6479  infcinf 9298  β„cr 10971  0cc0 10972   < clt 11110  β„•cn 12074  β„šcq 12789  0𝑝c0p 24939  Polycply 25451  degcdgr 25454  π”Έcaa 25580  degAAcdgraa 41228
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5243  ax-nul 5250  ax-pow 5308  ax-pr 5372  ax-un 7650  ax-resscn 11029  ax-pre-lttri 11046  ax-pre-lttrn 11047
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3349  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-br 5093  df-opab 5155  df-mpt 5176  df-id 5518  df-po 5532  df-so 5533  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-iota 6431  df-fun 6481  df-fn 6482  df-f 6483  df-f1 6484  df-fo 6485  df-f1o 6486  df-fv 6487  df-er 8569  df-en 8805  df-dom 8806  df-sdom 8807  df-sup 9299  df-inf 9300  df-pnf 11112  df-mnf 11113  df-ltxr 11115  df-dgraa 41230
This theorem is referenced by:  dgraalem  41233  dgraaub  41236
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