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Theorem dgraaval 42461
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 6894 . . . . . 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 3172 . . . 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 3434 . . 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 9474 . 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 42459 . 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 11298 . . 3 < Or ℝ
87infex 9490 . 2 inf({𝑑 ∈ β„• ∣ βˆƒπ‘ ∈ ((Polyβ€˜β„š) βˆ– {0𝑝})((degβ€˜π‘) = 𝑑 ∧ (π‘β€˜π΄) = 0)}, ℝ, < ) ∈ V
95, 6, 8fvmpt 6992 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 395   = wceq 1533   ∈ wcel 2098  βˆƒwrex 3064  {crab 3426   βˆ– cdif 3940  {csn 4623  β€˜cfv 6537  infcinf 9438  β„cr 11111  0cc0 11112   < clt 11252  β„•cn 12216  β„šcq 12936  0𝑝c0p 25553  Polycply 26073  degcdgr 26076  π”Έcaa 26204  degAAcdgraa 42457
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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  ax-resscn 11169  ax-pre-lttri 11186  ax-pre-lttrn 11187
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-rmo 3370  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-po 5581  df-so 5582  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-sup 9439  df-inf 9440  df-pnf 11254  df-mnf 11255  df-ltxr 11257  df-dgraa 42459
This theorem is referenced by:  dgraalem  42462  dgraaub  42465
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