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Mirrors > Home > MPE Home > Th. List > Mathboxes > dgraaval | Structured version Visualization version GIF version |
Description: Value of the degree function on an algebraic number. (Contributed by Stefan O'Rear, 25-Nov-2014.) (Revised by AV, 29-Sep-2020.) |
Ref | Expression |
---|---|
dgraaval | ⊢ (𝐴 ∈ 𝔸 → (degAA‘𝐴) = inf({𝑑 ∈ ℕ ∣ ∃𝑝 ∈ ((Poly‘ℚ) ∖ {0𝑝})((deg‘𝑝) = 𝑑 ∧ (𝑝‘𝐴) = 0)}, ℝ, < )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveqeq2 6665 | . . . . . 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 3297 | . . . 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 3472 | . . 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 8927 | . 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 39834 | . 2 ⊢ degAA = (𝑎 ∈ 𝔸 ↦ inf({𝑑 ∈ ℕ ∣ ∃𝑝 ∈ ((Poly‘ℚ) ∖ {0𝑝})((deg‘𝑝) = 𝑑 ∧ (𝑝‘𝑎) = 0)}, ℝ, < )) | |
7 | ltso 10707 | . . 3 ⊢ < Or ℝ | |
8 | 7 | infex 8943 | . 2 ⊢ inf({𝑑 ∈ ℕ ∣ ∃𝑝 ∈ ((Poly‘ℚ) ∖ {0𝑝})((deg‘𝑝) = 𝑑 ∧ (𝑝‘𝐴) = 0)}, ℝ, < ) ∈ V |
9 | 5, 6, 8 | fvmpt 6754 | 1 ⊢ (𝐴 ∈ 𝔸 → (degAA‘𝐴) = inf({𝑑 ∈ ℕ ∣ ∃𝑝 ∈ ((Poly‘ℚ) ∖ {0𝑝})((deg‘𝑝) = 𝑑 ∧ (𝑝‘𝐴) = 0)}, ℝ, < )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∃wrex 3139 {crab 3142 ∖ cdif 3921 {csn 4553 ‘cfv 6341 infcinf 8891 ℝcr 10522 0cc0 10523 < clt 10661 ℕcn 11624 ℚcq 12335 0𝑝c0p 24253 Polycply 24760 degcdgr 24763 𝔸caa 24889 degAAcdgraa 39832 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5189 ax-nul 5196 ax-pow 5252 ax-pr 5316 ax-un 7447 ax-resscn 10580 ax-pre-lttri 10597 ax-pre-lttrn 10598 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-rmo 3146 df-rab 3147 df-v 3488 df-sbc 3764 df-csb 3872 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-nul 4280 df-if 4454 df-pw 4527 df-sn 4554 df-pr 4556 df-op 4560 df-uni 4825 df-br 5053 df-opab 5115 df-mpt 5133 df-id 5446 df-po 5460 df-so 5461 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-rn 5552 df-res 5553 df-ima 5554 df-iota 6300 df-fun 6343 df-fn 6344 df-f 6345 df-f1 6346 df-fo 6347 df-f1o 6348 df-fv 6349 df-er 8275 df-en 8496 df-dom 8497 df-sdom 8498 df-sup 8892 df-inf 8893 df-pnf 10663 df-mnf 10664 df-ltxr 10666 df-dgraa 39834 |
This theorem is referenced by: dgraalem 39837 dgraaub 39840 |
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