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Mirrors > Home > MPE Home > Th. List > dgrval | Structured version Visualization version GIF version |
Description: Value of the degree function. (Contributed by Mario Carneiro, 22-Jul-2014.) |
Ref | Expression |
---|---|
dgrval.1 | ⊢ 𝐴 = (coeff‘𝐹) |
Ref | Expression |
---|---|
dgrval | ⊢ (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) = sup((◡𝐴 “ (ℂ ∖ {0})), ℕ0, < )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | plyssc 24362 | . . 3 ⊢ (Poly‘𝑆) ⊆ (Poly‘ℂ) | |
2 | 1 | sseli 3823 | . 2 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹 ∈ (Poly‘ℂ)) |
3 | fveq2 6437 | . . . . . . 7 ⊢ (𝑓 = 𝐹 → (coeff‘𝑓) = (coeff‘𝐹)) | |
4 | dgrval.1 | . . . . . . 7 ⊢ 𝐴 = (coeff‘𝐹) | |
5 | 3, 4 | syl6eqr 2879 | . . . . . 6 ⊢ (𝑓 = 𝐹 → (coeff‘𝑓) = 𝐴) |
6 | 5 | cnveqd 5534 | . . . . 5 ⊢ (𝑓 = 𝐹 → ◡(coeff‘𝑓) = ◡𝐴) |
7 | 6 | imaeq1d 5710 | . . . 4 ⊢ (𝑓 = 𝐹 → (◡(coeff‘𝑓) “ (ℂ ∖ {0})) = (◡𝐴 “ (ℂ ∖ {0}))) |
8 | 7 | supeq1d 8627 | . . 3 ⊢ (𝑓 = 𝐹 → sup((◡(coeff‘𝑓) “ (ℂ ∖ {0})), ℕ0, < ) = sup((◡𝐴 “ (ℂ ∖ {0})), ℕ0, < )) |
9 | df-dgr 24353 | . . 3 ⊢ deg = (𝑓 ∈ (Poly‘ℂ) ↦ sup((◡(coeff‘𝑓) “ (ℂ ∖ {0})), ℕ0, < )) | |
10 | nn0ssre 11629 | . . . . 5 ⊢ ℕ0 ⊆ ℝ | |
11 | ltso 10444 | . . . . 5 ⊢ < Or ℝ | |
12 | soss 5284 | . . . . 5 ⊢ (ℕ0 ⊆ ℝ → ( < Or ℝ → < Or ℕ0)) | |
13 | 10, 11, 12 | mp2 9 | . . . 4 ⊢ < Or ℕ0 |
14 | 13 | supex 8644 | . . 3 ⊢ sup((◡𝐴 “ (ℂ ∖ {0})), ℕ0, < ) ∈ V |
15 | 8, 9, 14 | fvmpt 6533 | . 2 ⊢ (𝐹 ∈ (Poly‘ℂ) → (deg‘𝐹) = sup((◡𝐴 “ (ℂ ∖ {0})), ℕ0, < )) |
16 | 2, 15 | syl 17 | 1 ⊢ (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) = sup((◡𝐴 “ (ℂ ∖ {0})), ℕ0, < )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1656 ∈ wcel 2164 ∖ cdif 3795 ⊆ wss 3798 {csn 4399 Or wor 5264 ◡ccnv 5345 “ cima 5349 ‘cfv 6127 supcsup 8621 ℂcc 10257 ℝcr 10258 0cc0 10259 < clt 10398 ℕ0cn0 11625 Polycply 24346 coeffccoe 24348 degcdgr 24349 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-i2m1 10327 ax-1ne0 10328 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-om 7332 df-1st 7433 df-2nd 7434 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-er 8014 df-map 8129 df-en 8229 df-dom 8230 df-sdom 8231 df-sup 8623 df-pnf 10400 df-mnf 10401 df-ltxr 10403 df-nn 11358 df-n0 11626 df-ply 24350 df-dgr 24353 |
This theorem is referenced by: dgrcl 24395 dgrub 24396 dgrlb 24398 coe11 24415 |
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