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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 24792 | . . 3 ⊢ (Poly‘𝑆) ⊆ (Poly‘ℂ) | |
2 | 1 | sseli 3965 | . 2 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹 ∈ (Poly‘ℂ)) |
3 | fveq2 6672 | . . . . . . 7 ⊢ (𝑓 = 𝐹 → (coeff‘𝑓) = (coeff‘𝐹)) | |
4 | dgrval.1 | . . . . . . 7 ⊢ 𝐴 = (coeff‘𝐹) | |
5 | 3, 4 | syl6eqr 2876 | . . . . . 6 ⊢ (𝑓 = 𝐹 → (coeff‘𝑓) = 𝐴) |
6 | 5 | cnveqd 5748 | . . . . 5 ⊢ (𝑓 = 𝐹 → ◡(coeff‘𝑓) = ◡𝐴) |
7 | 6 | imaeq1d 5930 | . . . 4 ⊢ (𝑓 = 𝐹 → (◡(coeff‘𝑓) “ (ℂ ∖ {0})) = (◡𝐴 “ (ℂ ∖ {0}))) |
8 | 7 | supeq1d 8912 | . . 3 ⊢ (𝑓 = 𝐹 → sup((◡(coeff‘𝑓) “ (ℂ ∖ {0})), ℕ0, < ) = sup((◡𝐴 “ (ℂ ∖ {0})), ℕ0, < )) |
9 | df-dgr 24783 | . . 3 ⊢ deg = (𝑓 ∈ (Poly‘ℂ) ↦ sup((◡(coeff‘𝑓) “ (ℂ ∖ {0})), ℕ0, < )) | |
10 | nn0ssre 11904 | . . . . 5 ⊢ ℕ0 ⊆ ℝ | |
11 | ltso 10723 | . . . . 5 ⊢ < Or ℝ | |
12 | soss 5495 | . . . . 5 ⊢ (ℕ0 ⊆ ℝ → ( < Or ℝ → < Or ℕ0)) | |
13 | 10, 11, 12 | mp2 9 | . . . 4 ⊢ < Or ℕ0 |
14 | 13 | supex 8929 | . . 3 ⊢ sup((◡𝐴 “ (ℂ ∖ {0})), ℕ0, < ) ∈ V |
15 | 8, 9, 14 | fvmpt 6770 | . 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 1537 ∈ wcel 2114 ∖ cdif 3935 ⊆ wss 3938 {csn 4569 Or wor 5475 ◡ccnv 5556 “ cima 5560 ‘cfv 6357 supcsup 8906 ℂcc 10537 ℝcr 10538 0cc0 10539 < clt 10677 ℕ0cn0 11900 Polycply 24776 coeffccoe 24778 degcdgr 24779 |
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 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-i2m1 10607 ax-1ne0 10608 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-sup 8908 df-pnf 10679 df-mnf 10680 df-ltxr 10682 df-nn 11641 df-n0 11901 df-ply 24780 df-dgr 24783 |
This theorem is referenced by: dgrcl 24825 dgrub 24826 dgrlb 24828 coe11 24845 |
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