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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 24771 | . . 3 ⊢ (Poly‘𝑆) ⊆ (Poly‘ℂ) | |
2 | 1 | sseli 3946 | . 2 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹 ∈ (Poly‘ℂ)) |
3 | fveq2 6651 | . . . . . . 7 ⊢ (𝑓 = 𝐹 → (coeff‘𝑓) = (coeff‘𝐹)) | |
4 | dgrval.1 | . . . . . . 7 ⊢ 𝐴 = (coeff‘𝐹) | |
5 | 3, 4 | syl6eqr 2873 | . . . . . 6 ⊢ (𝑓 = 𝐹 → (coeff‘𝑓) = 𝐴) |
6 | 5 | cnveqd 5727 | . . . . 5 ⊢ (𝑓 = 𝐹 → ◡(coeff‘𝑓) = ◡𝐴) |
7 | 6 | imaeq1d 5909 | . . . 4 ⊢ (𝑓 = 𝐹 → (◡(coeff‘𝑓) “ (ℂ ∖ {0})) = (◡𝐴 “ (ℂ ∖ {0}))) |
8 | 7 | supeq1d 8891 | . . 3 ⊢ (𝑓 = 𝐹 → sup((◡(coeff‘𝑓) “ (ℂ ∖ {0})), ℕ0, < ) = sup((◡𝐴 “ (ℂ ∖ {0})), ℕ0, < )) |
9 | df-dgr 24762 | . . 3 ⊢ deg = (𝑓 ∈ (Poly‘ℂ) ↦ sup((◡(coeff‘𝑓) “ (ℂ ∖ {0})), ℕ0, < )) | |
10 | nn0ssre 11883 | . . . . 5 ⊢ ℕ0 ⊆ ℝ | |
11 | ltso 10702 | . . . . 5 ⊢ < Or ℝ | |
12 | soss 5474 | . . . . 5 ⊢ (ℕ0 ⊆ ℝ → ( < Or ℝ → < Or ℕ0)) | |
13 | 10, 11, 12 | mp2 9 | . . . 4 ⊢ < Or ℕ0 |
14 | 13 | supex 8908 | . . 3 ⊢ sup((◡𝐴 “ (ℂ ∖ {0})), ℕ0, < ) ∈ V |
15 | 8, 9, 14 | fvmpt 6749 | . 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 3916 ⊆ wss 3919 {csn 4548 Or wor 5454 ◡ccnv 5535 “ cima 5539 ‘cfv 6336 supcsup 8885 ℂcc 10516 ℝcr 10517 0cc0 10518 < clt 10656 ℕ0cn0 11879 Polycply 24755 coeffccoe 24757 degcdgr 24758 |
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 2792 ax-rep 5171 ax-sep 5184 ax-nul 5191 ax-pow 5247 ax-pr 5311 ax-un 7442 ax-cnex 10574 ax-resscn 10575 ax-1cn 10576 ax-icn 10577 ax-addcl 10578 ax-addrcl 10579 ax-mulcl 10580 ax-mulrcl 10581 ax-i2m1 10586 ax-1ne0 10587 ax-rnegex 10589 ax-rrecex 10590 ax-cnre 10591 ax-pre-lttri 10592 ax-pre-lttrn 10593 |
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 2653 df-clab 2799 df-cleq 2813 df-clel 2891 df-nfc 2959 df-ne 3012 df-nel 3119 df-ral 3138 df-rex 3139 df-reu 3140 df-rmo 3141 df-rab 3142 df-v 3483 df-sbc 3759 df-csb 3867 df-dif 3922 df-un 3924 df-in 3926 df-ss 3935 df-pss 3937 df-nul 4275 df-if 4449 df-pw 4522 df-sn 4549 df-pr 4551 df-tp 4553 df-op 4555 df-uni 4820 df-iun 4902 df-br 5048 df-opab 5110 df-mpt 5128 df-tr 5154 df-id 5441 df-eprel 5446 df-po 5455 df-so 5456 df-fr 5495 df-we 5497 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6295 df-fun 6338 df-fn 6339 df-f 6340 df-f1 6341 df-fo 6342 df-f1o 6343 df-fv 6344 df-ov 7140 df-oprab 7141 df-mpo 7142 df-om 7562 df-1st 7670 df-2nd 7671 df-wrecs 7928 df-recs 7989 df-rdg 8027 df-er 8270 df-map 8389 df-en 8491 df-dom 8492 df-sdom 8493 df-sup 8887 df-pnf 10658 df-mnf 10659 df-ltxr 10661 df-nn 11620 df-n0 11880 df-ply 24759 df-dgr 24762 |
This theorem is referenced by: dgrcl 24804 dgrub 24805 dgrlb 24807 coe11 24824 |
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