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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 25094 | . . 3 ⊢ (Poly‘𝑆) ⊆ (Poly‘ℂ) | |
2 | 1 | sseli 3896 | . 2 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹 ∈ (Poly‘ℂ)) |
3 | fveq2 6717 | . . . . . . 7 ⊢ (𝑓 = 𝐹 → (coeff‘𝑓) = (coeff‘𝐹)) | |
4 | dgrval.1 | . . . . . . 7 ⊢ 𝐴 = (coeff‘𝐹) | |
5 | 3, 4 | eqtr4di 2796 | . . . . . 6 ⊢ (𝑓 = 𝐹 → (coeff‘𝑓) = 𝐴) |
6 | 5 | cnveqd 5744 | . . . . 5 ⊢ (𝑓 = 𝐹 → ◡(coeff‘𝑓) = ◡𝐴) |
7 | 6 | imaeq1d 5928 | . . . 4 ⊢ (𝑓 = 𝐹 → (◡(coeff‘𝑓) “ (ℂ ∖ {0})) = (◡𝐴 “ (ℂ ∖ {0}))) |
8 | 7 | supeq1d 9062 | . . 3 ⊢ (𝑓 = 𝐹 → sup((◡(coeff‘𝑓) “ (ℂ ∖ {0})), ℕ0, < ) = sup((◡𝐴 “ (ℂ ∖ {0})), ℕ0, < )) |
9 | df-dgr 25085 | . . 3 ⊢ deg = (𝑓 ∈ (Poly‘ℂ) ↦ sup((◡(coeff‘𝑓) “ (ℂ ∖ {0})), ℕ0, < )) | |
10 | nn0ssre 12094 | . . . . 5 ⊢ ℕ0 ⊆ ℝ | |
11 | ltso 10913 | . . . . 5 ⊢ < Or ℝ | |
12 | soss 5488 | . . . . 5 ⊢ (ℕ0 ⊆ ℝ → ( < Or ℝ → < Or ℕ0)) | |
13 | 10, 11, 12 | mp2 9 | . . . 4 ⊢ < Or ℕ0 |
14 | 13 | supex 9079 | . . 3 ⊢ sup((◡𝐴 “ (ℂ ∖ {0})), ℕ0, < ) ∈ V |
15 | 8, 9, 14 | fvmpt 6818 | . 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 1543 ∈ wcel 2110 ∖ cdif 3863 ⊆ wss 3866 {csn 4541 Or wor 5467 ◡ccnv 5550 “ cima 5554 ‘cfv 6380 supcsup 9056 ℂcc 10727 ℝcr 10728 0cc0 10729 < clt 10867 ℕ0cn0 12090 Polycply 25078 coeffccoe 25080 degcdgr 25081 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-i2m1 10797 ax-1ne0 10798 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-1st 7761 df-2nd 7762 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-er 8391 df-map 8510 df-en 8627 df-dom 8628 df-sdom 8629 df-sup 9058 df-pnf 10869 df-mnf 10870 df-ltxr 10872 df-nn 11831 df-n0 12091 df-ply 25082 df-dgr 25085 |
This theorem is referenced by: dgrcl 25127 dgrub 25128 dgrlb 25130 coe11 25147 |
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