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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 23760 | . . 3 ⊢ (Poly‘𝑆) ⊆ (Poly‘ℂ) | |
2 | 1 | sseli 3564 | . 2 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹 ∈ (Poly‘ℂ)) |
3 | fveq2 6103 | . . . . . . 7 ⊢ (𝑓 = 𝐹 → (coeff‘𝑓) = (coeff‘𝐹)) | |
4 | dgrval.1 | . . . . . . 7 ⊢ 𝐴 = (coeff‘𝐹) | |
5 | 3, 4 | syl6eqr 2662 | . . . . . 6 ⊢ (𝑓 = 𝐹 → (coeff‘𝑓) = 𝐴) |
6 | 5 | cnveqd 5220 | . . . . 5 ⊢ (𝑓 = 𝐹 → ◡(coeff‘𝑓) = ◡𝐴) |
7 | 6 | imaeq1d 5384 | . . . 4 ⊢ (𝑓 = 𝐹 → (◡(coeff‘𝑓) “ (ℂ ∖ {0})) = (◡𝐴 “ (ℂ ∖ {0}))) |
8 | 7 | supeq1d 8235 | . . 3 ⊢ (𝑓 = 𝐹 → sup((◡(coeff‘𝑓) “ (ℂ ∖ {0})), ℕ0, < ) = sup((◡𝐴 “ (ℂ ∖ {0})), ℕ0, < )) |
9 | df-dgr 23751 | . . 3 ⊢ deg = (𝑓 ∈ (Poly‘ℂ) ↦ sup((◡(coeff‘𝑓) “ (ℂ ∖ {0})), ℕ0, < )) | |
10 | nn0ssre 11173 | . . . . 5 ⊢ ℕ0 ⊆ ℝ | |
11 | ltso 9997 | . . . . 5 ⊢ < Or ℝ | |
12 | soss 4977 | . . . . 5 ⊢ (ℕ0 ⊆ ℝ → ( < Or ℝ → < Or ℕ0)) | |
13 | 10, 11, 12 | mp2 9 | . . . 4 ⊢ < Or ℕ0 |
14 | 13 | supex 8252 | . . 3 ⊢ sup((◡𝐴 “ (ℂ ∖ {0})), ℕ0, < ) ∈ V |
15 | 8, 9, 14 | fvmpt 6191 | . 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 1475 ∈ wcel 1977 ∖ cdif 3537 ⊆ wss 3540 {csn 4125 Or wor 4958 ◡ccnv 5037 “ cima 5041 ‘cfv 5804 supcsup 8229 ℂcc 9813 ℝcr 9814 0cc0 9815 < clt 9953 ℕ0cn0 11169 Polycply 23744 coeffccoe 23746 degcdgr 23747 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-i2m1 9883 ax-1ne0 9884 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-sup 8231 df-pnf 9955 df-mnf 9956 df-ltxr 9958 df-nn 10898 df-n0 11170 df-ply 23748 df-dgr 23751 |
This theorem is referenced by: dgrcl 23793 dgrub 23794 dgrlb 23796 coe11 23813 |
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