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Mirrors > Home > MPE Home > Th. List > deg1val | Structured version Visualization version GIF version |
Description: Value of the univariate degree as a supremum. (Contributed by Stefan O'Rear, 29-Mar-2015.) (Revised by AV, 25-Jul-2019.) |
Ref | Expression |
---|---|
deg1leb.d | ⊢ 𝐷 = ( deg1 ‘𝑅) |
deg1leb.p | ⊢ 𝑃 = (Poly1‘𝑅) |
deg1leb.b | ⊢ 𝐵 = (Base‘𝑃) |
deg1leb.y | ⊢ 0 = (0g‘𝑅) |
deg1leb.a | ⊢ 𝐴 = (coe1‘𝐹) |
Ref | Expression |
---|---|
deg1val | ⊢ (𝐹 ∈ 𝐵 → (𝐷‘𝐹) = sup((𝐴 supp 0 ), ℝ*, < )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | deg1leb.d | . . . 4 ⊢ 𝐷 = ( deg1 ‘𝑅) | |
2 | 1 | deg1fval 23644 | . . 3 ⊢ 𝐷 = (1𝑜 mDeg 𝑅) |
3 | eqid 2610 | . . 3 ⊢ (1𝑜 mPoly 𝑅) = (1𝑜 mPoly 𝑅) | |
4 | deg1leb.p | . . . 4 ⊢ 𝑃 = (Poly1‘𝑅) | |
5 | eqid 2610 | . . . 4 ⊢ (PwSer1‘𝑅) = (PwSer1‘𝑅) | |
6 | deg1leb.b | . . . 4 ⊢ 𝐵 = (Base‘𝑃) | |
7 | 4, 5, 6 | ply1bas 19386 | . . 3 ⊢ 𝐵 = (Base‘(1𝑜 mPoly 𝑅)) |
8 | deg1leb.y | . . 3 ⊢ 0 = (0g‘𝑅) | |
9 | psr1baslem 19376 | . . 3 ⊢ (ℕ0 ↑𝑚 1𝑜) = {𝑦 ∈ (ℕ0 ↑𝑚 1𝑜) ∣ (◡𝑦 “ ℕ) ∈ Fin} | |
10 | tdeglem2 23625 | . . 3 ⊢ (𝑥 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ (𝑥‘∅)) = (𝑥 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ (ℂfld Σg 𝑥)) | |
11 | 2, 3, 7, 8, 9, 10 | mdegval 23627 | . 2 ⊢ (𝐹 ∈ 𝐵 → (𝐷‘𝐹) = sup(((𝑥 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ (𝑥‘∅)) “ (𝐹 supp 0 )), ℝ*, < )) |
12 | fvex 6113 | . . . . . . . . 9 ⊢ (0g‘𝑅) ∈ V | |
13 | 8, 12 | eqeltri 2684 | . . . . . . . 8 ⊢ 0 ∈ V |
14 | suppimacnv 7193 | . . . . . . . 8 ⊢ ((𝐹 ∈ 𝐵 ∧ 0 ∈ V) → (𝐹 supp 0 ) = (◡𝐹 “ (V ∖ { 0 }))) | |
15 | 13, 14 | mpan2 703 | . . . . . . 7 ⊢ (𝐹 ∈ 𝐵 → (𝐹 supp 0 ) = (◡𝐹 “ (V ∖ { 0 }))) |
16 | 15 | imaeq2d 5385 | . . . . . 6 ⊢ (𝐹 ∈ 𝐵 → ((𝑥 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ (𝑥‘∅)) “ (𝐹 supp 0 )) = ((𝑥 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ (𝑥‘∅)) “ (◡𝐹 “ (V ∖ { 0 })))) |
17 | imaco 5557 | . . . . . 6 ⊢ (((𝑥 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ (𝑥‘∅)) ∘ ◡𝐹) “ (V ∖ { 0 })) = ((𝑥 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ (𝑥‘∅)) “ (◡𝐹 “ (V ∖ { 0 }))) | |
18 | 16, 17 | syl6eqr 2662 | . . . . 5 ⊢ (𝐹 ∈ 𝐵 → ((𝑥 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ (𝑥‘∅)) “ (𝐹 supp 0 )) = (((𝑥 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ (𝑥‘∅)) ∘ ◡𝐹) “ (V ∖ { 0 }))) |
19 | deg1leb.a | . . . . . . . . 9 ⊢ 𝐴 = (coe1‘𝐹) | |
20 | df1o2 7459 | . . . . . . . . . 10 ⊢ 1𝑜 = {∅} | |
21 | nn0ex 11175 | . . . . . . . . . 10 ⊢ ℕ0 ∈ V | |
22 | 0ex 4718 | . . . . . . . . . 10 ⊢ ∅ ∈ V | |
23 | eqid 2610 | . . . . . . . . . 10 ⊢ (𝑥 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ (𝑥‘∅)) = (𝑥 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ (𝑥‘∅)) | |
24 | 20, 21, 22, 23 | mapsncnv 7790 | . . . . . . . . 9 ⊢ ◡(𝑥 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ (𝑥‘∅)) = (𝑦 ∈ ℕ0 ↦ (1𝑜 × {𝑦})) |
25 | 19, 6, 4, 24 | coe1fval2 19401 | . . . . . . . 8 ⊢ (𝐹 ∈ 𝐵 → 𝐴 = (𝐹 ∘ ◡(𝑥 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ (𝑥‘∅)))) |
26 | 25 | cnveqd 5220 | . . . . . . 7 ⊢ (𝐹 ∈ 𝐵 → ◡𝐴 = ◡(𝐹 ∘ ◡(𝑥 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ (𝑥‘∅)))) |
27 | cnvco 5230 | . . . . . . . 8 ⊢ ◡(𝐹 ∘ ◡(𝑥 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ (𝑥‘∅))) = (◡◡(𝑥 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ (𝑥‘∅)) ∘ ◡𝐹) | |
28 | cocnvcnv1 5563 | . . . . . . . 8 ⊢ (◡◡(𝑥 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ (𝑥‘∅)) ∘ ◡𝐹) = ((𝑥 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ (𝑥‘∅)) ∘ ◡𝐹) | |
29 | 27, 28 | eqtri 2632 | . . . . . . 7 ⊢ ◡(𝐹 ∘ ◡(𝑥 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ (𝑥‘∅))) = ((𝑥 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ (𝑥‘∅)) ∘ ◡𝐹) |
30 | 26, 29 | syl6req 2661 | . . . . . 6 ⊢ (𝐹 ∈ 𝐵 → ((𝑥 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ (𝑥‘∅)) ∘ ◡𝐹) = ◡𝐴) |
31 | 30 | imaeq1d 5384 | . . . . 5 ⊢ (𝐹 ∈ 𝐵 → (((𝑥 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ (𝑥‘∅)) ∘ ◡𝐹) “ (V ∖ { 0 })) = (◡𝐴 “ (V ∖ { 0 }))) |
32 | 18, 31 | eqtrd 2644 | . . . 4 ⊢ (𝐹 ∈ 𝐵 → ((𝑥 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ (𝑥‘∅)) “ (𝐹 supp 0 )) = (◡𝐴 “ (V ∖ { 0 }))) |
33 | fvex 6113 | . . . . . 6 ⊢ (coe1‘𝐹) ∈ V | |
34 | 19, 33 | eqeltri 2684 | . . . . 5 ⊢ 𝐴 ∈ V |
35 | suppimacnv 7193 | . . . . . 6 ⊢ ((𝐴 ∈ V ∧ 0 ∈ V) → (𝐴 supp 0 ) = (◡𝐴 “ (V ∖ { 0 }))) | |
36 | 35 | eqcomd 2616 | . . . . 5 ⊢ ((𝐴 ∈ V ∧ 0 ∈ V) → (◡𝐴 “ (V ∖ { 0 })) = (𝐴 supp 0 )) |
37 | 34, 13, 36 | mp2an 704 | . . . 4 ⊢ (◡𝐴 “ (V ∖ { 0 })) = (𝐴 supp 0 ) |
38 | 32, 37 | syl6eq 2660 | . . 3 ⊢ (𝐹 ∈ 𝐵 → ((𝑥 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ (𝑥‘∅)) “ (𝐹 supp 0 )) = (𝐴 supp 0 )) |
39 | 38 | supeq1d 8235 | . 2 ⊢ (𝐹 ∈ 𝐵 → sup(((𝑥 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ (𝑥‘∅)) “ (𝐹 supp 0 )), ℝ*, < ) = sup((𝐴 supp 0 ), ℝ*, < )) |
40 | 11, 39 | eqtrd 2644 | 1 ⊢ (𝐹 ∈ 𝐵 → (𝐷‘𝐹) = sup((𝐴 supp 0 ), ℝ*, < )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∖ cdif 3537 ∅c0 3874 {csn 4125 ↦ cmpt 4643 ◡ccnv 5037 “ cima 5041 ∘ ccom 5042 ‘cfv 5804 (class class class)co 6549 supp csupp 7182 1𝑜c1o 7440 ↑𝑚 cmap 7744 supcsup 8229 ℝ*cxr 9952 < clt 9953 ℕ0cn0 11169 Basecbs 15695 0gc0g 15923 mPoly cmpl 19174 PwSer1cps1 19366 Poly1cpl1 19368 coe1cco1 19369 deg1 cdg1 23618 |
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-inf2 8421 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-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-addf 9894 ax-mulf 9895 |
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-int 4411 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-se 4998 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-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-sup 8231 df-oi 8298 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-fz 12198 df-fzo 12335 df-seq 12664 df-hash 12980 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-sca 15784 df-vsca 15785 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-0g 15925 df-gsum 15926 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-grp 17248 df-mulg 17364 df-cntz 17573 df-cmn 18018 df-mgp 18313 df-ring 18372 df-cring 18373 df-psr 19177 df-mpl 19179 df-opsr 19181 df-psr1 19371 df-ply1 19373 df-coe1 19374 df-cnfld 19568 df-mdeg 23619 df-deg1 23620 |
This theorem is referenced by: deg1mul3 23679 deg1mul3le 23680 |
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