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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 25396 | . . 3 ⊢ 𝐷 = (1o mDeg 𝑅) |
3 | eqid 2737 | . . 3 ⊢ (1o mPoly 𝑅) = (1o mPoly 𝑅) | |
4 | deg1leb.p | . . . 4 ⊢ 𝑃 = (Poly1‘𝑅) | |
5 | eqid 2737 | . . . 4 ⊢ (PwSer1‘𝑅) = (PwSer1‘𝑅) | |
6 | deg1leb.b | . . . 4 ⊢ 𝐵 = (Base‘𝑃) | |
7 | 4, 5, 6 | ply1bas 21517 | . . 3 ⊢ 𝐵 = (Base‘(1o mPoly 𝑅)) |
8 | deg1leb.y | . . 3 ⊢ 0 = (0g‘𝑅) | |
9 | psr1baslem 21507 | . . 3 ⊢ (ℕ0 ↑m 1o) = {𝑦 ∈ (ℕ0 ↑m 1o) ∣ (◡𝑦 “ ℕ) ∈ Fin} | |
10 | tdeglem2 25377 | . . 3 ⊢ (𝑥 ∈ (ℕ0 ↑m 1o) ↦ (𝑥‘∅)) = (𝑥 ∈ (ℕ0 ↑m 1o) ↦ (ℂfld Σg 𝑥)) | |
11 | 2, 3, 7, 8, 9, 10 | mdegval 25379 | . 2 ⊢ (𝐹 ∈ 𝐵 → (𝐷‘𝐹) = sup(((𝑥 ∈ (ℕ0 ↑m 1o) ↦ (𝑥‘∅)) “ (𝐹 supp 0 )), ℝ*, < )) |
12 | 8 | fvexi 6853 | . . . . . . . 8 ⊢ 0 ∈ V |
13 | suppimacnv 8097 | . . . . . . . 8 ⊢ ((𝐹 ∈ 𝐵 ∧ 0 ∈ V) → (𝐹 supp 0 ) = (◡𝐹 “ (V ∖ { 0 }))) | |
14 | 12, 13 | mpan2 689 | . . . . . . 7 ⊢ (𝐹 ∈ 𝐵 → (𝐹 supp 0 ) = (◡𝐹 “ (V ∖ { 0 }))) |
15 | 14 | imaeq2d 6011 | . . . . . 6 ⊢ (𝐹 ∈ 𝐵 → ((𝑥 ∈ (ℕ0 ↑m 1o) ↦ (𝑥‘∅)) “ (𝐹 supp 0 )) = ((𝑥 ∈ (ℕ0 ↑m 1o) ↦ (𝑥‘∅)) “ (◡𝐹 “ (V ∖ { 0 })))) |
16 | imaco 6201 | . . . . . 6 ⊢ (((𝑥 ∈ (ℕ0 ↑m 1o) ↦ (𝑥‘∅)) ∘ ◡𝐹) “ (V ∖ { 0 })) = ((𝑥 ∈ (ℕ0 ↑m 1o) ↦ (𝑥‘∅)) “ (◡𝐹 “ (V ∖ { 0 }))) | |
17 | 15, 16 | eqtr4di 2795 | . . . . 5 ⊢ (𝐹 ∈ 𝐵 → ((𝑥 ∈ (ℕ0 ↑m 1o) ↦ (𝑥‘∅)) “ (𝐹 supp 0 )) = (((𝑥 ∈ (ℕ0 ↑m 1o) ↦ (𝑥‘∅)) ∘ ◡𝐹) “ (V ∖ { 0 }))) |
18 | deg1leb.a | . . . . . . . . 9 ⊢ 𝐴 = (coe1‘𝐹) | |
19 | df1o2 8411 | . . . . . . . . . 10 ⊢ 1o = {∅} | |
20 | nn0ex 12377 | . . . . . . . . . 10 ⊢ ℕ0 ∈ V | |
21 | 0ex 5262 | . . . . . . . . . 10 ⊢ ∅ ∈ V | |
22 | eqid 2737 | . . . . . . . . . 10 ⊢ (𝑥 ∈ (ℕ0 ↑m 1o) ↦ (𝑥‘∅)) = (𝑥 ∈ (ℕ0 ↑m 1o) ↦ (𝑥‘∅)) | |
23 | 19, 20, 21, 22 | mapsncnv 8789 | . . . . . . . . 9 ⊢ ◡(𝑥 ∈ (ℕ0 ↑m 1o) ↦ (𝑥‘∅)) = (𝑦 ∈ ℕ0 ↦ (1o × {𝑦})) |
24 | 18, 6, 4, 23 | coe1fval2 21532 | . . . . . . . 8 ⊢ (𝐹 ∈ 𝐵 → 𝐴 = (𝐹 ∘ ◡(𝑥 ∈ (ℕ0 ↑m 1o) ↦ (𝑥‘∅)))) |
25 | 24 | cnveqd 5829 | . . . . . . 7 ⊢ (𝐹 ∈ 𝐵 → ◡𝐴 = ◡(𝐹 ∘ ◡(𝑥 ∈ (ℕ0 ↑m 1o) ↦ (𝑥‘∅)))) |
26 | cnvco 5839 | . . . . . . . 8 ⊢ ◡(𝐹 ∘ ◡(𝑥 ∈ (ℕ0 ↑m 1o) ↦ (𝑥‘∅))) = (◡◡(𝑥 ∈ (ℕ0 ↑m 1o) ↦ (𝑥‘∅)) ∘ ◡𝐹) | |
27 | cocnvcnv1 6207 | . . . . . . . 8 ⊢ (◡◡(𝑥 ∈ (ℕ0 ↑m 1o) ↦ (𝑥‘∅)) ∘ ◡𝐹) = ((𝑥 ∈ (ℕ0 ↑m 1o) ↦ (𝑥‘∅)) ∘ ◡𝐹) | |
28 | 26, 27 | eqtri 2765 | . . . . . . 7 ⊢ ◡(𝐹 ∘ ◡(𝑥 ∈ (ℕ0 ↑m 1o) ↦ (𝑥‘∅))) = ((𝑥 ∈ (ℕ0 ↑m 1o) ↦ (𝑥‘∅)) ∘ ◡𝐹) |
29 | 25, 28 | eqtr2di 2794 | . . . . . 6 ⊢ (𝐹 ∈ 𝐵 → ((𝑥 ∈ (ℕ0 ↑m 1o) ↦ (𝑥‘∅)) ∘ ◡𝐹) = ◡𝐴) |
30 | 29 | imaeq1d 6010 | . . . . 5 ⊢ (𝐹 ∈ 𝐵 → (((𝑥 ∈ (ℕ0 ↑m 1o) ↦ (𝑥‘∅)) ∘ ◡𝐹) “ (V ∖ { 0 })) = (◡𝐴 “ (V ∖ { 0 }))) |
31 | 17, 30 | eqtrd 2777 | . . . 4 ⊢ (𝐹 ∈ 𝐵 → ((𝑥 ∈ (ℕ0 ↑m 1o) ↦ (𝑥‘∅)) “ (𝐹 supp 0 )) = (◡𝐴 “ (V ∖ { 0 }))) |
32 | 18 | fvexi 6853 | . . . . 5 ⊢ 𝐴 ∈ V |
33 | suppimacnv 8097 | . . . . . 6 ⊢ ((𝐴 ∈ V ∧ 0 ∈ V) → (𝐴 supp 0 ) = (◡𝐴 “ (V ∖ { 0 }))) | |
34 | 33 | eqcomd 2743 | . . . . 5 ⊢ ((𝐴 ∈ V ∧ 0 ∈ V) → (◡𝐴 “ (V ∖ { 0 })) = (𝐴 supp 0 )) |
35 | 32, 12, 34 | mp2an 690 | . . . 4 ⊢ (◡𝐴 “ (V ∖ { 0 })) = (𝐴 supp 0 ) |
36 | 31, 35 | eqtrdi 2793 | . . 3 ⊢ (𝐹 ∈ 𝐵 → ((𝑥 ∈ (ℕ0 ↑m 1o) ↦ (𝑥‘∅)) “ (𝐹 supp 0 )) = (𝐴 supp 0 )) |
37 | 36 | supeq1d 9340 | . 2 ⊢ (𝐹 ∈ 𝐵 → sup(((𝑥 ∈ (ℕ0 ↑m 1o) ↦ (𝑥‘∅)) “ (𝐹 supp 0 )), ℝ*, < ) = sup((𝐴 supp 0 ), ℝ*, < )) |
38 | 11, 37 | eqtrd 2777 | 1 ⊢ (𝐹 ∈ 𝐵 → (𝐷‘𝐹) = sup((𝐴 supp 0 ), ℝ*, < )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 Vcvv 3443 ∖ cdif 3905 ∅c0 4280 {csn 4584 ↦ cmpt 5186 ◡ccnv 5630 “ cima 5634 ∘ ccom 5635 ‘cfv 6493 (class class class)co 7351 supp csupp 8084 1oc1o 8397 ↑m cmap 8723 supcsup 9334 ℝ*cxr 11146 < clt 11147 ℕ0cn0 12371 Basecbs 17042 0gc0g 17280 mPoly cmpl 21260 PwSer1cps1 21497 Poly1cpl1 21499 coe1cco1 21500 deg1 cdg1 25367 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-addf 11088 ax-mulf 11089 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4864 df-int 4906 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-se 5587 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-of 7609 df-om 7795 df-1st 7913 df-2nd 7914 df-supp 8085 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-rdg 8348 df-1o 8404 df-er 8606 df-map 8725 df-en 8842 df-dom 8843 df-sdom 8844 df-fin 8845 df-fsupp 9264 df-sup 9336 df-oi 9404 df-card 9833 df-pnf 11149 df-mnf 11150 df-xr 11151 df-ltxr 11152 df-le 11153 df-sub 11345 df-neg 11346 df-nn 12112 df-2 12174 df-3 12175 df-4 12176 df-5 12177 df-6 12178 df-7 12179 df-8 12180 df-9 12181 df-n0 12372 df-z 12458 df-dec 12577 df-uz 12722 df-fz 13379 df-fzo 13522 df-seq 13861 df-hash 14184 df-struct 16978 df-sets 16995 df-slot 17013 df-ndx 17025 df-base 17043 df-ress 17072 df-plusg 17105 df-mulr 17106 df-starv 17107 df-sca 17108 df-vsca 17109 df-tset 17111 df-ple 17112 df-ds 17114 df-unif 17115 df-0g 17282 df-gsum 17283 df-mgm 18456 df-sgrp 18505 df-mnd 18516 df-grp 18710 df-mulg 18831 df-cntz 19055 df-cmn 19522 df-mgp 19855 df-ring 19919 df-cring 19920 df-cnfld 20749 df-psr 21263 df-mpl 21265 df-opsr 21267 df-psr1 21502 df-ply1 21504 df-coe1 21505 df-mdeg 25368 df-deg1 25369 |
This theorem is referenced by: deg1mul3 25431 deg1mul3le 25432 ressdeg1 32091 |
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