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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 24240 | . . 3 ⊢ 𝐷 = (1o mDeg 𝑅) |
3 | eqid 2826 | . . 3 ⊢ (1o mPoly 𝑅) = (1o mPoly 𝑅) | |
4 | deg1leb.p | . . . 4 ⊢ 𝑃 = (Poly1‘𝑅) | |
5 | eqid 2826 | . . . 4 ⊢ (PwSer1‘𝑅) = (PwSer1‘𝑅) | |
6 | deg1leb.b | . . . 4 ⊢ 𝐵 = (Base‘𝑃) | |
7 | 4, 5, 6 | ply1bas 19926 | . . 3 ⊢ 𝐵 = (Base‘(1o mPoly 𝑅)) |
8 | deg1leb.y | . . 3 ⊢ 0 = (0g‘𝑅) | |
9 | psr1baslem 19916 | . . 3 ⊢ (ℕ0 ↑𝑚 1o) = {𝑦 ∈ (ℕ0 ↑𝑚 1o) ∣ (◡𝑦 “ ℕ) ∈ Fin} | |
10 | tdeglem2 24221 | . . 3 ⊢ (𝑥 ∈ (ℕ0 ↑𝑚 1o) ↦ (𝑥‘∅)) = (𝑥 ∈ (ℕ0 ↑𝑚 1o) ↦ (ℂfld Σg 𝑥)) | |
11 | 2, 3, 7, 8, 9, 10 | mdegval 24223 | . 2 ⊢ (𝐹 ∈ 𝐵 → (𝐷‘𝐹) = sup(((𝑥 ∈ (ℕ0 ↑𝑚 1o) ↦ (𝑥‘∅)) “ (𝐹 supp 0 )), ℝ*, < )) |
12 | 8 | fvexi 6448 | . . . . . . . 8 ⊢ 0 ∈ V |
13 | suppimacnv 7571 | . . . . . . . 8 ⊢ ((𝐹 ∈ 𝐵 ∧ 0 ∈ V) → (𝐹 supp 0 ) = (◡𝐹 “ (V ∖ { 0 }))) | |
14 | 12, 13 | mpan2 684 | . . . . . . 7 ⊢ (𝐹 ∈ 𝐵 → (𝐹 supp 0 ) = (◡𝐹 “ (V ∖ { 0 }))) |
15 | 14 | imaeq2d 5708 | . . . . . 6 ⊢ (𝐹 ∈ 𝐵 → ((𝑥 ∈ (ℕ0 ↑𝑚 1o) ↦ (𝑥‘∅)) “ (𝐹 supp 0 )) = ((𝑥 ∈ (ℕ0 ↑𝑚 1o) ↦ (𝑥‘∅)) “ (◡𝐹 “ (V ∖ { 0 })))) |
16 | imaco 5882 | . . . . . 6 ⊢ (((𝑥 ∈ (ℕ0 ↑𝑚 1o) ↦ (𝑥‘∅)) ∘ ◡𝐹) “ (V ∖ { 0 })) = ((𝑥 ∈ (ℕ0 ↑𝑚 1o) ↦ (𝑥‘∅)) “ (◡𝐹 “ (V ∖ { 0 }))) | |
17 | 15, 16 | syl6eqr 2880 | . . . . 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 7840 | . . . . . . . . . 10 ⊢ 1o = {∅} | |
20 | nn0ex 11626 | . . . . . . . . . 10 ⊢ ℕ0 ∈ V | |
21 | 0ex 5015 | . . . . . . . . . 10 ⊢ ∅ ∈ V | |
22 | eqid 2826 | . . . . . . . . . 10 ⊢ (𝑥 ∈ (ℕ0 ↑𝑚 1o) ↦ (𝑥‘∅)) = (𝑥 ∈ (ℕ0 ↑𝑚 1o) ↦ (𝑥‘∅)) | |
23 | 19, 20, 21, 22 | mapsncnv 8172 | . . . . . . . . 9 ⊢ ◡(𝑥 ∈ (ℕ0 ↑𝑚 1o) ↦ (𝑥‘∅)) = (𝑦 ∈ ℕ0 ↦ (1o × {𝑦})) |
24 | 18, 6, 4, 23 | coe1fval2 19941 | . . . . . . . 8 ⊢ (𝐹 ∈ 𝐵 → 𝐴 = (𝐹 ∘ ◡(𝑥 ∈ (ℕ0 ↑𝑚 1o) ↦ (𝑥‘∅)))) |
25 | 24 | cnveqd 5531 | . . . . . . 7 ⊢ (𝐹 ∈ 𝐵 → ◡𝐴 = ◡(𝐹 ∘ ◡(𝑥 ∈ (ℕ0 ↑𝑚 1o) ↦ (𝑥‘∅)))) |
26 | cnvco 5541 | . . . . . . . 8 ⊢ ◡(𝐹 ∘ ◡(𝑥 ∈ (ℕ0 ↑𝑚 1o) ↦ (𝑥‘∅))) = (◡◡(𝑥 ∈ (ℕ0 ↑𝑚 1o) ↦ (𝑥‘∅)) ∘ ◡𝐹) | |
27 | cocnvcnv1 5888 | . . . . . . . 8 ⊢ (◡◡(𝑥 ∈ (ℕ0 ↑𝑚 1o) ↦ (𝑥‘∅)) ∘ ◡𝐹) = ((𝑥 ∈ (ℕ0 ↑𝑚 1o) ↦ (𝑥‘∅)) ∘ ◡𝐹) | |
28 | 26, 27 | eqtri 2850 | . . . . . . 7 ⊢ ◡(𝐹 ∘ ◡(𝑥 ∈ (ℕ0 ↑𝑚 1o) ↦ (𝑥‘∅))) = ((𝑥 ∈ (ℕ0 ↑𝑚 1o) ↦ (𝑥‘∅)) ∘ ◡𝐹) |
29 | 25, 28 | syl6req 2879 | . . . . . 6 ⊢ (𝐹 ∈ 𝐵 → ((𝑥 ∈ (ℕ0 ↑𝑚 1o) ↦ (𝑥‘∅)) ∘ ◡𝐹) = ◡𝐴) |
30 | 29 | imaeq1d 5707 | . . . . 5 ⊢ (𝐹 ∈ 𝐵 → (((𝑥 ∈ (ℕ0 ↑𝑚 1o) ↦ (𝑥‘∅)) ∘ ◡𝐹) “ (V ∖ { 0 })) = (◡𝐴 “ (V ∖ { 0 }))) |
31 | 17, 30 | eqtrd 2862 | . . . 4 ⊢ (𝐹 ∈ 𝐵 → ((𝑥 ∈ (ℕ0 ↑𝑚 1o) ↦ (𝑥‘∅)) “ (𝐹 supp 0 )) = (◡𝐴 “ (V ∖ { 0 }))) |
32 | 18 | fvexi 6448 | . . . . 5 ⊢ 𝐴 ∈ V |
33 | suppimacnv 7571 | . . . . . 6 ⊢ ((𝐴 ∈ V ∧ 0 ∈ V) → (𝐴 supp 0 ) = (◡𝐴 “ (V ∖ { 0 }))) | |
34 | 33 | eqcomd 2832 | . . . . 5 ⊢ ((𝐴 ∈ V ∧ 0 ∈ V) → (◡𝐴 “ (V ∖ { 0 })) = (𝐴 supp 0 )) |
35 | 32, 12, 34 | mp2an 685 | . . . 4 ⊢ (◡𝐴 “ (V ∖ { 0 })) = (𝐴 supp 0 ) |
36 | 31, 35 | syl6eq 2878 | . . 3 ⊢ (𝐹 ∈ 𝐵 → ((𝑥 ∈ (ℕ0 ↑𝑚 1o) ↦ (𝑥‘∅)) “ (𝐹 supp 0 )) = (𝐴 supp 0 )) |
37 | 36 | supeq1d 8622 | . 2 ⊢ (𝐹 ∈ 𝐵 → sup(((𝑥 ∈ (ℕ0 ↑𝑚 1o) ↦ (𝑥‘∅)) “ (𝐹 supp 0 )), ℝ*, < ) = sup((𝐴 supp 0 ), ℝ*, < )) |
38 | 11, 37 | eqtrd 2862 | 1 ⊢ (𝐹 ∈ 𝐵 → (𝐷‘𝐹) = sup((𝐴 supp 0 ), ℝ*, < )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1658 ∈ wcel 2166 Vcvv 3415 ∖ cdif 3796 ∅c0 4145 {csn 4398 ↦ cmpt 4953 ◡ccnv 5342 “ cima 5346 ∘ ccom 5347 ‘cfv 6124 (class class class)co 6906 supp csupp 7560 1oc1o 7820 ↑𝑚 cmap 8123 supcsup 8616 ℝ*cxr 10391 < clt 10392 ℕ0cn0 11619 Basecbs 16223 0gc0g 16454 mPoly cmpl 19715 PwSer1cps1 19906 Poly1cpl1 19908 coe1cco1 19909 deg1 cdg1 24214 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-rep 4995 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210 ax-inf2 8816 ax-cnex 10309 ax-resscn 10310 ax-1cn 10311 ax-icn 10312 ax-addcl 10313 ax-addrcl 10314 ax-mulcl 10315 ax-mulrcl 10316 ax-mulcom 10317 ax-addass 10318 ax-mulass 10319 ax-distr 10320 ax-i2m1 10321 ax-1ne0 10322 ax-1rid 10323 ax-rnegex 10324 ax-rrecex 10325 ax-cnre 10326 ax-pre-lttri 10327 ax-pre-lttrn 10328 ax-pre-ltadd 10329 ax-pre-mulgt0 10330 ax-addf 10332 ax-mulf 10333 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-nel 3104 df-ral 3123 df-rex 3124 df-reu 3125 df-rmo 3126 df-rab 3127 df-v 3417 df-sbc 3664 df-csb 3759 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-pss 3815 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4660 df-int 4699 df-iun 4743 df-br 4875 df-opab 4937 df-mpt 4954 df-tr 4977 df-id 5251 df-eprel 5256 df-po 5264 df-so 5265 df-fr 5302 df-se 5303 df-we 5304 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 df-pred 5921 df-ord 5967 df-on 5968 df-lim 5969 df-suc 5970 df-iota 6087 df-fun 6126 df-fn 6127 df-f 6128 df-f1 6129 df-fo 6130 df-f1o 6131 df-fv 6132 df-isom 6133 df-riota 6867 df-ov 6909 df-oprab 6910 df-mpt2 6911 df-of 7158 df-om 7328 df-1st 7429 df-2nd 7430 df-supp 7561 df-wrecs 7673 df-recs 7735 df-rdg 7773 df-1o 7827 df-oadd 7831 df-er 8010 df-map 8125 df-en 8224 df-dom 8225 df-sdom 8226 df-fin 8227 df-fsupp 8546 df-sup 8618 df-oi 8685 df-card 9079 df-pnf 10394 df-mnf 10395 df-xr 10396 df-ltxr 10397 df-le 10398 df-sub 10588 df-neg 10589 df-nn 11352 df-2 11415 df-3 11416 df-4 11417 df-5 11418 df-6 11419 df-7 11420 df-8 11421 df-9 11422 df-n0 11620 df-z 11706 df-dec 11823 df-uz 11970 df-fz 12621 df-fzo 12762 df-seq 13097 df-hash 13412 df-struct 16225 df-ndx 16226 df-slot 16227 df-base 16229 df-sets 16230 df-ress 16231 df-plusg 16319 df-mulr 16320 df-starv 16321 df-sca 16322 df-vsca 16323 df-tset 16325 df-ple 16326 df-ds 16328 df-unif 16329 df-0g 16456 df-gsum 16457 df-mgm 17596 df-sgrp 17638 df-mnd 17649 df-grp 17780 df-mulg 17896 df-cntz 18101 df-cmn 18549 df-mgp 18845 df-ring 18904 df-cring 18905 df-psr 19718 df-mpl 19720 df-opsr 19722 df-psr1 19911 df-ply1 19913 df-coe1 19914 df-cnfld 20108 df-mdeg 24215 df-deg1 24216 |
This theorem is referenced by: deg1mul3 24275 deg1mul3le 24276 |
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