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Mirrors > Home > MPE Home > Th. List > deg1leb | Structured version Visualization version GIF version |
Description: Property of being of limited degree. (Contributed by Stefan O'Rear, 23-Mar-2015.) |
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 |
---|---|
deg1leb | ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ ℝ*) → ((𝐷‘𝐹) ≤ 𝐺 ↔ ∀𝑥 ∈ ℕ0 (𝐺 < 𝑥 → (𝐴‘𝑥) = 0 ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | deg1leb.d | . . . 4 ⊢ 𝐷 = ( deg1 ‘𝑅) | |
2 | 1 | deg1fval 24060 | . . 3 ⊢ 𝐷 = (1𝑜 mDeg 𝑅) |
3 | eqid 2771 | . . 3 ⊢ (1𝑜 mPoly 𝑅) = (1𝑜 mPoly 𝑅) | |
4 | deg1leb.p | . . . 4 ⊢ 𝑃 = (Poly1‘𝑅) | |
5 | eqid 2771 | . . . 4 ⊢ (PwSer1‘𝑅) = (PwSer1‘𝑅) | |
6 | deg1leb.b | . . . 4 ⊢ 𝐵 = (Base‘𝑃) | |
7 | 4, 5, 6 | ply1bas 19780 | . . 3 ⊢ 𝐵 = (Base‘(1𝑜 mPoly 𝑅)) |
8 | deg1leb.y | . . 3 ⊢ 0 = (0g‘𝑅) | |
9 | psr1baslem 19770 | . . 3 ⊢ (ℕ0 ↑𝑚 1𝑜) = {𝑎 ∈ (ℕ0 ↑𝑚 1𝑜) ∣ (◡𝑎 “ ℕ) ∈ Fin} | |
10 | tdeglem2 24041 | . . 3 ⊢ (𝑏 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ (𝑏‘∅)) = (𝑏 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ (ℂfld Σg 𝑏)) | |
11 | 2, 3, 7, 8, 9, 10 | mdegleb 24044 | . 2 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ ℝ*) → ((𝐷‘𝐹) ≤ 𝐺 ↔ ∀𝑦 ∈ (ℕ0 ↑𝑚 1𝑜)(𝐺 < ((𝑏 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ (𝑏‘∅))‘𝑦) → (𝐹‘𝑦) = 0 ))) |
12 | df1o2 7726 | . . . . 5 ⊢ 1𝑜 = {∅} | |
13 | nn0ex 11500 | . . . . 5 ⊢ ℕ0 ∈ V | |
14 | 0ex 4924 | . . . . 5 ⊢ ∅ ∈ V | |
15 | eqid 2771 | . . . . 5 ⊢ (𝑏 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ (𝑏‘∅)) = (𝑏 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ (𝑏‘∅)) | |
16 | 12, 13, 14, 15 | mapsnf1o2 8059 | . . . 4 ⊢ (𝑏 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ (𝑏‘∅)):(ℕ0 ↑𝑚 1𝑜)–1-1-onto→ℕ0 |
17 | f1ofo 6285 | . . . 4 ⊢ ((𝑏 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ (𝑏‘∅)):(ℕ0 ↑𝑚 1𝑜)–1-1-onto→ℕ0 → (𝑏 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ (𝑏‘∅)):(ℕ0 ↑𝑚 1𝑜)–onto→ℕ0) | |
18 | breq2 4790 | . . . . . 6 ⊢ (((𝑏 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ (𝑏‘∅))‘𝑦) = 𝑥 → (𝐺 < ((𝑏 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ (𝑏‘∅))‘𝑦) ↔ 𝐺 < 𝑥)) | |
19 | fveq2 6332 | . . . . . . 7 ⊢ (((𝑏 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ (𝑏‘∅))‘𝑦) = 𝑥 → (𝐴‘((𝑏 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ (𝑏‘∅))‘𝑦)) = (𝐴‘𝑥)) | |
20 | 19 | eqeq1d 2773 | . . . . . 6 ⊢ (((𝑏 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ (𝑏‘∅))‘𝑦) = 𝑥 → ((𝐴‘((𝑏 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ (𝑏‘∅))‘𝑦)) = 0 ↔ (𝐴‘𝑥) = 0 )) |
21 | 18, 20 | imbi12d 333 | . . . . 5 ⊢ (((𝑏 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ (𝑏‘∅))‘𝑦) = 𝑥 → ((𝐺 < ((𝑏 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ (𝑏‘∅))‘𝑦) → (𝐴‘((𝑏 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ (𝑏‘∅))‘𝑦)) = 0 ) ↔ (𝐺 < 𝑥 → (𝐴‘𝑥) = 0 ))) |
22 | 21 | cbvfo 6687 | . . . 4 ⊢ ((𝑏 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ (𝑏‘∅)):(ℕ0 ↑𝑚 1𝑜)–onto→ℕ0 → (∀𝑦 ∈ (ℕ0 ↑𝑚 1𝑜)(𝐺 < ((𝑏 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ (𝑏‘∅))‘𝑦) → (𝐴‘((𝑏 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ (𝑏‘∅))‘𝑦)) = 0 ) ↔ ∀𝑥 ∈ ℕ0 (𝐺 < 𝑥 → (𝐴‘𝑥) = 0 ))) |
23 | 16, 17, 22 | mp2b 10 | . . 3 ⊢ (∀𝑦 ∈ (ℕ0 ↑𝑚 1𝑜)(𝐺 < ((𝑏 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ (𝑏‘∅))‘𝑦) → (𝐴‘((𝑏 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ (𝑏‘∅))‘𝑦)) = 0 ) ↔ ∀𝑥 ∈ ℕ0 (𝐺 < 𝑥 → (𝐴‘𝑥) = 0 )) |
24 | fveq1 6331 | . . . . . . . . . 10 ⊢ (𝑏 = 𝑦 → (𝑏‘∅) = (𝑦‘∅)) | |
25 | fvex 6342 | . . . . . . . . . 10 ⊢ (𝑦‘∅) ∈ V | |
26 | 24, 15, 25 | fvmpt 6424 | . . . . . . . . 9 ⊢ (𝑦 ∈ (ℕ0 ↑𝑚 1𝑜) → ((𝑏 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ (𝑏‘∅))‘𝑦) = (𝑦‘∅)) |
27 | 26 | fveq2d 6336 | . . . . . . . 8 ⊢ (𝑦 ∈ (ℕ0 ↑𝑚 1𝑜) → (𝐴‘((𝑏 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ (𝑏‘∅))‘𝑦)) = (𝐴‘(𝑦‘∅))) |
28 | 27 | adantl 467 | . . . . . . 7 ⊢ (((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ ℝ*) ∧ 𝑦 ∈ (ℕ0 ↑𝑚 1𝑜)) → (𝐴‘((𝑏 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ (𝑏‘∅))‘𝑦)) = (𝐴‘(𝑦‘∅))) |
29 | deg1leb.a | . . . . . . . . 9 ⊢ 𝐴 = (coe1‘𝐹) | |
30 | 29 | fvcoe1 19792 | . . . . . . . 8 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝑦 ∈ (ℕ0 ↑𝑚 1𝑜)) → (𝐹‘𝑦) = (𝐴‘(𝑦‘∅))) |
31 | 30 | adantlr 694 | . . . . . . 7 ⊢ (((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ ℝ*) ∧ 𝑦 ∈ (ℕ0 ↑𝑚 1𝑜)) → (𝐹‘𝑦) = (𝐴‘(𝑦‘∅))) |
32 | 28, 31 | eqtr4d 2808 | . . . . . 6 ⊢ (((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ ℝ*) ∧ 𝑦 ∈ (ℕ0 ↑𝑚 1𝑜)) → (𝐴‘((𝑏 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ (𝑏‘∅))‘𝑦)) = (𝐹‘𝑦)) |
33 | 32 | eqeq1d 2773 | . . . . 5 ⊢ (((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ ℝ*) ∧ 𝑦 ∈ (ℕ0 ↑𝑚 1𝑜)) → ((𝐴‘((𝑏 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ (𝑏‘∅))‘𝑦)) = 0 ↔ (𝐹‘𝑦) = 0 )) |
34 | 33 | imbi2d 329 | . . . 4 ⊢ (((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ ℝ*) ∧ 𝑦 ∈ (ℕ0 ↑𝑚 1𝑜)) → ((𝐺 < ((𝑏 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ (𝑏‘∅))‘𝑦) → (𝐴‘((𝑏 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ (𝑏‘∅))‘𝑦)) = 0 ) ↔ (𝐺 < ((𝑏 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ (𝑏‘∅))‘𝑦) → (𝐹‘𝑦) = 0 ))) |
35 | 34 | ralbidva 3134 | . . 3 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ ℝ*) → (∀𝑦 ∈ (ℕ0 ↑𝑚 1𝑜)(𝐺 < ((𝑏 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ (𝑏‘∅))‘𝑦) → (𝐴‘((𝑏 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ (𝑏‘∅))‘𝑦)) = 0 ) ↔ ∀𝑦 ∈ (ℕ0 ↑𝑚 1𝑜)(𝐺 < ((𝑏 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ (𝑏‘∅))‘𝑦) → (𝐹‘𝑦) = 0 ))) |
36 | 23, 35 | syl5bbr 274 | . 2 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ ℝ*) → (∀𝑥 ∈ ℕ0 (𝐺 < 𝑥 → (𝐴‘𝑥) = 0 ) ↔ ∀𝑦 ∈ (ℕ0 ↑𝑚 1𝑜)(𝐺 < ((𝑏 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ (𝑏‘∅))‘𝑦) → (𝐹‘𝑦) = 0 ))) |
37 | 11, 36 | bitr4d 271 | 1 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ ℝ*) → ((𝐷‘𝐹) ≤ 𝐺 ↔ ∀𝑥 ∈ ℕ0 (𝐺 < 𝑥 → (𝐴‘𝑥) = 0 ))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 382 = wceq 1631 ∈ wcel 2145 ∀wral 3061 ∅c0 4063 class class class wbr 4786 ↦ cmpt 4863 –onto→wfo 6029 –1-1-onto→wf1o 6030 ‘cfv 6031 (class class class)co 6793 1𝑜c1o 7706 ↑𝑚 cmap 8009 ℝ*cxr 10275 < clt 10276 ≤ cle 10277 ℕ0cn0 11494 Basecbs 16064 0gc0g 16308 mPoly cmpl 19568 PwSer1cps1 19760 Poly1cpl1 19762 coe1cco1 19763 deg1 cdg1 24034 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 ax-inf2 8702 ax-cnex 10194 ax-resscn 10195 ax-1cn 10196 ax-icn 10197 ax-addcl 10198 ax-addrcl 10199 ax-mulcl 10200 ax-mulrcl 10201 ax-mulcom 10202 ax-addass 10203 ax-mulass 10204 ax-distr 10205 ax-i2m1 10206 ax-1ne0 10207 ax-1rid 10208 ax-rnegex 10209 ax-rrecex 10210 ax-cnre 10211 ax-pre-lttri 10212 ax-pre-lttrn 10213 ax-pre-ltadd 10214 ax-pre-mulgt0 10215 ax-pre-sup 10216 ax-addf 10217 ax-mulf 10218 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 835 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-int 4612 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-se 5209 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-isom 6040 df-riota 6754 df-ov 6796 df-oprab 6797 df-mpt2 6798 df-of 7044 df-om 7213 df-1st 7315 df-2nd 7316 df-supp 7447 df-wrecs 7559 df-recs 7621 df-rdg 7659 df-1o 7713 df-oadd 7717 df-er 7896 df-map 8011 df-en 8110 df-dom 8111 df-sdom 8112 df-fin 8113 df-fsupp 8432 df-sup 8504 df-oi 8571 df-card 8965 df-pnf 10278 df-mnf 10279 df-xr 10280 df-ltxr 10281 df-le 10282 df-sub 10470 df-neg 10471 df-nn 11223 df-2 11281 df-3 11282 df-4 11283 df-5 11284 df-6 11285 df-7 11286 df-8 11287 df-9 11288 df-n0 11495 df-z 11580 df-dec 11696 df-uz 11889 df-fz 12534 df-fzo 12674 df-seq 13009 df-hash 13322 df-struct 16066 df-ndx 16067 df-slot 16068 df-base 16070 df-sets 16071 df-ress 16072 df-plusg 16162 df-mulr 16163 df-starv 16164 df-sca 16165 df-vsca 16166 df-tset 16168 df-ple 16169 df-ds 16172 df-unif 16173 df-0g 16310 df-gsum 16311 df-mgm 17450 df-sgrp 17492 df-mnd 17503 df-submnd 17544 df-grp 17633 df-minusg 17634 df-mulg 17749 df-cntz 17957 df-cmn 18402 df-abl 18403 df-mgp 18698 df-ur 18710 df-ring 18757 df-cring 18758 df-psr 19571 df-mpl 19573 df-opsr 19575 df-psr1 19765 df-ply1 19767 df-coe1 19768 df-cnfld 19962 df-mdeg 24035 df-deg1 24036 |
This theorem is referenced by: deg1lt 24077 deg1tmle 24097 |
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