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Mirrors > Home > MPE Home > Th. List > deg1tmle | Structured version Visualization version GIF version |
Description: Limiting degree of a polynomial term. (Contributed by Stefan O'Rear, 27-Mar-2015.) |
Ref | Expression |
---|---|
deg1tm.d | ⊢ 𝐷 = ( deg1 ‘𝑅) |
deg1tm.k | ⊢ 𝐾 = (Base‘𝑅) |
deg1tm.p | ⊢ 𝑃 = (Poly1‘𝑅) |
deg1tm.x | ⊢ 𝑋 = (var1‘𝑅) |
deg1tm.m | ⊢ · = ( ·𝑠 ‘𝑃) |
deg1tm.n | ⊢ 𝑁 = (mulGrp‘𝑃) |
deg1tm.e | ⊢ ↑ = (.g‘𝑁) |
Ref | Expression |
---|---|
deg1tmle | ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐹 ∈ ℕ0) → (𝐷‘(𝐶 · (𝐹 ↑ 𝑋))) ≤ 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2771 | . . . . 5 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
2 | deg1tm.k | . . . . 5 ⊢ 𝐾 = (Base‘𝑅) | |
3 | deg1tm.p | . . . . 5 ⊢ 𝑃 = (Poly1‘𝑅) | |
4 | deg1tm.x | . . . . 5 ⊢ 𝑋 = (var1‘𝑅) | |
5 | deg1tm.m | . . . . 5 ⊢ · = ( ·𝑠 ‘𝑃) | |
6 | deg1tm.n | . . . . 5 ⊢ 𝑁 = (mulGrp‘𝑃) | |
7 | deg1tm.e | . . . . 5 ⊢ ↑ = (.g‘𝑁) | |
8 | simpl1 1172 | . . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐹 ∈ ℕ0) ∧ (𝑥 ∈ ℕ0 ∧ 𝐹 < 𝑥)) → 𝑅 ∈ Ring) | |
9 | simpl2 1173 | . . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐹 ∈ ℕ0) ∧ (𝑥 ∈ ℕ0 ∧ 𝐹 < 𝑥)) → 𝐶 ∈ 𝐾) | |
10 | simpl3 1174 | . . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐹 ∈ ℕ0) ∧ (𝑥 ∈ ℕ0 ∧ 𝐹 < 𝑥)) → 𝐹 ∈ ℕ0) | |
11 | simprl 759 | . . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐹 ∈ ℕ0) ∧ (𝑥 ∈ ℕ0 ∧ 𝐹 < 𝑥)) → 𝑥 ∈ ℕ0) | |
12 | 10 | nn0red 11766 | . . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐹 ∈ ℕ0) ∧ (𝑥 ∈ ℕ0 ∧ 𝐹 < 𝑥)) → 𝐹 ∈ ℝ) |
13 | simprr 761 | . . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐹 ∈ ℕ0) ∧ (𝑥 ∈ ℕ0 ∧ 𝐹 < 𝑥)) → 𝐹 < 𝑥) | |
14 | 12, 13 | ltned 10574 | . . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐹 ∈ ℕ0) ∧ (𝑥 ∈ ℕ0 ∧ 𝐹 < 𝑥)) → 𝐹 ≠ 𝑥) |
15 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14 | coe1tmfv2 20161 | . . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐹 ∈ ℕ0) ∧ (𝑥 ∈ ℕ0 ∧ 𝐹 < 𝑥)) → ((coe1‘(𝐶 · (𝐹 ↑ 𝑋)))‘𝑥) = (0g‘𝑅)) |
16 | 15 | expr 449 | . . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐹 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → (𝐹 < 𝑥 → ((coe1‘(𝐶 · (𝐹 ↑ 𝑋)))‘𝑥) = (0g‘𝑅))) |
17 | 16 | ralrimiva 3125 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐹 ∈ ℕ0) → ∀𝑥 ∈ ℕ0 (𝐹 < 𝑥 → ((coe1‘(𝐶 · (𝐹 ↑ 𝑋)))‘𝑥) = (0g‘𝑅))) |
18 | eqid 2771 | . . . 4 ⊢ (Base‘𝑃) = (Base‘𝑃) | |
19 | 2, 3, 4, 5, 6, 7, 18 | ply1tmcl 20158 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐹 ∈ ℕ0) → (𝐶 · (𝐹 ↑ 𝑋)) ∈ (Base‘𝑃)) |
20 | nn0re 11715 | . . . . 5 ⊢ (𝐹 ∈ ℕ0 → 𝐹 ∈ ℝ) | |
21 | 20 | rexrd 10488 | . . . 4 ⊢ (𝐹 ∈ ℕ0 → 𝐹 ∈ ℝ*) |
22 | 21 | 3ad2ant3 1116 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐹 ∈ ℕ0) → 𝐹 ∈ ℝ*) |
23 | deg1tm.d | . . . 4 ⊢ 𝐷 = ( deg1 ‘𝑅) | |
24 | eqid 2771 | . . . 4 ⊢ (coe1‘(𝐶 · (𝐹 ↑ 𝑋))) = (coe1‘(𝐶 · (𝐹 ↑ 𝑋))) | |
25 | 23, 3, 18, 1, 24 | deg1leb 24407 | . . 3 ⊢ (((𝐶 · (𝐹 ↑ 𝑋)) ∈ (Base‘𝑃) ∧ 𝐹 ∈ ℝ*) → ((𝐷‘(𝐶 · (𝐹 ↑ 𝑋))) ≤ 𝐹 ↔ ∀𝑥 ∈ ℕ0 (𝐹 < 𝑥 → ((coe1‘(𝐶 · (𝐹 ↑ 𝑋)))‘𝑥) = (0g‘𝑅)))) |
26 | 19, 22, 25 | syl2anc 576 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐹 ∈ ℕ0) → ((𝐷‘(𝐶 · (𝐹 ↑ 𝑋))) ≤ 𝐹 ↔ ∀𝑥 ∈ ℕ0 (𝐹 < 𝑥 → ((coe1‘(𝐶 · (𝐹 ↑ 𝑋)))‘𝑥) = (0g‘𝑅)))) |
27 | 17, 26 | mpbird 249 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐹 ∈ ℕ0) → (𝐷‘(𝐶 · (𝐹 ↑ 𝑋))) ≤ 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 ∧ w3a 1069 = wceq 1508 ∈ wcel 2051 ∀wral 3081 class class class wbr 4925 ‘cfv 6185 (class class class)co 6974 ℝ*cxr 10471 < clt 10472 ≤ cle 10473 ℕ0cn0 11705 Basecbs 16337 ·𝑠 cvsca 16423 0gc0g 16567 .gcmg 18023 mulGrpcmgp 18974 Ringcrg 19032 var1cv1 20062 Poly1cpl1 20063 coe1cco1 20064 deg1 cdg1 24366 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2743 ax-rep 5045 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 ax-cnex 10389 ax-resscn 10390 ax-1cn 10391 ax-icn 10392 ax-addcl 10393 ax-addrcl 10394 ax-mulcl 10395 ax-mulrcl 10396 ax-mulcom 10397 ax-addass 10398 ax-mulass 10399 ax-distr 10400 ax-i2m1 10401 ax-1ne0 10402 ax-1rid 10403 ax-rnegex 10404 ax-rrecex 10405 ax-cnre 10406 ax-pre-lttri 10407 ax-pre-lttrn 10408 ax-pre-ltadd 10409 ax-pre-mulgt0 10410 ax-pre-sup 10411 ax-addf 10412 ax-mulf 10413 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ne 2961 df-nel 3067 df-ral 3086 df-rex 3087 df-reu 3088 df-rmo 3089 df-rab 3090 df-v 3410 df-sbc 3675 df-csb 3780 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-pss 3838 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4709 df-int 4746 df-iun 4790 df-iin 4791 df-br 4926 df-opab 4988 df-mpt 5005 df-tr 5027 df-id 5308 df-eprel 5313 df-po 5322 df-so 5323 df-fr 5362 df-se 5363 df-we 5364 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-pred 5983 df-ord 6029 df-on 6030 df-lim 6031 df-suc 6032 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-isom 6194 df-riota 6935 df-ov 6977 df-oprab 6978 df-mpo 6979 df-of 7225 df-ofr 7226 df-om 7395 df-1st 7499 df-2nd 7500 df-supp 7632 df-wrecs 7748 df-recs 7810 df-rdg 7848 df-1o 7903 df-2o 7904 df-oadd 7907 df-er 8087 df-map 8206 df-pm 8207 df-ixp 8258 df-en 8305 df-dom 8306 df-sdom 8307 df-fin 8308 df-fsupp 8627 df-sup 8699 df-oi 8767 df-card 9160 df-pnf 10474 df-mnf 10475 df-xr 10476 df-ltxr 10477 df-le 10478 df-sub 10670 df-neg 10671 df-nn 11438 df-2 11501 df-3 11502 df-4 11503 df-5 11504 df-6 11505 df-7 11506 df-8 11507 df-9 11508 df-n0 11706 df-z 11792 df-dec 11910 df-uz 12057 df-fz 12707 df-fzo 12848 df-seq 13183 df-hash 13504 df-struct 16339 df-ndx 16340 df-slot 16341 df-base 16343 df-sets 16344 df-ress 16345 df-plusg 16432 df-mulr 16433 df-starv 16434 df-sca 16435 df-vsca 16436 df-tset 16438 df-ple 16439 df-ds 16441 df-unif 16442 df-0g 16569 df-gsum 16570 df-mre 16727 df-mrc 16728 df-acs 16730 df-mgm 17722 df-sgrp 17764 df-mnd 17775 df-mhm 17815 df-submnd 17816 df-grp 17906 df-minusg 17907 df-sbg 17908 df-mulg 18024 df-subg 18072 df-ghm 18139 df-cntz 18230 df-cmn 18680 df-abl 18681 df-mgp 18975 df-ur 18987 df-ring 19034 df-cring 19035 df-subrg 19268 df-lmod 19370 df-lss 19438 df-psr 19862 df-mvr 19863 df-mpl 19864 df-opsr 19866 df-psr1 20066 df-vr1 20067 df-ply1 20068 df-coe1 20069 df-cnfld 20263 df-mdeg 24367 df-deg1 24368 |
This theorem is referenced by: deg1tm 24430 deg1pwle 24431 ply1divex 24448 |
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