Nearby theorems
|
|
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
| Mirrors > Home > MPE Home > Th. List > deg1tm | Structured version Visualization version GIF version | ||
| Description: Exact 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) |
| deg1tm.z | ⊢ 0 = (0g‘𝑅) |
| Ref | Expression |
|---|---|
| deg1tm | ⊢ ((𝑅 ∈ Ring ∧ (𝐶 ∈ 𝐾 ∧ 𝐶 ≠ 0 ) ∧ 𝐹 ∈ ℕ0) → (𝐷‘(𝐶 · (𝐹 ↑ 𝑋))) = 𝐹) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | deg1tm.d | . . . 4 ⊢ 𝐷 = ( deg1 ‘𝑅) | |
| 2 | deg1tm.k | . . . 4 ⊢ 𝐾 = (Base‘𝑅) | |
| 3 | deg1tm.p | . . . 4 ⊢ 𝑃 = (Poly1‘𝑅) | |
| 4 | deg1tm.x | . . . 4 ⊢ 𝑋 = (var1‘𝑅) | |
| 5 | deg1tm.m | . . . 4 ⊢ · = ( ·𝑠 ‘𝑃) | |
| 6 | deg1tm.n | . . . 4 ⊢ 𝑁 = (mulGrp‘𝑃) | |
| 7 | deg1tm.e | . . . 4 ⊢ ↑ = (.g‘𝑁) | |
| 8 | 1, 2, 3, 4, 5, 6, 7 | deg1tmle 24325 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐹 ∈ ℕ0) → (𝐷‘(𝐶 · (𝐹 ↑ 𝑋))) ≤ 𝐹) |
| 9 | 8 | 3adant2r 1184 | . 2 ⊢ ((𝑅 ∈ Ring ∧ (𝐶 ∈ 𝐾 ∧ 𝐶 ≠ 0 ) ∧ 𝐹 ∈ ℕ0) → (𝐷‘(𝐶 · (𝐹 ↑ 𝑋))) ≤ 𝐹) |
| 10 | eqid 2778 | . . . . 5 ⊢ (Base‘𝑃) = (Base‘𝑃) | |
| 11 | 2, 3, 4, 5, 6, 7, 10 | ply1tmcl 20049 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐹 ∈ ℕ0) → (𝐶 · (𝐹 ↑ 𝑋)) ∈ (Base‘𝑃)) |
| 12 | 11 | 3adant2r 1184 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝐶 ∈ 𝐾 ∧ 𝐶 ≠ 0 ) ∧ 𝐹 ∈ ℕ0) → (𝐶 · (𝐹 ↑ 𝑋)) ∈ (Base‘𝑃)) |
| 13 | simp3 1129 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝐶 ∈ 𝐾 ∧ 𝐶 ≠ 0 ) ∧ 𝐹 ∈ ℕ0) → 𝐹 ∈ ℕ0) | |
| 14 | deg1tm.z | . . . . . 6 ⊢ 0 = (0g‘𝑅) | |
| 15 | 14, 2, 3, 4, 5, 6, 7 | coe1tmfv1 20051 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐹 ∈ ℕ0) → ((coe1‘(𝐶 · (𝐹 ↑ 𝑋)))‘𝐹) = 𝐶) |
| 16 | 15 | 3adant2r 1184 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝐶 ∈ 𝐾 ∧ 𝐶 ≠ 0 ) ∧ 𝐹 ∈ ℕ0) → ((coe1‘(𝐶 · (𝐹 ↑ 𝑋)))‘𝐹) = 𝐶) |
| 17 | simp2r 1214 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝐶 ∈ 𝐾 ∧ 𝐶 ≠ 0 ) ∧ 𝐹 ∈ ℕ0) → 𝐶 ≠ 0 ) | |
| 18 | 16, 17 | eqnetrd 3036 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝐶 ∈ 𝐾 ∧ 𝐶 ≠ 0 ) ∧ 𝐹 ∈ ℕ0) → ((coe1‘(𝐶 · (𝐹 ↑ 𝑋)))‘𝐹) ≠ 0 ) |
| 19 | eqid 2778 | . . . 4 ⊢ (coe1‘(𝐶 · (𝐹 ↑ 𝑋))) = (coe1‘(𝐶 · (𝐹 ↑ 𝑋))) | |
| 20 | 1, 3, 10, 14, 19 | deg1ge 24306 | . . 3 ⊢ (((𝐶 · (𝐹 ↑ 𝑋)) ∈ (Base‘𝑃) ∧ 𝐹 ∈ ℕ0 ∧ ((coe1‘(𝐶 · (𝐹 ↑ 𝑋)))‘𝐹) ≠ 0 ) → 𝐹 ≤ (𝐷‘(𝐶 · (𝐹 ↑ 𝑋)))) |
| 21 | 12, 13, 18, 20 | syl3anc 1439 | . 2 ⊢ ((𝑅 ∈ Ring ∧ (𝐶 ∈ 𝐾 ∧ 𝐶 ≠ 0 ) ∧ 𝐹 ∈ ℕ0) → 𝐹 ≤ (𝐷‘(𝐶 · (𝐹 ↑ 𝑋)))) |
| 22 | 1, 3, 10 | deg1xrcl 24290 | . . . 4 ⊢ ((𝐶 · (𝐹 ↑ 𝑋)) ∈ (Base‘𝑃) → (𝐷‘(𝐶 · (𝐹 ↑ 𝑋))) ∈ ℝ*) |
| 23 | 12, 22 | syl 17 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝐶 ∈ 𝐾 ∧ 𝐶 ≠ 0 ) ∧ 𝐹 ∈ ℕ0) → (𝐷‘(𝐶 · (𝐹 ↑ 𝑋))) ∈ ℝ*) |
| 24 | 13 | nn0red 11708 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝐶 ∈ 𝐾 ∧ 𝐶 ≠ 0 ) ∧ 𝐹 ∈ ℕ0) → 𝐹 ∈ ℝ) |
| 25 | 24 | rexrd 10428 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝐶 ∈ 𝐾 ∧ 𝐶 ≠ 0 ) ∧ 𝐹 ∈ ℕ0) → 𝐹 ∈ ℝ*) |
| 26 | xrletri3 12302 | . . 3 ⊢ (((𝐷‘(𝐶 · (𝐹 ↑ 𝑋))) ∈ ℝ* ∧ 𝐹 ∈ ℝ*) → ((𝐷‘(𝐶 · (𝐹 ↑ 𝑋))) = 𝐹 ↔ ((𝐷‘(𝐶 · (𝐹 ↑ 𝑋))) ≤ 𝐹 ∧ 𝐹 ≤ (𝐷‘(𝐶 · (𝐹 ↑ 𝑋)))))) | |
| 27 | 23, 25, 26 | syl2anc 579 | . 2 ⊢ ((𝑅 ∈ Ring ∧ (𝐶 ∈ 𝐾 ∧ 𝐶 ≠ 0 ) ∧ 𝐹 ∈ ℕ0) → ((𝐷‘(𝐶 · (𝐹 ↑ 𝑋))) = 𝐹 ↔ ((𝐷‘(𝐶 · (𝐹 ↑ 𝑋))) ≤ 𝐹 ∧ 𝐹 ≤ (𝐷‘(𝐶 · (𝐹 ↑ 𝑋)))))) |
| 28 | 9, 21, 27 | mpbir2and 703 | 1 ⊢ ((𝑅 ∈ Ring ∧ (𝐶 ∈ 𝐾 ∧ 𝐶 ≠ 0 ) ∧ 𝐹 ∈ ℕ0) → (𝐷‘(𝐶 · (𝐹 ↑ 𝑋))) = 𝐹) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 ∧ w3a 1071 = wceq 1601 ∈ wcel 2107 ≠ wne 2969 class class class wbr 4888 ‘cfv 6137 (class class class)co 6924 ℝ*cxr 10412 ≤ cle 10414 ℕ0cn0 11647 Basecbs 16266 ·𝑠 cvsca 16353 0gc0g 16497 .gcmg 17938 mulGrpcmgp 18887 Ringcrg 18945 var1cv1 19953 Poly1cpl1 19954 coe1cco1 19955 deg1 cdg1 24262 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-inf2 8837 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 ax-pre-sup 10352 ax-addf 10353 ax-mulf 10354 |
| This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-int 4713 df-iun 4757 df-iin 4758 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-se 5317 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-isom 6146 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-of 7176 df-ofr 7177 df-om 7346 df-1st 7447 df-2nd 7448 df-supp 7579 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-1o 7845 df-2o 7846 df-oadd 7849 df-er 8028 df-map 8144 df-pm 8145 df-ixp 8197 df-en 8244 df-dom 8245 df-sdom 8246 df-fin 8247 df-fsupp 8566 df-sup 8638 df-oi 8706 df-card 9100 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-nn 11380 df-2 11443 df-3 11444 df-4 11445 df-5 11446 df-6 11447 df-7 11448 df-8 11449 df-9 11450 df-n0 11648 df-z 11734 df-dec 11851 df-uz 11998 df-fz 12649 df-fzo 12790 df-seq 13125 df-hash 13442 df-struct 16268 df-ndx 16269 df-slot 16270 df-base 16272 df-sets 16273 df-ress 16274 df-plusg 16362 df-mulr 16363 df-starv 16364 df-sca 16365 df-vsca 16366 df-tset 16368 df-ple 16369 df-ds 16371 df-unif 16372 df-0g 16499 df-gsum 16500 df-mre 16643 df-mrc 16644 df-acs 16646 df-mgm 17639 df-sgrp 17681 df-mnd 17692 df-mhm 17732 df-submnd 17733 df-grp 17823 df-minusg 17824 df-sbg 17825 df-mulg 17939 df-subg 17986 df-ghm 18053 df-cntz 18144 df-cmn 18592 df-abl 18593 df-mgp 18888 df-ur 18900 df-ring 18947 df-cring 18948 df-subrg 19181 df-lmod 19268 df-lss 19336 df-psr 19764 df-mvr 19765 df-mpl 19766 df-opsr 19768 df-psr1 19957 df-vr1 19958 df-ply1 19959 df-coe1 19960 df-cnfld 20154 df-mdeg 24263 df-deg1 24264 |
| This theorem is referenced by: deg1pw 24328 fta1blem 24376 |
| Copyright terms: Public domain | W3C validator |