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Mirrors > Home > MPE Home > Th. List > cply1coe0 | Structured version Visualization version GIF version |
Description: All but the first coefficient of a constant polynomial ( i.e. a "lifted scalar") are zero. (Contributed by AV, 16-Nov-2019.) |
Ref | Expression |
---|---|
cply1coe0.k | ⊢ 𝐾 = (Base‘𝑅) |
cply1coe0.0 | ⊢ 0 = (0g‘𝑅) |
cply1coe0.p | ⊢ 𝑃 = (Poly1‘𝑅) |
cply1coe0.b | ⊢ 𝐵 = (Base‘𝑃) |
cply1coe0.a | ⊢ 𝐴 = (algSc‘𝑃) |
Ref | Expression |
---|---|
cply1coe0 | ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐾) → ∀𝑛 ∈ ℕ ((coe1‘(𝐴‘𝑆))‘𝑛) = 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cply1coe0.p | . . . . 5 ⊢ 𝑃 = (Poly1‘𝑅) | |
2 | cply1coe0.a | . . . . 5 ⊢ 𝐴 = (algSc‘𝑃) | |
3 | cply1coe0.k | . . . . 5 ⊢ 𝐾 = (Base‘𝑅) | |
4 | cply1coe0.0 | . . . . 5 ⊢ 0 = (0g‘𝑅) | |
5 | 1, 2, 3, 4 | coe1scl 21162 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐾) → (coe1‘(𝐴‘𝑆)) = (𝑘 ∈ ℕ0 ↦ if(𝑘 = 0, 𝑆, 0 ))) |
6 | 5 | adantr 484 | . . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐾) ∧ 𝑛 ∈ ℕ) → (coe1‘(𝐴‘𝑆)) = (𝑘 ∈ ℕ0 ↦ if(𝑘 = 0, 𝑆, 0 ))) |
7 | nnne0 11829 | . . . . . . . 8 ⊢ (𝑛 ∈ ℕ → 𝑛 ≠ 0) | |
8 | 7 | neneqd 2937 | . . . . . . 7 ⊢ (𝑛 ∈ ℕ → ¬ 𝑛 = 0) |
9 | 8 | adantl 485 | . . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐾) ∧ 𝑛 ∈ ℕ) → ¬ 𝑛 = 0) |
10 | 9 | adantr 484 | . . . . 5 ⊢ ((((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐾) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 = 𝑛) → ¬ 𝑛 = 0) |
11 | eqeq1 2740 | . . . . . . 7 ⊢ (𝑘 = 𝑛 → (𝑘 = 0 ↔ 𝑛 = 0)) | |
12 | 11 | notbid 321 | . . . . . 6 ⊢ (𝑘 = 𝑛 → (¬ 𝑘 = 0 ↔ ¬ 𝑛 = 0)) |
13 | 12 | adantl 485 | . . . . 5 ⊢ ((((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐾) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 = 𝑛) → (¬ 𝑘 = 0 ↔ ¬ 𝑛 = 0)) |
14 | 10, 13 | mpbird 260 | . . . 4 ⊢ ((((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐾) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 = 𝑛) → ¬ 𝑘 = 0) |
15 | 14 | iffalsed 4436 | . . 3 ⊢ ((((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐾) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 = 𝑛) → if(𝑘 = 0, 𝑆, 0 ) = 0 ) |
16 | nnnn0 12062 | . . . 4 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ0) | |
17 | 16 | adantl 485 | . . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐾) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ0) |
18 | 4 | fvexi 6709 | . . . 4 ⊢ 0 ∈ V |
19 | 18 | a1i 11 | . . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐾) ∧ 𝑛 ∈ ℕ) → 0 ∈ V) |
20 | 6, 15, 17, 19 | fvmptd 6803 | . 2 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐾) ∧ 𝑛 ∈ ℕ) → ((coe1‘(𝐴‘𝑆))‘𝑛) = 0 ) |
21 | 20 | ralrimiva 3095 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐾) → ∀𝑛 ∈ ℕ ((coe1‘(𝐴‘𝑆))‘𝑛) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ∀wral 3051 Vcvv 3398 ifcif 4425 ↦ cmpt 5120 ‘cfv 6358 0cc0 10694 ℕcn 11795 ℕ0cn0 12055 Basecbs 16666 0gc0g 16898 Ringcrg 19516 algSccascl 20768 Poly1cpl1 21052 coe1cco1 21053 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-int 4846 df-iun 4892 df-iin 4893 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-se 5495 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-of 7447 df-ofr 7448 df-om 7623 df-1st 7739 df-2nd 7740 df-supp 7882 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-er 8369 df-map 8488 df-pm 8489 df-ixp 8557 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-fsupp 8964 df-oi 9104 df-card 9520 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-nn 11796 df-2 11858 df-3 11859 df-4 11860 df-5 11861 df-6 11862 df-7 11863 df-8 11864 df-9 11865 df-n0 12056 df-z 12142 df-dec 12259 df-uz 12404 df-fz 13061 df-fzo 13204 df-seq 13540 df-hash 13862 df-struct 16668 df-ndx 16669 df-slot 16670 df-base 16672 df-sets 16673 df-ress 16674 df-plusg 16762 df-mulr 16763 df-sca 16765 df-vsca 16766 df-tset 16768 df-ple 16769 df-0g 16900 df-gsum 16901 df-mre 17043 df-mrc 17044 df-acs 17046 df-mgm 18068 df-sgrp 18117 df-mnd 18128 df-mhm 18172 df-submnd 18173 df-grp 18322 df-minusg 18323 df-sbg 18324 df-mulg 18443 df-subg 18494 df-ghm 18574 df-cntz 18665 df-cmn 19126 df-abl 19127 df-mgp 19459 df-ur 19471 df-ring 19518 df-subrg 19752 df-lmod 19855 df-lss 19923 df-ascl 20771 df-psr 20822 df-mvr 20823 df-mpl 20824 df-opsr 20826 df-psr1 21055 df-vr1 21056 df-ply1 21057 df-coe1 21058 |
This theorem is referenced by: cply1coe0bi 21175 1elcpmat 21566 |
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